Subgroups — C₂²×#16811431 #28297908

Order 13568.

11118 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133263	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811431	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133248	C₂	1	3	yes	maximal, complemented
4	C₂²×#13133242	C₂	1	3	yes	maximal, complemented
5	C₂²×#13133237	C₂	1	3	yes	maximal, complemented
6	C₂³×#11719924	C₂	1	3	yes	maximal, complemented
7	C₂²×#13133224	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×2₋¹⁺⁴×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133263	—	2	12	no	—
10	C₂×#13133263	C₂²	1	24	yes	complemented
11	C₂²×#11719949	—	2	1	no	—
12	C₂²×#11719945	C₂²	1	3	yes	—
13	C₂³×D₂₁₂	C₂²	1	3	yes	—
14	C₂²×D₄×D₅₃	—	2	3	no	—
15	#16811431	C₂²	1	256	yes	direct product factor
16	C₂×#13133248	C₂²	1	72	yes	complemented
17	C₂×#13133242	C₂²	1	72	yes	complemented
18	C₂×#13133237	C₂²	1	72	yes	complemented
19	C₂²×#11719925	C₂²	1	3	yes	complemented
20	C₂²×#11719923	C₂²	1	1	yes	characteristic, complemented
21	C₂²×#11719924	C₂²	1	3	yes	complemented
22	C₂²×#11719924	C₂²	1	6	yes	complemented
23	C₂²×#11719924	C₂²	1	72	yes	complemented
24	C₂²×#11719922	C₂²	1	3	yes	complemented
25	C₂×#13133224	C₂²	1	24	yes	complemented
26	C₂²×#11719917	C₂²	1	3	yes	complemented
27	C₂³×#88322	C₂²	1	3	yes	complemented
28	C₂×2₋¹⁺⁴×C₅₃	—	2	12	no	—
29	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	24	yes	complemented
30	C₂²×#41×C₅₃	—	2	3	no	—
31	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
32	C₂²×#41×C₅₃	C₂²	1	3	yes	complemented
33	C₂³×Q₈×C₅₃	—	2	1	no	—
34	C₂³×Q₈×C₅₃	C₂²	1	3	yes	complemented
35	C₂⁴×D₅₃	—	2	3	no	—
36	#13133263	—	2	96	no	—
37	#13133263	C₂³	1	64	yes	complemented
38	C₂×#11719949	—	2	24	no	—
39	C₂×#11719945	—	2	18	no	—
40	C₂×#11719945	C₂³	1	36	yes	—
41	C₂²×Dic₁₀₆	C₂³	1	1	yes	characteristic
42	C₂²×D₂₁₂	—	2	3	no	—
43	C₂²×D₂₁₂	—	2	18	no	—
44	C₂²×D₂₁₂	C₂³	1	3	yes	—
45	C₂²×D₂₁₂	C₂³	1	36	yes	—
46	C₂²×C₂²⋊D₅₃	—	2	3	no	—
47	C₂×D₄×D₅₃	—	2	72	no	—
48	C₂²×C₄×D₅₃	—	2	3	no	—
49	#13133248	C₂³	1	192	yes	complemented
50	#13133242	C₂³	1	192	yes	complemented
51	#13133237	C₂³	1	192	yes	complemented
52	C₂×#11719925	C₂³	1	36	yes	complemented
53	C₂×#11719923	C₂³	1	12	yes	complemented
54	C₂×#11719924	C₂³	1	36	yes	complemented
55	C₂×#11719924	C₂³	1	72	yes	complemented
56	C₂×#11719924	C₂³	1	192	yes	complemented
57	C₂×#11719922	C₂³	1	36	yes	complemented
58	#13133224	C₂³	1	64	yes	complemented
59	C₂×#11719917	C₂³	1	36	yes	complemented
60	C₂²×#88322	C₂³	1	1	yes	characteristic
61	C₂²×#88322	C₂³	1	3	yes	—
62	C₂²×#88322	C₂³	1	36	yes	complemented
63	2₋¹⁺⁴×C₅₃	—	2	96	no	—
64	2₋¹⁺⁴×C₅₃	C₂³	1	64	yes	complemented
65	C₂×#41×C₅₃	—	2	72	no	—
66	C₂×#41×C₅₃	—	2	18	no	—
67	C₂×#41×C₅₃	D₄	1	12	yes	—
68	C₂×#41×C₅₃	C₂³	1	12	yes	—
69	C₂×#41×C₅₃	C₂³	1	36	yes	complemented
70	C₂²×Q₈×C₅₃	—	2	24	no	—
71	C₂²×Q₈×C₅₃	—	2	3	no	—
72	C₂²×Q₈×C₅₃	—	2	18	no	—
73	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
74	C₂²×Q₈×C₅₃	C₂³	1	3	yes	—
75	C₂²×Q₈×C₅₃	C₂³	1	36	yes	complemented
76	C₂²×D₄×C₅₃	—	2	3	no	—
77	C₂²×D₄×C₅₃	D₄	1	3	yes	—
78	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₅₃	—	2	3	no	—
80	C₂³×C₄×C₅₃	—	2	3	no	—
81	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
82	C₂⁴×C₅₃	—	2	1	no	—
83	C₂⁴×C₅₃	C₂×D₄	1	3	yes	—
84	C₂³×D₅₃	—	2	3	no	—
85	C₂³×D₅₃	—	2	36	no	—
86	C₂³×D₅₃	—	8	12	no	—
87	#41×C₅₃	—	2	192	no	—
88	#41×C₅₃	—	2	72	no	—
89	#41×C₅₃	C₂×D₄	1	48	yes	—
90	#41×C₅₃	C₂⁴	1	16	yes	—
91	#41×C₅₃	C₂⁴	1	48	yes	complemented
92	C₂×Q₈×C₅₃	—	2	64	no	—
93	C₂×Q₈×C₅₃	—	2	9	no	—
94	C₂×Q₈×C₅₃	—	2	36	no	—
95	C₂×Q₈×C₅₃	—	2	72	no	—
96	C₂×Q₈×C₅₃	C₂×D₄	1	12	yes	—
97	C₂×Q₈×C₅₃	C₂⁴	1	18	yes	—
98	C₂×Q₈×C₅₃	C₂⁴	1	48	yes	complemented
99	C₂×D₄×C₅₃	—	2	36	no	—
100	C₂×D₄×C₅₃	—	2	3	no	—
101	C₂×D₄×C₅₃	C₂×D₄	1	36	yes	—
102	C₂×D₄×C₅₃	C₂⁴	1	6	yes	—
103	C₂²×C₄×C₅₃	—	2	3	no	—
104	C₂²×C₄×C₅₃	—	2	36	no	—
105	C₂²×C₄×C₅₃	—	2	36	no	—
106	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
107	C₂²×C₄×C₅₃	C₂×D₄	1	18	yes	—
108	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
110	C₂×C₂²⋊D₅₃	—	2	36	no	—
111	C₂²×Dic₅₃	—	2	1	no	—
112	#11719949	—	2	64	no	—
113	D₄×D₅₃	—	2	192	no	—
114	#11719945	—	2	72	no	—
115	#11719945	C₂⁴	1	48	yes	—
116	C₂×D₂₁₂	—	2	36	no	—
117	C₂×D₂₁₂	—	2	9	no	—
118	C₂×D₂₁₂	—	2	72	no	—
119	C₂×D₂₁₂	C₂⁴	1	18	yes	—
120	C₂×D₂₁₂	C₂⁴	1	48	yes	—
121	C₂×C₄×D₅₃	—	2	36	no	—
122	C₂×Dic₁₀₆	—	2	3	no	—
123	C₂×Dic₁₀₆	C₂⁴	1	6	yes	—
124	#11719925	C₂⁴	1	48	yes	complemented
125	#11719924	C₂⁴	1	48	yes	complemented
126	#11719924	C₂⁴	1	96	yes	complemented
127	#11719923	C₂⁴	1	16	yes	complemented
128	#11719922	C₂⁴	1	48	yes	complemented
129	#11719917	C₂⁴	1	48	yes	complemented
130	C₂×#88322	C₂⁴	1	6	yes	—
131	C₂×#88322	C₂⁴	1	18	yes	—
132	C₂×#88322	C₂⁴	1	48	yes	complemented
133	C₂³×C₅₃	—	2	12	no	—
134	C₂³×C₅₃	—	2	6	no	—
135	C₂³×C₅₃	—	4	4	no	—
136	C₂³×C₅₃	C₂²×D₄	1	18	yes	—
137	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
138	C₂²×D₅₃	—	2	18	no	—
139	C₂²×D₅₃	—	2	48	no	—
140	C₂²×D₅₃	—	4	12	no	—
141	C₂²×D₅₃	—	8	72	no	—
142	Q₈×C₅₃	—	2	18	no	—
143	Q₈×C₅₃	—	2	48	no	—
144	Q₈×C₅₃	C₂²×D₄	1	16	yes	—
145	Q₈×C₅₃	C₂⁵	1	12	yes	—
146	D₄×C₅₃	—	2	48	no	—
147	D₄×C₅₃	—	2	6	no	—
148	D₄×C₅₃	C₂²×D₄	1	48	yes	—
149	D₄×C₅₃	C₂⁵	1	4	yes	—
150	C₂×C₄×C₅₃	—	2	18	no	—
151	C₂×C₄×C₅₃	—	2	48	no	—
152	C₂×C₄×C₅₃	—	2	48	no	—
153	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
154	C₂×C₄×C₅₃	C₂²×D₄	1	18	yes	—
155	C₂×C₄×C₅₃	C₂²×D₄	1	36	yes	—
156	C₂×C₄×C₅₃	C₂⁵	1	3	yes	—
157	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
158	C₂²⋊D₅₃	—	2	48	no	—
159	C₂×Dic₅₃	—	2	6	no	—
160	D₂₁₂	—	2	18	no	—
161	D₂₁₂	—	2	48	no	—
162	D₂₁₂	C₂⁵	1	12	yes	—
163	C₄×D₅₃	—	2	48	no	—
164	Dic₁₀₆	—	2	6	no	—
165	Dic₁₀₆	C₂⁵	1	4	yes	—
166	#88322	C₂⁵	1	4	yes	—
167	#88322	C₂⁵	1	12	yes	—
168	C₂²×#577	—	53	1	no	maximal, Sylow
169	C₂²×C₅₃	—	2	36	no	—
170	C₂²×C₅₃	—	2	4	no	—
171	C₂²×C₅₃	—	4	12	no	—
172	C₂²×C₅₃	#577	1	4	yes	complemented
173	C₂²×C₅₃	C₂³×D₄	1	12	yes	—
174	C₂²×C₅₃	C₂³×D₄	1	3	yes	—
175	C₂×D₅₃	—	2	12	no	—
176	C₂×D₅₃	—	4	36	no	—
177	C₂×D₅₃	—	8	48	no	—
178	C₄×C₅₃	—	2	12	no	—
179	C₄×C₅₃	C₂³×D₄	1	3	yes	—
180	C₄×C₅₃	C₂³×D₄	1	12	yes	—
181	C₄×C₅₃	C₂⁶	1	1	yes	characteristic
182	Dic₅₃	—	2	4	no	—
183	C₂²×2₋¹⁺⁴	D₅₃	1	1	yes	characteristic, complemented
184	C₂²×H(5,𝔽₂)	—	53	1	no	—
185	C₂×#577	—	53	48	no	—
186	C₂²×#137	—	53	3	no	—
187	C₂²×AGL(1,ℤ/8)	—	53	3	no	—
188	C₂²×#135	—	53	3	no	—
189	C₂³×SD₁₆	—	53	3	no	—
190	C₂²×#131	—	53	1	no	—
191	C₂×C₅₃	—	2	12	no	—
192	C₂×C₅₃	—	4	4	no	—
193	C₂×C₅₃	C₂×#577	1	6	yes	complemented
194	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
195	D₅₃	—	4	12	no	—
196	C₂×2₋¹⁺⁴	—	2	12	no	—
197	C₂×2₋¹⁺⁴	C₂×D₅₃	1	24	yes	complemented
198	C₂×H(5,𝔽₂)	—	53	24	no	—
199	C₂×H(5,𝔽₂)	—	106	12	no	—
200	C₂²×#41	—	2	3	no	—
201	C₂²×#41	—	53	3	no	—
202	C₂²×#41	—	106	1	no	—
203	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
204	C₂²×#41	C₂×D₅₃	1	3	yes	complemented
205	C₂³×Q₈	—	2	1	no	—
206	C₂³×Q₈	C₂×D₅₃	1	3	yes	complemented
207	C₂³×D₄	—	53	3	no	—
208	C₂³×D₄	—	106	3	no	—
209	#577	—	53	256	no	—
210	C₂×#137	—	53	72	no	—
211	C₂×AGL(1,ℤ/8)	—	53	72	no	—
212	C₂×#135	—	53	72	no	—
213	C₂²×Q₁₆	—	53	3	no	—
214	C₂²×SD₁₆	—	53	3	no	—
215	C₂²×SD₁₆	—	53	6	no	—
216	C₂²×SD₁₆	—	53	72	no	—
217	C₂²×SD₁₆	—	53	1	no	—
218	C₂²×D₈	—	53	3	no	—
219	C₂×#131	—	53	24	no	—
220	C₂²×M₁₆	—	53	3	no	—
221	C₂³×C₈	—	53	3	no	—
222	C₅₃	C₂²×#577	1	1	yes	characteristic, complemented, Sylow
223	C₂⁵	—	106	3	no	—
224	2₋¹⁺⁴	—	2	96	no	—
225	2₋¹⁺⁴	C₂²×D₅₃	1	64	yes	complemented
226	H(5,𝔽₂)	—	53	64	no	—
227	H(5,𝔽₂)	—	106	96	no	—
228	C₂×#41	—	2	72	no	—
229	C₂×#41	—	2	18	no	—
230	C₂×#41	—	53	36	no	—
231	C₂×#41	—	106	24	no	—
232	C₂×#41	—	106	18	no	—
233	C₂×#41	C₂²⋊D₅₃	1	12	yes	—
234	C₂×#41	C₂²×D₅₃	1	12	yes	—
235	C₂×#41	C₂²×D₅₃	1	36	yes	complemented
236	C₂²×Q₈	—	2	24	no	—
237	C₂²×Q₈	—	2	3	no	—
238	C₂²×Q₈	—	2	18	no	—
239	C₂²×Q₈	—	53	1	no	—
240	C₂²×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
241	C₂²×Q₈	C₂²×D₅₃	1	3	yes	—
242	C₂²×Q₈	C₂²×D₅₃	1	36	yes	complemented
243	C₂²×D₄	—	2	3	no	—
244	C₂²×D₄	—	53	3	no	—
245	C₂²×D₄	—	53	36	no	—
246	C₂²×D₄	—	106	3	no	—
247	C₂²×D₄	—	106	72	no	—
248	C₂²×D₄	—	106	3	no	—
249	C₂²×D₄	—	106	18	no	—
250	C₂²×D₄	C₂²⋊D₅₃	1	3	yes	—
251	C₂²×D₄	C₂²×D₅₃	1	1	yes	characteristic
252	C₂³×C₄	—	2	3	no	—
253	C₂³×C₄	—	2	3	no	—
254	C₂³×C₄	—	106	3	no	—
255	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
256	#137	—	53	192	no	—
257	AGL(1,ℤ/8)	—	53	192	no	—
258	#135	—	53	192	no	—
259	C₂×Q₁₆	—	53	36	no	—
260	C₂×SD₁₆	—	53	36	no	—
261	C₂×SD₁₆	—	53	72	no	—
262	C₂×SD₁₆	—	53	192	no	—
263	C₂×SD₁₆	—	53	12	no	—
264	C₂×D₈	—	53	36	no	—
265	#131	—	53	64	no	—
266	C₂×M₁₆	—	53	36	no	—
267	C₂²×C₈	—	53	1	no	—
268	C₂²×C₈	—	53	3	no	—
269	C₂²×C₈	—	53	36	no	—
270	C₂⁴	—	2	1	no	—
271	C₂⁴	—	106	36	no	—
272	C₂⁴	—	106	3	no	—
273	C₂⁴	—	424	12	no	—
274	C₂⁴	C₂×C₂²⋊D₅₃	1	3	yes	—
275	#41	—	2	192	no	—
276	#41	—	2	72	no	—
277	#41	—	53	48	no	—
278	#41	—	106	64	no	—
279	#41	—	106	72	no	—
280	#41	C₂×C₂²⋊D₅₃	1	48	yes	—
281	#41	C₂³×D₅₃	1	16	yes	—
282	#41	C₂³×D₅₃	1	48	yes	complemented
283	C₂×Q₈	—	2	64	no	—
284	C₂×Q₈	—	2	9	no	—
285	C₂×Q₈	—	2	36	no	—
286	C₂×Q₈	—	2	72	no	—
287	C₂×Q₈	—	53	6	no	—
288	C₂×Q₈	—	106	3	no	—
289	C₂×Q₈	C₂×C₂²⋊D₅₃	1	12	yes	—
290	C₂×Q₈	C₂³×D₅₃	1	18	yes	—
291	C₂×Q₈	C₂³×D₅₃	1	48	yes	complemented
292	C₂×D₄	—	2	36	no	—
293	C₂×D₄	—	2	3	no	—
294	C₂×D₄	—	53	18	no	—
295	C₂×D₄	—	53	48	no	—
296	C₂×D₄	—	106	36	no	—
297	C₂×D₄	—	106	192	no	—
298	C₂×D₄	—	106	36	no	—
299	C₂×D₄	—	106	9	no	—
300	C₂×D₄	—	106	72	no	—
301	C₂×D₄	C₂×C₂²⋊D₅₃	1	36	yes	—
302	C₂×D₄	C₂³×D₅₃	1	6	yes	—
303	C₂²×C₄	—	2	36	no	—
304	C₂²×C₄	—	2	3	no	—
305	C₂²×C₄	—	2	36	no	—
306	C₂²×C₄	—	106	1	no	—
307	C₂²×C₄	—	106	36	no	—
308	C₂²×C₄	C₂×C₂²⋊D₅₃	1	18	yes	—
309	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
310	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
311	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
312	Q₁₆	—	53	48	no	—
313	SD₁₆	—	53	48	no	—
314	SD₁₆	—	53	96	no	—
315	SD₁₆	—	53	16	no	—
316	D₈	—	53	48	no	—
317	M₁₆	—	53	48	no	—
318	C₂×C₈	—	53	6	no	—
319	C₂×C₈	—	53	18	no	—
320	C₂×C₈	—	53	48	no	—
321	C₂³	—	2	12	no	—
322	C₂³	—	2	6	no	—
323	C₂³	—	4	4	no	—
324	C₂³	—	106	48	no	—
325	C₂³	—	106	18	no	—
326	C₂³	—	424	72	no	—
327	C₂³	—	212	12	no	—
328	C₂³	C₂²×C₂²⋊D₅₃	1	18	yes	—
329	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
330	Q₈	—	2	18	no	—
331	Q₈	—	2	48	no	—
332	Q₈	—	53	4	no	—
333	Q₈	—	106	6	no	—
334	Q₈	C₂²×C₂²⋊D₅₃	1	16	yes	—
335	Q₈	C₂⁴×D₅₃	1	12	yes	—
336	D₄	—	2	48	no	—
337	D₄	—	2	6	no	—
338	D₄	—	53	12	no	—
339	D₄	—	106	48	no	—
340	D₄	—	106	48	no	—
341	D₄	—	106	18	no	—
342	D₄	C₂²×C₂²⋊D₅₃	1	48	yes	—
343	D₄	C₂⁴×D₅₃	1	4	yes	—
344	C₂×C₄	—	2	18	no	—
345	C₂×C₄	—	2	48	no	—
346	C₂×C₄	—	2	48	no	—
347	C₂×C₄	—	106	6	no	—
348	C₂×C₄	—	106	48	no	—
349	C₂×C₄	C₂²×C₂²⋊D₅₃	1	18	yes	—
350	C₂×C₄	C₂²×C₂²⋊D₅₃	1	3	yes	—
351	C₂×C₄	C₂²×C₂²⋊D₅₃	1	36	yes	—
352	C₂×C₄	C₂⁴×D₅₃	1	3	yes	—
353	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
354	C₈	—	53	4	no	—
355	C₈	—	53	12	no	—
356	C₂²	—	2	36	no	—
357	C₂²	—	2	4	no	—
358	C₂²	—	4	12	no	—
359	C₂²	—	106	12	no	—
360	C₂²	—	424	48	no	—
361	C₂²	—	212	36	no	—
362	C₂²	#16811431	1	4	yes	direct product factor
363	C₂²	C₂³×C₂²⋊D₅₃	1	12	yes	—
364	C₂²	C₂³×C₂²⋊D₅₃	1	3	yes	—
365	C₄	—	2	12	no	—
366	C₄	—	106	4	no	—
367	C₄	C₂³×C₂²⋊D₅₃	1	12	yes	—
368	C₄	C₂³×C₂²⋊D₅₃	1	3	yes	—
369	C₄	C₂⁵×D₅₃	1	1	yes	characteristic
370	C₂	—	2	12	no	—
371	C₂	—	4	4	no	—
372	C₂	—	212	12	no	—
373	C₂	C₂×#16811431	1	6	yes	direct product factor
374	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₅₃
commutator, cosocle	C₄×C₅₃	C₂⁶
Frattini	C₄	C₂⁵×D₅₃
Fitting	C₂²×2₋¹⁺⁴×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×D₄
hypercenter	C₂²×2₋¹⁺⁴	D₅₃
hypocenter	C₅₃	C₂²×#577
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133263	C₂	semidirect
2. C₂×#16811431	C₂	direct (N × H)
3. C₂²×#13133248	C₂	semidirect
4. C₂²×#13133242	C₂	semidirect
5. C₂²×#13133237	C₂	semidirect
6. C₂³×#11719924	C₂	semidirect
7. C₂²×#13133224	C₂	semidirect
8. C₂²×2₋¹⁺⁴×C₅₃	C₂	semidirect
10. C₂×#13133263	C₂²	semidirect
15. #16811431	C₂²	direct (N × H)
16. C₂×#13133248	C₂²	semidirect
17. C₂×#13133242	C₂²	semidirect
18. C₂×#13133237	C₂²	semidirect
19. C₂²×#11719925	C₂²	semidirect
20. C₂²×#11719923	C₂²	semidirect
21. C₂²×#11719924	C₂²	semidirect
22. C₂²×#11719924	C₂²	semidirect
23. C₂²×#11719924	C₂²	semidirect
24. C₂²×#11719922	C₂²	semidirect
25. C₂×#13133224	C₂²	semidirect
26. C₂²×#11719917	C₂²	semidirect
27. C₂³×#88322	C₂²	semidirect
29. C₂×2₋¹⁺⁴×C₅₃	C₂²	semidirect
32. C₂²×#41×C₅₃	C₂²	semidirect
34. C₂³×Q₈×C₅₃	C₂²	semidirect
37. #13133263	C₂³	semidirect
49. #13133248	C₂³	semidirect
50. #13133242	C₂³	semidirect
51. #13133237	C₂³	semidirect
52. C₂×#11719925	C₂³	semidirect
53. C₂×#11719923	C₂³	semidirect
54. C₂×#11719924	C₂³	semidirect
55. C₂×#11719924	C₂³	semidirect
56. C₂×#11719924	C₂³	semidirect
57. C₂×#11719922	C₂³	semidirect
58. #13133224	C₂³	semidirect
59. C₂×#11719917	C₂³	semidirect
62. C₂²×#88322	C₂³	semidirect
64. 2₋¹⁺⁴×C₅₃	C₂³	semidirect
69. C₂×#41×C₅₃	C₂³	semidirect
75. C₂²×Q₈×C₅₃	C₂³	semidirect
91. #41×C₅₃	C₂⁴	semidirect
98. C₂×Q₈×C₅₃	C₂⁴	semidirect
124. #11719925	C₂⁴	semidirect
125. #11719924	C₂⁴	semidirect
126. #11719924	C₂⁴	semidirect
127. #11719923	C₂⁴	semidirect
128. #11719922	C₂⁴	semidirect
129. #11719917	C₂⁴	semidirect
132. C₂×#88322	C₂⁴	semidirect
172. C₂²×C₅₃	#577	semidirect
183. C₂²×2₋¹⁺⁴	D₅₃	semidirect
193. C₂×C₅₃	C₂×#577	semidirect
197. C₂×2₋¹⁺⁴	C₂×D₅₃	semidirect
204. C₂²×#41	C₂×D₅₃	semidirect
206. C₂³×Q₈	C₂×D₅₃	semidirect
222. C₅₃	C₂²×#577	semidirect
225. 2₋¹⁺⁴	C₂²×D₅₃	semidirect
235. C₂×#41	C₂²×D₅₃	semidirect
242. C₂²×Q₈	C₂²×D₅₃	semidirect
282. #41	C₂³×D₅₃	semidirect
291. C₂×Q₈	C₂³×D₅₃	semidirect
362. C₂²	#16811431	direct (N × H)
373. C₂	C₂×#16811431	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×#13133263	2
C₂×#16811431	2
C₂²×#13133248	2
C₂²×#13133242	2
C₂²×#13133237	2
C₂³×#11719924	2
C₂²×#13133224	2
C₂²×2₋¹⁺⁴×C₅₃	2
C₂²×#577	53

Subgroups — C22×#16811431 #28297908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811431 #28297908