Subgroups — C₂×#20288479 #28291986

Order 13568.

12378 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133262	C₂	1	2	yes	maximal, complemented
2	#20288479	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16811277	C₂	1	4	yes	maximal, complemented
4	C₂×#16811268	C₂	1	8	yes	maximal, complemented
5	C₂×D₄×D₂₁₂	C₂	1	4	yes	maximal, complemented
6	C₂×#16811204	C₂	1	2	yes	maximal, complemented
7	C₂×#16811185	C₂	1	4	yes	maximal, complemented
8	C₂×#16811173	C₂	1	2	yes	maximal, complemented
9	C₂²×#13133191	C₂	1	4	yes	maximal, complemented
10	C₂²×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#13133262	—	2	32	no	—
12	C₂×#13133262	C₂²	1	32	yes	complemented
13	C₂²×#11719945	C₂²	1	4	yes	complemented
14	C₂²×#11719947	C₂²	1	8	yes	complemented
15	C₂³×D₂₁₂	C₂²	1	4	yes	complemented
16	C₂³×C₂²⋊D₅₃	C₂²	1	8	yes	complemented
17	C₂²×D₄×D₅₃	C₂²	1	8	yes	complemented
18	#16811277	C₂²	1	64	yes	complemented
19	#16811268	C₂²	1	128	yes	complemented
20	D₄×D₂₁₂	C₂²	1	64	yes	complemented
21	#16811204	C₂²	1	32	yes	complemented
22	#16811185	C₂²	1	64	yes	complemented
23	#16811173	C₂²	1	32	yes	complemented
24	C₂×#13133217	C₂²	1	2	yes	complemented
25	C₂×#13133214	C₂²	1	2	yes	complemented
26	C₂×#13133205	C₂²	1	4	yes	complemented
27	C₂×#13133201	C₂²	1	16	yes	complemented
28	C₂×#13133197	C₂²	1	8	yes	complemented
29	C₂×#13133192	C₂²	1	8	yes	complemented
30	C₂×#13133191	C₂²	1	4	yes	complemented
31	C₂×#13133191	C₂²	1	8	yes	complemented
32	C₂×#13133191	C₂²	1	8	yes	complemented
33	C₂×#13133191	C₂²	1	8	yes	complemented
34	C₂×#13133191	C₂²	1	4	yes	complemented
35	C₂×#13133191	C₂²	1	64	yes	complemented
36	C₂×#13133187	C₂²	1	16	yes	complemented
37	C₂×C₄×D₂₁₂	C₂²	1	4	yes	complemented
38	C₂×C₄×D₂₁₂	C₂²	1	4	yes	complemented
39	C₂²×C₄⋊Dic₅₃	C₂²	1	4	yes	complemented
40	C₂²×#11719921	C₂²	1	8	yes	complemented
41	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
42	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	complemented
43	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
44	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
45	C₂×C₄×D₄×C₅₃	C₂²	1	16	yes	complemented
46	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
47	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
48	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
49	C₂⁴×D₅₃	C₂³	1	8	yes	—
50	#13133262	—	2	256	no	—
51	C₂×#11719945	—	2	32	no	—
52	C₂×#11719945	C₂³	1	32	yes	complemented
53	C₂×#11719947	—	2	64	no	—
54	C₂×#11719947	C₂³	1	64	yes	complemented
55	C₂²×Dic₁₀₆	C₂³	1	4	yes	—
56	C₂²×D₂₁₂	—	4	16	no	—
57	C₂²×D₂₁₂	C₂³	1	4	yes	—
58	C₂²×D₂₁₂	C₂³	1	4	yes	—
59	C₂²×D₂₁₂	C₂³	1	4	yes	—
60	C₂²×D₂₁₂	C₂³	1	8	yes	—
61	C₂²×D₂₁₂	C₂³	1	32	yes	complemented
62	C₂²×C₂²⋊D₅₃	—	2	64	no	—
63	C₂²×C₂²⋊D₅₃	C₂³	1	16	yes	—
64	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
65	C₂²×C₂²⋊D₅₃	C₂³	1	64	yes	complemented
66	C₂×D₄×D₅₃	—	2	64	no	—
67	C₂×D₄×D₅₃	C₂³	1	64	yes	complemented
68	C₂³×Dic₅₃	C₂³	1	8	yes	—
69	C₂²×C₄×D₅₃	C₂³	1	8	yes	—
70	#13133217	C₂³	1	16	yes	complemented
71	#13133214	C₂³	1	16	yes	complemented
72	#13133205	C₂³	1	32	yes	complemented
73	#13133201	C₂³	1	128	yes	complemented
74	#13133197	C₂³	1	64	yes	complemented
75	#13133192	C₂³	1	64	yes	complemented
76	#13133191	C₂³	1	64	yes	complemented
77	#13133191	C₂³	1	64	yes	complemented
78	#13133191	C₂³	1	32	yes	complemented
79	#13133191	C₂³	1	64	yes	complemented
80	#13133191	C₂³	1	32	yes	complemented
81	#13133187	C₂³	1	128	yes	complemented
82	C₄×D₂₁₂	C₂³	1	32	yes	complemented
83	C₄×D₂₁₂	C₂³	1	32	yes	complemented
84	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
85	C₂×C₄⋊Dic₅₃	C₂³	1	8	yes	—
86	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
87	C₂×C₄⋊Dic₅₃	C₂³	1	32	yes	complemented
88	C₂×#11719921	C₂³	1	8	yes	—
89	C₂×#11719921	C₂³	1	16	yes	—
90	C₂×#11719921	C₂³	1	8	yes	—
91	C₂×#11719921	C₂³	1	64	yes	complemented
92	C₂⁵×C₅₃	C₂³	1	2	yes	—
93	C₂²×D₄×C₅₃	D₄	1	16	yes	complemented
94	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
95	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
96	C₂³×C₄×C₅₃	—	4	8	no	—
97	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
98	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
99	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
100	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
101	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
102	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
103	C₂³×C₄×C₅₃	C₂³	1	16	yes	complemented
104	C₄×D₄×C₅₃	C₂³	1	32	yes	complemented
105	C₄×D₄×C₅₃	C₂³	1	32	yes	complemented
106	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	complemented
108	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
109	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	complemented
110	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
111	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
112	C₂×C₄²×C₅₃	C₂³	1	8	yes	complemented
113	C₂⁴×C₅₃	—	4	4	no	—
114	C₂⁴×C₅₃	—	4	4	no	—
115	C₂⁴×C₅₃	C₂×D₄	1	16	yes	complemented
116	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
117	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
118	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
119	C₂³×D₅₃	—	2	32	no	—
120	C₂³×D₅₃	—	4	16	no	—
121	C₂³×D₅₃	—	4	16	no	—
122	C₂³×D₅₃	C₂⁴	1	8	yes	—
123	C₂³×D₅₃	C₂⁴	1	32	yes	—
124	C₂×D₄×C₅₃	C₂×D₄	1	32	yes	complemented
125	C₂×D₄×C₅₃	C₂×D₄	1	64	yes	complemented
126	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
127	C₂²×C₄×C₅₃	—	2	4	no	—
128	C₂²×C₄×C₅₃	—	2	16	no	—
129	C₂²×C₄×C₅₃	—	4	32	no	—
130	C₂²×C₄×C₅₃	—	4	4	no	—
131	C₂²×C₄×C₅₃	—	4	8	no	—
132	C₂²×C₄×C₅₃	—	4	2	no	—
133	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	complemented
134	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
135	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
136	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
137	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
138	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
139	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
140	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
141	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
142	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
143	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
145	C₂×C₂²⋊D₅₃	—	2	64	no	—
146	C₂×C₂²⋊D₅₃	—	2	32	no	—
147	C₂×C₂²⋊D₅₃	—	2	256	no	—
148	C₂×C₂²⋊D₅₃	C₂⁴	1	64	yes	—
149	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
150	C₂²×Dic₅₃	—	2	32	no	—
151	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
152	C₂²×Dic₅₃	C₂⁴	1	32	yes	—
153	#11719947	—	2	256	no	—
154	D₄×D₅₃	—	2	256	no	—
155	#11719945	—	2	128	no	—
156	C₂×D₂₁₂	—	2	16	no	—
157	C₂×D₂₁₂	—	2	16	no	—
158	C₂×D₂₁₂	—	4	64	no	—
159	C₂×D₂₁₂	—	4	16	no	—
160	C₂×D₂₁₂	—	4	8	no	—
161	C₂×D₂₁₂	C₂⁴	1	16	yes	—
162	C₂×D₂₁₂	C₂⁴	1	16	yes	—
163	C₂×D₂₁₂	C₂⁴	1	16	yes	—
164	C₂×D₂₁₂	C₂⁴	1	32	yes	—
165	C₂×C₄×D₅₃	—	2	32	no	—
166	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
167	C₂×Dic₁₀₆	—	2	16	no	—
168	C₂×Dic₁₀₆	C₂⁴	1	16	yes	—
169	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
170	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
171	C₄²×C₅₃	C₂⁴	1	8	yes	—
172	C₄²×C₅₃	C₂⁴	1	8	yes	—
173	#11719921	C₂⁴	1	32	yes	—
174	#11719921	C₂⁴	1	64	yes	—
175	#11719921	C₂⁴	1	32	yes	—
176	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
177	C₄⋊Dic₅₃	C₂⁴	1	32	yes	—
178	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
179	C₂³×C₅₃	—	2	2	no	—
180	C₂³×C₅₃	—	2	8	no	—
181	C₂³×C₅₃	—	2	2	no	—
182	C₂³×C₅₃	—	2	8	no	—
183	C₂³×C₅₃	—	4	8	no	—
184	C₂³×C₅₃	—	4	32	no	—
185	C₂³×C₅₃	—	4	8	no	—
186	C₂³×C₅₃	C₂²×D₄	1	16	yes	—
187	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
188	C₂³×C₅₃	C₂²×D₄	1	32	yes	complemented
189	C₂³×C₅₃	C₂⁵	1	8	yes	—
190	C₂³×C₅₃	C₂⁵	1	2	yes	—
191	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
192	C₂²×D₅₃	—	2	16	no	—
193	C₂²×D₅₃	—	2	16	no	—
194	C₂²×D₅₃	—	2	16	no	—
195	C₂²×D₅₃	—	2	64	no	—
196	C₂²×D₅₃	—	4	32	no	—
197	C₂²×D₅₃	—	4	32	no	—
198	C₂²×D₅₃	—	8	64	no	—
199	C₂²×D₅₃	C₂⁵	1	16	yes	—
200	D₄×C₅₃	C₂²×D₄	1	64	yes	complemented
201	C₂×C₄×C₅₃	—	2	8	no	—
202	C₂×C₄×C₅₃	—	2	2	no	—
203	C₂×C₄×C₅₃	—	2	4	no	—
204	C₂×C₄×C₅₃	—	2	32	no	—
205	C₂×C₄×C₅₃	—	2	8	no	—
206	C₂×C₄×C₅₃	—	4	8	no	—
207	C₂×C₄×C₅₃	—	4	16	no	—
208	C₂×C₄×C₅₃	—	4	4	no	—
209	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
210	C₂×C₄×C₅₃	C₂²×D₄	1	16	yes	complemented
211	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	complemented
212	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
213	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
214	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
215	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
216	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
217	C₂²⋊D₅₃	—	2	64	no	—
218	C₂²⋊D₅₃	—	2	128	no	—
219	C₂×Dic₅₃	—	2	16	no	—
220	C₂×Dic₅₃	—	2	64	no	—
221	C₂×Dic₅₃	C₂⁵	1	16	yes	—
222	D₂₁₂	—	2	32	no	—
223	D₂₁₂	—	2	32	no	—
224	D₂₁₂	—	4	32	no	—
225	D₂₁₂	—	4	16	no	—
226	C₄×D₅₃	—	2	64	no	—
227	Dic₁₀₆	—	2	32	no	—
228	C₂×#3496	—	53	1	no	maximal, Sylow
229	C₂²×C₅₃	—	2	2	no	—
230	C₂²×C₅₃	—	2	8	no	—
231	C₂²×C₅₃	—	2	2	no	—
232	C₂²×C₅₃	—	2	8	no	—
233	C₂²×C₅₃	—	2	32	no	—
234	C₂²×C₅₃	—	2	8	no	—
235	C₂²×C₅₃	—	4	32	no	—
236	C₂²×C₅₃	C₂³×D₄	1	16	yes	—
237	C₂²×C₅₃	C₂³×D₄	1	4	yes	—
238	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
239	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
240	C₂²×C₅₃	#585	1	2	yes	—
241	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
242	C₂×D₅₃	—	2	16	no	—
243	C₂×D₅₃	—	2	16	no	—
244	C₂×D₅₃	—	2	16	no	—
245	C₂×D₅₃	—	4	32	no	—
246	C₂×D₅₃	—	8	64	no	—
247	C₄×C₅₃	—	2	8	no	—
248	C₄×C₅₃	—	2	2	no	—
249	C₄×C₅₃	—	2	4	no	—
250	C₄×C₅₃	C₂³×D₄	1	8	yes	—
251	C₄×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	complemented
252	Dic₅₃	—	2	16	no	—
253	C₂²×H(5,𝔽₂)	—	53	2	no	—
254	#3496	—	53	32	no	—
255	C₂×#547	—	53	4	no	—
256	C₂×#546	—	53	8	no	—
257	C₂×D₄²	—	53	4	no	—
258	C₂×#537	—	53	2	no	—
259	C₂×#534	—	53	4	no	—
260	C₂×#532	—	53	2	no	—
261	C₂²×#121	—	53	4	no	—
262	C₂²×C₄×D₄	D₅₃	1	1	yes	characteristic, complemented
263	C₂×C₅₃	—	2	16	no	—
264	C₂×C₅₃	—	2	4	no	—
265	C₂×C₅₃	#3496	1	4	yes	complemented
266	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
267	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
268	C₂×C₅₃	C₂×#585	1	1	yes	characteristic
269	D₅₃	—	4	16	no	—
270	C₂×H(5,𝔽₂)	—	53	32	no	—
271	C₂×H(5,𝔽₂)	—	106	32	no	—
272	C₂²×#41	—	53	8	no	—
273	C₂²×#41	—	53	4	no	—
274	C₂³×D₄	—	53	8	no	—
275	C₂³×D₄	—	53	8	no	—
276	C₂³×D₄	—	53	4	no	—
277	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
278	C₂⁴×C₄	C₂×D₅₃	1	2	yes	complemented
279	#547	—	53	64	no	—
280	#546	—	53	128	no	—
281	D₄²	—	53	64	no	—
282	#537	—	53	32	no	—
283	#534	—	53	64	no	—
284	#532	—	53	32	no	—
285	C₂×C₄⋊Q₈	—	53	2	no	—
286	C₂×#127	—	53	2	no	—
287	C₂×#124	—	53	4	no	—
288	C₂×#123	—	53	16	no	—
289	C₂×C₂²⋊Q₈	—	53	8	no	—
290	C₂×#121	—	53	4	no	—
291	C₂×#121	—	53	8	no	—
292	C₂×#121	—	53	8	no	—
293	C₂×#121	—	53	8	no	—
294	C₂×#121	—	53	4	no	—
295	C₂×#121	—	53	64	no	—
296	C₂×#121	—	53	8	no	—
297	C₂×#120	—	53	16	no	—
298	C₂×C₄×D₄	—	53	4	no	—
299	C₂×C₄×D₄	—	53	4	no	—
300	C₂×C₄×D₄	C₂×D₅₃	1	4	yes	complemented
301	C₂×C₄×D₄	C₂×D₅₃	1	4	yes	complemented
302	C₂×C₄×D₄	C₂×D₅₃	1	16	yes	complemented
303	C₂²×C₄⋊C₄	—	53	4	no	—
304	C₂²×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
305	C₂²×C₂²⋊C₄	—	53	8	no	—
306	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	complemented
307	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
308	C₅₃	C₂×#3496	1	1	yes	characteristic, complemented, Sylow
309	C₂⁵	—	53	8	no	—
310	C₂⁵	C₂²×D₅₃	1	2	yes	—
311	H(5,𝔽₂)	—	106	256	no	—
312	C₂×#41	—	53	64	no	—
313	C₂×#41	—	53	32	no	—
314	C₂×#41	—	106	64	no	—
315	C₂×#41	—	106	32	no	—
316	C₂²×Q₈	—	53	4	no	—
317	C₂²×D₄	—	53	16	no	—
318	C₂²×D₄	—	53	8	no	—
319	C₂²×D₄	—	53	64	no	—
320	C₂²×D₄	—	53	64	no	—
321	C₂²×D₄	—	53	4	no	—
322	C₂²×D₄	—	53	4	no	—
323	C₂²×D₄	—	53	4	no	—
324	C₂²×D₄	—	53	8	no	—
325	C₂²×D₄	—	53	32	no	—
326	C₂²×D₄	—	106	64	no	—
327	C₂²×D₄	—	106	64	no	—
328	C₂²×D₄	—	212	16	no	—
329	C₂²×D₄	D₂₁₂	1	16	yes	complemented
330	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
331	C₂²×D₄	C₂²×D₅₃	1	8	yes	—
332	C₂³×C₄	—	4	8	no	—
333	C₂³×C₄	—	53	8	no	—
334	C₂³×C₄	—	53	8	no	—
335	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
336	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
337	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
338	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
339	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
340	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
341	C₂³×C₄	C₂²×D₅₃	1	16	yes	complemented
342	C₄⋊Q₈	—	53	16	no	—
343	#127	—	53	16	no	—
344	#124	—	53	32	no	—
345	#123	—	53	128	no	—
346	C₂²⋊Q₈	—	53	64	no	—
347	#121	—	53	32	no	—
348	#121	—	53	64	no	—
349	#121	—	53	64	no	—
350	#121	—	53	64	no	—
351	#121	—	53	32	no	—
352	#121	—	53	64	no	—
353	#120	—	53	128	no	—
354	C₄×D₄	—	53	32	no	—
355	C₄×D₄	—	53	32	no	—
356	C₄×D₄	C₂²×D₅₃	1	32	yes	complemented
357	C₄×D₄	C₂²×D₅₃	1	32	yes	complemented
358	C₂×C₄⋊C₄	—	53	4	no	—
359	C₂×C₄⋊C₄	—	53	8	no	—
360	C₂×C₄⋊C₄	—	53	32	no	—
361	C₂×C₄⋊C₄	—	53	4	no	—
362	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
363	C₂×C₄⋊C₄	C₂²×D₅₃	1	8	yes	complemented
364	C₂×C₂²⋊C₄	—	53	8	no	—
365	C₂×C₂²⋊C₄	—	53	16	no	—
366	C₂×C₂²⋊C₄	—	53	64	no	—
367	C₂×C₂²⋊C₄	—	53	8	no	—
368	C₂×C₂²⋊C₄	C₂²×D₅₃	1	8	yes	—
369	C₂×C₂²⋊C₄	C₂²×D₅₃	1	16	yes	complemented
370	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
371	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
372	C₂×C₄²	C₂²×D₅₃	1	8	yes	complemented
373	C₂⁴	—	4	4	no	—
374	C₂⁴	—	4	4	no	—
375	C₂⁴	—	53	8	no	—
376	C₂⁴	—	53	32	no	—
377	C₂⁴	—	106	32	no	—
378	C₂⁴	—	212	16	no	—
379	C₂⁴	—	212	16	no	—
380	C₂⁴	C₂×D₂₁₂	1	16	yes	complemented
381	C₂⁴	C₂³×D₅₃	1	4	yes	—
382	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
383	C₂⁴	C₂³×D₅₃	1	8	yes	—
384	#41	—	106	256	no	—
385	#41	—	106	128	no	—
386	C₂×Q₈	—	53	16	no	—
387	C₂×Q₈	—	106	16	no	—
388	C₂×D₄	—	53	64	no	—
389	C₂×D₄	—	53	32	no	—
390	C₂×D₄	—	53	16	no	—
391	C₂×D₄	—	53	16	no	—
392	C₂×D₄	—	53	16	no	—
393	C₂×D₄	—	53	32	no	—
394	C₂×D₄	—	106	64	no	—
395	C₂×D₄	—	106	32	no	—
396	C₂×D₄	—	106	256	no	—
397	C₂×D₄	—	106	256	no	—
398	C₂×D₄	—	106	16	no	—
399	C₂×D₄	—	106	16	no	—
400	C₂×D₄	—	212	64	no	—
401	C₂×D₄	—	212	8	no	—
402	C₂×D₄	—	212	16	no	—
403	C₂×D₄	C₂×D₂₁₂	1	32	yes	complemented
404	C₂×D₄	C₂×D₂₁₂	1	64	yes	complemented
405	C₂×D₄	C₂³×D₅₃	1	16	yes	—
406	C₂²×C₄	—	2	4	no	—
407	C₂²×C₄	—	2	16	no	—
408	C₂²×C₄	—	4	32	no	—
409	C₂²×C₄	—	4	8	no	—
410	C₂²×C₄	—	4	4	no	—
411	C₂²×C₄	—	4	2	no	—
412	C₂²×C₄	—	53	32	no	—
413	C₂²×C₄	—	53	8	no	—
414	C₂²×C₄	—	53	32	no	—
415	C₂²×C₄	—	106	32	no	—
416	C₂²×C₄	—	106	32	no	—
417	C₂²×C₄	C₂×D₂₁₂	1	8	yes	complemented
418	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
419	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
420	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
421	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
422	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
423	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
424	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
425	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
426	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
427	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
428	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
429	C₄⋊C₄	—	53	16	no	—
430	C₄⋊C₄	—	53	32	no	—
431	C₄⋊C₄	—	53	16	no	—
432	C₄⋊C₄	C₂³×D₅₃	1	16	yes	—
433	C₂²⋊C₄	—	53	32	no	—
434	C₂²⋊C₄	—	53	64	no	—
435	C₂²⋊C₄	—	53	32	no	—
436	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	—
437	C₄²	C₂³×D₅₃	1	8	yes	—
438	C₄²	C₂³×D₅₃	1	8	yes	—
439	C₂³	—	2	2	no	—
440	C₂³	—	2	8	no	—
441	C₂³	—	2	2	no	—
442	C₂³	—	2	8	no	—
443	C₂³	—	4	8	no	—
444	C₂³	—	4	32	no	—
445	C₂³	—	4	8	no	—
446	C₂³	—	53	16	no	—
447	C₂³	—	106	64	no	—
448	C₂³	—	106	16	no	—
449	C₂³	—	106	16	no	—
450	C₂³	—	106	16	no	—
451	C₂³	—	424	64	no	—
452	C₂³	—	212	32	no	—
453	C₂³	—	212	32	no	—
454	C₂³	C₂²×D₂₁₂	1	16	yes	—
455	C₂³	C₂²×D₂₁₂	1	4	yes	—
456	C₂³	C₂²×D₂₁₂	1	32	yes	complemented
457	C₂³	C₂⁴×D₅₃	1	8	yes	—
458	C₂³	C₂⁴×D₅₃	1	2	yes	—
459	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
460	Q₈	—	106	32	no	—
461	D₄	—	106	128	no	—
462	D₄	—	106	64	no	—
463	D₄	—	106	32	no	—
464	D₄	—	106	32	no	—
465	D₄	—	212	16	no	—
466	D₄	—	212	32	no	—
467	D₄	C₂²×D₂₁₂	1	64	yes	complemented
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	8	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	8	no	—
472	C₂×C₄	—	2	32	no	—
473	C₂×C₄	—	4	16	no	—
474	C₂×C₄	—	4	8	no	—
475	C₂×C₄	—	4	4	no	—
476	C₂×C₄	—	53	16	no	—
477	C₂×C₄	—	106	16	no	—
478	C₂×C₄	—	106	64	no	—
479	C₂×C₄	—	106	64	no	—
480	C₂×C₄	C₂²×D₂₁₂	1	8	yes	—
481	C₂×C₄	C₂²×D₂₁₂	1	16	yes	complemented
482	C₂×C₄	#13133262	1	4	yes	complemented
483	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
484	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
485	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
486	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
487	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
488	C₂²	—	2	8	no	—
489	C₂²	—	2	32	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	8	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	2	8	no	—
494	C₂²	—	4	32	no	—
495	C₂²	—	106	16	no	—
496	C₂²	—	106	16	no	—
497	C₂²	—	106	16	no	—
498	C₂²	—	424	64	no	—
499	C₂²	—	212	32	no	—
500	C₂²	C₂³×D₂₁₂	1	16	yes	—
501	C₂²	C₂³×D₂₁₂	1	4	yes	—
502	C₂²	C₂³×D₂₁₂	1	2	yes	—
503	C₂²	C₂×#13133262	1	2	yes	—
504	C₂²	#16811459	1	2	yes	—
505	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
506	C₄	—	2	2	no	—
507	C₄	—	2	8	no	—
508	C₄	—	2	4	no	—
509	C₄	—	106	16	no	—
510	C₄	C₂³×D₂₁₂	1	8	yes	—
511	C₄	C₂×#13133262	1	4	yes	complemented
512	C₂	—	2	4	no	—
513	C₂	—	2	16	no	—
514	C₂	—	212	16	no	—
515	C₂	#20288479	1	4	yes	direct product factor
516	C₂	C₂⁴×D₂₁₂	1	1	yes	characteristic
517	C₂	C₂²×#13133262	1	1	yes	characteristic
518	C₂	C₂×#16811459	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×C₄×D₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂²×C₄×D₄	D₅₃
hypocenter	C₅₃	C₂×#3496
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133262	C₂	semidirect
2. #20288479	C₂	direct (N × H)
3. C₂×#16811277	C₂	semidirect
4. C₂×#16811268	C₂	semidirect
5. C₂×D₄×D₂₁₂	C₂	semidirect
6. C₂×#16811204	C₂	semidirect
7. C₂×#16811185	C₂	semidirect
8. C₂×#16811173	C₂	semidirect
9. C₂²×#13133191	C₂	semidirect
10. C₂²×C₄×D₄×C₅₃	C₂	semidirect
12. C₂×#13133262	C₂²	semidirect
13. C₂²×#11719945	C₂²	semidirect
14. C₂²×#11719947	C₂²	semidirect
15. C₂³×D₂₁₂	C₂²	semidirect
16. C₂³×C₂²⋊D₅₃	C₂²	semidirect
17. C₂²×D₄×D₅₃	C₂²	semidirect
18. #16811277	C₂²	semidirect
19. #16811268	C₂²	semidirect
20. D₄×D₂₁₂	C₂²	semidirect
21. #16811204	C₂²	semidirect
22. #16811185	C₂²	semidirect
23. #16811173	C₂²	semidirect
24. C₂×#13133217	C₂²	semidirect
25. C₂×#13133214	C₂²	semidirect
26. C₂×#13133205	C₂²	semidirect
27. C₂×#13133201	C₂²	semidirect
28. C₂×#13133197	C₂²	semidirect
29. C₂×#13133192	C₂²	semidirect
30. C₂×#13133191	C₂²	semidirect
31. C₂×#13133191	C₂²	semidirect
32. C₂×#13133191	C₂²	semidirect
33. C₂×#13133191	C₂²	semidirect
34. C₂×#13133191	C₂²	semidirect
35. C₂×#13133191	C₂²	semidirect
36. C₂×#13133187	C₂²	semidirect
37. C₂×C₄×D₂₁₂	C₂²	semidirect
38. C₂×C₄×D₂₁₂	C₂²	semidirect
39. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
40. C₂²×#11719921	C₂²	semidirect
42. C₂⁴×C₄×C₅₃	C₂²	semidirect
43. C₂×C₄×D₄×C₅₃	C₂²	semidirect
44. C₂×C₄×D₄×C₅₃	C₂²	semidirect
45. C₂×C₄×D₄×C₅₃	C₂²	semidirect
46. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
47. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
48. C₂²×C₄²×C₅₃	C₂²	semidirect
52. C₂×#11719945	C₂³	semidirect
54. C₂×#11719947	C₂³	semidirect
61. C₂²×D₂₁₂	C₂³	semidirect
65. C₂²×C₂²⋊D₅₃	C₂³	semidirect
67. C₂×D₄×D₅₃	C₂³	semidirect
70. #13133217	C₂³	semidirect
71. #13133214	C₂³	semidirect
72. #13133205	C₂³	semidirect
73. #13133201	C₂³	semidirect
74. #13133197	C₂³	semidirect
75. #13133192	C₂³	semidirect
76. #13133191	C₂³	semidirect
77. #13133191	C₂³	semidirect
78. #13133191	C₂³	semidirect
79. #13133191	C₂³	semidirect
80. #13133191	C₂³	semidirect
81. #13133187	C₂³	semidirect
82. C₄×D₂₁₂	C₂³	semidirect
83. C₄×D₂₁₂	C₂³	semidirect
87. C₂×C₄⋊Dic₅₃	C₂³	semidirect
91. C₂×#11719921	C₂³	semidirect
93. C₂²×D₄×C₅₃	D₄	semidirect
103. C₂³×C₄×C₅₃	C₂³	semidirect
104. C₄×D₄×C₅₃	C₂³	semidirect
105. C₄×D₄×C₅₃	C₂³	semidirect
107. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
109. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
112. C₂×C₄²×C₅₃	C₂³	semidirect
115. C₂⁴×C₅₃	C₂×D₄	semidirect
124. C₂×D₄×C₅₃	C₂×D₄	semidirect
125. C₂×D₄×C₅₃	C₂×D₄	semidirect
133. C₂²×C₄×C₅₃	C₂×D₄	semidirect
188. C₂³×C₅₃	C₂²×D₄	semidirect
200. D₄×C₅₃	C₂²×D₄	semidirect
210. C₂×C₄×C₅₃	C₂²×D₄	semidirect
211. C₂×C₄×C₅₃	H(5,𝔽₂)	semidirect
251. C₄×C₅₃	C₂×H(5,𝔽₂)	semidirect
262. C₂²×C₄×D₄	D₅₃	semidirect
265. C₂×C₅₃	#3496	semidirect
278. C₂⁴×C₄	C₂×D₅₃	semidirect
300. C₂×C₄×D₄	C₂×D₅₃	semidirect
301. C₂×C₄×D₄	C₂×D₅₃	semidirect
302. C₂×C₄×D₄	C₂×D₅₃	semidirect
304. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
306. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
307. C₂²×C₄²	C₂×D₅₃	semidirect
308. C₅₃	C₂×#3496	semidirect
329. C₂²×D₄	D₂₁₂	semidirect
341. C₂³×C₄	C₂²×D₅₃	semidirect
356. C₄×D₄	C₂²×D₅₃	semidirect
357. C₄×D₄	C₂²×D₅₃	semidirect
363. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
369. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
372. C₂×C₄²	C₂²×D₅₃	semidirect
380. C₂⁴	C₂×D₂₁₂	semidirect
403. C₂×D₄	C₂×D₂₁₂	semidirect
404. C₂×D₄	C₂×D₂₁₂	semidirect
417. C₂²×C₄	C₂×D₂₁₂	semidirect
456. C₂³	C₂²×D₂₁₂	semidirect
467. D₄	C₂²×D₂₁₂	semidirect
481. C₂×C₄	C₂²×D₂₁₂	semidirect
482. C₂×C₄	#13133262	semidirect
511. C₄	C₂×#13133262	semidirect
515. C₂	#20288479	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133262	2
#20288479	2
C₂×#16811277	2
C₂×#16811268	2
C₂×D₄×D₂₁₂	2
C₂×#16811204	2
C₂×#16811185	2
C₂×#16811173	2
C₂²×#13133191	2
C₂²×C₄×D₄×C₅₃	2
C₂×#3496	53

Subgroups — C2×#20288479 #28291986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288479 #28291986