Subgroups — C₂²×#16811426 #28297902

Order 13568.

11462 conjugacy classes of subgroups.

The subgroup lattice has 382 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×#16811426	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133243	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×#11719922	C₂	1	3	yes	maximal, complemented
7	C₂²×#13133224	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×H(5,𝔽₂)×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	complemented
13	C₂³×D₂₁₂	C₂²	1	3	yes	complemented
14	C₂²×D₄×D₅₃	—	2	3	no	—
15	#16811426	C₂²	1	256	yes	direct product factor
16	C₂×#13133243	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	1	yes	characteristic, complemented
20	C₂²×#11719923	C₂²	1	3	yes	complemented
21	C₂²×#11719924	C₂²	1	3	yes	complemented
22	C₂²×#11719922	C₂²	1	3	yes	complemented
23	C₂²×#11719922	C₂²	1	6	yes	complemented
24	C₂²×#11719922	C₂²	1	72	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×H(5,𝔽₂)×C₅₃	—	2	12	no	—
29	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	24	yes	complemented
30	C₂²×#41×C₅₃	—	2	1	no	—
31	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
32	C₂²×#41×C₅₃	C₂²	1	3	yes	complemented
33	C₂³×D₄×C₅₃	—	2	3	no	—
34	C₂³×D₄×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	complemented
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	complemented
46	C₂²×C₂²⋊D₅₃	—	2	3	no	—
47	C₂×D₄×D₅₃	—	2	72	no	—
48	C₂²×C₄×D₅₃	—	2	3	no	—
49	#13133243	C₂³	1	192	yes	complemented
50	#13133242	C₂³	1	192	yes	complemented
51	#13133237	C₂³	1	192	yes	complemented
52	C₂×#11719925	C₂³	1	12	yes	complemented
53	C₂×#11719923	C₂³	1	36	yes	complemented
54	C₂×#11719924	C₂³	1	36	yes	complemented
55	C₂×#11719922	C₂³	1	36	yes	complemented
56	C₂×#11719922	C₂³	1	72	yes	complemented
57	C₂×#11719922	C₂³	1	192	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	C₂⁵×C₅₃	—	2	3	no	—
64	H(5,𝔽₂)×C₅₃	—	2	96	no	—
65	H(5,𝔽₂)×C₅₃	C₂³	1	64	yes	complemented
66	C₂×#41×C₅₃	—	2	24	no	—
67	C₂×#41×C₅₃	—	2	18	no	—
68	C₂×#41×C₅₃	D₄	1	12	yes	—
69	C₂×#41×C₅₃	C₂³	1	12	yes	—
70	C₂×#41×C₅₃	C₂³	1	36	yes	complemented
71	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
72	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
73	C₂²×D₄×C₅₃	—	2	3	no	—
74	C₂²×D₄×C₅₃	—	2	72	no	—
75	C₂²×D₄×C₅₃	—	2	3	no	—
76	C₂²×D₄×C₅₃	—	2	18	no	—
77	C₂²×D₄×C₅₃	D₄	1	3	yes	—
78	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
79	C₂²×D₄×C₅₃	C₂³	1	36	yes	complemented
80	C₂³×C₄×C₅₃	—	2	3	no	—
81	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
82	C₂⁴×C₅₃	—	2	3	no	—
83	C₂⁴×C₅₃	—	2	36	no	—
84	C₂⁴×C₅₃	—	8	12	no	—
85	C₂⁴×C₅₃	C₂×D₄	1	3	yes	—
86	C₂³×D₅₃	—	2	3	no	—
87	C₂³×D₅₃	—	2	36	no	—
88	C₂³×D₅₃	—	8	12	no	—
89	#41×C₅₃	—	2	64	no	—
90	#41×C₅₃	—	2	72	no	—
91	#41×C₅₃	C₂×D₄	1	48	yes	—
92	#41×C₅₃	C₂⁴	1	16	yes	—
93	#41×C₅₃	C₂⁴	1	48	yes	complemented
94	C₂×Q₈×C₅₃	—	2	3	no	—
95	C₂×Q₈×C₅₃	C₂×D₄	1	12	yes	—
96	C₂×Q₈×C₅₃	C₂⁴	1	6	yes	—
97	C₂×D₄×C₅₃	—	2	36	no	—
98	C₂×D₄×C₅₃	—	2	192	no	—
99	C₂×D₄×C₅₃	—	2	9	no	—
100	C₂×D₄×C₅₃	—	2	36	no	—
101	C₂×D₄×C₅₃	—	2	72	no	—
102	C₂×D₄×C₅₃	C₂×D₄	1	36	yes	—
103	C₂×D₄×C₅₃	C₂⁴	1	18	yes	—
104	C₂×D₄×C₅₃	C₂⁴	1	48	yes	complemented
105	C₂²×C₄×C₅₃	—	2	1	no	—
106	C₂²×C₄×C₅₃	—	2	36	no	—
107	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
108	C₂²×C₄×C₅₃	C₂×D₄	1	18	yes	—
109	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
111	C₂×C₂²⋊D₅₃	—	2	36	no	—
112	C₂²×Dic₅₃	—	2	1	no	—
113	#11719949	—	2	64	no	—
114	D₄×D₅₃	—	2	192	no	—
115	#11719945	—	2	72	no	—
116	#11719945	C₂⁴	1	48	yes	complemented
117	C₂×D₂₁₂	—	2	36	no	—
118	C₂×D₂₁₂	—	2	9	no	—
119	C₂×D₂₁₂	—	2	72	no	—
120	C₂×D₂₁₂	C₂⁴	1	18	yes	—
121	C₂×D₂₁₂	C₂⁴	1	48	yes	complemented
122	C₂×C₄×D₅₃	—	2	36	no	—
123	C₂×Dic₁₀₆	—	2	3	no	—
124	C₂×Dic₁₀₆	C₂⁴	1	6	yes	—
125	#11719925	C₂⁴	1	16	yes	complemented
126	#11719924	C₂⁴	1	48	yes	complemented
127	#11719923	C₂⁴	1	48	yes	complemented
128	#11719922	C₂⁴	1	48	yes	complemented
129	#11719922	C₂⁴	1	96	yes	complemented
130	#11719917	C₂⁴	1	48	yes	complemented
131	C₂×#88322	C₂⁴	1	6	yes	—
132	C₂×#88322	C₂⁴	1	18	yes	—
133	C₂×#88322	C₂⁴	1	48	yes	complemented
134	C₂³×C₅₃	—	2	12	no	—
135	C₂³×C₅₃	—	2	18	no	—
136	C₂³×C₅₃	—	2	48	no	—
137	C₂³×C₅₃	—	4	12	no	—
138	C₂³×C₅₃	—	8	72	no	—
139	C₂³×C₅₃	C₂²×D₄	1	18	yes	—
140	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
141	C₂²×D₅₃	—	2	18	no	—
142	C₂²×D₅₃	—	2	48	no	—
143	C₂²×D₅₃	—	4	12	no	—
144	C₂²×D₅₃	—	8	72	no	—
145	Q₈×C₅₃	—	2	6	no	—
146	Q₈×C₅₃	C₂²×D₄	1	16	yes	—
147	Q₈×C₅₃	C₂⁵	1	4	yes	—
148	D₄×C₅₃	—	2	48	no	—
149	D₄×C₅₃	—	2	18	no	—
150	D₄×C₅₃	—	2	48	no	—
151	D₄×C₅₃	C₂²×D₄	1	48	yes	—
152	D₄×C₅₃	C₂⁵	1	12	yes	—
153	C₂×C₄×C₅₃	—	2	6	no	—
154	C₂×C₄×C₅₃	—	2	48	no	—
155	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
156	C₂×C₄×C₅₃	C₂²×D₄	1	18	yes	—
157	C₂×C₄×C₅₃	C₂²×D₄	1	36	yes	—
158	C₂×C₄×C₅₃	C₂⁵	1	3	yes	—
159	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
160	C₂²⋊D₅₃	—	2	48	no	—
161	C₂×Dic₅₃	—	2	6	no	—
162	D₂₁₂	—	2	18	no	—
163	D₂₁₂	—	2	48	no	—
164	D₂₁₂	C₂⁵	1	12	yes	—
165	C₄×D₅₃	—	2	48	no	—
166	Dic₁₀₆	—	2	6	no	—
167	Dic₁₀₆	C₂⁵	1	4	yes	—
168	#88322	C₂⁵	1	4	yes	—
169	#88322	C₂⁵	1	12	yes	—
170	C₂²×#576	—	53	1	no	maximal, Sylow
171	C₂²×C₅₃	—	2	36	no	—
172	C₂²×C₅₃	—	2	12	no	—
173	C₂²×C₅₃	—	4	36	no	—
174	C₂²×C₅₃	—	8	48	no	—
175	C₂²×C₅₃	#576	1	4	yes	complemented
176	C₂²×C₅₃	C₂³×D₄	1	12	yes	—
177	C₂²×C₅₃	C₂³×D₄	1	3	yes	—
178	C₂×D₅₃	—	2	12	no	—
179	C₂×D₅₃	—	4	36	no	—
180	C₂×D₅₃	—	8	48	no	—
181	C₄×C₅₃	—	2	4	no	—
182	C₄×C₅₃	C₂³×D₄	1	3	yes	—
183	C₄×C₅₃	C₂³×D₄	1	12	yes	—
184	C₄×C₅₃	C₂⁶	1	1	yes	characteristic
185	Dic₅₃	—	2	4	no	—
186	C₂²×H(5,𝔽₂)	—	53	1	no	—
187	C₂²×H(5,𝔽₂)	D₅₃	1	1	yes	characteristic, complemented
188	C₂×#576	—	53	48	no	—
189	C₂²×AGL(1,ℤ/8)	—	53	3	no	—
190	C₂²×AGL(1,ℤ/8)	—	53	3	no	—
191	C₂²×#135	—	53	3	no	—
192	C₂³×D₈	—	53	3	no	—
193	C₂²×#131	—	53	1	no	—
194	C₂×C₅₃	—	2	12	no	—
195	C₂×C₅₃	—	4	12	no	—
196	C₂×C₅₃	C₂×#576	1	6	yes	complemented
197	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
198	D₅₃	—	4	12	no	—
199	C₂×H(5,𝔽₂)	—	2	12	no	—
200	C₂×H(5,𝔽₂)	—	53	24	no	—
201	C₂×H(5,𝔽₂)	—	106	12	no	—
202	C₂×H(5,𝔽₂)	C₂×D₅₃	1	24	yes	complemented
203	C₂²×#41	—	2	1	no	—
204	C₂²×#41	—	53	3	no	—
205	C₂²×#41	—	106	1	no	—
206	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
207	C₂²×#41	C₂×D₅₃	1	3	yes	complemented
208	C₂³×D₄	—	2	3	no	—
209	C₂³×D₄	—	53	3	no	—
210	C₂³×D₄	—	106	3	no	—
211	C₂³×D₄	C₂×D₅₃	1	3	yes	complemented
212	#576	—	53	256	no	—
213	C₂×AGL(1,ℤ/8)	—	53	72	no	—
214	C₂×AGL(1,ℤ/8)	—	53	72	no	—
215	C₂×#135	—	53	72	no	—
216	C₂²×Q₁₆	—	53	1	no	—
217	C₂²×SD₁₆	—	53	3	no	—
218	C₂²×SD₁₆	—	53	3	no	—
219	C₂²×D₈	—	53	3	no	—
220	C₂²×D₈	—	53	6	no	—
221	C₂²×D₈	—	53	72	no	—
222	C₂×#131	—	53	24	no	—
223	C₂²×M₁₆	—	53	3	no	—
224	C₂³×C₈	—	53	3	no	—
225	C₅₃	C₂²×#576	1	1	yes	characteristic, complemented, Sylow
226	C₂⁵	—	2	3	no	—
227	C₂⁵	—	106	3	no	—
228	H(5,𝔽₂)	—	2	96	no	—
229	H(5,𝔽₂)	—	53	64	no	—
230	H(5,𝔽₂)	—	106	96	no	—
231	H(5,𝔽₂)	C₂²×D₅₃	1	64	yes	complemented
232	C₂×#41	—	2	24	no	—
233	C₂×#41	—	2	18	no	—
234	C₂×#41	—	53	36	no	—
235	C₂×#41	—	106	24	no	—
236	C₂×#41	—	106	18	no	—
237	C₂×#41	C₂²⋊D₅₃	1	12	yes	—
238	C₂×#41	C₂²×D₅₃	1	12	yes	—
239	C₂×#41	C₂²×D₅₃	1	36	yes	complemented
240	C₂²×Q₈	—	53	1	no	—
241	C₂²×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
242	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
243	C₂²×D₄	—	2	3	no	—
244	C₂²×D₄	—	2	72	no	—
245	C₂²×D₄	—	2	3	no	—
246	C₂²×D₄	—	2	18	no	—
247	C₂²×D₄	—	53	3	no	—
248	C₂²×D₄	—	53	36	no	—
249	C₂²×D₄	—	106	3	no	—
250	C₂²×D₄	—	106	72	no	—
251	C₂²×D₄	—	106	3	no	—
252	C₂²×D₄	—	106	18	no	—
253	C₂²×D₄	C₂²⋊D₅₃	1	3	yes	—
254	C₂²×D₄	C₂²×D₅₃	1	3	yes	—
255	C₂²×D₄	C₂²×D₅₃	1	36	yes	complemented
256	C₂³×C₄	—	2	3	no	—
257	C₂³×C₄	—	106	3	no	—
258	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
259	AGL(1,ℤ/8)	—	53	192	no	—
260	AGL(1,ℤ/8)	—	53	192	no	—
261	#135	—	53	192	no	—
262	C₂×Q₁₆	—	53	12	no	—
263	C₂×SD₁₆	—	53	36	no	—
264	C₂×SD₁₆	—	53	36	no	—
265	C₂×D₈	—	53	36	no	—
266	C₂×D₈	—	53	72	no	—
267	C₂×D₈	—	53	192	no	—
268	#131	—	53	64	no	—
269	C₂×M₁₆	—	53	36	no	—
270	C₂²×C₈	—	53	1	no	—
271	C₂²×C₈	—	53	3	no	—
272	C₂²×C₈	—	53	36	no	—
273	C₂⁴	—	2	3	no	—
274	C₂⁴	—	2	36	no	—
275	C₂⁴	—	8	12	no	—
276	C₂⁴	—	106	36	no	—
277	C₂⁴	—	106	3	no	—
278	C₂⁴	—	424	12	no	—
279	C₂⁴	C₂×C₂²⋊D₅₃	1	3	yes	—
280	#41	—	2	64	no	—
281	#41	—	2	72	no	—
282	#41	—	53	48	no	—
283	#41	—	106	64	no	—
284	#41	—	106	72	no	—
285	#41	C₂×C₂²⋊D₅₃	1	48	yes	—
286	#41	C₂³×D₅₃	1	16	yes	—
287	#41	C₂³×D₅₃	1	48	yes	complemented
288	C₂×Q₈	—	2	3	no	—
289	C₂×Q₈	—	53	6	no	—
290	C₂×Q₈	—	106	3	no	—
291	C₂×Q₈	C₂×C₂²⋊D₅₃	1	12	yes	—
292	C₂×Q₈	C₂³×D₅₃	1	6	yes	—
293	C₂×D₄	—	2	36	no	—
294	C₂×D₄	—	2	192	no	—
295	C₂×D₄	—	2	9	no	—
296	C₂×D₄	—	2	36	no	—
297	C₂×D₄	—	2	72	no	—
298	C₂×D₄	—	53	18	no	—
299	C₂×D₄	—	53	48	no	—
300	C₂×D₄	—	106	36	no	—
301	C₂×D₄	—	106	192	no	—
302	C₂×D₄	—	106	36	no	—
303	C₂×D₄	—	106	9	no	—
304	C₂×D₄	—	106	72	no	—
305	C₂×D₄	C₂×C₂²⋊D₅₃	1	36	yes	—
306	C₂×D₄	C₂³×D₅₃	1	18	yes	—
307	C₂×D₄	C₂³×D₅₃	1	48	yes	complemented
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	2	36	no	—
310	C₂²×C₄	—	106	1	no	—
311	C₂²×C₄	—	106	36	no	—
312	C₂²×C₄	C₂×C₂²⋊D₅₃	1	18	yes	—
313	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
314	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
315	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
316	Q₁₆	—	53	16	no	—
317	SD₁₆	—	53	48	no	—
318	SD₁₆	—	53	48	no	—
319	D₈	—	53	48	no	—
320	D₈	—	53	96	no	—
321	M₁₆	—	53	48	no	—
322	C₂×C₈	—	53	6	no	—
323	C₂×C₈	—	53	18	no	—
324	C₂×C₈	—	53	48	no	—
325	C₂³	—	2	12	no	—
326	C₂³	—	2	18	no	—
327	C₂³	—	2	48	no	—
328	C₂³	—	4	12	no	—
329	C₂³	—	8	72	no	—
330	C₂³	—	106	48	no	—
331	C₂³	—	106	18	no	—
332	C₂³	—	424	72	no	—
333	C₂³	—	212	12	no	—
334	C₂³	C₂²×C₂²⋊D₅₃	1	18	yes	—
335	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
336	Q₈	—	2	6	no	—
337	Q₈	—	53	4	no	—
338	Q₈	—	106	6	no	—
339	Q₈	C₂²×C₂²⋊D₅₃	1	16	yes	—
340	Q₈	C₂⁴×D₅₃	1	4	yes	—
341	D₄	—	2	48	no	—
342	D₄	—	2	18	no	—
343	D₄	—	2	48	no	—
344	D₄	—	53	12	no	—
345	D₄	—	106	48	no	—
346	D₄	—	106	48	no	—
347	D₄	—	106	18	no	—
348	D₄	C₂²×C₂²⋊D₅₃	1	48	yes	—
349	D₄	C₂⁴×D₅₃	1	12	yes	—
350	C₂×C₄	—	2	6	no	—
351	C₂×C₄	—	2	48	no	—
352	C₂×C₄	—	106	6	no	—
353	C₂×C₄	—	106	48	no	—
354	C₂×C₄	C₂²×C₂²⋊D₅₃	1	18	yes	—
355	C₂×C₄	C₂²×C₂²⋊D₅₃	1	3	yes	—
356	C₂×C₄	C₂²×C₂²⋊D₅₃	1	36	yes	—
357	C₂×C₄	C₂⁴×D₅₃	1	3	yes	—
358	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
359	C₈	—	53	4	no	—
360	C₈	—	53	12	no	—
361	C₂²	—	2	36	no	—
362	C₂²	—	2	12	no	—
363	C₂²	—	4	36	no	—
364	C₂²	—	8	48	no	—
365	C₂²	—	106	12	no	—
366	C₂²	—	424	48	no	—
367	C₂²	—	212	36	no	—
368	C₂²	#16811426	1	4	yes	direct product factor
369	C₂²	C₂³×C₂²⋊D₅₃	1	12	yes	—
370	C₂²	C₂³×C₂²⋊D₅₃	1	3	yes	—
371	C₄	—	2	4	no	—
372	C₄	—	106	4	no	—
373	C₄	C₂³×C₂²⋊D₅₃	1	12	yes	—
374	C₄	C₂³×C₂²⋊D₅₃	1	3	yes	—
375	C₄	C₂⁵×D₅₃	1	1	yes	characteristic
376	C₂	—	2	12	no	—
377	C₂	—	4	12	no	—
378	C₂	—	212	12	no	—
379	C₂	C₂×#16811426	1	6	yes	direct product factor
380	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×H(5,𝔽₂)×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×D₄
hypercenter	C₂²×H(5,𝔽₂)	D₅₃
hypocenter	C₅₃	C₂²×#576
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133263	C₂	semidirect
2. C₂×#16811426	C₂	direct (N × H)
3. C₂²×#13133243	C₂	semidirect
4. C₂²×#13133242	C₂	semidirect
5. C₂²×#13133237	C₂	semidirect
6. C₂³×#11719922	C₂	semidirect
7. C₂²×#13133224	C₂	semidirect
8. C₂²×H(5,𝔽₂)×C₅₃	C₂	semidirect
10. C₂×#13133263	C₂²	semidirect
12. C₂²×#11719945	C₂²	semidirect
13. C₂³×D₂₁₂	C₂²	semidirect
15. #16811426	C₂²	direct (N × H)
16. C₂×#13133243	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×#11719922	C₂²	semidirect
23. C₂²×#11719922	C₂²	semidirect
24. C₂²×#11719922	C₂²	semidirect
25. C₂×#13133224	C₂²	semidirect
26. C₂²×#11719917	C₂²	semidirect
27. C₂³×#88322	C₂²	semidirect
29. C₂×H(5,𝔽₂)×C₅₃	C₂²	semidirect
32. C₂²×#41×C₅₃	C₂²	semidirect
34. C₂³×D₄×C₅₃	C₂²	semidirect
37. #13133263	C₂³	semidirect
40. C₂×#11719945	C₂³	semidirect
45. C₂²×D₂₁₂	C₂³	semidirect
49. #13133243	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×#11719922	C₂³	semidirect
56. C₂×#11719922	C₂³	semidirect
57. C₂×#11719922	C₂³	semidirect
58. #13133224	C₂³	semidirect
59. C₂×#11719917	C₂³	semidirect
62. C₂²×#88322	C₂³	semidirect
65. H(5,𝔽₂)×C₅₃	C₂³	semidirect
70. C₂×#41×C₅₃	C₂³	semidirect
79. C₂²×D₄×C₅₃	C₂³	semidirect
93. #41×C₅₃	C₂⁴	semidirect
104. C₂×D₄×C₅₃	C₂⁴	semidirect
116. #11719945	C₂⁴	semidirect
121. C₂×D₂₁₂	C₂⁴	semidirect
125. #11719925	C₂⁴	semidirect
126. #11719924	C₂⁴	semidirect
127. #11719923	C₂⁴	semidirect
128. #11719922	C₂⁴	semidirect
129. #11719922	C₂⁴	semidirect
130. #11719917	C₂⁴	semidirect
133. C₂×#88322	C₂⁴	semidirect
175. C₂²×C₅₃	#576	semidirect
187. C₂²×H(5,𝔽₂)	D₅₃	semidirect
196. C₂×C₅₃	C₂×#576	semidirect
202. C₂×H(5,𝔽₂)	C₂×D₅₃	semidirect
207. C₂²×#41	C₂×D₅₃	semidirect
211. C₂³×D₄	C₂×D₅₃	semidirect
225. C₅₃	C₂²×#576	semidirect
231. H(5,𝔽₂)	C₂²×D₅₃	semidirect
239. C₂×#41	C₂²×D₅₃	semidirect
255. C₂²×D₄	C₂²×D₅₃	semidirect
287. #41	C₂³×D₅₃	semidirect
307. C₂×D₄	C₂³×D₅₃	semidirect
368. C₂²	#16811426	direct (N × H)
379. C₂	C₂×#16811426	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×#16811426	2
C₂²×#13133243	2
C₂²×#13133242	2
C₂²×#13133237	2
C₂³×#11719922	2
C₂²×#13133224	2
C₂²×H(5,𝔽₂)×C₅₃	2
C₂²×#576	53

Subgroups — C22×#16811426 #28297902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811426 #28297902