Subgroups — C₂×C₈×#11719922 #27353447

Order 13568.

910 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719922	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×#11719922	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810375	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810350	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11719922	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₅₃	—	2	1	no	—
11	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
12	C₄×#11719922	C₄	1	4	yes	—
13	C₄×#11719922	C₂²	1	4	yes	—
14	C₈×D₂₁₂	—	2	2	no	—
15	C₈×D₂₁₂	C₂²	1	4	yes	complemented
16	C₂×#13133150	C₄	1	1	yes	characteristic
17	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
18	C₂×#13133138	C₄	1	1	yes	characteristic
19	C₂×#13133137	C₄	1	1	yes	characteristic
20	C₂×#13133129	—	2	1	no	—
21	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
22	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
23	#16810375	C₂²	1	4	yes	complemented
24	D₄⋊#88322	C₂²	1	4	yes	complemented
25	#16810350	C₂²	1	4	yes	complemented
26	C₈×#88322	C₂²	1	4	yes	complemented
27	C₂³×C₈×C₅₃	—	2	1	no	—
28	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₈×D₄×C₅₃	—	2	2	no	—
30	C₈×D₄×C₅₃	C₂²	1	4	yes	complemented
31	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
33	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂²×D₂₁₂	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₅₃	—	2	1	no	—
36	C₂×#11719922	C₈	1	8	yes	direct product factor
37	C₂×#11719922	C₂×C₄	1	4	yes	—
38	C₂²×#88322	—	2	1	no	—
39	C₂²×#88322	—	2	1	no	—
40	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
41	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₅₃	—	2	4	no	—
43	C₂×C₈×D₅₃	—	4	4	no	—
44	C₄×D₂₁₂	—	2	1	no	—
45	C₄×D₂₁₂	—	2	1	no	—
46	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
47	C₄×D₂₁₂	C₂³	1	2	yes	—
48	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×#11719921	—	2	1	no	—
50	#13133150	C₂×C₄	1	4	yes	—
51	C₄⋊#88322	—	2	1	no	—
52	C₄⋊#88322	C₂³	1	2	yes	—
53	#13133138	C₂×C₄	1	4	yes	—
54	#13133137	C₂×C₄	1	4	yes	—
55	#13133129	—	2	4	no	—
56	C₄×#88322	—	2	1	no	—
57	C₄×#88322	C₂×C₄	1	2	yes	—
58	C₄×#88322	C₂³	1	2	yes	—
59	C₄×#88322	C₂³	1	2	yes	—
60	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂³×C₄×C₅₃	—	2	1	no	—
62	C₂²×C₈×C₅₃	—	2	4	no	—
63	C₂²×C₈×C₅₃	—	2	1	no	—
64	C₂²×C₈×C₅₃	—	4	4	no	—
65	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
67	C₄×D₄×C₅₃	—	2	1	no	—
68	C₄×D₄×C₅₃	—	2	1	no	—
69	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
70	C₄×D₄×C₅₃	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
73	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₅₃	—	2	1	no	—
75	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
76	C₂²⋊C₈×C₅₃	—	2	4	no	—
77	C₄×C₈×C₅₃	D₄	1	2	yes	—
78	C₄×C₈×C₅₃	C₂³	1	2	yes	—
79	C₂⁴×C₅₃	—	2	1	no	—
80	C₂³×D₅₃	—	2	1	no	—
81	C₂×D₄×C₅₃	—	2	1	no	—
82	C₂×D₄×C₅₃	—	2	2	no	—
83	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	complemented
84	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₅₃	—	2	1	no	—
86	C₂²×C₄×C₅₃	—	2	2	no	—
87	C₂²×C₄×C₅₃	—	2	2	no	—
88	C₂²×C₄×C₅₃	—	4	2	no	—
89	C₂²×C₄×C₅₃	—	4	2	no	—
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
93	C₂²×Dic₅₃	—	2	1	no	—
94	C₂×D₂₁₂	—	2	1	no	—
95	C₂×D₂₁₂	—	2	2	no	—
96	C₂×D₂₁₂	C₂×C₈	1	4	yes	complemented
97	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
98	C₂×C₄×D₅₃	—	2	2	no	—
99	C₂×C₄×D₅₃	—	2	2	no	—
100	C₂×C₄×D₅₃	—	4	2	no	—
101	C₂×C₄×D₅₃	—	4	2	no	—
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	—	4	2	no	—
105	C₂×C₈×C₅₃	—	4	8	no	—
106	C₂×C₈×C₅₃	D₈	1	4	yes	complemented
107	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
109	C₄⋊C₄×C₅₃	—	2	1	no	—
110	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
111	C₂²⋊C₄×C₅₃	—	2	2	no	—
112	C₂²⋊C₄×C₅₃	—	2	2	no	—
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅₃	#41	1	1	yes	characteristic
116	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
117	#11719922	C₂×C₈	1	16	yes	direct product factor
118	#11719921	—	2	2	no	—
119	#11719921	—	2	2	no	—
120	C₄⋊Dic₅₃	—	2	1	no	—
121	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
122	C₂×#88322	—	2	4	no	—
123	C₂×#88322	—	2	2	no	—
124	C₂×#88322	—	2	2	no	—
125	C₂×#88322	—	2	2	no	—
126	C₂×#88322	—	4	2	no	—
127	C₂×#88322	C₂×C₈	1	4	yes	complemented
128	C₂×#88322	C₂²×C₄	1	2	yes	—
129	C₂×#88322	C₂²×C₄	1	2	yes	—
130	C₈×D₅₃	—	4	8	no	—
131	C₂³×C₅₃	—	2	2	no	—
132	C₂³×C₅₃	—	2	4	no	—
133	C₂³×C₅₃	—	4	2	no	—
134	C₂³×C₅₃	—	4	2	no	—
135	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
136	C₂²×D₅₃	—	2	2	no	—
137	C₂²×D₅₃	—	2	4	no	—
138	C₂²×D₅₃	—	4	2	no	—
139	C₂²×D₅₃	—	4	2	no	—
140	D₄×C₅₃	—	2	2	no	—
141	D₄×C₅₃	—	2	4	no	—
142	D₄×C₅₃	C₂²×C₈	1	4	yes	complemented
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	4	2	no	—
146	C₂×C₄×C₅₃	—	4	2	no	—
147	C₂×C₄×C₅₃	—	4	2	no	—
148	C₂×C₄×C₅₃	—	4	4	no	—
149	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
153	C₂×C₄×C₅₃	#131	1	2	yes	—
154	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
155	C₂×C₄×C₅₃	#135	1	2	yes	—
156	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
159	C₂×Dic₅₃	—	2	2	no	—
160	C₂×Dic₅₃	—	4	2	no	—
161	D₂₁₂	—	2	2	no	—
162	D₂₁₂	—	2	4	no	—
163	D₂₁₂	C₂²×C₈	1	4	yes	complemented
164	C₄×D₅₃	—	4	2	no	—
165	C₄×D₅₃	—	4	2	no	—
166	C₄×D₅₃	—	4	4	no	—
167	C₈×C₅₃	—	2	2	no	—
168	C₈×C₅₃	—	4	2	no	—
169	C₈×C₅₃	C₂×D₈	1	4	yes	complemented
170	#88322	—	2	4	no	—
171	#88322	—	2	2	no	—
172	#88322	—	4	2	no	—
173	#88322	C₂²×C₈	1	4	yes	complemented
174	C₂×C₈×D₈	—	53	1	no	maximal, Sylow
175	C₂²×C₅₃	—	2	4	no	—
176	C₂²×C₅₃	—	4	8	no	—
177	C₂²×C₅₃	—	4	2	no	—
178	C₂²×C₅₃	—	4	2	no	—
179	C₂²×C₅₃	C₈×D₄	1	2	yes	—
180	C₂²×C₅₃	C₄×D₈	1	2	yes	—
181	C₂²×C₅₃	#443	1	2	yes	—
182	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
183	C₂×D₅₃	—	2	4	no	—
184	C₂×D₅₃	—	4	8	no	—
185	C₂×D₅₃	—	4	2	no	—
186	C₂×D₅₃	—	4	2	no	—
187	C₄×C₅₃	—	2	1	no	—
188	C₄×C₅₃	—	2	1	no	—
189	C₄×C₅₃	—	4	2	no	—
190	C₄×C₅₃	C₈×D₄	1	2	yes	—
191	C₄×C₅₃	C₄×D₈	1	2	yes	—
192	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
193	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
194	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
195	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
196	Dic₅₃	—	4	2	no	—
197	C₂×C₄×D₈	—	53	1	no	—
198	C₂×C₈×D₄	—	53	1	no	—
199	C₂×C₈×D₄	D₅₃	1	1	yes	characteristic, complemented
200	C₈×D₈	—	53	8	no	—
201	C₂×#335	—	53	1	no	—
202	C₂×D₄⋊C₈	—	53	1	no	—
203	C₂×D₄⋊C₈	—	53	1	no	—
204	C₂×C₈²	—	53	1	no	—
205	C₂×C₅₃	—	4	4	no	—
206	C₂×C₅₃	C₈×D₈	1	4	yes	complemented
207	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
208	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
209	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
210	D₅₃	—	4	4	no	—
211	C₂²×D₈	—	53	1	no	—
212	C₂³×C₈	—	2	1	no	—
213	C₂³×C₈	—	106	1	no	—
214	C₂×C₄×D₄	—	53	1	no	—
215	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
216	C₄×D₈	—	53	4	no	—
217	C₄×D₈	—	53	4	no	—
218	C₈×D₄	—	2	2	no	—
219	C₈×D₄	—	53	4	no	—
220	C₈×D₄	—	106	2	no	—
221	C₈×D₄	C₂×D₅₃	1	4	yes	complemented
222	C₂×#107	—	53	1	no	—
223	C₂×C₄⋊C₈	—	53	1	no	—
224	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
225	C₂×#102	—	53	1	no	—
226	C₂×#102	—	53	1	no	—
227	C₂×C₂²⋊C₈	—	2	1	no	—
228	C₂×C₂²⋊C₈	—	106	1	no	—
229	C₂×C₄×C₈	—	53	1	no	—
230	C₂×C₄×C₈	—	53	1	no	—
231	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
232	#335	—	53	4	no	—
233	D₄⋊C₈	—	53	4	no	—
234	D₄⋊C₈	—	53	4	no	—
235	C₈²	—	53	4	no	—
236	C₅₃	C₂×C₈×D₈	1	1	yes	characteristic, complemented, Sylow
237	C₂²×D₄	—	53	1	no	—
238	C₂²×D₄	C₄×D₅₃	1	1	yes	characteristic
239	C₂³×C₄	—	2	1	no	—
240	C₂³×C₄	—	106	1	no	—
241	C₂×D₈	—	53	4	no	—
242	C₂×D₈	—	53	8	no	—
243	C₂²×C₈	—	2	1	no	—
244	C₂²×C₈	—	2	4	no	—
245	C₂²×C₈	—	4	4	no	—
246	C₂²×C₈	—	53	1	no	—
247	C₂²×C₈	—	53	1	no	—
248	C₂²×C₈	—	106	1	no	—
249	C₂²×C₈	—	106	1	no	—
250	C₂²×C₈	—	106	4	no	—
251	C₂²×C₈	—	212	4	no	—
252	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
253	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
254	C₄×D₄	—	2	1	no	—
255	C₄×D₄	—	2	1	no	—
256	C₄×D₄	—	53	2	no	—
257	C₄×D₄	—	53	2	no	—
258	C₄×D₄	—	106	1	no	—
259	C₄×D₄	—	106	1	no	—
260	C₄×D₄	C₄×D₅₃	1	2	yes	—
261	C₄×D₄	C₂²×D₅₃	1	2	yes	—
262	C₂×C₄⋊C₄	—	53	1	no	—
263	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
264	C₂×C₂²⋊C₄	—	2	1	no	—
265	C₂×C₂²⋊C₄	—	106	1	no	—
266	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
267	#107	—	53	4	no	—
268	C₄⋊C₈	—	2	1	no	—
269	C₄⋊C₈	—	53	2	no	—
270	C₄⋊C₈	—	106	1	no	—
271	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
272	#102	—	53	4	no	—
273	#102	—	53	4	no	—
274	C₂²⋊C₈	—	2	4	no	—
275	C₂²⋊C₈	—	106	4	no	—
276	C₄×C₈	—	53	2	no	—
277	C₄×C₈	—	53	2	no	—
278	C₄×C₈	—	53	2	no	—
279	C₄×C₈	—	106	1	no	—
280	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
281	C₄×C₈	C₂²×D₅₃	1	2	yes	—
282	C₂⁴	—	2	1	no	—
283	C₂⁴	—	106	1	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	53	4	no	—
287	C₂×D₄	—	53	2	no	—
288	C₂×D₄	—	106	1	no	—
289	C₂×D₄	—	106	2	no	—
290	C₂×D₄	C₈×D₅₃	1	4	yes	complemented
291	C₂×D₄	C₂×C₄×D₅₃	1	2	yes	—
292	C₂²×C₄	—	2	1	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	4	2	no	—
296	C₂²×C₄	—	4	2	no	—
297	C₂²×C₄	—	106	1	no	—
298	C₂²×C₄	—	106	2	no	—
299	C₂²×C₄	—	106	2	no	—
300	C₂²×C₄	—	212	2	no	—
301	C₂²×C₄	—	212	2	no	—
302	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
303	C₂²×C₄	#11719945	1	1	yes	characteristic
304	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
305	D₈	—	53	16	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	4	2	no	—
309	C₂×C₈	—	4	8	no	—
310	C₂×C₈	—	53	4	no	—
311	C₂×C₈	—	53	2	no	—
312	C₂×C₈	—	53	2	no	—
313	C₂×C₈	—	106	4	no	—
314	C₂×C₈	—	106	2	no	—
315	C₂×C₈	—	106	2	no	—
316	C₂×C₈	—	106	2	no	—
317	C₂×C₈	—	212	2	no	—
318	C₂×C₈	—	212	8	no	—
319	C₂×C₈	#11719922	1	4	yes	direct product factor
320	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
321	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	53	2	no	—
324	C₄⋊C₄	—	106	1	no	—
325	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
326	C₂²⋊C₄	—	2	2	no	—
327	C₂²⋊C₄	—	2	2	no	—
328	C₂²⋊C₄	—	106	2	no	—
329	C₂²⋊C₄	—	106	2	no	—
330	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
331	C₄²	#11719945	1	1	yes	characteristic
332	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
333	C₄²	C₂³×D₅₃	1	1	yes	characteristic
334	C₂³	—	2	2	no	—
335	C₂³	—	2	4	no	—
336	C₂³	—	4	2	no	—
337	C₂³	—	4	2	no	—
338	C₂³	—	106	2	no	—
339	C₂³	—	106	4	no	—
340	C₂³	—	212	2	no	—
341	C₂³	—	212	2	no	—
342	C₂³	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
343	D₄	—	2	2	no	—
344	D₄	—	2	4	no	—
345	D₄	—	53	4	no	—
346	D₄	—	106	2	no	—
347	D₄	—	106	4	no	—
348	D₄	C₂×C₈×D₅₃	1	4	yes	complemented
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	4	no	—
355	C₂×C₄	—	106	2	no	—
356	C₂×C₄	—	212	2	no	—
357	C₂×C₄	—	212	2	no	—
358	C₂×C₄	—	212	2	no	—
359	C₂×C₄	—	212	4	no	—
360	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
361	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
362	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
363	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
364	C₂×C₄	#13133223	1	2	yes	—
365	C₂×C₄	C₂×#11719922	1	2	yes	—
366	C₂×C₄	#13133237	1	2	yes	—
367	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
368	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
369	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
370	C₈	—	2	2	no	—
371	C₈	—	4	2	no	—
372	C₈	—	53	4	no	—
373	C₈	—	106	4	no	—
374	C₈	—	106	2	no	—
375	C₈	—	212	2	no	—
376	C₈	C₂×#11719922	1	4	yes	direct product factor
377	C₂²	—	2	4	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	4	8	no	—
380	C₂²	—	4	2	no	—
381	C₂²	—	106	4	no	—
382	C₂²	—	212	2	no	—
383	C₂²	—	212	8	no	—
384	C₂²	—	212	2	no	—
385	C₂²	C₈×C₂²⋊D₅₃	1	2	yes	—
386	C₂²	C₄×#11719922	1	2	yes	—
387	C₂²	#16810738	1	2	yes	—
388	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
389	C₄	—	2	1	no	—
390	C₄	—	2	1	no	—
391	C₄	—	4	2	no	—
392	C₄	—	212	2	no	—
393	C₄	C₈×C₂²⋊D₅₃	1	2	yes	—
394	C₄	C₄×#11719922	1	2	yes	—
395	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
396	C₄	C₂×#13133223	1	1	yes	characteristic
397	C₄	C₂²×#11719922	1	1	yes	characteristic
398	C₄	C₂×#13133237	1	1	yes	characteristic
399	C₂	—	4	4	no	—
400	C₂	—	212	4	no	—
401	C₂	C₈×#11719922	1	4	yes	direct product factor
402	C₂	C₂×C₈×C₂²⋊D₅₃	1	1	yes	characteristic
403	C₂	C₂×C₄×#11719922	1	1	yes	characteristic
404	C₂	C₂×#16810738	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₈×D₄×C₅₃	C₂
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂²×#11719922	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×C₈×D₄	D₅₃
hypocenter	C₅₃	C₂×C₈×D₈
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×D₂₁₂	C₂	semidirect
3. C₈×#11719922	C₂	direct (N × H)
4. C₂×#16810375	C₂	semidirect
5. C₂×D₄⋊#88322	C₂	semidirect
6. C₂×#16810350	C₂	semidirect
7. C₂×C₈×#88322	C₂	semidirect
8. C₂×C₈×D₄×C₅₃	C₂	semidirect
15. C₈×D₂₁₂	C₂²	semidirect
23. #16810375	C₂²	semidirect
24. D₄⋊#88322	C₂²	semidirect
25. #16810350	C₂²	semidirect
26. C₈×#88322	C₂²	semidirect
30. C₈×D₄×C₅₃	C₂²	semidirect
36. C₂×#11719922	C₈	direct (N × H)
83. C₂×D₄×C₅₃	C₂×C₈	semidirect
96. C₂×D₂₁₂	C₂×C₈	semidirect
106. C₂×C₈×C₅₃	D₈	semidirect
117. #11719922	C₂×C₈	direct (N × H)
127. C₂×#88322	C₂×C₈	semidirect
142. D₄×C₅₃	C₂²×C₈	semidirect
163. D₂₁₂	C₂²×C₈	semidirect
169. C₈×C₅₃	C₂×D₈	semidirect
173. #88322	C₂²×C₈	semidirect
199. C₂×C₈×D₄	D₅₃	semidirect
206. C₂×C₅₃	C₈×D₈	semidirect
221. C₈×D₄	C₂×D₅₃	semidirect
236. C₅₃	C₂×C₈×D₈	semidirect
290. C₂×D₄	C₈×D₅₃	semidirect
319. C₂×C₈	#11719922	direct (N × H)
348. D₄	C₂×C₈×D₅₃	semidirect
376. C₈	C₂×#11719922	direct (N × H)
401. C₂	C₈×#11719922	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×C₄×#11719922	2
C₂×C₈×D₂₁₂	2
C₈×#11719922	2
C₂×#16810375	2
C₂×D₄⋊#88322	2
C₂×#16810350	2
C₂×C₈×#88322	2
C₂×C₈×D₄×C₅₃	2
C₂×C₈×D₈	53

Subgroups — C2×C8×#11719922 #27353447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×#11719922 #27353447