Subgroups — C₂²×#16811459 #28297985

Order 13568.

60366 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811459	C₂	1	96	yes	maximal, direct product factor
2	C₂²×#13133265	C₂	1	2	yes	maximal, complemented
3	C₂²×#13133264	C₂	1	4	yes	maximal, complemented
4	C₂²×#13133263	C₂	1	4	yes	maximal, complemented
5	C₂²×#13133262	C₂	1	6	yes	maximal, complemented
6	C₂³×#11719945	C₂	1	6	yes	maximal, complemented
7	C₂²×#41×D₅₃	C₂	1	8	yes	maximal, complemented
8	C₂³×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811459	C₂²	1	1024	yes	direct product factor
10	C₂×#13133265	C₂²	1	96	yes	complemented
11	C₂×#13133264	C₂²	1	192	yes	complemented
12	C₂×#13133263	C₂²	1	192	yes	complemented
13	C₂×#13133262	C₂²	1	288	yes	complemented
14	C₂²×#11719949	C₂²	1	8	yes	complemented
15	C₂²×#11719945	C₂²	1	12	yes	complemented
16	C₂²×#11719945	C₂²	1	4	yes	complemented
17	C₂²×#11719945	C₂²	1	12	yes	complemented
18	C₂²×#11719945	C₂²	1	12	yes	complemented
19	C₂²×#11719945	C₂²	1	288	yes	complemented
20	C₂²×#11719947	C₂²	1	24	yes	complemented
21	C₂×#41×D₅₃	C₂²	1	384	yes	complemented
22	C₂³×Dic₁₀₆	C₂²	1	6	yes	complemented
23	C₂²×Q₈×D₅₃	C₂²	1	8	yes	complemented
24	C₂³×D₂₁₂	C₂²	1	6	yes	complemented
25	C₂³×C₂²⋊D₅₃	C₂²	1	12	yes	complemented
26	C₂²×D₄×D₅₃	C₂²	1	24	yes	complemented
27	C₂³×C₄×D₅₃	C₂²	1	12	yes	complemented
28	C₂²×#41×C₅₃	C₂²	1	4	yes	complemented
29	C₂²×#41×C₅₃	C₂²	1	4	yes	complemented
30	C₂²×#41×C₅₃	C₂²	1	48	yes	complemented
31	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
32	C₂³×D₄×C₅₃	C₂²	1	3	yes	complemented
33	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	complemented
34	C₂⁴×D₅₃	C₂³	1	12	yes	—
35	#13133265	C₂³	1	512	yes	complemented
36	#13133264	C₂³	1	1024	yes	complemented
37	#13133263	C₂³	1	1024	yes	complemented
38	#13133262	C₂³	1	1536	yes	complemented
39	C₂×#11719949	C₂³	1	192	yes	complemented
40	C₂×#11719945	C₂³	1	288	yes	complemented
41	C₂×#11719945	C₂³	1	96	yes	complemented
42	C₂×#11719945	C₂³	1	288	yes	complemented
43	C₂×#11719945	C₂³	1	288	yes	complemented
44	C₂×#11719945	C₂³	1	1536	yes	complemented
45	C₂×#11719947	C₂³	1	576	yes	complemented
46	#41×D₅₃	C₂³	1	2048	yes	complemented
47	C₂²×Dic₁₀₆	C₂³	1	12	yes	—
48	C₂²×Dic₁₀₆	C₂³	1	4	yes	—
49	C₂²×Dic₁₀₆	C₂³	1	144	yes	complemented
50	C₂×Q₈×D₅₃	C₂³	1	192	yes	complemented
51	C₂²×D₂₁₂	C₂³	1	12	yes	—
52	C₂²×D₂₁₂	C₂³	1	4	yes	—
53	C₂²×D₂₁₂	C₂³	1	144	yes	complemented
54	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
55	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
56	C₂²×C₂²⋊D₅₃	C₂³	1	288	yes	complemented
57	C₂×D₄×D₅₃	C₂³	1	576	yes	complemented
58	C₂³×Dic₅₃	C₂³	1	12	yes	—
59	C₂²×C₄×D₅₃	C₂³	1	8	yes	—
60	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
61	C₂²×C₄×D₅₃	C₂³	1	288	yes	complemented
62	C₂⁵×C₅₃	C₂³	1	3	yes	—
63	C₂×#41×C₅₃	C₂³	1	96	yes	complemented
64	C₂×#41×C₅₃	C₂³	1	96	yes	complemented
65	C₂×#41×C₅₃	C₂³	1	256	yes	complemented
66	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
67	C₂²×Q₈×C₅₃	C₂³	1	24	yes	complemented
68	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
69	C₂²×D₄×C₅₃	C₂³	1	72	yes	complemented
70	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
72	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
73	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
74	C₂³×C₄×C₅₃	C₂³	1	72	yes	complemented
75	C₂⁴×C₅₃	—	4	12	no	—
76	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
77	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
78	C₂⁴×C₅₃	C₂⁴	1	36	yes	—
79	C₂³×D₅₃	—	4	48	no	—
80	C₂³×D₅₃	C₂⁴	1	8	yes	—
81	C₂³×D₅₃	C₂⁴	1	144	yes	—
82	#41×C₅₃	C₂⁴	1	256	yes	complemented
83	#41×C₅₃	C₂⁴	1	256	yes	complemented
84	C₂×Q₈×C₅₃	C₂⁴	1	48	yes	—
85	C₂×Q₈×C₅₃	C₂⁴	1	64	yes	complemented
86	C₂×D₄×C₅₃	C₂⁴	1	144	yes	—
87	C₂×D₄×C₅₃	C₂⁴	1	192	yes	complemented
88	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
90	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
91	C₂²×C₄×C₅₃	C₂⁴	1	36	yes	—
92	C₂²×C₄×C₅₃	C₂⁴	1	72	yes	—
93	C₂²×C₄×C₅₃	C₂⁴	1	72	yes	—
94	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂⁴	1	192	yes	complemented
96	C₂×C₂²⋊D₅₃	C₂⁴	1	288	yes	—
97	C₂×C₂²⋊D₅₃	C₂⁴	1	96	yes	—
98	C₂×C₂²⋊D₅₃	C₂⁴	1	768	yes	complemented
99	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
100	C₂²×Dic₅₃	C₂⁴	1	144	yes	—
101	#11719949	C₂⁴	1	512	yes	complemented
102	Q₈×D₅₃	C₂⁴	1	512	yes	complemented
103	#11719947	C₂⁴	1	1536	yes	complemented
104	D₄×D₅₃	C₂⁴	1	1536	yes	complemented
105	#11719945	C₂⁴	1	768	yes	complemented
106	#11719945	C₂⁴	1	256	yes	complemented
107	#11719945	C₂⁴	1	768	yes	complemented
108	#11719945	C₂⁴	1	768	yes	complemented
109	C₂×D₂₁₂	C₂⁴	1	144	yes	—
110	C₂×D₂₁₂	C₂⁴	1	48	yes	—
111	C₂×D₂₁₂	C₂⁴	1	384	yes	complemented
112	C₂×C₄×D₅₃	C₂⁴	1	96	yes	—
113	C₂×C₄×D₅₃	C₂⁴	1	288	yes	—
114	C₂×C₄×D₅₃	C₂⁴	1	768	yes	complemented
115	C₂×Dic₁₀₆	C₂⁴	1	144	yes	—
116	C₂×Dic₁₀₆	C₂⁴	1	48	yes	—
117	C₂×Dic₁₀₆	C₂⁴	1	384	yes	complemented
118	C₂³×C₅₃	—	2	24	no	—
119	C₂³×C₅₃	—	2	4	no	—
120	C₂³×C₅₃	—	4	72	no	—
121	C₂³×C₅₃	C₂⁵	1	36	yes	—
122	C₂³×C₅₃	C₂⁵	1	6	yes	—
123	C₂³×C₅₃	C₂⁵	1	48	yes	—
124	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
125	C₂²×D₅₃	—	2	32	no	—
126	C₂²×D₅₃	—	4	288	no	—
127	C₂²×D₅₃	C₂⁵	1	48	yes	—
128	C₂²×D₅₃	C₂⁵	1	192	yes	—
129	Q₈×C₅₃	C₂⁵	1	64	yes	—
130	D₄×C₅₃	C₂⁵	1	192	yes	—
131	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
132	C₂×C₄×C₅₃	C₂⁵	1	36	yes	—
133	C₂×C₄×C₅₃	C₂⁵	1	96	yes	—
134	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
135	C₂×C₄×C₅₃	C₂⁵	1	96	yes	—
136	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
137	C₂×C₄×C₅₃	C₂⁵	1	48	yes	—
138	C₂²⋊D₅₃	C₂⁵	1	128	yes	—
139	C₂²⋊D₅₃	C₂⁵	1	384	yes	—
140	C₂×Dic₅₃	C₂⁵	1	48	yes	—
141	C₂×Dic₅₃	C₂⁵	1	192	yes	—
142	D₂₁₂	C₂⁵	1	64	yes	—
143	D₂₁₂	C₂⁵	1	192	yes	—
144	C₄×D₅₃	C₂⁵	1	128	yes	—
145	C₄×D₅₃	C₂⁵	1	384	yes	—
146	Dic₁₀₆	C₂⁵	1	64	yes	—
147	Dic₁₀₆	C₂⁵	1	192	yes	—
148	C₂²×#585	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	—	2	72	no	—
150	C₂²×C₅₃	—	2	12	no	—
151	C₂²×C₅₃	—	4	48	no	—
152	C₂²×C₅₃	#585	1	4	yes	complemented
153	C₂²×C₅₃	C₂⁶	1	24	yes	—
154	C₂²×C₅₃	C₂⁶	1	4	yes	—
155	C₂²×C₅₃	C₂⁶	1	3	yes	—
156	C₂×D₅₃	—	2	96	no	—
157	C₂×D₅₃	—	4	192	no	—
158	C₂×D₅₃	C₂⁶	1	32	yes	—
159	C₄×C₅₃	C₂⁶	1	4	yes	—
160	C₄×C₅₃	C₂⁶	1	24	yes	—
161	C₄×C₅₃	C₂⁶	1	4	yes	—
162	Dic₅₃	C₂⁶	1	32	yes	—
163	C₂×#585	—	53	96	no	—
164	C₂²×2₋¹⁺⁴	—	53	4	no	—
165	C₂²×2₋¹⁺⁴	—	53	2	no	—
166	C₂²×H(5,𝔽₂)	—	53	4	no	—
167	C₂²×H(5,𝔽₂)	—	53	6	no	—
168	C₂³×#41	—	53	8	no	—
169	C₂³×#41	—	53	6	no	—
170	C₂³×#41	D₅₃	1	1	yes	characteristic, complemented
171	C₂×C₅₃	—	2	24	no	—
172	C₂×C₅₃	—	2	4	no	—
173	C₂×C₅₃	C₂×#585	1	6	yes	complemented
174	C₂×C₅₃	C₂⁷	1	1	yes	characteristic
175	D₅₃	—	2	32	no	—
176	#585	—	53	1024	no	—
177	C₂×2₋¹⁺⁴	—	53	192	no	—
178	C₂×2₋¹⁺⁴	—	53	96	no	—
179	C₂×H(5,𝔽₂)	—	53	192	no	—
180	C₂×H(5,𝔽₂)	—	53	288	no	—
181	C₂²×#41	—	53	384	no	—
182	C₂²×#41	—	53	8	no	—
183	C₂²×#41	—	53	24	no	—
184	C₂²×#41	—	53	12	no	—
185	C₂²×#41	—	53	4	no	—
186	C₂²×#41	—	53	12	no	—
187	C₂²×#41	—	53	12	no	—
188	C₂²×#41	—	53	288	no	—
189	C₂²×#41	C₂×D₅₃	1	4	yes	complemented
190	C₂²×#41	C₂×D₅₃	1	4	yes	complemented
191	C₂²×#41	C₂×D₅₃	1	48	yes	complemented
192	C₂³×Q₈	—	53	8	no	—
193	C₂³×Q₈	—	53	6	no	—
194	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
195	C₂³×D₄	—	53	12	no	—
196	C₂³×D₄	—	53	24	no	—
197	C₂³×D₄	—	53	6	no	—
198	C₂³×D₄	C₂×D₅₃	1	3	yes	complemented
199	C₂⁴×C₄	—	53	12	no	—
200	C₂⁴×C₄	C₂×D₅₃	1	3	yes	complemented
201	C₅₃	C₂²×#585	1	1	yes	characteristic, complemented, Sylow
202	C₂⁵	—	53	12	no	—
203	C₂⁵	C₂²×D₅₃	1	3	yes	—
204	2₋¹⁺⁴	—	53	1024	no	—
205	2₋¹⁺⁴	—	53	512	no	—
206	H(5,𝔽₂)	—	53	1024	no	—
207	H(5,𝔽₂)	—	53	1536	no	—
208	C₂×#41	—	53	2048	no	—
209	C₂×#41	—	53	192	no	—
210	C₂×#41	—	53	576	no	—
211	C₂×#41	—	53	288	no	—
212	C₂×#41	—	53	96	no	—
213	C₂×#41	—	53	288	no	—
214	C₂×#41	—	53	288	no	—
215	C₂×#41	—	53	1536	no	—
216	C₂×#41	C₂²×D₅₃	1	96	yes	complemented
217	C₂×#41	C₂²×D₅₃	1	96	yes	complemented
218	C₂×#41	C₂²×D₅₃	1	256	yes	complemented
219	C₂²×Q₈	—	53	192	no	—
220	C₂²×Q₈	—	53	12	no	—
221	C₂²×Q₈	—	53	4	no	—
222	C₂²×Q₈	—	53	144	no	—
223	C₂²×Q₈	C₂²×D₅₃	1	4	yes	—
224	C₂²×Q₈	C₂²×D₅₃	1	24	yes	complemented
225	C₂²×D₄	—	53	24	no	—
226	C₂²×D₄	—	53	8	no	—
227	C₂²×D₄	—	53	288	no	—
228	C₂²×D₄	—	53	576	no	—
229	C₂²×D₄	—	53	12	no	—
230	C₂²×D₄	—	53	4	no	—
231	C₂²×D₄	—	53	144	no	—
232	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
233	C₂²×D₄	C₂²×D₅₃	1	72	yes	complemented
234	C₂³×C₄	—	53	12	no	—
235	C₂³×C₄	—	53	24	no	—
236	C₂³×C₄	—	53	288	no	—
237	C₂³×C₄	—	53	8	no	—
238	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
239	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
240	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
241	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
242	C₂³×C₄	C₂²×D₅₃	1	72	yes	complemented
243	C₂⁴	—	4	12	no	—
244	C₂⁴	—	53	8	no	—
245	C₂⁴	—	53	144	no	—
246	C₂⁴	—	212	48	no	—
247	C₂⁴	C₂³×D₅₃	1	6	yes	—
248	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
249	C₂⁴	C₂³×D₅₃	1	36	yes	—
250	#41	—	53	512	no	—
251	#41	—	53	1536	no	—
252	#41	—	53	768	no	—
253	#41	—	53	256	no	—
254	#41	—	53	768	no	—
255	#41	—	53	768	no	—
256	#41	C₂³×D₅₃	1	256	yes	complemented
257	#41	C₂³×D₅₃	1	256	yes	complemented
258	C₂×Q₈	—	53	512	no	—
259	C₂×Q₈	—	53	144	no	—
260	C₂×Q₈	—	53	48	no	—
261	C₂×Q₈	—	53	384	no	—
262	C₂×Q₈	C₂³×D₅₃	1	48	yes	—
263	C₂×Q₈	C₂³×D₅₃	1	64	yes	complemented
264	C₂×D₄	—	53	288	no	—
265	C₂×D₄	—	53	96	no	—
266	C₂×D₄	—	53	768	no	—
267	C₂×D₄	—	53	1536	no	—
268	C₂×D₄	—	53	144	no	—
269	C₂×D₄	—	53	48	no	—
270	C₂×D₄	—	53	384	no	—
271	C₂×D₄	C₂³×D₅₃	1	144	yes	—
272	C₂×D₄	C₂³×D₅₃	1	192	yes	complemented
273	C₂²×C₄	—	53	8	no	—
274	C₂²×C₄	—	53	144	no	—
275	C₂²×C₄	—	53	96	no	—
276	C₂²×C₄	—	53	288	no	—
277	C₂²×C₄	—	53	768	no	—
278	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
279	C₂²×C₄	C₂³×D₅₃	1	72	yes	—
280	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
281	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
282	C₂²×C₄	C₂³×D₅₃	1	72	yes	—
283	C₂²×C₄	C₂³×D₅₃	1	36	yes	—
284	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
285	C₂²×C₄	C₂³×D₅₃	1	192	yes	complemented
286	C₂³	—	2	4	no	—
287	C₂³	—	2	24	no	—
288	C₂³	—	4	72	no	—
289	C₂³	—	53	48	no	—
290	C₂³	—	53	192	no	—
291	C₂³	—	106	32	no	—
292	C₂³	—	212	288	no	—
293	C₂³	C₂⁴×D₅₃	1	36	yes	—
294	C₂³	C₂⁴×D₅₃	1	6	yes	—
295	C₂³	C₂⁴×D₅₃	1	48	yes	—
296	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
297	Q₈	—	53	192	no	—
298	Q₈	—	53	64	no	—
299	Q₈	C₂⁴×D₅₃	1	64	yes	—
300	D₄	—	53	384	no	—
301	D₄	—	53	128	no	—
302	D₄	—	53	192	no	—
303	D₄	—	53	64	no	—
304	D₄	C₂⁴×D₅₃	1	192	yes	—
305	C₂×C₄	—	53	48	no	—
306	C₂×C₄	—	53	192	no	—
307	C₂×C₄	—	53	128	no	—
308	C₂×C₄	—	53	384	no	—
309	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
310	C₂×C₄	C₂⁴×D₅₃	1	36	yes	—
311	C₂×C₄	C₂⁴×D₅₃	1	96	yes	—
312	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
313	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
314	C₂×C₄	C₂⁴×D₅₃	1	96	yes	—
315	C₂×C₄	C₂⁴×D₅₃	1	48	yes	—
316	C₂²	—	2	12	no	—
317	C₂²	—	2	72	no	—
318	C₂²	—	4	48	no	—
319	C₂²	—	53	32	no	—
320	C₂²	—	106	96	no	—
321	C₂²	—	212	192	no	—
322	C₂²	#16811459	1	4	yes	direct product factor
323	C₂²	C₂⁵×D₅₃	1	24	yes	—
324	C₂²	C₂⁵×D₅₃	1	4	yes	—
325	C₂²	C₂⁵×D₅₃	1	3	yes	—
326	C₄	—	53	32	no	—
327	C₄	C₂⁵×D₅₃	1	4	yes	—
328	C₄	C₂⁵×D₅₃	1	24	yes	—
329	C₄	C₂⁵×D₅₃	1	4	yes	—
330	C₂	—	2	4	no	—
331	C₂	—	2	24	no	—
332	C₂	—	106	32	no	—
333	C₂	C₂×#16811459	1	6	yes	direct product factor
334	C₂	C₂⁶×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₅₃
commutator, cosocle	C₂×C₅₃	C₂⁷
Frattini	C₂	C₂⁶×D₅₃
Fitting	C₂³×#41×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂³×#41	D₅₃
hypocenter	C₅₃	C₂²×#585
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#16811459	C₂	direct (N × H)
2. C₂²×#13133265	C₂	semidirect
3. C₂²×#13133264	C₂	semidirect
4. C₂²×#13133263	C₂	semidirect
5. C₂²×#13133262	C₂	semidirect
6. C₂³×#11719945	C₂	semidirect
7. C₂²×#41×D₅₃	C₂	semidirect
8. C₂³×#41×C₅₃	C₂	semidirect
9. #16811459	C₂²	direct (N × H)
10. C₂×#13133265	C₂²	semidirect
11. C₂×#13133264	C₂²	semidirect
12. C₂×#13133263	C₂²	semidirect
13. C₂×#13133262	C₂²	semidirect
14. C₂²×#11719949	C₂²	semidirect
15. C₂²×#11719945	C₂²	semidirect
16. C₂²×#11719945	C₂²	semidirect
17. C₂²×#11719945	C₂²	semidirect
18. C₂²×#11719945	C₂²	semidirect
19. C₂²×#11719945	C₂²	semidirect
20. C₂²×#11719947	C₂²	semidirect
21. C₂×#41×D₅₃	C₂²	semidirect
22. C₂³×Dic₁₀₆	C₂²	semidirect
23. C₂²×Q₈×D₅₃	C₂²	semidirect
24. C₂³×D₂₁₂	C₂²	semidirect
25. C₂³×C₂²⋊D₅₃	C₂²	semidirect
26. C₂²×D₄×D₅₃	C₂²	semidirect
27. C₂³×C₄×D₅₃	C₂²	semidirect
28. C₂²×#41×C₅₃	C₂²	semidirect
29. C₂²×#41×C₅₃	C₂²	semidirect
30. C₂²×#41×C₅₃	C₂²	semidirect
31. C₂³×Q₈×C₅₃	C₂²	semidirect
32. C₂³×D₄×C₅₃	C₂²	semidirect
33. C₂⁴×C₄×C₅₃	C₂²	semidirect
35. #13133265	C₂³	semidirect
36. #13133264	C₂³	semidirect
37. #13133263	C₂³	semidirect
38. #13133262	C₂³	semidirect
39. C₂×#11719949	C₂³	semidirect
40. C₂×#11719945	C₂³	semidirect
41. C₂×#11719945	C₂³	semidirect
42. C₂×#11719945	C₂³	semidirect
43. C₂×#11719945	C₂³	semidirect
44. C₂×#11719945	C₂³	semidirect
45. C₂×#11719947	C₂³	semidirect
46. #41×D₅₃	C₂³	semidirect
49. C₂²×Dic₁₀₆	C₂³	semidirect
50. C₂×Q₈×D₅₃	C₂³	semidirect
53. C₂²×D₂₁₂	C₂³	semidirect
56. C₂²×C₂²⋊D₅₃	C₂³	semidirect
57. C₂×D₄×D₅₃	C₂³	semidirect
61. C₂²×C₄×D₅₃	C₂³	semidirect
63. C₂×#41×C₅₃	C₂³	semidirect
64. C₂×#41×C₅₃	C₂³	semidirect
65. C₂×#41×C₅₃	C₂³	semidirect
67. C₂²×Q₈×C₅₃	C₂³	semidirect
69. C₂²×D₄×C₅₃	C₂³	semidirect
74. C₂³×C₄×C₅₃	C₂³	semidirect
82. #41×C₅₃	C₂⁴	semidirect
83. #41×C₅₃	C₂⁴	semidirect
85. C₂×Q₈×C₅₃	C₂⁴	semidirect
87. C₂×D₄×C₅₃	C₂⁴	semidirect
95. C₂²×C₄×C₅₃	C₂⁴	semidirect
98. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
101. #11719949	C₂⁴	semidirect
102. Q₈×D₅₃	C₂⁴	semidirect
103. #11719947	C₂⁴	semidirect
104. D₄×D₅₃	C₂⁴	semidirect
105. #11719945	C₂⁴	semidirect
106. #11719945	C₂⁴	semidirect
107. #11719945	C₂⁴	semidirect
108. #11719945	C₂⁴	semidirect
111. C₂×D₂₁₂	C₂⁴	semidirect
114. C₂×C₄×D₅₃	C₂⁴	semidirect
117. C₂×Dic₁₀₆	C₂⁴	semidirect
152. C₂²×C₅₃	#585	semidirect
170. C₂³×#41	D₅₃	semidirect
173. C₂×C₅₃	C₂×#585	semidirect
189. C₂²×#41	C₂×D₅₃	semidirect
190. C₂²×#41	C₂×D₅₃	semidirect
191. C₂²×#41	C₂×D₅₃	semidirect
194. C₂³×Q₈	C₂×D₅₃	semidirect
198. C₂³×D₄	C₂×D₅₃	semidirect
200. C₂⁴×C₄	C₂×D₅₃	semidirect
201. C₅₃	C₂²×#585	semidirect
216. C₂×#41	C₂²×D₅₃	semidirect
217. C₂×#41	C₂²×D₅₃	semidirect
218. C₂×#41	C₂²×D₅₃	semidirect
224. C₂²×Q₈	C₂²×D₅₃	semidirect
233. C₂²×D₄	C₂²×D₅₃	semidirect
242. C₂³×C₄	C₂²×D₅₃	semidirect
256. #41	C₂³×D₅₃	semidirect
257. #41	C₂³×D₅₃	semidirect
263. C₂×Q₈	C₂³×D₅₃	semidirect
272. C₂×D₄	C₂³×D₅₃	semidirect
285. C₂²×C₄	C₂³×D₅₃	semidirect
322. C₂²	#16811459	direct (N × H)
333. C₂	C₂×#16811459	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×#16811459	2
C₂²×#13133265	2
C₂²×#13133264	2
C₂²×#13133263	2
C₂²×#13133262	2
C₂³×#11719945	2
C₂²×#41×D₅₃	2
C₂³×#41×C₅₃	2
C₂²×#585	53

Subgroups — C22×#16811459 #28297985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811459 #28297985