Subgroups — C₂²×#16811288 #28291007

Order 13568.

11798 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	C₂³×Q₈×D₅₃	C₂	1	2	yes	maximal, complemented
2	C₂×#16811288	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133216	C₂	1	2	yes	maximal, complemented
4	C₂²×#13133195	C₂	1	4	yes	maximal, complemented
5	C₂²×#13133193	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₂₁₂	C₂	1	1	yes	maximal, characteristic
8	C₂²×#13133179	C₂	1	2	yes	maximal
9	C₂²×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Dic₁₀₆	C₂²	1	4	yes	—
11	C₂³×Dic₁₀₆	C₂²	1	1	yes	characteristic
12	C₂²×Q₈×D₅₃	—	2	32	no	—
13	C₂²×Q₈×D₅₃	C₂²	1	48	yes	complemented
14	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
15	C₂³×C₄×D₅₃	C₂²	1	4	yes	—
16	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
17	#16811288	C₂²	1	256	yes	direct product factor
18	C₂×#13133216	C₂²	1	48	yes	complemented
19	C₂×#13133195	C₂²	1	96	yes	complemented
20	C₂×#13133193	C₂²	1	48	yes	complemented
21	C₂×C₄×Dic₁₀₆	C₂²	1	24	yes	complemented
22	C₂×C₄×D₂₁₂	C₂²	1	24	yes	—
23	C₂×#13133179	C₂²	1	48	yes	—
24	C₂²×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
26	C₂²×#11719919	C₂²	1	2	yes	—
27	C₂²×#11719919	C₂²	1	4	yes	—
28	C₂²×#11719921	C₂²	1	4	yes	—
29	C₂²×#11719921	C₂²	1	2	yes	—
30	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	—
31	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
32	C₂×C₄⋊Q₈×C₅₃	C₂²	1	24	yes	complemented
33	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
35	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
36	C₂⁴×D₅₃	C₂³	1	2	yes	—
37	C₂²×Dic₁₀₆	—	2	32	no	—
38	C₂²×Dic₁₀₆	D₄	1	16	yes	complemented
39	C₂²×Dic₁₀₆	C₂³	1	48	yes	—
40	C₂²×Dic₁₀₆	C₂³	1	12	yes	—
41	C₂×Q₈×D₅₃	—	2	384	no	—
42	C₂×Q₈×D₅₃	C₂³	1	128	yes	complemented
43	C₂²×D₂₁₂	Q₈	1	16	yes	complemented
44	C₂²×D₂₁₂	C₂³	1	12	yes	—
45	C₂³×Dic₅₃	C₂³	1	2	yes	—
46	C₂³×Dic₅₃	C₂³	1	2	yes	—
47	C₂³×Dic₅₃	C₂³	1	2	yes	—
48	C₂²×C₄×D₅₃	—	2	16	no	—
49	C₂²×C₄×D₅₃	—	2	32	no	—
50	C₂²×C₄×D₅₃	C₂³	1	48	yes	—
51	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
52	#13133216	C₂³	1	128	yes	complemented
53	#13133195	C₂³	1	256	yes	complemented
54	#13133193	C₂³	1	128	yes	complemented
55	C₄×Dic₁₀₆	C₂³	1	64	yes	complemented
56	C₄×D₂₁₂	C₂³	1	64	yes	—
57	#13133179	C₂³	1	128	yes	—
58	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
59	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
60	C₂×#11719919	C₂³	1	24	yes	—
61	C₂×#11719919	C₂³	1	48	yes	—
62	C₂×#11719921	C₂³	1	48	yes	—
63	C₂×#11719921	C₂³	1	24	yes	—
64	C₂×C₄×Dic₅₃	C₂³	1	24	yes	—
65	C₂²×Q₈×C₅₃	—	2	16	no	—
66	C₂²×Q₈×C₅₃	C₂³	1	24	yes	—
67	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
69	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₅₃	C₂³	1	64	yes	complemented
72	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
73	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
74	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
75	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
76	C₂³×D₅₃	—	2	8	no	—
77	C₂³×D₅₃	—	2	8	no	—
78	C₂³×D₅₃	C₂×Q₈	1	16	yes	complemented
79	C₂³×D₅₃	C₂⁴	1	12	yes	—
80	C₂×Q₈×C₅₃	—	2	96	no	—
81	C₂×Q₈×C₅₃	C₂⁴	1	32	yes	—
82	C₂²×C₄×C₅₃	—	2	16	no	—
83	C₂²×C₄×C₅₃	—	2	4	no	—
84	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
85	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	complemented
86	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
87	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
88	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
89	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
90	C₂²×Dic₅₃	—	2	8	no	—
91	C₂²×Dic₅₃	—	2	8	no	—
92	C₂²×Dic₅₃	C₂×D₄	1	16	yes	—
93	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
94	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
95	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
96	Q₈×D₅₃	—	2	512	no	—
97	C₂×D₂₁₂	C₂×Q₈	1	96	yes	complemented
98	C₂×D₂₁₂	C₂⁴	1	16	yes	—
99	C₂×C₄×D₅₃	—	2	96	no	—
100	C₂×C₄×D₅₃	—	2	192	no	—
101	C₂×C₄×D₅₃	C₂⁴	1	64	yes	—
102	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
103	C₂×Dic₁₀₆	—	2	192	no	—
104	C₂×Dic₁₀₆	C₂×D₄	1	96	yes	complemented
105	C₂×Dic₁₀₆	C₂⁴	1	64	yes	—
106	C₂×Dic₁₀₆	C₂⁴	1	16	yes	—
107	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	—
108	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	—
109	C₄²×C₅₃	C₂⁴	1	16	yes	—
110	#11719921	C₂⁴	1	64	yes	—
111	#11719921	C₂⁴	1	32	yes	—
112	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
113	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
114	#11719919	C₂⁴	1	32	yes	—
115	#11719919	C₂⁴	1	64	yes	—
116	C₄×Dic₅₃	C₂⁴	1	32	yes	—
117	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
118	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
119	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
120	C₂³×C₅₃	C₂⁵	1	3	yes	—
121	C₂²×D₅₃	—	2	64	no	—
122	C₂²×D₅₃	—	2	24	no	—
123	C₂²×D₅₃	—	2	24	no	—
124	C₂²×D₅₃	C₂²×Q₈	1	48	yes	complemented
125	C₂²×D₅₃	C₂⁵	1	8	yes	—
126	Q₈×C₅₃	—	2	64	no	—
127	C₂×C₄×C₅₃	—	2	48	no	—
128	C₂×C₄×C₅₃	—	2	12	no	—
129	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
130	C₂×C₄×C₅₃	C₂²×Q₈	1	24	yes	complemented
131	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
132	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
133	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
134	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
135	C₂×Dic₅₃	—	2	24	no	—
136	C₂×Dic₅₃	—	2	24	no	—
137	C₂×Dic₅₃	C₂²×D₄	1	48	yes	—
138	C₂×Dic₅₃	C₂⁵	1	8	yes	—
139	C₂×Dic₅₃	C₂⁵	1	8	yes	—
140	C₂×Dic₅₃	C₂⁵	1	8	yes	—
141	D₂₁₂	C₂²×Q₈	1	64	yes	complemented
142	C₄×D₅₃	—	2	64	no	—
143	C₄×D₅₃	—	2	128	no	—
144	Dic₁₀₆	—	2	128	no	—
145	Dic₁₀₆	C₂²×D₄	1	64	yes	complemented
146	C₂²×D₄×Q₈	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	D₄×Q₈	1	16	yes	complemented
148	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
149	C₂²×C₅₃	C₂³×Q₈	1	6	yes	—
150	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	6	yes	—
151	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
152	C₂×D₅₃	—	2	8	no	—
153	C₂×D₅₃	—	2	8	no	—
154	C₂×D₅₃	—	2	96	no	—
155	C₂×D₅₃	C₂³×Q₈	1	16	yes	complemented
156	C₄×C₅₃	—	2	16	no	—
157	C₄×C₅₃	—	2	4	no	—
158	C₄×C₅₃	C₂³×D₄	1	8	yes	—
159	C₄×C₅₃	C₂³×Q₈	1	8	yes	complemented
160	Dic₅₃	—	2	8	no	—
161	Dic₅₃	—	2	8	no	—
162	Dic₅₃	C₂³×D₄	1	16	yes	—
163	C₂⁴×Q₈	—	53	2	no	—
164	C₂×D₄×Q₈	—	53	48	no	—
165	C₂²×C₄⋊Q₈	—	53	2	no	—
166	C₂²×C₄⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
167	C₂²×C₂²⋊Q₈	—	53	2	no	—
168	C₂²×C₂²⋊Q₈	—	53	4	no	—
169	C₂²×C₄×Q₈	—	53	1	no	—
170	C₂²×C₄×D₄	—	53	1	no	—
171	C₂²×C₄×D₄	—	53	2	no	—
172	C₂×C₅₃	C₂×D₄×Q₈	1	12	yes	complemented
173	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
174	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
175	C₂×C₅₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
176	D₅₃	—	2	16	no	—
177	C₂³×Q₈	—	53	48	no	—
178	C₂³×Q₈	—	53	1	no	—
179	C₂³×Q₈	—	53	4	no	—
180	C₂³×Q₈	—	106	32	no	—
181	C₂³×Q₈	C₂×D₅₃	1	2	yes	—
182	C₂³×D₄	—	53	1	no	—
183	C₂⁴×C₄	—	53	4	no	—
184	C₂⁴×C₄	—	53	2	no	—
185	D₄×Q₈	—	53	256	no	—
186	C₂×C₄⋊Q₈	—	53	48	no	—
187	C₂×C₄⋊Q₈	C₂×D₅₃	1	24	yes	complemented
188	C₂×C₂²⋊Q₈	—	53	48	no	—
189	C₂×C₂²⋊Q₈	—	53	96	no	—
190	C₂×C₄×Q₈	—	53	24	no	—
191	C₂×C₄×D₄	—	53	48	no	—
192	C₂×C₄×D₄	—	53	24	no	—
193	C₂²×C₄⋊C₄	—	53	2	no	—
194	C₂²×C₄⋊C₄	—	53	4	no	—
195	C₂²×C₄⋊C₄	—	53	1	no	—
196	C₂²×C₄⋊C₄	—	53	1	no	—
197	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
198	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
199	C₂²×C₂²⋊C₄	—	53	4	no	—
200	C₂²×C₂²⋊C₄	—	53	2	no	—
201	C₂²×C₄²	—	53	2	no	—
202	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
203	C₅₃	C₂²×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
204	C₂⁵	—	53	2	no	—
205	C₂²×Q₈	—	2	16	no	—
206	C₂²×Q₈	—	53	128	no	—
207	C₂²×Q₈	—	53	16	no	—
208	C₂²×Q₈	—	53	12	no	—
209	C₂²×Q₈	—	53	48	no	—
210	C₂²×Q₈	—	106	32	no	—
211	C₂²×Q₈	—	106	384	no	—
212	C₂²×Q₈	C₂²×D₅₃	1	24	yes	—
213	C₂²×D₄	—	53	16	no	—
214	C₂²×D₄	—	53	12	no	—
215	C₂³×C₄	—	53	2	no	—
216	C₂³×C₄	—	53	2	no	—
217	C₂³×C₄	—	53	2	no	—
218	C₂³×C₄	—	53	48	no	—
219	C₂³×C₄	—	53	24	no	—
220	C₂³×C₄	—	106	32	no	—
221	C₂³×C₄	—	106	16	no	—
222	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
223	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
224	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
225	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
226	C₄⋊Q₈	—	53	128	no	—
227	C₄⋊Q₈	C₂²×D₅₃	1	64	yes	complemented
228	C₂²⋊Q₈	—	53	128	no	—
229	C₂²⋊Q₈	—	53	256	no	—
230	C₄×Q₈	—	53	64	no	—
231	C₄×D₄	—	53	128	no	—
232	C₄×D₄	—	53	64	no	—
233	C₂×C₄⋊C₄	—	53	12	no	—
234	C₂×C₄⋊C₄	—	53	24	no	—
235	C₂×C₄⋊C₄	—	53	12	no	—
236	C₂×C₄⋊C₄	—	53	48	no	—
237	C₂×C₄⋊C₄	C₂²×D₅₃	1	24	yes	—
238	C₂×C₄⋊C₄	C₂²×D₅₃	1	24	yes	—
239	C₂×C₂²⋊C₄	—	53	48	no	—
240	C₂×C₂²⋊C₄	—	53	24	no	—
241	C₂×C₄²	—	53	24	no	—
242	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
243	C₂⁴	—	53	16	no	—
244	C₂⁴	—	53	12	no	—
245	C₂⁴	—	106	8	no	—
246	C₂⁴	—	106	8	no	—
247	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
248	C₂×Q₈	—	2	96	no	—
249	C₂×Q₈	—	53	96	no	—
250	C₂×Q₈	—	53	16	no	—
251	C₂×Q₈	—	53	64	no	—
252	C₂×Q₈	—	106	192	no	—
253	C₂×Q₈	—	106	512	no	—
254	C₂×Q₈	C₂³×D₅₃	1	32	yes	—
255	C₂×D₄	—	53	96	no	—
256	C₂×D₄	—	53	16	no	—
257	C₂²×C₄	—	2	16	no	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	53	16	no	—
260	C₂²×C₄	—	53	12	no	—
261	C₂²×C₄	—	53	12	no	—
262	C₂²×C₄	—	53	12	no	—
263	C₂²×C₄	—	53	64	no	—
264	C₂²×C₄	—	53	32	no	—
265	C₂²×C₄	—	106	192	no	—
266	C₂²×C₄	—	106	8	no	—
267	C₂²×C₄	—	106	8	no	—
268	C₂²×C₄	—	106	96	no	—
269	C₂²×C₄	D₄×D₅₃	1	8	yes	—
270	C₂²×C₄	Q₈×D₅₃	1	8	yes	complemented
271	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
272	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
273	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
274	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
275	C₄⋊C₄	—	53	16	no	—
276	C₄⋊C₄	—	53	32	no	—
277	C₄⋊C₄	—	53	16	no	—
278	C₄⋊C₄	—	53	64	no	—
279	C₄⋊C₄	C₂³×D₅₃	1	32	yes	—
280	C₄⋊C₄	C₂³×D₅₃	1	32	yes	—
281	C₂²⋊C₄	—	53	64	no	—
282	C₂²⋊C₄	—	53	32	no	—
283	C₄²	—	53	32	no	—
284	C₄²	C₂³×D₅₃	1	16	yes	—
285	C₂³	—	53	48	no	—
286	C₂³	—	53	8	no	—
287	C₂³	—	106	64	no	—
288	C₂³	—	106	24	no	—
289	C₂³	—	106	24	no	—
290	C₂³	C₂×D₄×D₅₃	1	4	yes	—
291	C₂³	C₂×Q₈×D₅₃	1	4	yes	—
292	C₂³	#13133265	1	4	yes	—
293	C₂³	C₂⁴×D₅₃	1	3	yes	—
294	Q₈	—	2	64	no	—
295	Q₈	—	53	64	no	—
296	Q₈	—	106	128	no	—
297	D₄	—	53	64	no	—
298	C₂×C₄	—	2	48	no	—
299	C₂×C₄	—	2	12	no	—
300	C₂×C₄	—	53	48	no	—
301	C₂×C₄	—	53	8	no	—
302	C₂×C₄	—	53	8	no	—
303	C₂×C₄	—	53	8	no	—
304	C₂×C₄	—	106	128	no	—
305	C₂×C₄	—	106	24	no	—
306	C₂×C₄	—	106	24	no	—
307	C₂×C₄	—	106	64	no	—
308	C₂×C₄	C₂×D₄×D₅₃	1	24	yes	—
309	C₂×C₄	C₂×Q₈×D₅₃	1	24	yes	complemented
310	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
311	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
312	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
313	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
314	C₂²	—	53	16	no	—
315	C₂²	—	106	96	no	—
316	C₂²	—	106	8	no	—
317	C₂²	—	106	8	no	—
318	C₂²	#16811288	1	16	yes	direct product factor
319	C₂²	C₂²×D₄×D₅₃	1	6	yes	—
320	C₂²	C₂²×Q₈×D₅₃	1	6	yes	—
321	C₂²	C₂×#13133265	1	6	yes	—
322	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
323	C₄	—	2	16	no	—
324	C₄	—	2	4	no	—
325	C₄	—	53	16	no	—
326	C₄	—	106	8	no	—
327	C₄	—	106	8	no	—
328	C₄	C₂²×D₄×D₅₃	1	8	yes	—
329	C₄	C₂²×Q₈×D₅₃	1	8	yes	complemented
330	C₂	—	106	16	no	—
331	C₂	C₂×#16811288	1	12	yes	direct product factor
332	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
333	C₂	C₂³×Q₈×D₅₃	1	1	yes	characteristic
334	C₂	C₂²×#13133265	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×C₄⋊Q₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂³×D₂₁₂	C₂²
hypercenter	C₂²×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	C₂²×D₄×Q₈
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₈×D₅₃	C₂	semidirect
2. C₂×#16811288	C₂	direct (N × H)
3. C₂²×#13133216	C₂	semidirect
4. C₂²×#13133195	C₂	semidirect
5. C₂²×#13133193	C₂	semidirect
6. C₂²×C₄×Dic₁₀₆	C₂	semidirect
9. C₂²×C₄⋊Q₈×C₅₃	C₂	semidirect
13. C₂²×Q₈×D₅₃	C₂²	semidirect
17. #16811288	C₂²	direct (N × H)
18. C₂×#13133216	C₂²	semidirect
19. C₂×#13133195	C₂²	semidirect
20. C₂×#13133193	C₂²	semidirect
21. C₂×C₄×Dic₁₀₆	C₂²	semidirect
32. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
38. C₂²×Dic₁₀₆	D₄	semidirect
42. C₂×Q₈×D₅₃	C₂³	semidirect
43. C₂²×D₂₁₂	Q₈	semidirect
52. #13133216	C₂³	semidirect
53. #13133195	C₂³	semidirect
54. #13133193	C₂³	semidirect
55. C₄×Dic₁₀₆	C₂³	semidirect
71. C₄⋊Q₈×C₅₃	C₂³	semidirect
78. C₂³×D₅₃	C₂×Q₈	semidirect
85. C₂²×C₄×C₅₃	C₂×Q₈	semidirect
97. C₂×D₂₁₂	C₂×Q₈	semidirect
104. C₂×Dic₁₀₆	C₂×D₄	semidirect
124. C₂²×D₅₃	C₂²×Q₈	semidirect
130. C₂×C₄×C₅₃	C₂²×Q₈	semidirect
141. D₂₁₂	C₂²×Q₈	semidirect
145. Dic₁₀₆	C₂²×D₄	semidirect
147. C₂²×C₅₃	D₄×Q₈	semidirect
155. C₂×D₅₃	C₂³×Q₈	semidirect
159. C₄×C₅₃	C₂³×Q₈	semidirect
166. C₂²×C₄⋊Q₈	D₅₃	semidirect
172. C₂×C₅₃	C₂×D₄×Q₈	semidirect
187. C₂×C₄⋊Q₈	C₂×D₅₃	semidirect
203. C₅₃	C₂²×D₄×Q₈	semidirect
227. C₄⋊Q₈	C₂²×D₅₃	semidirect
270. C₂²×C₄	Q₈×D₅₃	semidirect
309. C₂×C₄	C₂×Q₈×D₅₃	semidirect
318. C₂²	#16811288	direct (N × H)
329. C₄	C₂²×Q₈×D₅₃	semidirect
331. C₂	C₂×#16811288	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₈×D₅₃	2
C₂×#16811288	2
C₂²×#13133216	2
C₂²×#13133195	2
C₂²×#13133193	2
C₂²×C₄×Dic₁₀₆	2
C₂²×C₄×D₂₁₂	2
C₂²×#13133179	2
C₂²×C₄⋊Q₈×C₅₃	2
C₂²×D₄×Q₈	53

Subgroups — C22×#16811288 #28291007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811288 #28291007