Subgroups — #38745345

Order 25216.

568 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27160972	C₂	1	2	yes	maximal, complemented
2	#27160965	C₂	1	2	yes	maximal, complemented
3	#27160959	C₂	1	2	yes	maximal
4	#27160954	C₂	1	2	yes	maximal, complemented
5	#27160947	C₂	1	1	yes	maximal, characteristic, complemented
6	#27160945	C₂	1	1	yes	maximal, characteristic, complemented
7	#27160942	C₂	1	1	yes	maximal, characteristic, complemented
8	#27160939	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊Dic₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#20263233	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₁₉₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₁₉₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊Dic₁₉₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊Dic₁₉₇	C₂²	1	2	yes	—
18	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
19	#20263233	C₂²	1	1	yes	characteristic
20	#20263233	C₂²	1	1	yes	characteristic
21	#20263233	C₂²	1	2	yes	—
22	#20263233	C₂²	1	1	yes	characteristic
23	#20263233	C₂²	1	1	yes	characteristic
24	#20263233	C₂²	1	2	yes	—
25	#20263233	C₂²	1	2	yes	—
26	#20263233	C₂²	1	2	yes	—
27	#20263233	C₂²	1	2	yes	—
28	C₂²×D₄×C₁₉₇	C₂²	1	1	yes	characteristic
29	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
30	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₁₉₇	C₄	1	8	yes	complemented
32	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
36	C₂²×Dic₁₉₇	—	2	4	no	—
37	C₂²×Dic₁₉₇	—	4	2	no	—
38	C₂²×Dic₁₉₇	—	4	2	no	—
39	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
40	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
41	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
42	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
43	C₄⋊Dic₁₉₇	—	4	2	no	—
44	C₄⋊Dic₁₉₇	—	4	2	no	—
45	C₂²⋊Dic₁₉₇	—	2	4	no	—
46	C₂²⋊Dic₁₉₇	—	4	2	no	—
47	C₂²⋊Dic₁₉₇	—	4	2	no	—
48	C₄×Dic₁₉₇	—	4	2	no	—
49	C₂⁴×C₁₉₇	C₂³	1	1	yes	characteristic
50	C₂⁴×C₁₉₇	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₁₉₇	—	2	4	no	—
52	C₂×D₄×C₁₉₇	C₂×C₄	1	4	yes	—
53	C₂²×C₄×C₁₉₇	—	4	2	no	—
54	C₂²×C₄×C₁₉₇	—	4	2	no	—
55	C₂²×C₄×C₁₉₇	C₂×C₄	1	4	yes	—
56	C₂²×C₄×C₁₉₇	C₂×C₄	1	4	yes	complemented
57	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₁₉₇	C₂×C₄	1	4	yes	complemented
63	C₂²⋊C₄×C₁₉₇	C₂×C₄	1	4	yes	—
64	C₂²⋊C₄×C₁₉₇	C₂×C₄	1	4	yes	complemented
65	C₄²×C₁₉₇	C₂×C₄	1	4	yes	complemented
66	C₂×Dic₁₉₇	—	2	2	no	—
67	C₂×Dic₁₉₇	—	2	2	no	—
68	C₂×Dic₁₉₇	—	2	2	no	—
69	C₂×Dic₁₉₇	—	2	2	no	—
70	C₂×Dic₁₉₇	—	2	2	no	—
71	C₂×Dic₁₉₇	—	2	2	no	—
72	C₂×Dic₁₉₇	—	2	2	no	—
73	C₂×Dic₁₉₇	—	2	2	no	—
74	C₂×Dic₁₉₇	—	2	2	no	—
75	C₂×Dic₁₉₇	—	2	2	no	—
76	C₂×Dic₁₉₇	—	2	2	no	—
77	C₂×Dic₁₉₇	—	2	2	no	—
78	C₂×Dic₁₉₇	—	4	8	no	—
79	C₂³×C₁₉₇	—	2	2	no	—
80	C₂³×C₁₉₇	—	2	1	no	—
81	C₂³×C₁₉₇	—	2	1	no	—
82	C₂³×C₁₉₇	—	2	2	no	—
83	C₂³×C₁₉₇	—	2	2	no	—
84	C₂³×C₁₉₇	—	2	1	no	—
85	C₂³×C₁₉₇	—	2	1	no	—
86	C₂³×C₁₉₇	C₂²×C₄	1	2	yes	—
87	C₂³×C₁₉₇	C₂²×C₄	1	2	yes	—
88	C₂³×C₁₉₇	#41	1	4	yes	—
89	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
90	D₄×C₁₉₇	—	2	8	no	—
91	C₂×C₄×C₁₉₇	—	2	2	no	—
92	C₂×C₄×C₁₉₇	—	2	2	no	—
93	C₂×C₄×C₁₉₇	—	2	2	no	—
94	C₂×C₄×C₁₉₇	—	2	2	no	—
95	C₂×C₄×C₁₉₇	—	2	1	no	—
96	C₂×C₄×C₁₉₇	—	2	1	no	—
97	C₂×C₄×C₁₉₇	—	4	4	no	—
98	C₂×C₄×C₁₉₇	—	4	4	no	—
99	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
104	C₂²×C₁₉₇	—	2	2	no	—
105	C₂²×C₁₉₇	—	2	2	no	—
106	C₂²×C₁₉₇	—	2	4	no	—
107	C₂²×C₁₉₇	—	2	4	no	—
108	C₂²×C₁₉₇	—	2	2	no	—
109	C₂²×C₁₉₇	—	2	2	no	—
110	C₂²×C₁₉₇	—	2	2	no	—
111	C₂²×C₁₉₇	—	4	2	no	—
112	C₂²×C₁₉₇	—	4	2	no	—
113	C₂²×C₁₉₇	#117	1	4	yes	—
114	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
115	C₂²×C₁₉₇	C₂×#41	1	2	yes	—
116	C₂²×C₁₉₇	H(5,𝔽₂)	1	2	yes	—
117	C₂²×C₁₉₇	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²×C₁₉₇	2₋¹⁺⁴	1	1	yes	characteristic
119	C₄×C₁₉₇	—	2	2	no	—
120	C₄×C₁₉₇	—	2	2	no	—
121	C₄×C₁₉₇	—	2	2	no	—
122	C₄×C₁₉₇	—	2	2	no	—
123	C₄×C₁₉₇	—	2	2	no	—
124	Dic₁₉₇	—	4	2	no	—
125	Dic₁₉₇	—	4	2	no	—
126	Dic₁₉₇	—	4	2	no	—
127	Dic₁₉₇	—	4	2	no	—
128	C₂×C₁₉₇	—	2	4	no	—
129	C₂×C₁₉₇	—	4	2	no	—
130	C₂×C₁₉₇	C₂×#117	1	1	yes	characteristic
131	C₂×C₁₉₇	#518	1	1	yes	characteristic
132	C₂×C₁₉₇	#520	1	1	yes	characteristic
133	C₂×C₁₉₇	#538	1	2	yes	—
134	C₂×C₁₉₇	#539	1	2	yes	—
135	C₁₉₇	#2318	1	1	yes	characteristic, complemented, Sylow
136	#2318	—	197	1	no	maximal, Sylow
137	C₂×C₄×D₄	D₁₉₇	1	1	yes	characteristic
138	#388	—	197	2	no	—
139	#387	—	197	2	no	—
140	#386	—	197	2	no	—
141	#385	—	197	2	no	—
142	#383	—	197	1	no	—
143	#382	—	197	1	no	—
144	#381	—	197	1	no	—
145	#380	—	197	1	no	—
146	C₄×C₂²⋊C₄	—	197	1	no	—
147	C₂×#95	—	197	1	no	—
148	C₂²×D₄	C₂×D₁₉₇	1	1	yes	characteristic
149	C₂³×C₄	—	197	2	no	—
150	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
151	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
152	C₄×D₄	Dic₁₉₇	1	8	yes	complemented
153	C₂×C₄⋊C₄	—	197	2	no	—
154	C₂×C₄⋊C₄	—	197	2	no	—
155	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	—	197	2	no	—
157	C₂×C₂²⋊C₄	—	197	2	no	—
158	C₂×C₂²⋊C₄	—	197	2	no	—
159	C₂×C₂²⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
161	C₂×C₄²	—	197	2	no	—
162	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
163	#95	—	197	1	no	—
164	#95	—	197	1	no	—
165	#95	—	197	2	no	—
166	#95	—	197	1	no	—
167	#95	—	197	1	no	—
168	#95	—	197	2	no	—
169	#95	—	197	2	no	—
170	#95	—	197	2	no	—
171	#95	—	197	2	no	—
172	C₂⁴	C₂²×D₁₉₇	1	1	yes	characteristic
173	C₂⁴	C₂²×D₁₉₇	1	1	yes	characteristic
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	C₂×Dic₁₉₇	1	4	yes	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	788	2	no	—
179	C₂²×C₄	—	788	2	no	—
180	C₂²×C₄	—	394	4	no	—
181	C₂²×C₄	—	197	2	no	—
182	C₂²×C₄	—	197	2	no	—
183	C₂²×C₄	—	197	2	no	—
184	C₂²×C₄	—	197	2	no	—
185	C₂²×C₄	C₂×Dic₁₉₇	1	4	yes	—
186	C₂²×C₄	C₂×Dic₁₉₇	1	4	yes	complemented
187	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
188	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
192	C₄⋊C₄	—	788	2	no	—
193	C₄⋊C₄	—	788	2	no	—
194	C₄⋊C₄	C₂×Dic₁₉₇	1	4	yes	complemented
195	C₂²⋊C₄	—	788	2	no	—
196	C₂²⋊C₄	—	788	2	no	—
197	C₂²⋊C₄	—	394	4	no	—
198	C₂²⋊C₄	C₂×Dic₁₉₇	1	4	yes	—
199	C₂²⋊C₄	C₂×Dic₁₉₇	1	4	yes	complemented
200	C₄²	—	788	2	no	—
201	C₄²	C₂×Dic₁₉₇	1	4	yes	complemented
202	C₂³	—	2	2	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	1	no	—
209	C₂³	C₂²×Dic₁₉₇	1	2	yes	—
210	C₂³	C₂²×Dic₁₉₇	1	2	yes	—
211	C₂³	#15691358	1	4	yes	—
212	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
213	D₄	—	2	8	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	788	8	no	—
223	C₂×C₄	—	394	2	no	—
224	C₂×C₄	—	394	2	no	—
225	C₂×C₄	—	394	2	no	—
226	C₂×C₄	—	394	2	no	—
227	C₂×C₄	—	394	2	no	—
228	C₂×C₄	—	394	2	no	—
229	C₂×C₄	—	394	2	no	—
230	C₂×C₄	—	394	2	no	—
231	C₂×C₄	—	394	2	no	—
232	C₂×C₄	—	394	2	no	—
233	C₂×C₄	—	394	2	no	—
234	C₂×C₄	—	394	2	no	—
235	C₂×C₄	C₂²×Dic₁₉₇	1	2	yes	—
236	C₂×C₄	C₂²×Dic₁₉₇	1	2	yes	—
237	C₂×C₄	C₂²×Dic₁₉₇	1	2	yes	—
238	C₂×C₄	C₂²×Dic₁₉₇	1	2	yes	—
239	C₂×C₄	C₂²×Dic₁₉₇	1	2	yes	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	#20263285	1	4	yes	—
250	C₂²	C₂³×Dic₁₉₇	1	1	yes	characteristic
251	C₂²	C₂×#15691358	1	2	yes	—
252	C₂²	#20263371	1	2	yes	—
253	C₂²	#20263372	1	1	yes	characteristic
254	C₂²	#20263373	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	788	2	no	—
261	C₄	—	788	2	no	—
262	C₄	—	788	2	no	—
263	C₄	—	788	2	no	—
264	C₂	—	2	4	no	—
265	C₂	—	4	2	no	—
266	C₂	C₂×#20263285	1	1	yes	characteristic
267	C₂	#27161554	1	1	yes	characteristic
268	C₂	#27161562	1	1	yes	characteristic
269	C₂	#27161662	1	2	yes	—
270	C₂	#27161671	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₇
commutator	C₂²×C₁₉₇	C₂³×C₄
Fitting	C₂×C₄×D₄×C₁₉₇	C₂
socle, cosocle	C₂³×C₁₉₇	C₂⁴
prime radical	C₂²×D₄×C₁₉₇	C₂²
hypercenter	C₂×C₄×D₄	D₁₉₇
hypocenter	C₁₉₇	#2318
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #27160972	C₂	semidirect
2. #27160965	C₂	semidirect
4. #27160954	C₂	semidirect
5. #27160947	C₂	semidirect
6. #27160945	C₂	semidirect
7. #27160942	C₂	semidirect
8. #27160939	C₂	semidirect
9. C₄×C₂²⋊Dic₁₉₇	C₂	semidirect
10. C₂×#20263233	C₂	semidirect
31. C₄×D₄×C₁₉₇	C₄	semidirect
56. C₂²×C₄×C₁₉₇	C₂×C₄	semidirect
62. C₄⋊C₄×C₁₉₇	C₂×C₄	semidirect
64. C₂²⋊C₄×C₁₉₇	C₂×C₄	semidirect
65. C₄²×C₁₉₇	C₂×C₄	semidirect
135. C₁₉₇	#2318	semidirect
152. C₄×D₄	Dic₁₉₇	semidirect
186. C₂²×C₄	C₂×Dic₁₉₇	semidirect
194. C₄⋊C₄	C₂×Dic₁₉₇	semidirect
199. C₂²⋊C₄	C₂×Dic₁₉₇	semidirect
201. C₄²	C₂×Dic₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
#27160972	2
#27160965	2
#27160959	2
#27160954	2
#27160947	2
#27160945	2
#27160942	2
#27160939	2
C₄×C₂²⋊Dic₁₉₇	2
C₂×#20263233	2
C₂×C₄×D₄×C₁₉₇	2
#2318	197