Subgroups — #38754399

Order 25216.

526 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263348	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₁₅₇₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#27161725	C₂	1	2	yes	maximal, complemented
4	#27161463	C₂	1	1	yes	maximal, characteristic, complemented
5	#27161320	C₂	1	1	yes	maximal, characteristic, complemented
6	#27161315	C₂	1	1	yes	maximal, characteristic, complemented
7	#27161295	C₂	1	2	yes	maximal, complemented
8	#27161232	C₂	1	2	yes	maximal, complemented
9	#27161226	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₅₇₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15691358	C₂²	1	2	yes	complemented
13	C₂²×D₇₈₈	C₂²	1	2	yes	—
14	#20263348	—	2	4	no	—
15	C₂×Dic₇₈₈	C₂²	1	1	yes	characteristic, complemented
16	C₂×#15691347	C₂²	1	2	yes	complemented
17	C₂×#15691347	C₂²	1	2	yes	complemented
18	C₂×D₁₅₇₆	—	2	4	no	—
19	C₂×D₁₅₇₆	C₂²	1	1	yes	characteristic
20	C₂×D₁₅₇₆	C₂²	1	2	yes	—
21	C₂×D₁₅₇₆	C₂²	1	1	yes	characteristic
22	#20263314	C₂²	1	2	yes	complemented
23	#20263311	—	2	2	no	—
24	#20263305	C₂²	1	2	yes	—
25	#20263300	C₂²	1	2	yes	complemented
26	#20263299	—	2	2	no	—
27	#20263294	—	2	2	no	—
28	C₄×D₇₈₈	C₂²	1	2	yes	complemented
29	C₂²⋊C₄×D₁₉₇	—	2	2	no	—
30	#20263260	C₂²	1	1	yes	characteristic, complemented
31	#20263251	C₂²	1	2	yes	complemented
32	#20263248	C₂²	1	2	yes	complemented
33	#20263248	C₂²	1	2	yes	complemented
34	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
35	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
36	C₄×D₄×C₁₉₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
38	C₂²⋊C₈×C₁₉₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
41	C₂³×D₁₉₇	—	2	2	no	—
42	#15691358	—	2	4	no	—
43	#15691358	—	4	2	no	—
44	C₂×Dic₃₉₄	C₂³	1	2	yes	—
45	C₂×D₇₈₈	—	2	4	no	—
46	C₂×D₇₈₈	—	2	2	no	—
47	C₂×D₇₈₈	—	4	2	no	—
48	C₂×D₇₈₈	C₂³	1	2	yes	—
49	C₂×D₇₈₈	C₂³	1	2	yes	—
50	C₂×C₂²⋊D₁₉₇	—	2	2	no	—
51	C₂×C₂²⋊D₁₉₇	—	2	2	no	—
52	C₂×C₄×D₁₉₇	—	2	2	no	—
53	C₂×C₄×D₁₉₇	—	2	2	no	—
54	Dic₇₈₈	—	2	2	no	—
55	#15691347	—	2	4	no	—
56	#15691347	—	4	2	no	—
57	D₁₅₇₆	—	2	4	no	—
58	D₁₅₇₆	—	2	2	no	—
59	D₁₅₇₆	D₄	1	4	yes	complemented
60	C₄⋊Dic₁₉₇	C₂³	1	2	yes	—
61	#15691337	—	2	2	no	—
62	C₂²⋊Dic₁₉₇	—	2	2	no	—
63	#15691335	—	2	2	no	—
64	#15691335	—	2	2	no	—
65	#15691335	—	2	2	no	—
66	#15691335	—	2	2	no	—
67	C₂×D₄×C₁₉₇	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₉₇	—	2	2	no	—
71	C₂×C₈×C₁₉₇	—	2	2	no	—
72	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₉₇	D₄	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₉₇	D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₉₇	D₄	1	1	yes	characteristic
79	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
80	C₂²×D₁₉₇	—	2	2	no	—
81	C₂²×D₁₉₇	—	2	2	no	—
82	C₂²×D₁₉₇	—	4	4	no	—
83	C₂²×D₁₉₇	—	4	4	no	—
84	C₂²×D₁₉₇	—	8	2	no	—
85	Dic₃₉₄	—	2	2	no	—
86	Dic₃₉₄	—	2	2	no	—
87	D₇₈₈	—	2	2	no	—
88	D₇₈₈	—	2	2	no	—
89	D₇₈₈	—	2	2	no	—
90	D₇₈₈	—	4	4	no	—
91	D₇₈₈	C₂×D₄	1	4	yes	—
92	C₂²⋊D₁₉₇	—	4	4	no	—
93	C₂²⋊D₁₉₇	—	8	2	no	—
94	C₂×Dic₁₉₇	—	2	2	no	—
95	C₂×Dic₁₉₇	—	2	2	no	—
96	C₄×D₁₉₇	—	4	4	no	—
97	C₄×D₁₉₇	—	8	2	no	—
98	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
99	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
100	D₄×C₁₉₇	—	2	2	no	—
101	C₂×C₄×C₁₉₇	—	2	1	no	—
102	C₂×C₄×C₁₉₇	—	2	1	no	—
103	C₂×C₄×C₁₉₇	—	2	1	no	—
104	C₂×C₄×C₁₉₇	—	2	1	no	—
105	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
110	C₈×C₁₉₇	—	2	1	no	—
111	C₈×C₁₉₇	—	2	1	no	—
112	C₈×C₁₉₇	—	2	1	no	—
113	C₈×C₁₉₇	C₂×D₄	1	2	yes	—
114	C₂²×C₁₉₇	—	2	1	no	—
115	C₂²×C₁₉₇	—	2	1	no	—
116	C₂²×C₁₉₇	—	2	1	no	—
117	C₂²×C₁₉₇	—	2	1	no	—
118	C₂²×C₁₉₇	—	2	1	no	—
119	C₂²×C₁₉₇	#135	1	2	yes	complemented
120	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₁₉₇	—	2	4	no	—
122	C₂×D₁₉₇	—	4	2	no	—
123	C₂×D₁₉₇	—	4	2	no	—
124	C₂×D₁₉₇	—	4	2	no	—
125	C₂×D₁₉₇	—	4	2	no	—
126	C₂×D₁₉₇	—	4	2	no	—
127	C₂×D₁₉₇	—	8	8	no	—
128	C₄×C₁₉₇	—	2	1	no	—
129	C₄×C₁₉₇	—	2	1	no	—
130	C₄×C₁₉₇	—	2	1	no	—
131	C₄×C₁₉₇	—	2	1	no	—
132	C₄×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
133	C₄×C₁₉₇	H(5,𝔽₂)	1	1	yes	characteristic
134	Dic₁₉₇	—	4	2	no	—
135	Dic₁₉₇	—	4	2	no	—
136	C₂×C₁₉₇	—	2	2	no	—
137	C₂×C₁₉₇	—	4	1	no	—
138	C₂×C₁₉₇	D₄²	1	1	yes	characteristic
139	C₂×C₁₉₇	C₂×#135	1	1	yes	characteristic
140	C₂×C₁₉₇	#576	1	1	yes	characteristic
141	D₁₉₇	—	4	4	no	—
142	D₁₉₇	—	8	2	no	—
143	C₁₉₇	#3280	1	1	yes	characteristic, complemented, Sylow
144	#3280	—	197	1	no	maximal, Sylow
145	C₂×#135	—	197	1	no	—
146	C₂²×D₈	—	197	1	no	—
147	#546	—	197	2	no	—
148	#495	—	197	1	no	—
149	#467	—	197	1	no	—
150	#466	—	197	1	no	—
151	#463	—	197	2	no	—
152	#450	—	197	2	no	—
153	#449	—	197	2	no	—
154	C₄×D₈	—	197	1	no	—
155	C₈×D₄	D₁₉₇	1	1	yes	characteristic, complemented
156	C₂×#41	—	197	2	no	—
157	C₂²×D₄	—	197	2	no	—
158	#135	—	394	4	no	—
159	C₂×Q₁₆	—	197	1	no	—
160	C₂×SD₁₆	—	197	2	no	—
161	C₂×SD₁₆	—	197	2	no	—
162	C₂×D₈	—	394	4	no	—
163	C₂×D₈	—	197	2	no	—
164	C₂×D₈	—	197	1	no	—
165	C₂×D₈	—	197	1	no	—
166	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
167	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic, complemented
168	#124	—	197	2	no	—
169	#123	—	394	2	no	—
170	C₂²⋊Q₈	—	197	2	no	—
171	#121	—	394	2	no	—
172	#121	—	197	2	no	—
173	#120	—	394	2	no	—
174	C₄×D₄	—	197	2	no	—
175	C₄×D₄	C₂×D₁₉₇	1	1	yes	characteristic
176	C₂×C₂²⋊C₄	—	394	2	no	—
177	#107	—	197	1	no	—
178	C₄⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic, complemented
179	#103	—	197	2	no	—
180	#102	—	197	2	no	—
181	#102	—	197	2	no	—
182	C₂²⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic
183	C₂²⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic, complemented
184	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic, complemented
185	C₂⁴	—	394	2	no	—
186	#41	—	788	2	no	—
187	#41	—	394	4	no	—
188	C₂×Q₈	—	197	2	no	—
189	C₂×D₄	—	788	2	no	—
190	C₂×D₄	—	394	4	no	—
191	C₂×D₄	—	394	2	no	—
192	C₂×D₄	—	394	2	no	—
193	C₂×D₄	—	394	2	no	—
194	C₂×D₄	—	197	2	no	—
195	C₂×D₄	—	197	2	no	—
196	C₂×D₄	D₇₈₈	1	1	yes	characteristic
197	C₂²×C₄	—	394	2	no	—
198	C₂²×C₄	—	394	2	no	—
199	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
201	Q₁₆	—	394	2	no	—
202	SD₁₆	—	788	2	no	—
203	SD₁₆	—	394	4	no	—
204	D₈	—	394	4	no	—
205	D₈	—	394	2	no	—
206	D₈	—	197	4	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
211	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
212	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
213	C₄⋊C₄	—	394	2	no	—
214	C₄⋊C₄	—	197	2	no	—
215	C₄⋊C₄	D₇₈₈	1	1	yes	characteristic
216	C₂²⋊C₄	—	394	2	no	—
217	C₂²⋊C₄	—	394	2	no	—
218	C₂²⋊C₄	—	394	2	no	—
219	C₂²⋊C₄	—	394	2	no	—
220	C₂²⋊C₄	—	394	2	no	—
221	C₂²⋊C₄	D₇₈₈	1	1	yes	characteristic
222	C₂²⋊C₄	D₇₈₈	1	1	yes	characteristic
223	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
224	C₂³	—	788	4	no	—
225	C₂³	—	788	4	no	—
226	C₂³	—	1576	2	no	—
227	C₂³	—	394	2	no	—
228	C₂³	—	394	2	no	—
229	C₂³	C₂×D₇₈₈	1	1	yes	characteristic
230	C₂³	C₂×D₇₈₈	1	1	yes	characteristic
231	Q₈	—	394	2	no	—
232	Q₈	—	394	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	788	4	no	—
235	D₄	—	788	4	no	—
236	D₄	—	1576	2	no	—
237	D₄	—	394	2	no	—
238	D₄	—	394	2	no	—
239	D₄	—	394	2	no	—
240	D₄	—	197	4	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	788	4	no	—
246	C₂×C₄	—	1576	2	no	—
247	C₂×C₄	—	394	2	no	—
248	C₂×C₄	—	394	2	no	—
249	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
250	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
251	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
252	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
253	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	D₄×D₁₉₇	1	2	yes	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	788	2	no	—
264	C₂²	—	788	2	no	—
265	C₂²	—	788	2	no	—
266	C₂²	—	788	2	no	—
267	C₂²	—	788	2	no	—
268	C₂²	—	1576	8	no	—
269	C₂²	—	394	4	no	—
270	C₂²	#20263348	1	2	yes	complemented
271	C₂²	C₂²×D₇₈₈	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	788	2	no	—
277	C₄	—	788	2	no	—
278	C₄	C₂×D₄×D₁₉₇	1	1	yes	characteristic
279	C₄	#20263372	1	1	yes	characteristic
280	C₂	—	2	2	no	—
281	C₂	—	4	1	no	—
282	C₂	—	788	4	no	—
283	C₂	—	1576	2	no	—
284	C₂	D₄×D₇₈₈	1	1	yes	characteristic
285	C₂	C₂×#20263348	1	1	yes	characteristic
286	C₂	#27161884	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₈₈
commutator, cosocle	C₂×C₄×C₁₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	C₈×D₄×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂²×D₄
hypercenter	C₈×D₄	D₁₉₇
hypocenter	C₁₉₇	#3280
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#20263348	C₂	semidirect
2. C₂²×D₁₅₇₆	C₂	semidirect
3. #27161725	C₂	semidirect
4. #27161463	C₂	semidirect
5. #27161320	C₂	semidirect
6. #27161315	C₂	semidirect
7. #27161295	C₂	semidirect
8. #27161232	C₂	semidirect
9. #27161226	C₂	semidirect
10. C₄×D₁₅₇₆	C₂	semidirect
11. C₈×D₄×C₁₉₇	C₂	semidirect
12. C₂×#15691358	C₂²	semidirect
15. C₂×Dic₇₈₈	C₂²	semidirect
16. C₂×#15691347	C₂²	semidirect
17. C₂×#15691347	C₂²	semidirect
22. #20263314	C₂²	semidirect
25. #20263300	C₂²	semidirect
28. C₄×D₇₈₈	C₂²	semidirect
30. #20263260	C₂²	semidirect
31. #20263251	C₂²	semidirect
32. #20263248	C₂²	semidirect
33. #20263248	C₂²	semidirect
35. C₂²×C₈×C₁₉₇	C₂²	semidirect
37. C₄⋊C₈×C₁₉₇	C₂²	semidirect
39. C₂²⋊C₈×C₁₉₇	C₂²	semidirect
40. C₄×C₈×C₁₉₇	C₂²	semidirect
59. D₁₅₇₆	D₄	semidirect
119. C₂²×C₁₉₇	#135	semidirect
143. C₁₉₇	#3280	semidirect
155. C₈×D₄	D₁₉₇	semidirect
167. C₂²×C₈	C₂×D₁₉₇	semidirect
178. C₄⋊C₈	C₂×D₁₉₇	semidirect
183. C₂²⋊C₈	C₂×D₁₉₇	semidirect
184. C₄×C₈	C₂×D₁₉₇	semidirect
270. C₂²	#20263348	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263348	2
C₂²×D₁₅₇₆	2
#27161725	2
#27161463	2
#27161320	2
#27161315	2
#27161295	2
#27161232	2
#27161226	2
C₄×D₁₅₇₆	2
C₈×D₄×C₁₉₇	2
#3280	197