Subgroups — #38752262

Order 25216.

472 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263356	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#20263346	C₂	1	1	yes	maximal, characteristic
3	#27161655	C₂	1	1	yes	maximal, characteristic, complemented
4	#27161301	C₂	1	2	yes	maximal, complemented
5	#27161294	C₂	1	2	yes	maximal, complemented
6	#27161289	C₂	1	2	yes	maximal, complemented
7	#27161284	C₂	1	2	yes	maximal, complemented
8	#27161106	C₂	1	1	yes	maximal, characteristic, complemented
9	#27161078	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#20263249	C₂	1	1	yes	maximal, characteristic, complemented
11	#520×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15691358	C₂²	1	1	yes	characteristic
13	C₂²×Dic₃₉₄	C₂²	1	1	yes	characteristic
14	#20263356	—	2	4	no	—
15	#20263346	—	2	4	no	—
16	C₂×#15691348	C₂²	1	1	yes	characteristic
17	C₂×#15691348	C₂²	1	2	yes	—
18	C₂×#15691346	C₂²	1	1	yes	characteristic
19	C₂×#15691346	C₂²	1	2	yes	—
20	C₂×#15691345	C₂²	1	1	yes	characteristic
21	C₂×#15691343	C₂²	1	1	yes	characteristic
22	C₂×#15691342	C₂²	1	1	yes	characteristic
23	C₂²×#13096802	C₂²	1	1	yes	characteristic
24	#20263326	C₂²	1	2	yes	—
25	#20263314	C₂²	1	2	yes	—
26	#20263310	—	2	2	no	—
27	#20263306	—	2	1	no	—
28	#20263306	—	2	1	no	—
29	C₄⋊#13096802	C₂²	1	2	yes	—
30	C₄⋊#13096802	C₂²	1	2	yes	—
31	#20263249	C₂²	1	2	yes	—
32	#20263249	C₂²	1	2	yes	—
33	#20263249	C₂²	1	2	yes	—
34	#20263246	C₂²	1	2	yes	—
35	C₂×#41×C₁₉₇	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₉₇	C₂²	1	2	yes	—
37	C₄×D₄×C₁₉₇	—	2	2	no	—
38	C₄×D₄×C₁₉₇	C₂²	1	2	yes	—
39	#117×C₁₉₇	—	2	1	no	—
40	#117×C₁₉₇	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₉₇	—	2	1	no	—
42	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
43	#15691358	—	2	2	no	—
44	#15691358	—	2	2	no	—
45	C₂×Dic₃₉₄	—	2	1	no	—
46	C₂×Dic₃₉₄	—	2	2	no	—
47	C₂×Dic₃₉₄	—	2	2	no	—
48	C₂×Dic₃₉₄	C₂³	1	2	yes	—
49	C₂×Dic₃₉₄	C₂³	1	1	yes	characteristic
50	C₂×D₇₈₈	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₉₇	—	2	1	no	—
52	C₂²×Dic₁₉₇	—	2	1	no	—
53	C₂×C₄×D₁₉₇	—	2	1	no	—
54	#15691348	—	2	2	no	—
55	#15691348	—	2	4	no	—
56	#15691346	—	2	2	no	—
57	#15691346	—	2	4	no	—
58	#15691345	—	2	2	no	—
59	#15691343	—	2	2	no	—
60	#15691342	—	2	2	no	—
61	C₂×#13096802	—	2	2	no	—
62	C₂×#13096802	C₂³	1	2	yes	—
63	C₂×#13096802	C₂³	1	1	yes	characteristic
64	C₂×#13096802	C₂³	1	1	yes	characteristic
65	C₄⋊Dic₁₉₇	—	2	2	no	—
66	C₄⋊Dic₁₉₇	—	2	2	no	—
67	#15691335	—	2	2	no	—
68	#15691335	—	2	2	no	—
69	#41×C₁₉₇	—	2	2	no	—
70	#41×C₁₉₇	D₄	1	4	yes	—
71	C₂×Q₈×C₁₉₇	C₂³	1	1	yes	characteristic
72	C₂×D₄×C₁₉₇	—	2	1	no	—
73	C₂×D₄×C₁₉₇	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₁₉₇	—	2	1	no	—
75	C₂²×C₄×C₁₉₇	—	2	1	no	—
76	C₂²×C₄×C₁₉₇	—	2	2	no	—
77	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₉₇	—	2	1	no	—
79	C₄⋊C₄×C₁₉₇	—	2	2	no	—
80	C₄⋊C₄×C₁₉₇	D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₉₇	D₄	1	1	yes	characteristic
82	C₄⋊C₄×C₁₉₇	C₂³	1	2	yes	—
83	C₂²⋊C₄×C₁₉₇	—	2	2	no	—
84	C₂²⋊C₄×C₁₉₇	D₄	1	2	yes	—
85	C₄²×C₁₉₇	—	2	2	no	—
86	C₄²×C₁₉₇	C₂³	1	2	yes	—
87	C₂²×D₁₉₇	—	2	1	no	—
88	Dic₃₉₄	—	2	2	no	—
89	Dic₃₉₄	—	2	1	no	—
90	Dic₃₉₄	—	2	2	no	—
91	Dic₃₉₄	—	2	1	no	—
92	Dic₃₉₄	—	4	2	no	—
93	D₇₈₈	—	2	1	no	—
94	D₇₈₈	—	2	1	no	—
95	C₂²⋊D₁₉₇	—	4	2	no	—
96	C₂×Dic₁₉₇	—	2	2	no	—
97	C₂×Dic₁₉₇	—	2	1	no	—
98	C₂×Dic₁₉₇	—	4	2	no	—
99	C₄×D₁₉₇	—	4	2	no	—
100	#13096802	—	2	1	no	—
101	#13096802	—	2	1	no	—
102	#13096802	—	2	2	no	—
103	C₂³×C₁₉₇	—	2	1	no	—
104	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
105	Q₈×C₁₉₇	—	2	1	no	—
106	Q₈×C₁₉₇	C₂×D₄	1	2	yes	—
107	D₄×C₁₉₇	—	2	4	no	—
108	D₄×C₁₉₇	—	2	1	no	—
109	D₄×C₁₉₇	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₉₇	—	2	1	no	—
111	C₂×C₄×C₁₉₇	—	2	4	no	—
112	C₂×C₄×C₁₉₇	—	2	2	no	—
113	C₂×C₄×C₁₉₇	—	8	2	no	—
114	C₂×C₄×C₁₉₇	—	8	1	no	—
115	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₁₉₇	#41	1	2	yes	—
120	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
121	C₂²×C₁₉₇	—	2	1	no	—
122	C₂²×C₁₉₇	—	2	1	no	—
123	C₂²×C₁₉₇	—	2	2	no	—
124	C₂²×C₁₉₇	—	4	1	no	—
125	C₂²×C₁₉₇	—	4	1	no	—
126	C₂²×C₁₉₇	#121	1	2	yes	—
127	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×D₁₉₇	—	4	1	no	—
129	C₂×D₁₉₇	—	4	1	no	—
130	C₂×D₁₉₇	—	4	1	no	—
131	C₄×C₁₉₇	—	2	2	no	—
132	C₄×C₁₉₇	—	2	2	no	—
133	C₄×C₁₉₇	—	2	2	no	—
134	C₄×C₁₉₇	—	4	2	no	—
135	C₄×C₁₉₇	#121	1	2	yes	—
136	C₄×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
137	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
138	Dic₁₉₇	—	4	1	no	—
139	Dic₁₉₇	—	4	2	no	—
140	C₂×C₁₉₇	—	2	2	no	—
141	C₂×C₁₉₇	—	4	2	no	—
142	C₂×C₁₉₇	C₂×#121	1	1	yes	characteristic
143	C₂×C₁₉₇	#577	1	1	yes	characteristic
144	C₂×C₁₉₇	#578	1	1	yes	characteristic
145	D₁₉₇	—	8	1	no	—
146	C₁₉₇	#3046	1	1	yes	characteristic, complemented, Sylow
147	#3046	—	197	1	no	maximal, Sylow
148	C₂×#137	—	197	1	no	—
149	C₂×#135	—	197	1	no	—
150	#536	—	197	1	no	—
151	#520	D₁₉₇	1	1	yes	characteristic, complemented
152	#464	—	197	2	no	—
153	#463	—	197	2	no	—
154	#462	—	197	2	no	—
155	#461	—	197	2	no	—
156	#424	—	197	1	no	—
157	#417	—	197	1	no	—
158	C₂×#103	—	197	1	no	—
159	C₂×#41	—	197	1	no	—
160	C₂×#41	C₂×D₁₉₇	1	1	yes	characteristic
161	C₂²×Q₈	—	197	1	no	—
162	#137	—	394	4	no	—
163	#135	—	394	4	no	—
164	C₂×Q₁₆	—	197	1	no	—
165	C₂×Q₁₆	—	197	2	no	—
166	C₂×SD₁₆	—	197	1	no	—
167	C₂×SD₁₆	—	197	1	no	—
168	C₂×SD₁₆	—	197	2	no	—
169	C₂×D₈	—	197	1	no	—
170	C₂×M₁₆	—	197	1	no	—
171	C₂²×C₈	—	197	1	no	—
172	C₄⋊Q₈	—	197	2	no	—
173	#124	—	197	2	no	—
174	#123	—	394	2	no	—
175	C₂²⋊Q₈	—	394	1	no	—
176	C₂²⋊Q₈	—	394	1	no	—
177	C₄×Q₈	C₂×D₁₉₇	1	2	yes	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	C₂×D₁₉₇	1	2	yes	—
180	#117	—	2	1	no	—
181	#117	C₂×D₁₉₇	1	1	yes	characteristic
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
184	C₄⋊C₈	—	197	2	no	—
185	C₄⋊C₈	—	197	2	no	—
186	#103	—	197	2	no	—
187	#103	—	197	2	no	—
188	#103	—	197	2	no	—
189	#102	—	197	2	no	—
190	#41	—	2	2	no	—
191	#41	—	394	2	no	—
192	#41	—	394	2	no	—
193	#41	D₇₈₈	1	4	yes	—
194	C₂×Q₈	—	394	2	no	—
195	C₂×Q₈	—	394	2	no	—
196	C₂×Q₈	—	394	1	no	—
197	C₂×Q₈	—	197	2	no	—
198	C₂×Q₈	—	197	1	no	—
199	C₂×Q₈	C₂²×D₁₉₇	1	1	yes	characteristic
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	394	1	no	—
202	C₂×D₄	—	197	1	no	—
203	C₂×D₄	C₂²×D₁₉₇	1	1	yes	characteristic
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	394	1	no	—
208	C₂²×C₄	—	394	1	no	—
209	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
210	Q₁₆	—	394	2	no	—
211	Q₁₆	—	394	4	no	—
212	SD₁₆	—	394	2	no	—
213	SD₁₆	—	394	2	no	—
214	SD₁₆	—	394	4	no	—
215	D₈	—	394	2	no	—
216	M₁₆	—	394	2	no	—
217	C₂×C₈	—	394	2	no	—
218	C₂×C₈	—	197	2	no	—
219	C₂×C₈	—	197	1	no	—
220	C₂×C₈	—	197	1	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	394	2	no	—
224	C₄⋊C₄	—	394	2	no	—
225	C₄⋊C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
226	C₄⋊C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
227	C₄⋊C₄	C₂²×D₁₉₇	1	2	yes	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	394	2	no	—
230	C₂²⋊C₄	—	394	2	no	—
231	C₂²⋊C₄	C₂²⋊D₁₉₇	1	2	yes	—
232	C₄²	—	2	2	no	—
233	C₄²	C₂²×D₁₉₇	1	2	yes	—
234	C₂³	—	2	1	no	—
235	C₂³	—	394	1	no	—
236	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
237	Q₈	—	2	1	no	—
238	Q₈	—	788	2	no	—
239	Q₈	—	394	2	no	—
240	Q₈	—	394	1	no	—
241	Q₈	—	394	2	no	—
242	Q₈	—	394	1	no	—
243	Q₈	C₂×D₇₈₈	1	2	yes	—
244	D₄	—	2	4	no	—
245	D₄	—	2	1	no	—
246	D₄	—	788	2	no	—
247	D₄	—	394	1	no	—
248	D₄	—	394	1	no	—
249	D₄	C₂×D₇₈₈	1	2	yes	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	788	2	no	—
256	C₂×C₄	—	788	2	no	—
257	C₂×C₄	—	394	2	no	—
258	C₂×C₄	—	394	1	no	—
259	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
260	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
261	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
262	C₂×C₄	C₂×D₇₈₈	1	2	yes	—
263	C₂×C₄	#15691358	1	2	yes	—
264	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
265	C₈	—	394	1	no	—
266	C₈	—	394	1	no	—
267	C₈	—	394	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	788	1	no	—
274	C₂²	—	788	1	no	—
275	C₂²	—	788	1	no	—
276	C₂²	#20263300	1	2	yes	—
277	C₂²	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	2	no	—
282	C₄	—	788	1	no	—
283	C₄	—	788	2	no	—
284	C₄	#20263300	1	2	yes	—
285	C₄	C₂²×D₇₈₈	1	1	yes	characteristic
286	C₄	C₂×#15691358	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	4	2	no	—
289	C₂	—	1576	1	no	—
290	C₂	C₂×#20263300	1	1	yes	characteristic
291	C₂	#27161890	1	1	yes	characteristic
292	C₂	#27161894	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	#520×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂²×D₄
prime radical	C₂×#20263346	C₂
hypercenter	#520	D₁₉₇
hypocenter	C₁₉₇	#3046
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
1. C₂×#20263356	C₂	semidirect
3. #27161655	C₂	semidirect
4. #27161301	C₂	semidirect
5. #27161294	C₂	semidirect
6. #27161289	C₂	semidirect
7. #27161284	C₂	semidirect
8. #27161106	C₂	semidirect
9. #27161078	C₂	semidirect
10. C₂×#20263249	C₂	semidirect
11. #520×C₁₉₇	C₂	semidirect
146. C₁₉₇	#3046	semidirect
151. #520	D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263356	2
C₂×#20263346	2
#27161655	2
#27161301	2
#27161294	2
#27161289	2
#27161284	2
#27161106	2
#27161078	2
C₂×#20263249	2
#520×C₁₉₇	2
#3046	197