Subgroups — C₂²×#16811210 #28290821

Order 13568.

9566 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811210	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13133212	C₂	1	2	yes	maximal, complemented
3	C₂²×#13133203	C₂	1	2	yes	maximal, complemented
4	C₂²×#13133201	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133196	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13133189	C₂	1	2	yes	maximal, complemented
7	C₂²×#13133185	C₂	1	2	yes	maximal, complemented
8	C₂²×D₄×Dic₅₃	C₂	1	2	yes	maximal, complemented
9	C₂³×#11719921	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#121×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
12	C₂³×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
13	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
14	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
15	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂⁴×Dic₅₃	C₂²	1	2	yes	complemented
17	#16811210	C₂²	1	256	yes	direct product factor
18	C₂×#13133212	C₂²	1	48	yes	complemented
19	C₂×#13133203	C₂²	1	48	yes	complemented
20	C₂×#13133201	C₂²	1	48	yes	complemented
21	C₂×#13133196	C₂²	1	24	yes	complemented
22	C₂×#13133189	C₂²	1	48	yes	complemented
23	C₂×#13133185	C₂²	1	48	yes	complemented
24	C₂×D₄×Dic₅₃	C₂²	1	48	yes	complemented
25	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
26	C₂²×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
27	C₂²×#11719919	C₂²	1	2	yes	complemented
28	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	—
29	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	complemented
30	C₂²×#11719921	C₂²	1	2	yes	—
31	C₂²×#11719921	C₂²	1	2	yes	—
32	C₂²×#11719921	C₂²	1	2	yes	—
33	C₂²×#11719921	C₂²	1	2	yes	—
34	C₂²×#11719921	C₂²	1	24	yes	complemented
35	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	complemented
36	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
37	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
38	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×#121×C₅₃	C₂²	1	24	yes	complemented
40	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
41	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
42	C₂⁴×D₅₃	—	2	8	no	—
43	C₂⁴×D₅₃	—	4	4	no	—
44	C₂⁴×D₅₃	—	4	4	no	—
45	C₂⁴×D₅₃	C₂³	1	2	yes	—
46	C₂⁴×D₅₃	C₂³	1	12	yes	—
47	C₂²×Dic₁₀₆	—	4	4	no	—
48	C₂²×Dic₁₀₆	C₂³	1	12	yes	complemented
49	C₂²×C₂²⋊D₅₃	—	4	8	no	—
50	C₂²×C₂²⋊D₅₃	—	4	8	no	—
51	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
52	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
53	C₂³×Dic₅₃	—	4	4	no	—
54	C₂³×Dic₅₃	—	4	8	no	—
55	C₂³×Dic₅₃	C₂³	1	2	yes	—
56	C₂³×Dic₅₃	C₂³	1	2	yes	—
57	C₂³×Dic₅₃	C₂³	1	2	yes	—
58	C₂³×Dic₅₃	C₂³	1	12	yes	—
59	C₂³×Dic₅₃	C₂³	1	24	yes	complemented
60	#13133212	C₂³	1	128	yes	complemented
61	#13133203	C₂³	1	128	yes	complemented
62	#13133201	C₂³	1	128	yes	complemented
63	#13133196	C₂³	1	64	yes	complemented
64	#13133189	C₂³	1	128	yes	complemented
65	#13133185	C₂³	1	128	yes	complemented
66	D₄×Dic₅₃	C₂³	1	128	yes	complemented
67	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	complemented
68	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	complemented
69	C₂×#11719919	C₂³	1	24	yes	complemented
70	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	—
71	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	complemented
72	C₂×#11719921	C₂³	1	24	yes	—
73	C₂×#11719921	C₂³	1	24	yes	—
74	C₂×#11719921	C₂³	1	24	yes	—
75	C₂×#11719921	C₂³	1	24	yes	—
76	C₂×#11719921	C₂³	1	64	yes	complemented
77	C₂×C₄×Dic₅₃	C₂³	1	24	yes	complemented
78	C₂⁵×C₅₃	C₂³	1	2	yes	—
79	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
80	C₂²×D₄×C₅₃	—	2	16	no	—
81	C₂²×D₄×C₅₃	—	4	4	no	—
82	C₂²×D₄×C₅₃	C₂³	1	24	yes	—
83	C₂²×D₄×C₅₃	C₂³	1	12	yes	complemented
84	C₂³×C₄×C₅₃	—	4	4	no	—
85	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
87	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
89	#121×C₅₃	C₂³	1	64	yes	complemented
90	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	complemented
91	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	complemented
92	C₂⁴×C₅₃	—	2	4	no	—
93	C₂⁴×C₅₃	—	2	4	no	—
94	C₂⁴×C₅₃	—	2	8	no	—
95	C₂⁴×C₅₃	—	2	8	no	—
96	C₂⁴×C₅₃	—	2	8	no	—
97	C₂⁴×C₅₃	#41	1	8	yes	—
98	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
99	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
100	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
101	C₂³×D₅₃	—	2	48	no	—
102	C₂³×D₅₃	—	2	8	no	—
103	C₂³×D₅₃	—	2	8	no	—
104	C₂³×D₅₃	—	2	8	no	—
105	C₂³×D₅₃	—	4	24	no	—
106	C₂³×D₅₃	—	4	24	no	—
107	C₂³×D₅₃	—	4	64	no	—
108	C₂³×D₅₃	C₂⁴	1	12	yes	—
109	C₂³×D₅₃	C₂⁴	1	16	yes	—
110	C₂×D₄×C₅₃	—	2	96	no	—
111	C₂×D₄×C₅₃	—	4	24	no	—
112	C₂×D₄×C₅₃	C₂⁴	1	32	yes	—
113	C₂×D₄×C₅₃	C₂⁴	1	16	yes	complemented
114	C₂²×C₄×C₅₃	—	2	8	no	—
115	C₂²×C₄×C₅₃	—	2	4	no	—
116	C₂²×C₄×C₅₃	—	2	4	no	—
117	C₂²×C₄×C₅₃	—	4	24	no	—
118	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
119	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
120	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
121	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
122	C₂×C₂²⋊D₅₃	—	4	48	no	—
123	C₂×C₂²⋊D₅₃	—	4	48	no	—
124	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
125	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
126	C₂²×Dic₅₃	—	2	8	no	—
127	C₂²×Dic₅₃	—	2	8	no	—
128	C₂²×Dic₅₃	—	2	8	no	—
129	C₂²×Dic₅₃	—	4	24	no	—
130	C₂²×Dic₅₃	—	4	48	no	—
131	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
132	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
133	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
134	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
135	C₂²×Dic₅₃	C₂⁴	1	32	yes	complemented
136	C₂×Dic₁₀₆	—	4	24	no	—
137	C₂×Dic₁₀₆	C₂⁴	1	16	yes	complemented
138	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
139	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
140	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
141	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	complemented
142	#11719921	C₂⁴	1	32	yes	—
143	#11719921	C₂⁴	1	32	yes	—
144	#11719921	C₂⁴	1	32	yes	—
145	#11719921	C₂⁴	1	32	yes	—
146	C₄⋊Dic₅₃	C₂⁴	1	32	yes	complemented
147	C₄⋊Dic₅₃	C₂⁴	1	16	yes	complemented
148	#11719919	C₂⁴	1	32	yes	complemented
149	C₄×Dic₅₃	C₂⁴	1	32	yes	complemented
150	C₂³×C₅₃	—	2	32	no	—
151	C₂³×C₅₃	—	2	12	no	—
152	C₂³×C₅₃	—	2	12	no	—
153	C₂³×C₅₃	—	2	24	no	—
154	C₂³×C₅₃	—	2	24	no	—
155	C₂³×C₅₃	—	2	24	no	—
156	C₂³×C₅₃	—	4	32	no	—
157	C₂³×C₅₃	C₂×#41	1	24	yes	—
158	C₂³×C₅₃	C₂×#41	1	4	yes	—
159	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
160	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
161	C₂³×C₅₃	C₂⁵	1	8	yes	—
162	C₂³×C₅₃	C₂⁵	1	4	yes	—
163	C₂³×C₅₃	C₂⁵	1	3	yes	—
164	C₂²×D₅₃	—	2	32	no	—
165	C₂²×D₅₃	—	2	24	no	—
166	C₂²×D₅₃	—	2	24	no	—
167	C₂²×D₅₃	—	2	24	no	—
168	C₂²×D₅₃	—	4	16	no	—
169	C₂²×D₅₃	—	4	16	no	—
170	C₂²×D₅₃	—	4	192	no	—
171	C₂²×D₅₃	—	4	32	no	—
172	C₂²×D₅₃	C₂⁵	1	8	yes	—
173	D₄×C₅₃	—	2	64	no	—
174	D₄×C₅₃	—	4	16	no	—
175	C₂×C₄×C₅₃	—	2	24	no	—
176	C₂×C₄×C₅₃	—	2	12	no	—
177	C₂×C₄×C₅₃	—	2	12	no	—
178	C₂×C₄×C₅₃	—	4	16	no	—
179	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
180	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
181	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
182	C₂²⋊D₅₃	—	4	32	no	—
183	C₂²⋊D₅₃	—	4	32	no	—
184	C₂×Dic₅₃	—	2	24	no	—
185	C₂×Dic₅₃	—	2	24	no	—
186	C₂×Dic₅₃	—	2	24	no	—
187	C₂×Dic₅₃	—	4	16	no	—
188	C₂×Dic₅₃	—	4	32	no	—
189	C₂×Dic₅₃	C₂⁵	1	8	yes	—
190	C₂×Dic₅₃	C₂⁵	1	8	yes	—
191	C₂×Dic₅₃	C₂⁵	1	8	yes	—
192	Dic₁₀₆	—	4	16	no	—
193	C₂²×#538	—	53	1	no	maximal, Sylow
194	C₂²×C₅₃	—	2	4	no	—
195	C₂²×C₅₃	—	2	4	no	—
196	C₂²×C₅₃	—	2	48	no	—
197	C₂²×C₅₃	—	2	8	no	—
198	C₂²×C₅₃	—	2	8	no	—
199	C₂²×C₅₃	—	2	8	no	—
200	C₂²×C₅₃	—	4	48	no	—
201	C₂²×C₅₃	#538	1	16	yes	complemented
202	C₂²×C₅₃	C₂²×#41	1	8	yes	—
203	C₂²×C₅₃	C₂²×#41	1	6	yes	—
204	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
205	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
206	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
207	C₂×D₅₃	—	2	8	no	—
208	C₂×D₅₃	—	2	8	no	—
209	C₂×D₅₃	—	2	8	no	—
210	C₂×D₅₃	—	4	64	no	—
211	C₂×D₅₃	—	4	48	no	—
212	C₄×C₅₃	—	2	8	no	—
213	C₄×C₅₃	—	2	4	no	—
214	C₄×C₅₃	—	2	4	no	—
215	Dic₅₃	—	2	8	no	—
216	Dic₅₃	—	2	8	no	—
217	Dic₅₃	—	2	8	no	—
218	C₂×#538	—	53	48	no	—
219	C₂²×#126	—	53	2	no	—
220	C₂²×#124	—	53	2	no	—
221	C₂²×#123	—	53	2	no	—
222	C₂²×C₂²⋊Q₈	—	53	1	no	—
223	C₂²×#121	—	53	2	no	—
224	C₂²×#121	D₅₃	1	1	yes	characteristic, complemented
225	C₂²×#120	—	53	2	no	—
226	C₂²×C₄×D₄	—	53	2	no	—
227	C₂³×C₂²⋊C₄	—	53	1	no	—
228	C₂×C₅₃	—	2	8	no	—
229	C₂×C₅₃	—	4	8	no	—
230	C₂×C₅₃	C₂×#538	1	12	yes	complemented
231	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
232	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
233	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
234	D₅₃	—	4	8	no	—
235	C₂⁶	—	53	1	no	—
236	C₂³×Q₈	—	53	1	no	—
237	C₂³×D₄	—	53	2	no	—
238	C₂³×D₄	—	53	2	no	—
239	C₂³×D₄	C₂×D₅₃	1	2	yes	—
240	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
241	C₂⁴×C₄	—	53	1	no	—
242	C₂⁴×C₄	—	53	2	no	—
243	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
244	#538	—	53	256	no	—
245	C₂×#126	—	53	48	no	—
246	C₂×#124	—	53	48	no	—
247	C₂×#123	—	53	48	no	—
248	C₂×C₂²⋊Q₈	—	53	24	no	—
249	C₂×#121	—	53	48	no	—
250	C₂×#121	C₂×D₅₃	1	24	yes	complemented
251	C₂×#120	—	53	48	no	—
252	C₂×C₄×D₄	—	53	48	no	—
253	C₂²×C₄⋊C₄	—	53	2	no	—
254	C₂²×C₄⋊C₄	—	53	2	no	—
255	C₂²×C₄⋊C₄	—	53	1	no	—
256	C₂²×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
257	C₂²×C₂²⋊C₄	—	53	2	no	—
258	C₂²×C₂²⋊C₄	—	53	2	no	—
259	C₂²×C₂²⋊C₄	—	53	2	no	—
260	C₂²×C₂²⋊C₄	—	53	2	no	—
261	C₂²×C₂²⋊C₄	—	53	2	no	—
262	C₂²×C₂²⋊C₄	—	53	2	no	—
263	C₂²×C₂²⋊C₄	—	53	24	no	—
264	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	complemented
265	C₂²×C₄²	—	53	2	no	—
266	C₅₃	C₂²×#538	1	1	yes	characteristic, complemented, Sylow
267	C₂⁵	—	53	2	no	—
268	C₂⁵	—	53	12	no	—
269	C₂⁵	—	106	8	no	—
270	C₂⁵	—	212	4	no	—
271	C₂⁵	—	212	4	no	—
272	C₂⁵	C₂²×D₅₃	1	2	yes	—
273	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
274	C₂²×Q₈	—	53	12	no	—
275	C₂²×Q₈	—	212	4	no	—
276	C₂²×D₄	—	2	16	no	—
277	C₂²×D₄	—	4	4	no	—
278	C₂²×D₄	—	53	24	no	—
279	C₂²×D₄	—	53	24	no	—
280	C₂²×D₄	—	212	8	no	—
281	C₂²×D₄	—	212	8	no	—
282	C₂²×D₄	C₂²×D₅₃	1	24	yes	—
283	C₂²×D₄	C₂²×D₅₃	1	12	yes	complemented
284	C₂³×C₄	—	4	4	no	—
285	C₂³×C₄	—	53	2	no	—
286	C₂³×C₄	—	53	2	no	—
287	C₂³×C₄	—	53	2	no	—
288	C₂³×C₄	—	53	12	no	—
289	C₂³×C₄	—	53	24	no	—
290	C₂³×C₄	—	212	4	no	—
291	C₂³×C₄	—	212	8	no	—
292	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
293	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
294	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
295	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
296	#126	—	53	128	no	—
297	#124	—	53	128	no	—
298	#123	—	53	128	no	—
299	C₂²⋊Q₈	—	53	64	no	—
300	#121	—	53	128	no	—
301	#121	C₂²×D₅₃	1	64	yes	complemented
302	#120	—	53	128	no	—
303	C₄×D₄	—	53	128	no	—
304	C₂×C₄⋊C₄	—	53	24	no	—
305	C₂×C₄⋊C₄	—	53	24	no	—
306	C₂×C₄⋊C₄	—	53	12	no	—
307	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	complemented
308	C₂×C₂²⋊C₄	—	53	24	no	—
309	C₂×C₂²⋊C₄	—	53	24	no	—
310	C₂×C₂²⋊C₄	—	53	24	no	—
311	C₂×C₂²⋊C₄	—	53	24	no	—
312	C₂×C₂²⋊C₄	—	53	64	no	—
313	C₂×C₂²⋊C₄	—	53	24	no	—
314	C₂×C₂²⋊C₄	—	53	24	no	—
315	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	complemented
316	C₂×C₄²	—	53	24	no	—
317	C₂⁴	—	2	4	no	—
318	C₂⁴	—	2	4	no	—
319	C₂⁴	—	2	8	no	—
320	C₂⁴	—	2	8	no	—
321	C₂⁴	—	2	8	no	—
322	C₂⁴	—	53	12	no	—
323	C₂⁴	—	53	16	no	—
324	C₂⁴	—	106	48	no	—
325	C₂⁴	—	106	8	no	—
326	C₂⁴	—	106	8	no	—
327	C₂⁴	—	106	8	no	—
328	C₂⁴	—	212	24	no	—
329	C₂⁴	—	212	24	no	—
330	C₂⁴	—	212	64	no	—
331	C₂⁴	#11719947	1	8	yes	—
332	C₂⁴	C₂³×D₅₃	1	12	yes	—
333	C₂⁴	C₂³×D₅₃	1	6	yes	—
334	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
335	C₂×Q₈	—	53	16	no	—
336	C₂×Q₈	—	212	24	no	—
337	C₂×D₄	—	2	96	no	—
338	C₂×D₄	—	4	24	no	—
339	C₂×D₄	—	53	32	no	—
340	C₂×D₄	—	53	32	no	—
341	C₂×D₄	—	212	48	no	—
342	C₂×D₄	—	212	48	no	—
343	C₂×D₄	C₂³×D₅₃	1	32	yes	—
344	C₂×D₄	C₂³×D₅₃	1	16	yes	complemented
345	C₂²×C₄	—	2	8	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	4	no	—
348	C₂²×C₄	—	4	24	no	—
349	C₂²×C₄	—	53	12	no	—
350	C₂²×C₄	—	53	12	no	—
351	C₂²×C₄	—	53	12	no	—
352	C₂²×C₄	—	53	16	no	—
353	C₂²×C₄	—	53	32	no	—
354	C₂²×C₄	—	106	8	no	—
355	C₂²×C₄	—	106	8	no	—
356	C₂²×C₄	—	106	8	no	—
357	C₂²×C₄	—	212	24	no	—
358	C₂²×C₄	—	212	48	no	—
359	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
360	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
361	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
362	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
363	C₄⋊C₄	—	53	32	no	—
364	C₄⋊C₄	—	53	32	no	—
365	C₄⋊C₄	—	53	16	no	—
366	C₄⋊C₄	C₂³×D₅₃	1	16	yes	complemented
367	C₂²⋊C₄	—	53	32	no	—
368	C₂²⋊C₄	—	53	32	no	—
369	C₂²⋊C₄	—	53	32	no	—
370	C₂²⋊C₄	—	53	32	no	—
371	C₂²⋊C₄	—	53	32	no	—
372	C₂²⋊C₄	—	53	32	no	—
373	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	complemented
374	C₄²	—	53	32	no	—
375	C₂³	—	2	12	no	—
376	C₂³	—	2	12	no	—
377	C₂³	—	2	32	no	—
378	C₂³	—	2	24	no	—
379	C₂³	—	2	24	no	—
380	C₂³	—	2	24	no	—
381	C₂³	—	4	32	no	—
382	C₂³	—	53	8	no	—
383	C₂³	—	106	32	no	—
384	C₂³	—	106	24	no	—
385	C₂³	—	106	24	no	—
386	C₂³	—	106	24	no	—
387	C₂³	—	212	16	no	—
388	C₂³	—	212	16	no	—
389	C₂³	—	212	192	no	—
390	C₂³	—	212	32	no	—
391	C₂³	C₂×#11719947	1	24	yes	—
392	C₂³	C₂×#11719947	1	4	yes	—
393	C₂³	#13133262	1	4	yes	—
394	C₂³	#13133263	1	4	yes	—
395	C₂³	C₂⁴×D₅₃	1	8	yes	—
396	C₂³	C₂⁴×D₅₃	1	4	yes	—
397	C₂³	C₂⁴×D₅₃	1	3	yes	—
398	Q₈	—	212	16	no	—
399	D₄	—	2	64	no	—
400	D₄	—	4	16	no	—
401	D₄	—	212	32	no	—
402	D₄	—	212	32	no	—
403	C₂×C₄	—	2	24	no	—
404	C₂×C₄	—	2	12	no	—
405	C₂×C₄	—	2	12	no	—
406	C₂×C₄	—	4	16	no	—
407	C₂×C₄	—	53	8	no	—
408	C₂×C₄	—	53	8	no	—
409	C₂×C₄	—	53	8	no	—
410	C₂×C₄	—	106	24	no	—
411	C₂×C₄	—	106	24	no	—
412	C₂×C₄	—	106	24	no	—
413	C₂×C₄	—	212	16	no	—
414	C₂×C₄	—	212	32	no	—
415	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
416	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
417	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
418	C₂²	—	2	48	no	—
419	C₂²	—	2	4	no	—
420	C₂²	—	2	4	no	—
421	C₂²	—	2	8	no	—
422	C₂²	—	2	8	no	—
423	C₂²	—	2	8	no	—
424	C₂²	—	4	48	no	—
425	C₂²	—	106	8	no	—
426	C₂²	—	106	8	no	—
427	C₂²	—	106	8	no	—
428	C₂²	—	212	64	no	—
429	C₂²	—	212	48	no	—
430	C₂²	#16811210	1	16	yes	direct product factor
431	C₂²	C₂²×#11719947	1	8	yes	—
432	C₂²	C₂²×#11719947	1	6	yes	—
433	C₂²	C₂×#13133262	1	6	yes	—
434	C₂²	C₂×#13133263	1	6	yes	—
435	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
436	C₄	—	2	8	no	—
437	C₄	—	2	4	no	—
438	C₄	—	2	4	no	—
439	C₄	—	106	8	no	—
440	C₄	—	106	8	no	—
441	C₄	—	106	8	no	—
442	C₂	—	2	8	no	—
443	C₂	—	4	8	no	—
444	C₂	—	212	8	no	—
445	C₂	C₂×#16811210	1	12	yes	direct product factor
446	C₂	C₂³×#11719947	1	1	yes	characteristic
447	C₂	C₂²×#13133262	1	1	yes	characteristic
448	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×#121×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂²×#121	D₅₃
hypocenter	C₅₃	C₂²×#538
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#121

Complements

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

N	H ≅ G/N	type
1. C₂×#16811210	C₂	direct (N × H)
2. C₂²×#13133212	C₂	semidirect
3. C₂²×#13133203	C₂	semidirect
4. C₂²×#13133201	C₂	semidirect
5. C₂²×#13133196	C₂	semidirect
6. C₂²×#13133189	C₂	semidirect
7. C₂²×#13133185	C₂	semidirect
8. C₂²×D₄×Dic₅₃	C₂	semidirect
9. C₂³×#11719921	C₂	semidirect
10. C₂²×#121×C₅₃	C₂	semidirect
12. C₂³×Dic₁₀₆	C₂²	semidirect
16. C₂⁴×Dic₅₃	C₂²	semidirect
17. #16811210	C₂²	direct (N × H)
18. C₂×#13133212	C₂²	semidirect
19. C₂×#13133203	C₂²	semidirect
20. C₂×#13133201	C₂²	semidirect
21. C₂×#13133196	C₂²	semidirect
22. C₂×#13133189	C₂²	semidirect
23. C₂×#13133185	C₂²	semidirect
24. C₂×D₄×Dic₅₃	C₂²	semidirect
25. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
26. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
27. C₂²×#11719919	C₂²	semidirect
29. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
34. C₂²×#11719921	C₂²	semidirect
35. C₂²×C₄×Dic₅₃	C₂²	semidirect
37. C₂³×D₄×C₅₃	C₂²	semidirect
39. C₂×#121×C₅₃	C₂²	semidirect
40. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
41. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
48. C₂²×Dic₁₀₆	C₂³	semidirect
59. C₂³×Dic₅₃	C₂³	semidirect
60. #13133212	C₂³	semidirect
61. #13133203	C₂³	semidirect
62. #13133201	C₂³	semidirect
63. #13133196	C₂³	semidirect
64. #13133189	C₂³	semidirect
65. #13133185	C₂³	semidirect
66. D₄×Dic₅₃	C₂³	semidirect
67. C₂×C₄⋊Dic₅₃	C₂³	semidirect
68. C₂×C₄⋊Dic₅₃	C₂³	semidirect
69. C₂×#11719919	C₂³	semidirect
71. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
76. C₂×#11719921	C₂³	semidirect
77. C₂×C₄×Dic₅₃	C₂³	semidirect
83. C₂²×D₄×C₅₃	C₂³	semidirect
89. #121×C₅₃	C₂³	semidirect
90. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
91. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
113. C₂×D₄×C₅₃	C₂⁴	semidirect
135. C₂²×Dic₅₃	C₂⁴	semidirect
137. C₂×Dic₁₀₆	C₂⁴	semidirect
138. C₄⋊C₄×C₅₃	C₂⁴	semidirect
139. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
141. C₂²⋊Dic₅₃	C₂⁴	semidirect
146. C₄⋊Dic₅₃	C₂⁴	semidirect
147. C₄⋊Dic₅₃	C₂⁴	semidirect
148. #11719919	C₂⁴	semidirect
149. C₄×Dic₅₃	C₂⁴	semidirect
201. C₂²×C₅₃	#538	semidirect
224. C₂²×#121	D₅₃	semidirect
230. C₂×C₅₃	C₂×#538	semidirect
240. C₂³×D₄	C₂×D₅₃	semidirect
250. C₂×#121	C₂×D₅₃	semidirect
256. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
264. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
266. C₅₃	C₂²×#538	semidirect
283. C₂²×D₄	C₂²×D₅₃	semidirect
301. #121	C₂²×D₅₃	semidirect
307. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
315. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
344. C₂×D₄	C₂³×D₅₃	semidirect
366. C₄⋊C₄	C₂³×D₅₃	semidirect
373. C₂²⋊C₄	C₂³×D₅₃	semidirect
430. C₂²	#16811210	direct (N × H)
445. C₂	C₂×#16811210	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×#16811210	2
C₂²×#13133212	2
C₂²×#13133203	2
C₂²×#13133201	2
C₂²×#13133196	2
C₂²×#13133189	2
C₂²×#13133185	2
C₂²×D₄×Dic₅₃	2
C₂³×#11719921	2
C₂²×#121×C₅₃	2
C₂²×#538	53

Subgroups — C22×#16811210 #28290821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811210 #28290821