Subgroups — C₂²×#16811333 #28291065

Order 13568.

8454 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811333	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13133217	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13133209	C₂	1	2	yes	maximal, complemented
4	C₂²×#13133195	C₂	1	4	yes	maximal, complemented
5	C₂²×#13133193	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×D₂₁₂	C₂	1	2	yes	maximal
7	C₂²×C₄×D₂₁₂	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄⋊C₄×D₅₃	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Dic₁₀₆	C₂²	1	2	yes	—
11	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
12	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
13	C₂³×C₄×D₅₃	C₂²	1	4	yes	—
14	#16811333	C₂²	1	256	yes	direct product factor
15	C₂×#13133217	C₂²	1	24	yes	complemented
16	C₂×#13133209	C₂²	1	48	yes	complemented
17	C₂×#13133195	C₂²	1	96	yes	complemented
18	C₂×#13133193	C₂²	1	48	yes	complemented
19	C₂×C₄×D₂₁₂	C₂²	1	48	yes	—
20	C₂×C₄×D₂₁₂	C₂²	1	24	yes	—
21	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
22	C₂²×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
24	C₂²×#11719919	C₂²	1	4	yes	—
25	C₂²×#11719919	C₂²	1	4	yes	—
26	C₂×C₄⋊C₄×D₅₃	C₂²	1	48	yes	complemented
27	C₂²×#11719921	C₂²	1	4	yes	—
28	C₂²×#11719921	C₂²	1	2	yes	—
29	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₅₃	C₂²	1	24	yes	complemented
31	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
32	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
33	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
34	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
35	C₂⁴×D₅₃	C₂³	1	2	yes	—
36	C₂²×Dic₁₀₆	—	4	8	no	—
37	C₂²×Dic₁₀₆	C₂³	1	24	yes	—
38	C₂²×D₂₁₂	Q₈	1	16	yes	complemented
39	C₂²×D₂₁₂	C₂³	1	12	yes	—
40	C₂³×Dic₅₃	C₂³	1	2	yes	—
41	C₂³×Dic₅₃	C₂³	1	4	yes	—
42	C₂²×C₄×D₅₃	—	2	16	no	—
43	C₂²×C₄×D₅₃	—	4	16	no	—
44	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
45	C₂²×C₄×D₅₃	C₂³	1	48	yes	—
46	#13133217	C₂³	1	64	yes	complemented
47	#13133209	C₂³	1	128	yes	complemented
48	#13133195	C₂³	1	256	yes	complemented
49	#13133193	C₂³	1	128	yes	complemented
50	C₄×D₂₁₂	C₂³	1	64	yes	—
51	C₄×D₂₁₂	C₂³	1	128	yes	—
52	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	—
53	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
54	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	—
55	C₂×#11719919	C₂³	1	48	yes	—
56	C₂×#11719919	C₂³	1	48	yes	—
57	C₄⋊C₄×D₅₃	C₂³	1	128	yes	complemented
58	C₂×#11719921	C₂³	1	48	yes	—
59	C₂×#11719921	C₂³	1	24	yes	—
60	C₂²×Q₈×C₅₃	—	2	8	no	—
61	C₂²×Q₈×C₅₃	C₂³	1	12	yes	—
62	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
64	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
65	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×Q₈×C₅₃	C₂³	1	64	yes	complemented
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
69	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
70	C₂×C₄²×C₅₃	C₂³	1	24	yes	—
71	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
72	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₂³×D₅₃	—	2	8	no	—
74	C₂³×D₅₃	—	2	8	no	—
75	C₂³×D₅₃	C₂×Q₈	1	16	yes	—
76	C₂³×D₅₃	C₂⁴	1	12	yes	—
77	C₂×Q₈×C₅₃	—	2	48	no	—
78	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	—
79	C₂²×C₄×C₅₃	—	2	8	no	—
80	C₂²×C₄×C₅₃	—	2	8	no	—
81	C₂²×C₄×C₅₃	—	2	4	no	—
82	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
83	C₂²×C₄×C₅₃	#41	1	8	yes	—
84	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
85	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
86	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
87	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
88	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
89	C₂²×Dic₅₃	—	2	8	no	—
90	C₂²×Dic₅₃	—	2	16	no	—
91	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
92	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
93	C₂×D₂₁₂	C₂×Q₈	1	96	yes	complemented
94	C₂×D₂₁₂	C₂⁴	1	16	yes	—
95	C₂×C₄×D₅₃	—	2	96	no	—
96	C₂×C₄×D₅₃	—	4	96	no	—
97	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
98	C₂×C₄×D₅₃	C₂⁴	1	64	yes	—
99	C₂×Dic₁₀₆	—	4	48	no	—
100	C₂×Dic₁₀₆	C₂⁴	1	32	yes	—
101	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
102	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	—
103	C₄²×C₅₃	C₂⁴	1	32	yes	—
104	C₄²×C₅₃	C₂⁴	1	16	yes	—
105	#11719921	C₂⁴	1	64	yes	—
106	#11719921	C₂⁴	1	32	yes	—
107	C₄⋊Dic₅₃	C₂⁴	1	32	yes	—
108	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
109	C₄⋊Dic₅₃	C₂⁴	1	32	yes	—
110	#11719919	C₂⁴	1	64	yes	—
111	#11719919	C₂⁴	1	64	yes	—
112	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
113	C₂³×C₅₃	C₂×#41	1	4	yes	—
114	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
115	C₂³×C₅₃	C₂⁵	1	3	yes	—
116	C₂²×D₅₃	—	2	64	no	—
117	C₂²×D₅₃	—	2	24	no	—
118	C₂²×D₅₃	—	2	24	no	—
119	C₂²×D₅₃	C₂²×Q₈	1	48	yes	—
120	C₂²×D₅₃	C₂⁵	1	8	yes	—
121	Q₈×C₅₃	—	2	32	no	—
122	C₂×C₄×C₅₃	—	2	24	no	—
123	C₂×C₄×C₅₃	—	2	12	no	—
124	C₂×C₄×C₅₃	—	2	24	no	—
125	C₂×C₄×C₅₃	C₂²×Q₈	1	24	yes	—
126	C₂×C₄×C₅₃	C₂×#41	1	24	yes	—
127	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
128	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
129	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
130	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
131	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
132	C₂×Dic₅₃	—	2	24	no	—
133	C₂×Dic₅₃	—	2	48	no	—
134	C₂×Dic₅₃	C₂⁵	1	16	yes	—
135	C₂×Dic₅₃	C₂⁵	1	8	yes	—
136	D₂₁₂	C₂²×Q₈	1	64	yes	complemented
137	C₄×D₅₃	—	2	64	no	—
138	C₄×D₅₃	—	4	64	no	—
139	Dic₁₀₆	—	4	32	no	—
140	C₂²×#554	—	53	1	no	maximal, Sylow
141	C₂²×C₅₃	#554	1	16	yes	complemented
142	C₂²×C₅₃	C₂³×Q₈	1	6	yes	—
143	C₂²×C₅₃	C₂²×#41	1	6	yes	—
144	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
145	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
146	C₂×D₅₃	—	2	8	no	—
147	C₂×D₅₃	—	2	8	no	—
148	C₂×D₅₃	—	2	96	no	—
149	C₂×D₅₃	C₂³×Q₈	1	16	yes	—
150	C₄×C₅₃	—	2	8	no	—
151	C₄×C₅₃	—	2	4	no	—
152	C₄×C₅₃	—	2	8	no	—
153	C₄×C₅₃	C₂³×Q₈	1	8	yes	—
154	C₄×C₅₃	C₂²×#41	1	8	yes	—
155	Dic₅₃	—	2	8	no	—
156	Dic₅₃	—	2	16	no	—
157	C₂×#554	—	53	48	no	—
158	C₂²×C₄⋊Q₈	—	53	1	no	—
159	C₂²×#125	—	53	2	no	—
160	C₂²×C₂²⋊Q₈	—	53	2	no	—
161	C₂²×C₂²⋊Q₈	—	53	4	no	—
162	C₂²×C₄×Q₈	D₅₃	1	1	yes	characteristic, complemented
163	C₂²×C₄×D₄	—	53	1	no	—
164	C₂²×C₄×D₄	—	53	2	no	—
165	C₂³×C₄⋊C₄	—	53	2	no	—
166	C₂×C₅₃	C₂×#554	1	12	yes	complemented
167	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
168	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
169	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
170	D₅₃	—	2	16	no	—
171	C₂³×Q₈	—	53	2	no	—
172	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic
173	C₂³×D₄	—	53	1	no	—
174	C₂⁴×C₄	—	53	2	no	—
175	C₂⁴×C₄	—	53	4	no	—
176	#554	—	53	256	no	—
177	C₂×C₄⋊Q₈	—	53	24	no	—
178	C₂×#125	—	53	48	no	—
179	C₂×C₂²⋊Q₈	—	53	48	no	—
180	C₂×C₂²⋊Q₈	—	53	96	no	—
181	C₂×C₄×Q₈	C₂×D₅₃	1	24	yes	complemented
182	C₂×C₄×D₄	—	53	48	no	—
183	C₂×C₄×D₄	—	53	24	no	—
184	C₂²×C₄⋊C₄	—	53	4	no	—
185	C₂²×C₄⋊C₄	—	53	4	no	—
186	C₂²×C₄⋊C₄	—	53	2	no	—
187	C₂²×C₄⋊C₄	—	53	1	no	—
188	C₂²×C₄⋊C₄	—	53	2	no	—
189	C₂²×C₄⋊C₄	—	53	48	no	—
190	C₂²×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
191	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
192	C₂²×C₂²⋊C₄	—	53	2	no	—
193	C₂²×C₂²⋊C₄	—	53	4	no	—
194	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
195	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
196	C₅₃	C₂²×#554	1	1	yes	characteristic, complemented, Sylow
197	C₂⁵	—	53	2	no	—
198	C₂²×Q₈	—	2	8	no	—
199	C₂²×Q₈	—	53	24	no	—
200	C₂²×Q₈	—	212	8	no	—
201	C₂²×Q₈	C₂²×D₅₃	1	12	yes	—
202	C₂²×D₄	—	53	16	no	—
203	C₂²×D₄	—	53	12	no	—
204	C₂³×C₄	—	53	2	no	—
205	C₂³×C₄	—	53	4	no	—
206	C₂³×C₄	—	53	24	no	—
207	C₂³×C₄	—	53	48	no	—
208	C₂³×C₄	—	106	16	no	—
209	C₂³×C₄	—	212	16	no	—
210	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
211	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
212	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
213	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
214	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
215	C₄⋊Q₈	—	53	64	no	—
216	#125	—	53	128	no	—
217	C₂²⋊Q₈	—	53	128	no	—
218	C₂²⋊Q₈	—	53	256	no	—
219	C₄×Q₈	C₂²×D₅₃	1	64	yes	complemented
220	C₄×D₄	—	53	128	no	—
221	C₄×D₄	—	53	64	no	—
222	C₂×C₄⋊C₄	—	53	48	no	—
223	C₂×C₄⋊C₄	—	53	24	no	—
224	C₂×C₄⋊C₄	—	53	12	no	—
225	C₂×C₄⋊C₄	—	53	24	no	—
226	C₂×C₄⋊C₄	—	53	48	no	—
227	C₂×C₄⋊C₄	—	53	128	no	—
228	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	—
229	C₂×C₄⋊C₄	C₂²×D₅₃	1	24	yes	—
230	C₂×C₂²⋊C₄	—	53	24	no	—
231	C₂×C₂²⋊C₄	—	53	48	no	—
232	C₂×C₄²	C₂²×D₅₃	1	24	yes	—
233	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
234	C₂⁴	—	53	16	no	—
235	C₂⁴	—	53	12	no	—
236	C₂⁴	—	106	8	no	—
237	C₂⁴	—	106	8	no	—
238	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
239	C₂×Q₈	—	2	48	no	—
240	C₂×Q₈	—	53	32	no	—
241	C₂×Q₈	—	212	48	no	—
242	C₂×Q₈	C₂³×D₅₃	1	16	yes	—
243	C₂×D₄	—	53	96	no	—
244	C₂×D₄	—	53	16	no	—
245	C₂²×C₄	—	2	8	no	—
246	C₂²×C₄	—	2	8	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	53	12	no	—
249	C₂²×C₄	—	53	24	no	—
250	C₂²×C₄	—	53	32	no	—
251	C₂²×C₄	—	53	64	no	—
252	C₂²×C₄	—	106	8	no	—
253	C₂²×C₄	—	106	16	no	—
254	C₂²×C₄	—	106	96	no	—
255	C₂²×C₄	—	212	96	no	—
256	C₂²×C₄	#11719945	1	8	yes	—
257	C₂²×C₄	Q₈×D₅₃	1	8	yes	—
258	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
259	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
260	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
261	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
262	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
263	C₄⋊C₄	—	53	64	no	—
264	C₄⋊C₄	—	53	32	no	—
265	C₄⋊C₄	—	53	16	no	—
266	C₄⋊C₄	—	53	32	no	—
267	C₄⋊C₄	—	53	64	no	—
268	C₄⋊C₄	C₂³×D₅₃	1	16	yes	—
269	C₄⋊C₄	C₂³×D₅₃	1	32	yes	—
270	C₂²⋊C₄	—	53	32	no	—
271	C₂²⋊C₄	—	53	64	no	—
272	C₄²	C₂³×D₅₃	1	32	yes	—
273	C₄²	C₂³×D₅₃	1	16	yes	—
274	C₂³	—	53	48	no	—
275	C₂³	—	53	8	no	—
276	C₂³	—	106	64	no	—
277	C₂³	—	106	24	no	—
278	C₂³	—	106	24	no	—
279	C₂³	C₂×Q₈×D₅₃	1	4	yes	—
280	C₂³	C₂×#11719945	1	4	yes	—
281	C₂³	#13133263	1	4	yes	—
282	C₂³	C₂⁴×D₅₃	1	3	yes	—
283	Q₈	—	2	32	no	—
284	Q₈	—	212	32	no	—
285	D₄	—	53	64	no	—
286	C₂×C₄	—	2	24	no	—
287	C₂×C₄	—	2	24	no	—
288	C₂×C₄	—	2	12	no	—
289	C₂×C₄	—	53	8	no	—
290	C₂×C₄	—	53	16	no	—
291	C₂×C₄	—	106	24	no	—
292	C₂×C₄	—	106	48	no	—
293	C₂×C₄	—	106	64	no	—
294	C₂×C₄	—	212	64	no	—
295	C₂×C₄	C₂×Q₈×D₅₃	1	24	yes	—
296	C₂×C₄	C₂×#11719945	1	24	yes	—
297	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
298	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
299	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
300	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
301	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
302	C₂²	—	53	16	no	—
303	C₂²	—	106	8	no	—
304	C₂²	—	106	8	no	—
305	C₂²	—	106	96	no	—
306	C₂²	#16811333	1	16	yes	direct product factor
307	C₂²	C₂²×Q₈×D₅₃	1	6	yes	—
308	C₂²	C₂²×#11719945	1	6	yes	—
309	C₂²	C₂×#13133263	1	6	yes	—
310	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
311	C₄	—	2	8	no	—
312	C₄	—	2	8	no	—
313	C₄	—	2	4	no	—
314	C₄	—	106	8	no	—
315	C₄	—	106	16	no	—
316	C₄	C₂²×Q₈×D₅₃	1	8	yes	—
317	C₄	C₂²×#11719945	1	8	yes	—
318	C₂	—	106	16	no	—
319	C₂	C₂×#16811333	1	12	yes	direct product factor
320	C₂	C₂³×Q₈×D₅₃	1	1	yes	characteristic
321	C₂	C₂³×#11719945	1	1	yes	characteristic
322	C₂	C₂²×#13133263	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×C₄×Q₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂³×D₂₁₂	C₂²
hypercenter	C₂²×C₄×Q₈	D₅₃
hypocenter	C₅₃	C₂²×#554
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16811333	C₂	direct (N × H)
2. C₂²×#13133217	C₂	semidirect
3. C₂²×#13133209	C₂	semidirect
4. C₂²×#13133195	C₂	semidirect
5. C₂²×#13133193	C₂	semidirect
8. C₂²×C₄⋊C₄×D₅₃	C₂	semidirect
9. C₂²×C₄×Q₈×C₅₃	C₂	semidirect
14. #16811333	C₂²	direct (N × H)
15. C₂×#13133217	C₂²	semidirect
16. C₂×#13133209	C₂²	semidirect
17. C₂×#13133195	C₂²	semidirect
18. C₂×#13133193	C₂²	semidirect
26. C₂×C₄⋊C₄×D₅₃	C₂²	semidirect
30. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
38. C₂²×D₂₁₂	Q₈	semidirect
46. #13133217	C₂³	semidirect
47. #13133209	C₂³	semidirect
48. #13133195	C₂³	semidirect
49. #13133193	C₂³	semidirect
57. C₄⋊C₄×D₅₃	C₂³	semidirect
67. C₄×Q₈×C₅₃	C₂³	semidirect
93. C₂×D₂₁₂	C₂×Q₈	semidirect
136. D₂₁₂	C₂²×Q₈	semidirect
141. C₂²×C₅₃	#554	semidirect
162. C₂²×C₄×Q₈	D₅₃	semidirect
166. C₂×C₅₃	C₂×#554	semidirect
181. C₂×C₄×Q₈	C₂×D₅₃	semidirect
196. C₅₃	C₂²×#554	semidirect
219. C₄×Q₈	C₂²×D₅₃	semidirect
306. C₂²	#16811333	direct (N × H)
319. C₂	C₂×#16811333	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16811333	2
C₂²×#13133217	2
C₂²×#13133209	2
C₂²×#13133195	2
C₂²×#13133193	2
C₂²×C₄×D₂₁₂	2
C₂²×C₄⋊C₄×D₅₃	2
C₂²×C₄×Q₈×C₅₃	2
C₂²×#554	53

Subgroups — C22×#16811333 #28291065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811333 #28291065