Subgroups — C₂×D₄×#11719949 #28291056

Order 13568.

12314 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719949	C₂	1	2	yes	maximal, complemented
2	D₄×#11719949	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16811295	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16811284	C₂	1	2	yes	maximal, complemented
5	C₂×#16811276	C₂	1	3	yes	maximal, complemented
6	C₂×#16811264	C₂	1	6	yes	maximal, complemented
7	C₂×D₄×D₂₁₂	C₂	1	3	yes	maximal, complemented
8	C₂×#16811174	C₂	1	3	yes	maximal, complemented
9	C₂×#16811136	C₂	1	6	yes	maximal, complemented
10	C₂×C₄×#11719949	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄×D₅₃	C₂	1	3	yes	maximal, complemented
12	C₂×D₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#11719949	—	2	32	no	—
14	C₂²×#11719949	C₂²	1	1	yes	characteristic
15	C₂²×#11719949	C₂²	1	6	yes	complemented
16	C₂²×#11719949	C₂²	1	2	yes	complemented
17	C₂²×#11719949	C₂²	1	6	yes	complemented
18	C₂²×#11719949	C₂²	1	32	yes	complemented
19	C₂³×D₂₁₂	C₂²	1	6	yes	complemented
20	C₂²×D₄×D₅₃	C₂²	1	3	yes	—
21	C₂³×C₄×D₅₃	C₂²	1	6	yes	complemented
22	#16811295	C₂²	1	16	yes	complemented
23	#16811284	C₂²	1	32	yes	complemented
24	#16811276	C₂²	1	48	yes	complemented
25	#16811264	C₂²	1	96	yes	complemented
26	D₄×D₂₁₂	C₂²	1	48	yes	complemented
27	#16811174	C₂²	1	48	yes	complemented
28	C₂×#13133214	C₂²	1	3	yes	complemented
29	C₂×#13133206	C₂²	1	6	yes	complemented
30	#16811136	C₂²	1	96	yes	complemented
31	C₂×#13133202	C₂²	1	12	yes	complemented
32	C₂×#13133193	C₂²	1	6	yes	complemented
33	C₂×#13133192	C₂²	1	6	yes	complemented
34	C₂×#13133192	C₂²	1	6	yes	complemented
35	C₂×#13133192	C₂²	1	6	yes	complemented
36	C₂×#13133191	C₂²	1	6	yes	complemented
37	C₂×#13133187	C₂²	1	12	yes	complemented
38	C₄×#11719949	C₂²	1	16	yes	complemented
39	C₂×Q₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₂×C₄×D₂₁₂	C₂²	1	3	yes	complemented
41	C₂×C₄×D₂₁₂	C₂²	1	3	yes	complemented
42	C₂×D₄×Dic₅₃	C₂²	1	1	yes	characteristic
43	C₂×#13133179	C₂²	1	3	yes	complemented
44	C₂×#13133179	C₂²	1	3	yes	complemented
45	C₂×#13133179	C₂²	1	6	yes	complemented
46	C₂×C₄×C₂²⋊D₅₃	C₂²	1	6	yes	complemented
47	C₂×#13133177	C₂²	1	6	yes	complemented
48	C₄×D₄×D₅₃	C₂²	1	48	yes	complemented
49	C₂×#13133175	C₂²	1	3	yes	complemented
50	C₂×C₄⋊C₄×D₅₃	C₂²	1	3	yes	complemented
51	C₂×C₂²⋊C₄×D₅₃	C₂²	1	6	yes	complemented
52	C₂×C₄²×D₅₃	C₂²	1	3	yes	complemented
53	C₂³×Q₈×C₅₃	C₂²	1	2	yes	complemented
54	D₄×Q₈×C₅₃	C₂²	1	16	yes	complemented
55	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	complemented
56	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	6	yes	complemented
57	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
58	C₂×C₄×D₄×C₅₃	C₂²	1	3	yes	—
59	C₂⁴×D₅₃	C₂³	1	6	yes	—
60	C₂×#11719949	—	2	16	no	—
61	C₂×#11719949	—	2	256	no	—
62	C₂×#11719949	—	2	48	no	—
63	C₂×#11719949	—	2	48	no	—
64	C₂×#11719949	D₄	1	16	yes	direct product factor
65	C₂×#11719949	C₂³	1	8	yes	—
66	C₂×#11719949	C₂³	1	48	yes	complemented
67	C₂×#11719949	C₂³	1	16	yes	complemented
68	C₂×#11719949	C₂³	1	48	yes	complemented
69	C₂²×D₂₁₂	—	2	48	no	—
70	C₂²×D₂₁₂	C₂³	1	3	yes	—
71	C₂²×D₂₁₂	C₂³	1	12	yes	—
72	C₂²×D₂₁₂	C₂³	1	6	yes	—
73	C₂²×D₂₁₂	C₂³	1	6	yes	—
74	C₂²×D₂₁₂	C₂³	1	3	yes	—
75	C₂²×D₂₁₂	C₂³	1	48	yes	complemented
76	C₂²×C₂²⋊D₅₃	C₂³	1	6	yes	—
77	C₂×D₄×D₅₃	—	2	24	no	—
78	C₂×D₄×D₅₃	C₂³	1	24	yes	—
79	C₂³×Dic₅₃	C₂³	1	2	yes	—
80	C₂²×C₄×D₅₃	—	2	48	no	—
81	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
82	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
83	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
84	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
85	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
86	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
87	C₂²×C₄×D₅₃	C₂³	1	48	yes	complemented
88	#13133214	C₂³	1	24	yes	complemented
89	#13133206	C₂³	1	48	yes	complemented
90	#13133202	C₂³	1	96	yes	complemented
91	#13133193	C₂³	1	48	yes	complemented
92	#13133192	C₂³	1	48	yes	complemented
93	#13133192	C₂³	1	48	yes	complemented
94	#13133192	C₂³	1	48	yes	complemented
95	#13133191	C₂³	1	48	yes	complemented
96	#13133187	C₂³	1	96	yes	complemented
97	Q₈×Dic₅₃	C₂³	1	8	yes	complemented
98	C₄×D₂₁₂	C₂³	1	24	yes	complemented
99	C₄×D₂₁₂	C₂³	1	24	yes	complemented
100	D₄×Dic₅₃	C₂³	1	8	yes	—
101	#13133179	C₂³	1	24	yes	complemented
102	#13133179	C₂³	1	24	yes	complemented
103	#13133179	C₂³	1	48	yes	complemented
104	C₄×C₂²⋊D₅₃	C₂³	1	48	yes	complemented
105	#13133177	C₂³	1	48	yes	complemented
106	#13133175	C₂³	1	24	yes	complemented
107	C₂×C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊Dic₅₃	C₂³	1	3	yes	—
109	C₂×#11719919	C₂³	1	6	yes	—
110	C₄⋊C₄×D₅₃	C₂³	1	24	yes	complemented
111	C₂×C₂²⋊Dic₅₃	C₂³	1	2	yes	—
112	C₂×#11719921	C₂³	1	6	yes	—
113	C₂×#11719921	C₂³	1	6	yes	—
114	C₂×#11719921	C₂³	1	6	yes	—
115	C₂×#11719921	C₂³	1	12	yes	—
116	C₂²⋊C₄×D₅₃	C₂³	1	48	yes	complemented
117	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
118	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
119	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
120	C₄²×D₅₃	C₂³	1	24	yes	complemented
121	C₂²×Q₈×C₅₃	—	2	16	no	—
122	C₂²×Q₈×C₅₃	C₂³	1	6	yes	—
123	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
124	C₂²×Q₈×C₅₃	C₂³	1	16	yes	complemented
125	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
126	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
127	C₄⋊Q₈×C₅₃	C₂³	1	24	yes	complemented
128	C₂²⋊Q₈×C₅₃	C₂³	1	48	yes	complemented
129	C₄×Q₈×C₅₃	C₂³	1	8	yes	complemented
130	C₄×D₄×C₅₃	C₂³	1	24	yes	—
131	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
132	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
133	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
134	C₂×C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
135	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
136	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
137	C₂³×D₅₃	—	2	24	no	—
138	C₂³×D₅₃	—	2	24	no	—
139	C₂³×D₅₃	—	4	12	no	—
140	C₂³×D₅₃	C₂⁴	1	6	yes	—
141	C₂³×D₅₃	C₂⁴	1	24	yes	—
142	C₂³×D₅₃	C₂⁴	1	3	yes	—
143	C₂×Q₈×C₅₃	—	2	64	no	—
144	C₂×Q₈×C₅₃	—	2	24	no	—
145	C₂×Q₈×C₅₃	C₂×D₄	1	8	yes	complemented
146	C₂×Q₈×C₅₃	C₂⁴	1	24	yes	—
147	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
148	C₂×D₄×C₅₃	#41	1	8	yes	complemented
149	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₅₃	—	2	24	no	—
151	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
152	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
154	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
156	C₂×C₂²⋊D₅₃	—	2	24	no	—
157	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
158	C₂²×Dic₅₃	—	2	8	no	—
159	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
160	C₂²×Dic₅₃	C₂⁴	1	3	yes	—
161	C₂²×Dic₅₃	C₂⁴	1	2	yes	—
162	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
163	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
164	#11719949	—	2	64	no	—
165	#11719949	—	2	192	no	—
166	#11719949	—	2	192	no	—
167	#11719949	C₂×D₄	1	64	yes	direct product factor
168	D₄×D₅₃	—	2	96	no	—
169	C₂×D₂₁₂	—	2	48	no	—
170	C₂×D₂₁₂	—	2	24	no	—
171	C₂×D₂₁₂	—	2	12	no	—
172	C₂×D₂₁₂	—	2	192	no	—
173	C₂×D₂₁₂	—	4	12	no	—
174	C₂×D₂₁₂	C₂×D₄	1	24	yes	complemented
175	C₂×D₂₁₂	C₂⁴	1	12	yes	—
176	C₂×D₂₁₂	C₂⁴	1	48	yes	—
177	C₂×D₂₁₂	C₂⁴	1	24	yes	—
178	C₂×D₂₁₂	C₂⁴	1	24	yes	—
179	C₂×D₂₁₂	C₂⁴	1	12	yes	—
180	C₂×C₄×D₅₃	—	2	24	no	—
181	C₂×C₄×D₅₃	—	2	24	no	—
182	C₂×C₄×D₅₃	—	2	12	no	—
183	C₂×C₄×D₅₃	—	2	24	no	—
184	C₂×C₄×D₅₃	—	2	192	no	—
185	C₂×C₄×D₅₃	—	4	6	no	—
186	C₂×C₄×D₅₃	C₂×D₄	1	24	yes	complemented
187	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
188	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
189	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
190	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
191	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
192	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
193	C₄⋊C₄×C₅₃	C₂⁴	1	12	yes	—
194	C₄⋊C₄×C₅₃	C₂⁴	1	12	yes	—
195	C₄⋊C₄×C₅₃	C₂⁴	1	24	yes	—
196	C₂²⋊C₄×C₅₃	C₂⁴	1	24	yes	—
197	C₄²×C₅₃	C₂⁴	1	12	yes	—
198	C₂²⋊Dic₅₃	C₂⁴	1	8	yes	—
199	#11719921	C₂⁴	1	24	yes	—
200	#11719921	C₂⁴	1	24	yes	—
201	#11719921	C₂⁴	1	24	yes	—
202	#11719921	C₂⁴	1	48	yes	—
203	C₄⋊Dic₅₃	C₂⁴	1	4	yes	—
204	C₄⋊Dic₅₃	C₂⁴	1	12	yes	—
205	#11719919	C₂⁴	1	24	yes	—
206	C₄×Dic₅₃	C₂⁴	1	4	yes	—
207	C₄×Dic₅₃	C₂⁴	1	12	yes	—
208	C₄×Dic₅₃	C₂⁴	1	12	yes	—
209	C₂³×C₅₃	—	2	4	no	—
210	C₂³×C₅₃	—	2	4	no	—
211	C₂³×C₅₃	C₂×#41	1	8	yes	complemented
212	C₂³×C₅₃	C₂⁵	1	4	yes	—
213	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
214	C₂²×D₅₃	—	2	6	no	—
215	C₂²×D₅₃	—	2	6	no	—
216	C₂²×D₅₃	—	2	12	no	—
217	C₂²×D₅₃	—	2	12	no	—
218	C₂²×D₅₃	—	2	48	no	—
219	C₂²×D₅₃	—	2	12	no	—
220	C₂²×D₅₃	—	2	48	no	—
221	C₂²×D₅₃	—	4	24	no	—
222	C₂²×D₅₃	—	4	96	no	—
223	C₂²×D₅₃	C₂²×D₄	1	12	yes	—
224	C₂²×D₅₃	C₂⁵	1	12	yes	—
225	C₂²×D₅₃	C₂⁵	1	6	yes	—
226	Q₈×C₅₃	—	2	48	no	—
227	Q₈×C₅₃	C₂²×D₄	1	16	yes	complemented
228	D₄×C₅₃	C₂×#41	1	16	yes	complemented
229	C₂×C₄×C₅₃	—	2	12	no	—
230	C₂×C₄×C₅₃	—	2	6	no	—
231	C₂×C₄×C₅₃	—	2	48	no	—
232	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
233	C₂×C₄×C₅₃	C₂×#41	1	4	yes	complemented
234	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
235	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
236	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
237	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
238	C₂²⋊D₅₃	—	2	48	no	—
239	C₂×Dic₅₃	—	2	4	no	—
240	C₂×Dic₅₃	—	2	6	no	—
241	C₂×Dic₅₃	—	2	2	no	—
242	C₂×Dic₅₃	—	2	16	no	—
243	C₂×Dic₅₃	C₂²×D₄	1	4	yes	—
244	C₂×Dic₅₃	C₂⁵	1	4	yes	—
245	C₂×Dic₅₃	C₂⁵	1	2	yes	—
246	C₂×Dic₅₃	C₂⁵	1	6	yes	—
247	C₂×Dic₅₃	C₂⁵	1	2	yes	—
248	D₂₁₂	—	2	96	no	—
249	D₂₁₂	—	2	48	no	—
250	D₂₁₂	—	2	24	no	—
251	D₂₁₂	—	4	24	no	—
252	D₂₁₂	C₂²×D₄	1	48	yes	complemented
253	C₄×D₅₃	—	2	48	no	—
254	C₄×D₅₃	—	2	48	no	—
255	C₄×D₅₃	—	2	24	no	—
256	C₄×D₅₃	—	2	48	no	—
257	C₄×D₅₃	—	4	12	no	—
258	C₄×D₅₃	C₂²×D₄	1	48	yes	complemented
259	C₂×D₄×#41	—	53	1	no	maximal, Sylow
260	C₂²×C₅₃	—	2	4	no	—
261	C₂²×C₅₃	—	2	4	no	—
262	C₂²×C₅₃	—	2	16	no	—
263	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
264	C₂²×C₅₃	C₂²×#41	1	2	yes	—
265	C₂²×C₅₃	C₂²×#41	1	8	yes	complemented
266	C₂²×C₅₃	#585	1	2	yes	—
267	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
268	C₂×D₅₃	—	2	6	no	—
269	C₂×D₅₃	—	2	6	no	—
270	C₂×D₅₃	—	2	24	no	—
271	C₂×D₅₃	—	2	12	no	—
272	C₂×D₅₃	—	2	12	no	—
273	C₂×D₅₃	—	2	12	no	—
274	C₂×D₅₃	—	4	96	no	—
275	C₂×D₅₃	—	4	24	no	—
276	C₂×D₅₃	C₂³×D₄	1	12	yes	—
277	C₄×C₅₃	—	2	12	no	—
278	C₄×C₅₃	—	2	6	no	—
279	C₄×C₅₃	C₂³×D₄	1	12	yes	—
280	C₄×C₅₃	C₂²×#41	1	4	yes	complemented
281	Dic₅₃	—	2	4	no	—
282	Dic₅₃	—	2	6	no	—
283	Dic₅₃	—	2	2	no	—
284	Dic₅₃	C₂³×D₄	1	4	yes	—
285	C₂³×#41	—	53	2	no	—
286	D₄×#41	—	53	32	no	—
287	C₂×#550	—	53	1	no	—
288	C₂×D₄×Q₈	D₅₃	1	1	yes	characteristic, complemented
289	C₂×#548	—	53	2	no	—
290	C₂×#547	—	53	3	no	—
291	C₂×#546	—	53	6	no	—
292	C₂×D₄²	—	53	3	no	—
293	C₂×#532	—	53	3	no	—
294	C₂×#525	—	53	6	no	—
295	C₂×C₄×#41	—	53	1	no	—
296	C₂²×C₄×D₄	—	53	3	no	—
297	C₂×C₅₃	—	2	8	no	—
298	C₂×C₅₃	D₄×#41	1	4	yes	complemented
299	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
300	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
301	C₂×C₅₃	C₂×#585	1	1	yes	characteristic
302	D₅₃	—	2	12	no	—
303	D₅₃	—	4	12	no	—
304	C₂²×#41	—	53	1	no	—
305	C₂²×#41	—	53	6	no	—
306	C₂²×#41	—	53	6	no	—
307	C₂²×#41	—	53	2	no	—
308	C₂²×#41	—	53	32	no	—
309	C₂²×#41	—	106	32	no	—
310	C₂³×Q₈	C₂×D₅₃	1	2	yes	complemented
311	C₂³×D₄	—	53	3	no	—
312	C₂³×D₄	—	53	6	no	—
313	C₂⁴×C₄	—	53	6	no	—
314	#550	—	53	16	no	—
315	D₄×Q₈	C₂×D₅₃	1	16	yes	complemented
316	#548	—	53	32	no	—
317	#547	—	53	48	no	—
318	#546	—	53	96	no	—
319	D₄²	—	53	48	no	—
320	#532	—	53	48	no	—
321	C₂×C₄⋊Q₈	C₂×D₅₃	1	3	yes	complemented
322	C₂×#127	—	53	3	no	—
323	C₂×#124	—	53	6	no	—
324	#525	—	53	96	no	—
325	C₂×#123	—	53	12	no	—
326	C₂×C₂²⋊Q₈	—	53	6	no	—
327	C₂×C₂²⋊Q₈	C₂×D₅₃	1	6	yes	complemented
328	C₂×#121	—	53	6	no	—
329	C₂×#121	—	53	6	no	—
330	C₂×#121	—	53	6	no	—
331	C₂×#121	—	53	6	no	—
332	C₂×#120	—	53	12	no	—
333	C₄×#41	—	53	16	no	—
334	C₂×C₄×Q₈	—	53	1	no	—
335	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
336	C₂×C₄×D₄	—	53	1	no	—
337	C₂×C₄×D₄	—	53	6	no	—
338	C₂×C₄×D₄	—	53	48	no	—
339	C₂×C₄×D₄	—	53	3	no	—
340	C₂×C₄×D₄	—	53	3	no	—
341	C₂×C₄×D₄	—	53	6	no	—
342	C₂×C₄×D₄	—	53	6	no	—
343	C₂×C₄×D₄	—	53	3	no	—
344	C₂×C₄×D₄	—	53	3	no	—
345	C₂×C₄×D₄	C₂×D₅₃	1	3	yes	—
346	C₂×#117	—	53	3	no	—
347	C₂²×C₄⋊C₄	—	53	3	no	—
348	C₂²×C₂²⋊C₄	—	53	6	no	—
349	C₂²×C₄²	—	53	3	no	—
350	C₅₃	C₂×D₄×#41	1	1	yes	characteristic, complemented, Sylow
351	C₂⁵	—	53	6	no	—
352	C₂×#41	—	53	16	no	—
353	C₂×#41	—	53	8	no	—
354	C₂×#41	—	53	48	no	—
355	C₂×#41	—	53	48	no	—
356	C₂×#41	—	53	16	no	—
357	C₂×#41	—	106	48	no	—
358	C₂×#41	—	106	48	no	—
359	C₂×#41	—	106	256	no	—
360	C₂×#41	—	106	16	no	—
361	C₂²×Q₈	—	2	16	no	—
362	C₂²×Q₈	C₂²×D₅₃	1	6	yes	—
363	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
364	C₂²×Q₈	C₂²×D₅₃	1	16	yes	complemented
365	C₂²×D₄	—	53	6	no	—
366	C₂²×D₄	—	53	24	no	—
367	C₂²×D₄	—	53	3	no	—
368	C₂²×D₄	—	53	3	no	—
369	C₂²×D₄	—	53	12	no	—
370	C₂²×D₄	—	53	6	no	—
371	C₂²×D₄	—	53	48	no	—
372	C₂²×D₄	—	53	6	no	—
373	C₂²×D₄	—	106	48	no	—
374	C₂²×D₄	—	106	24	no	—
375	C₂²×D₄	C₂²×D₅₃	1	1	yes	characteristic
376	C₂³×C₄	—	53	3	no	—
377	C₂³×C₄	—	53	2	no	—
378	C₂³×C₄	—	53	6	no	—
379	C₂³×C₄	—	53	3	no	—
380	C₂³×C₄	—	53	6	no	—
381	C₂³×C₄	—	53	6	no	—
382	C₂³×C₄	—	53	48	no	—
383	C₂³×C₄	—	53	3	no	—
384	C₂³×C₄	—	106	48	no	—
385	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
386	C₄⋊Q₈	C₂²×D₅₃	1	24	yes	complemented
387	#127	—	53	24	no	—
388	#124	—	53	48	no	—
389	#123	—	53	96	no	—
390	C₂²⋊Q₈	—	53	48	no	—
391	C₂²⋊Q₈	C₂²×D₅₃	1	48	yes	complemented
392	#121	—	53	48	no	—
393	#121	—	53	48	no	—
394	#121	—	53	48	no	—
395	#121	—	53	48	no	—
396	#120	—	53	96	no	—
397	C₄×Q₈	—	53	8	no	—
398	C₄×Q₈	C₂²×D₅₃	1	8	yes	complemented
399	C₄×D₄	—	53	8	no	—
400	C₄×D₄	—	53	48	no	—
401	C₄×D₄	—	53	24	no	—
402	C₄×D₄	—	53	24	no	—
403	C₄×D₄	—	53	48	no	—
404	C₄×D₄	—	53	48	no	—
405	C₄×D₄	—	53	24	no	—
406	C₄×D₄	—	53	24	no	—
407	C₄×D₄	C₂²×D₅₃	1	24	yes	—
408	#117	—	53	24	no	—
409	C₂×C₄⋊C₄	—	53	1	no	—
410	C₂×C₄⋊C₄	—	53	6	no	—
411	C₂×C₄⋊C₄	—	53	3	no	—
412	C₂×C₄⋊C₄	—	53	24	no	—
413	C₂×C₄⋊C₄	C₂²×D₅₃	1	3	yes	—
414	C₂×C₄⋊C₄	C₂²×D₅₃	1	3	yes	—
415	C₂×C₄⋊C₄	C₂²×D₅₃	1	6	yes	—
416	C₂×C₂²⋊C₄	—	53	2	no	—
417	C₂×C₂²⋊C₄	—	53	6	no	—
418	C₂×C₂²⋊C₄	—	53	6	no	—
419	C₂×C₂²⋊C₄	—	53	6	no	—
420	C₂×C₂²⋊C₄	—	53	12	no	—
421	C₂×C₂²⋊C₄	—	53	48	no	—
422	C₂×C₂²⋊C₄	C₂²×D₅₃	1	6	yes	—
423	C₂×C₄²	—	53	3	no	—
424	C₂×C₄²	—	53	1	no	—
425	C₂×C₄²	—	53	3	no	—
426	C₂×C₄²	—	53	24	no	—
427	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
428	C₂⁴	—	53	3	no	—
429	C₂⁴	—	53	6	no	—
430	C₂⁴	—	53	24	no	—
431	C₂⁴	—	106	24	no	—
432	C₂⁴	—	106	24	no	—
433	C₂⁴	—	212	12	no	—
434	C₂⁴	C₂³×D₅₃	1	2	yes	—
435	#41	—	53	64	no	—
436	#41	—	106	192	no	—
437	#41	—	106	192	no	—
438	#41	—	106	64	no	—
439	C₂×Q₈	—	2	64	no	—
440	C₂×Q₈	—	2	24	no	—
441	C₂×Q₈	D₄×D₅₃	1	8	yes	complemented
442	C₂×Q₈	C₂³×D₅₃	1	24	yes	—
443	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
444	C₂×D₄	—	53	24	no	—
445	C₂×D₄	—	53	24	no	—
446	C₂×D₄	—	53	12	no	—
447	C₂×D₄	—	53	12	no	—
448	C₂×D₄	—	53	48	no	—
449	C₂×D₄	—	53	24	no	—
450	C₂×D₄	—	53	24	no	—
451	C₂×D₄	—	106	192	no	—
452	C₂×D₄	—	106	48	no	—
453	C₂×D₄	—	106	24	no	—
454	C₂×D₄	—	106	96	no	—
455	C₂×D₄	—	106	24	no	—
456	C₂×D₄	—	106	12	no	—
457	C₂×D₄	—	212	12	no	—
458	C₂×D₄	#11719949	1	8	yes	direct product factor
459	C₂×D₄	C₂³×D₅₃	1	4	yes	—
460	C₂²×C₄	—	2	24	no	—
461	C₂²×C₄	—	53	1	no	—
462	C₂²×C₄	—	53	24	no	—
463	C₂²×C₄	—	53	12	no	—
464	C₂²×C₄	—	53	8	no	—
465	C₂²×C₄	—	53	1	no	—
466	C₂²×C₄	—	53	3	no	—
467	C₂²×C₄	—	53	2	no	—
468	C₂²×C₄	—	53	12	no	—
469	C₂²×C₄	—	53	24	no	—
470	C₂²×C₄	—	53	24	no	—
471	C₂²×C₄	—	53	24	no	—
472	C₂²×C₄	—	53	12	no	—
473	C₂²×C₄	—	106	24	no	—
474	C₂²×C₄	—	106	192	no	—
475	C₂²×C₄	—	106	24	no	—
476	C₂²×C₄	—	106	8	no	—
477	C₂²×C₄	—	106	24	no	—
478	C₂²×C₄	—	106	12	no	—
479	C₂²×C₄	—	212	6	no	—
480	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
481	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
482	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
483	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
484	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
485	C₄⋊C₄	—	53	4	no	—
486	C₄⋊C₄	—	53	24	no	—
487	C₄⋊C₄	—	53	12	no	—
488	C₄⋊C₄	C₂³×D₅₃	1	12	yes	—
489	C₄⋊C₄	C₂³×D₅₃	1	12	yes	—
490	C₄⋊C₄	C₂³×D₅₃	1	24	yes	—
491	C₂²⋊C₄	—	53	8	no	—
492	C₂²⋊C₄	—	53	24	no	—
493	C₂²⋊C₄	—	53	24	no	—
494	C₂²⋊C₄	—	53	24	no	—
495	C₂²⋊C₄	—	53	48	no	—
496	C₂²⋊C₄	C₂³×D₅₃	1	24	yes	—
497	C₄²	—	53	4	no	—
498	C₄²	—	53	12	no	—
499	C₄²	—	53	12	no	—
500	C₄²	C₂³×D₅₃	1	12	yes	—
501	C₂³	—	2	4	no	—
502	C₂³	—	2	4	no	—
503	C₂³	—	53	12	no	—
504	C₂³	—	53	6	no	—
505	C₂³	—	53	12	no	—
506	C₂³	—	106	48	no	—
507	C₂³	—	106	6	no	—
508	C₂³	—	106	6	no	—
509	C₂³	—	106	12	no	—
510	C₂³	—	106	12	no	—
511	C₂³	—	106	48	no	—
512	C₂³	—	106	12	no	—
513	C₂³	—	212	24	no	—
514	C₂³	—	212	96	no	—
515	C₂³	C₂×#11719949	1	8	yes	complemented
516	C₂³	C₂⁴×D₅₃	1	4	yes	—
517	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
518	Q₈	—	2	48	no	—
519	Q₈	C₂×D₄×D₅₃	1	16	yes	complemented
520	D₄	—	53	48	no	—
521	D₄	—	106	96	no	—
522	D₄	—	106	48	no	—
523	D₄	—	106	48	no	—
524	D₄	—	106	24	no	—
525	D₄	—	212	24	no	—
526	D₄	C₂×#11719949	1	16	yes	direct product factor
527	C₂×C₄	—	2	12	no	—
528	C₂×C₄	—	2	6	no	—
529	C₂×C₄	—	2	48	no	—
530	C₂×C₄	—	53	4	no	—
531	C₂×C₄	—	53	2	no	—
532	C₂×C₄	—	53	48	no	—
533	C₂×C₄	—	53	2	no	—
534	C₂×C₄	—	53	6	no	—
535	C₂×C₄	—	53	4	no	—
536	C₂×C₄	—	106	48	no	—
537	C₂×C₄	—	106	48	no	—
538	C₂×C₄	—	106	16	no	—
539	C₂×C₄	—	106	4	no	—
540	C₂×C₄	—	106	6	no	—
541	C₂×C₄	—	106	2	no	—
542	C₂×C₄	—	106	48	no	—
543	C₂×C₄	—	106	24	no	—
544	C₂×C₄	—	212	12	no	—
545	C₂×C₄	C₂×D₄×D₅₃	1	12	yes	—
546	C₂×C₄	C₂×#11719949	1	4	yes	complemented
547	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
548	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
549	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
550	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
551	C₂²	—	2	16	no	—
552	C₂²	—	2	4	no	—
553	C₂²	—	2	4	no	—
554	C₂²	—	53	12	no	—
555	C₂²	—	106	24	no	—
556	C₂²	—	106	6	no	—
557	C₂²	—	106	6	no	—
558	C₂²	—	106	12	no	—
559	C₂²	—	106	12	no	—
560	C₂²	—	106	12	no	—
561	C₂²	—	212	96	no	—
562	C₂²	—	212	24	no	—
563	C₂²	C₂²×D₄×D₅₃	1	2	yes	—
564	C₂²	C₂²×#11719949	1	2	yes	—
565	C₂²	C₂²×#11719949	1	8	yes	complemented
566	C₂²	#16811460	1	2	yes	—
567	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
568	C₄	—	2	12	no	—
569	C₄	—	2	6	no	—
570	C₄	—	53	4	no	—
571	C₄	—	106	4	no	—
572	C₄	—	106	6	no	—
573	C₄	—	106	2	no	—
574	C₄	C₂²×D₄×D₅₃	1	12	yes	—
575	C₄	C₂²×#11719949	1	4	yes	complemented
576	C₂	—	2	8	no	—
577	C₂	—	106	12	no	—
578	C₂	—	212	12	no	—
579	C₂	D₄×#11719949	1	4	yes	direct product factor
580	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
581	C₂	C₂³×#11719949	1	1	yes	characteristic
582	C₂	C₂×#16811460	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×D₄×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×D₄×Q₈	D₅₃
hypocenter	C₅₃	C₂×D₄×#41
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×D₄×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂³×#11719949	C₂	semidirect
2. D₄×#11719949	C₂	direct (N × H)
3. C₂×#16811295	C₂	semidirect
4. C₂×#16811284	C₂	semidirect
5. C₂×#16811276	C₂	semidirect
6. C₂×#16811264	C₂	semidirect
7. C₂×D₄×D₂₁₂	C₂	semidirect
8. C₂×#16811174	C₂	semidirect
9. C₂×#16811136	C₂	semidirect
10. C₂×C₄×#11719949	C₂	semidirect
11. C₂×C₄×D₄×D₅₃	C₂	semidirect
12. C₂×D₄×Q₈×C₅₃	C₂	semidirect
15. C₂²×#11719949	C₂²	semidirect
16. C₂²×#11719949	C₂²	semidirect
17. C₂²×#11719949	C₂²	semidirect
18. C₂²×#11719949	C₂²	semidirect
19. C₂³×D₂₁₂	C₂²	semidirect
21. C₂³×C₄×D₅₃	C₂²	semidirect
22. #16811295	C₂²	semidirect
23. #16811284	C₂²	semidirect
24. #16811276	C₂²	semidirect
25. #16811264	C₂²	semidirect
26. D₄×D₂₁₂	C₂²	semidirect
27. #16811174	C₂²	semidirect
28. C₂×#13133214	C₂²	semidirect
29. C₂×#13133206	C₂²	semidirect
30. #16811136	C₂²	semidirect
31. C₂×#13133202	C₂²	semidirect
32. C₂×#13133193	C₂²	semidirect
33. C₂×#13133192	C₂²	semidirect
34. C₂×#13133192	C₂²	semidirect
35. C₂×#13133192	C₂²	semidirect
36. C₂×#13133191	C₂²	semidirect
37. C₂×#13133187	C₂²	semidirect
38. C₄×#11719949	C₂²	semidirect
39. C₂×Q₈×Dic₅₃	C₂²	semidirect
40. C₂×C₄×D₂₁₂	C₂²	semidirect
41. C₂×C₄×D₂₁₂	C₂²	semidirect
43. C₂×#13133179	C₂²	semidirect
44. C₂×#13133179	C₂²	semidirect
45. C₂×#13133179	C₂²	semidirect
46. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
47. C₂×#13133177	C₂²	semidirect
48. C₄×D₄×D₅₃	C₂²	semidirect
49. C₂×#13133175	C₂²	semidirect
50. C₂×C₄⋊C₄×D₅₃	C₂²	semidirect
51. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
52. C₂×C₄²×D₅₃	C₂²	semidirect
53. C₂³×Q₈×C₅₃	C₂²	semidirect
54. D₄×Q₈×C₅₃	C₂²	semidirect
55. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
56. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
57. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
64. C₂×#11719949	D₄	direct (N × H)
66. C₂×#11719949	C₂³	semidirect
67. C₂×#11719949	C₂³	semidirect
68. C₂×#11719949	C₂³	semidirect
75. C₂²×D₂₁₂	C₂³	semidirect
87. C₂²×C₄×D₅₃	C₂³	semidirect
88. #13133214	C₂³	semidirect
89. #13133206	C₂³	semidirect
90. #13133202	C₂³	semidirect
91. #13133193	C₂³	semidirect
92. #13133192	C₂³	semidirect
93. #13133192	C₂³	semidirect
94. #13133192	C₂³	semidirect
95. #13133191	C₂³	semidirect
96. #13133187	C₂³	semidirect
97. Q₈×Dic₅₃	C₂³	semidirect
98. C₄×D₂₁₂	C₂³	semidirect
99. C₄×D₂₁₂	C₂³	semidirect
101. #13133179	C₂³	semidirect
102. #13133179	C₂³	semidirect
103. #13133179	C₂³	semidirect
104. C₄×C₂²⋊D₅₃	C₂³	semidirect
105. #13133177	C₂³	semidirect
106. #13133175	C₂³	semidirect
110. C₄⋊C₄×D₅₃	C₂³	semidirect
116. C₂²⋊C₄×D₅₃	C₂³	semidirect
120. C₄²×D₅₃	C₂³	semidirect
124. C₂²×Q₈×C₅₃	C₂³	semidirect
127. C₄⋊Q₈×C₅₃	C₂³	semidirect
128. C₂²⋊Q₈×C₅₃	C₂³	semidirect
129. C₄×Q₈×C₅₃	C₂³	semidirect
145. C₂×Q₈×C₅₃	C₂×D₄	semidirect
148. C₂×D₄×C₅₃	#41	semidirect
167. #11719949	C₂×D₄	direct (N × H)
174. C₂×D₂₁₂	C₂×D₄	semidirect
186. C₂×C₄×D₅₃	C₂×D₄	semidirect
211. C₂³×C₅₃	C₂×#41	semidirect
227. Q₈×C₅₃	C₂²×D₄	semidirect
228. D₄×C₅₃	C₂×#41	semidirect
233. C₂×C₄×C₅₃	C₂×#41	semidirect
252. D₂₁₂	C₂²×D₄	semidirect
258. C₄×D₅₃	C₂²×D₄	semidirect
265. C₂²×C₅₃	C₂²×#41	semidirect
280. C₄×C₅₃	C₂²×#41	semidirect
288. C₂×D₄×Q₈	D₅₃	semidirect
298. C₂×C₅₃	D₄×#41	semidirect
310. C₂³×Q₈	C₂×D₅₃	semidirect
315. D₄×Q₈	C₂×D₅₃	semidirect
321. C₂×C₄⋊Q₈	C₂×D₅₃	semidirect
327. C₂×C₂²⋊Q₈	C₂×D₅₃	semidirect
335. C₂×C₄×Q₈	C₂×D₅₃	semidirect
350. C₅₃	C₂×D₄×#41	semidirect
364. C₂²×Q₈	C₂²×D₅₃	semidirect
386. C₄⋊Q₈	C₂²×D₅₃	semidirect
391. C₂²⋊Q₈	C₂²×D₅₃	semidirect
398. C₄×Q₈	C₂²×D₅₃	semidirect
441. C₂×Q₈	D₄×D₅₃	semidirect
458. C₂×D₄	#11719949	direct (N × H)
515. C₂³	C₂×#11719949	semidirect
519. Q₈	C₂×D₄×D₅₃	semidirect
526. D₄	C₂×#11719949	direct (N × H)
546. C₂×C₄	C₂×#11719949	semidirect
565. C₂²	C₂²×#11719949	semidirect
575. C₄	C₂²×#11719949	semidirect
579. C₂	D₄×#11719949	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#11719949	2
D₄×#11719949	2
C₂×#16811295	2
C₂×#16811284	2
C₂×#16811276	2
C₂×#16811264	2
C₂×D₄×D₂₁₂	2
C₂×#16811174	2
C₂×#16811136	2
C₂×C₄×#11719949	2
C₂×C₄×D₄×D₅₃	2
C₂×D₄×Q₈×C₅₃	2
C₂×D₄×#41	53

Subgroups — C2×D4×#11719949 #28291056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×#11719949 #28291056