Subgroups — #38740941

Order 25216.

656 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₁₉₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#20263233	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₉₇	C₂²	1	2	yes	—
6	C₂³×Dic₁₉₇	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₉₇	—	2	2	no	—
8	C₂×C₄⋊Dic₁₉₇	—	2	2	no	—
9	C₂×C₄⋊Dic₁₉₇	C₄	1	4	yes	complemented
10	C₂×C₄×Dic₁₉₇	—	2	2	no	—
11	C₂×C₄×Dic₁₉₇	—	2	2	no	—
12	C₂×C₄×Dic₁₉₇	C₄	1	4	yes	complemented
13	#20263233	—	2	8	no	—
14	C₂³×C₄×C₁₉₇	C₂²	1	2	yes	—
15	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₉₇	—	2	2	no	—
17	C₂×C₄⋊C₄×C₁₉₇	—	2	2	no	—
18	C₂×C₄⋊C₄×C₁₉₇	C₄	1	4	yes	complemented
19	C₂²×Dic₁₉₇	—	2	2	no	—
20	C₂²×Dic₁₉₇	—	2	2	no	—
21	C₂²×Dic₁₉₇	—	2	4	no	—
22	C₂²×Dic₁₉₇	—	2	2	no	—
23	C₂²×Dic₁₉₇	—	2	2	no	—
24	C₂²×Dic₁₉₇	—	2	2	no	—
25	C₂²×Dic₁₉₇	—	2	2	no	—
26	C₂²×Dic₁₉₇	—	2	4	no	—
27	C₂²×Dic₁₉₇	C₂×C₄	1	4	yes	—
28	C₂²×Dic₁₉₇	C₂×C₄	1	4	yes	—
29	C₂²×Dic₁₉₇	D₄	1	4	yes	—
30	C₂²×Dic₁₉₇	Q₈	1	4	yes	—
31	C₄⋊Dic₁₉₇	—	2	8	no	—
32	C₄×Dic₁₉₇	—	2	8	no	—
33	C₂⁴×C₁₉₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₁₉₇	—	2	2	no	—
35	C₂²×C₄×C₁₉₇	—	2	2	no	—
36	C₂²×C₄×C₁₉₇	—	2	4	no	—
37	C₂²×C₄×C₁₉₇	—	2	4	no	—
38	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₁₉₇	C₂×C₄	1	4	yes	—
41	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
43	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
44	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
45	C₂²×C₄×C₁₉₇	Q₈	1	2	yes	—
46	C₄⋊C₄×C₁₉₇	—	2	8	no	—
47	C₂×Dic₁₉₇	—	2	4	no	—
48	C₂×Dic₁₉₇	—	2	4	no	—
49	C₂×Dic₁₉₇	—	2	4	no	—
50	C₂×Dic₁₉₇	—	2	4	no	—
51	C₂×Dic₁₉₇	—	2	4	no	—
52	C₂×Dic₁₉₇	—	2	4	no	—
53	C₂×Dic₁₉₇	—	2	4	no	—
54	C₂×Dic₁₉₇	—	2	8	no	—
55	C₂×Dic₁₉₇	—	4	4	no	—
56	C₂×Dic₁₉₇	—	4	2	no	—
57	C₂×Dic₁₉₇	—	4	2	no	—
58	C₂×Dic₁₉₇	C₄⋊C₄	1	8	yes	complemented
59	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
60	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
61	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
62	C₂³×C₁₉₇	C₂×D₄	1	2	yes	—
63	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
64	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
65	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
66	C₂³×C₁₉₇	C₂×Q₈	1	2	yes	—
67	C₂³×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
68	C₂³×C₁₉₇	#41	1	2	yes	—
69	C₂³×C₁₉₇	#41	1	2	yes	—
70	C₂×C₄×C₁₉₇	—	2	4	no	—
71	C₂×C₄×C₁₉₇	—	2	8	no	—
72	C₂×C₄×C₁₉₇	—	4	4	no	—
73	C₂×C₄×C₁₉₇	—	4	2	no	—
74	C₂×C₄×C₁₉₇	—	4	2	no	—
75	C₂×C₄×C₁₉₇	C₄²	1	4	yes	—
76	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	4	yes	—
77	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	4	yes	—
78	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	4	yes	—
79	C₂×C₄×C₁₉₇	C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₁₉₇	C₄⋊C₄	1	4	yes	—
81	C₂×C₄×C₁₉₇	C₄⋊C₄	1	4	yes	—
82	C₂²×C₁₉₇	C₂×C₄²	1	1	yes	characteristic
83	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₁₉₇	#117	1	2	yes	—
91	C₂²×C₁₉₇	C₄×D₄	1	2	yes	—
92	C₂²×C₁₉₇	C₄×D₄	1	2	yes	—
93	C₂²×C₁₉₇	C₄×Q₈	1	2	yes	—
94	C₂²×C₁₉₇	C₄×Q₈	1	2	yes	—
95	C₂²×C₁₉₇	#121	1	2	yes	—
96	C₂²×C₁₉₇	#121	1	2	yes	—
97	C₂²×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
98	C₂²×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
99	C₂²×C₁₉₇	#124	1	1	yes	characteristic
100	C₂²×C₁₉₇	#124	1	1	yes	characteristic
101	C₂²×C₁₉₇	#125	1	2	yes	—
102	C₂²×C₁₉₇	#127	1	1	yes	characteristic
103	C₂²×C₁₉₇	C₄⋊Q₈	1	1	yes	characteristic
104	C₂²×C₁₉₇	C₄⋊Q₈	1	2	yes	—
105	C₄×C₁₉₇	—	4	4	no	—
106	C₄×C₁₉₇	#95	1	8	yes	complemented
107	Dic₁₉₇	—	2	8	no	—
108	Dic₁₉₇	—	4	4	no	—
109	C₂×C₁₉₇	C₂×#95	1	1	yes	characteristic
110	C₂×C₁₉₇	C₄×C₄⋊C₄	1	2	yes	—
111	C₂×C₁₉₇	#380	1	2	yes	—
112	C₂×C₁₉₇	#382	1	1	yes	characteristic
113	C₂×C₁₉₇	#384	1	1	yes	characteristic
114	C₂×C₁₉₇	#389	1	2	yes	—
115	C₂×C₁₉₇	#389	1	2	yes	—
116	C₂×C₁₉₇	#390	1	1	yes	characteristic
117	C₂×C₁₉₇	#390	1	1	yes	characteristic
118	C₂×C₁₉₇	#391	1	1	yes	characteristic
119	C₂×C₁₉₇	#391	1	1	yes	characteristic
120	C₁₉₇	C₄⋊#95	1	1	yes	characteristic, complemented, Sylow
121	C₄⋊#95	—	197	1	no	maximal, Sylow
122	C₂²×C₄⋊C₄	—	197	1	no	—
123	C₂²×C₄⋊C₄	D₁₉₇	1	1	yes	characteristic
124	C₂²×C₄²	—	197	1	no	—
125	C₂×#95	—	197	4	no	—
126	C₂³×C₄	—	197	2	no	—
127	C₂³×C₄	—	197	2	no	—
128	C₂³×C₄	C₂×D₁₉₇	1	2	yes	—
129	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	394	2	no	—
133	C₂×C₄⋊C₄	—	394	2	no	—
134	C₂×C₄⋊C₄	—	197	4	no	—
135	C₂×C₄⋊C₄	Dic₁₉₇	1	4	yes	complemented
136	C₂×C₄²	—	394	2	no	—
137	C₂×C₄²	—	394	2	no	—
138	C₂×C₄²	—	197	4	no	—
139	#95	—	394	8	no	—
140	C₂⁴	C₂²×D₁₉₇	1	1	yes	characteristic
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	394	4	no	—
146	C₂²×C₄	—	394	2	no	—
147	C₂²×C₄	—	394	2	no	—
148	C₂²×C₄	—	394	4	no	—
149	C₂²×C₄	—	394	2	no	—
150	C₂²×C₄	—	394	2	no	—
151	C₂²×C₄	—	394	2	no	—
152	C₂²×C₄	—	394	2	no	—
153	C₂²×C₄	—	197	4	no	—
154	C₂²×C₄	—	197	4	no	—
155	C₂²×C₄	—	197	4	no	—
156	C₂²×C₄	—	197	4	no	—
157	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
158	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
159	C₂²×C₄	C₂×Dic₁₉₇	1	4	yes	—
160	C₂²×C₄	C₂×Dic₁₉₇	1	2	yes	—
161	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
162	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
163	C₂²×C₄	D₇₈₈	1	2	yes	—
164	C₂²×C₄	Dic₃₉₄	1	2	yes	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	—	394	8	no	—
167	C₄²	—	394	8	no	—
168	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
169	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
170	C₂³	C₂²×Dic₁₉₇	1	1	yes	characteristic
171	C₂³	D₄×D₁₉₇	1	2	yes	—
172	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
173	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
174	C₂³	C₂×D₇₈₈	1	1	yes	characteristic
175	C₂³	Q₈×D₁₉₇	1	2	yes	—
176	C₂³	C₂×Dic₃₉₄	1	1	yes	characteristic
177	C₂³	#15691357	1	2	yes	—
178	C₂³	#15691359	1	2	yes	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	788	4	no	—
185	C₂×C₄	—	788	2	no	—
186	C₂×C₄	—	788	2	no	—
187	C₂×C₄	—	394	4	no	—
188	C₂×C₄	—	394	8	no	—
189	C₂×C₄	—	394	4	no	—
190	C₂×C₄	—	394	4	no	—
191	C₂×C₄	—	394	4	no	—
192	C₂×C₄	—	394	4	no	—
193	C₂×C₄	—	394	4	no	—
194	C₂×C₄	—	394	4	no	—
195	C₂×C₄	—	197	8	no	—
196	C₂×C₄	C₄×Dic₁₉₇	1	4	yes	—
197	C₂×C₄	#15691335	1	4	yes	—
198	C₂×C₄	#15691335	1	4	yes	—
199	C₂×C₄	C₂²⋊Dic₁₉₇	1	4	yes	—
200	C₂×C₄	#15691337	1	4	yes	—
201	C₂×C₄	#15691337	1	4	yes	—
202	C₂×C₄	C₄⋊Dic₁₉₇	1	4	yes	—
203	C₂²	C₂×C₄×Dic₁₉₇	1	1	yes	characteristic
204	C₂²	C₂×#15691335	1	1	yes	characteristic
205	C₂²	C₂×#15691335	1	1	yes	characteristic
206	C₂²	C₂×C₂²⋊Dic₁₉₇	1	1	yes	characteristic
207	C₂²	C₄⋊C₄×D₁₉₇	1	2	yes	—
208	C₂²	C₂×#15691337	1	1	yes	characteristic
209	C₂²	C₂×#15691337	1	1	yes	characteristic
210	C₂²	C₂×C₄⋊Dic₁₉₇	1	1	yes	characteristic
211	C₂²	#20263284	1	2	yes	—
212	C₂²	#20263288	1	2	yes	—
213	C₂²	D₄×Dic₁₉₇	1	2	yes	—
214	C₂²	#20263291	1	2	yes	—
215	C₂²	Q₈×Dic₁₉₇	1	2	yes	—
216	C₂²	#20263297	1	2	yes	—
217	C₂²	#20263301	1	2	yes	—
218	C₂²	#20263302	1	2	yes	—
219	C₂²	#20263306	1	2	yes	—
220	C₂²	#20263313	1	1	yes	characteristic
221	C₂²	#20263315	1	1	yes	characteristic
222	C₂²	#20263317	1	2	yes	—
223	C₂²	#20263322	1	1	yes	characteristic
224	C₂²	#20263324	1	1	yes	characteristic
225	C₂²	#20263325	1	2	yes	—
226	C₄	—	4	4	no	—
227	C₄	—	788	4	no	—
228	C₄	—	394	8	no	—
229	C₄	#20263233	1	8	yes	complemented
230	C₂	C₂×#20263233	1	1	yes	characteristic
231	C₂	C₄⋊C₄×Dic₁₉₇	1	2	yes	—
232	C₂	#27160936	1	2	yes	—
233	C₂	#27160944	1	1	yes	characteristic
234	C₂	#27160948	1	1	yes	characteristic
235	C₂	#27160975	1	2	yes	—
236	C₂	#27160978	1	2	yes	—
237	C₂	#27160980	1	1	yes	characteristic
238	C₂	#27160982	1	1	yes	characteristic
239	C₂	#27160984	1	1	yes	characteristic
240	C₂	#27160986	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₁₉₇
commutator	C₂²×C₁₉₇	C₂×C₄²
Fitting	C₂²×C₄⋊C₄×C₁₉₇	C₂
socle, prime radical, cosocle	C₂⁴×C₁₉₇	C₂³
hypercenter	C₂²×C₄⋊C₄	D₁₉₇
hypocenter	C₁₉₇	C₄⋊#95
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. C₂×C₄⋊Dic₁₉₇	C₄	semidirect
12. C₂×C₄×Dic₁₉₇	C₄	semidirect
18. C₂×C₄⋊C₄×C₁₉₇	C₄	semidirect
58. C₂×Dic₁₉₇	C₄⋊C₄	semidirect
106. C₄×C₁₉₇	#95	semidirect
120. C₁₉₇	C₄⋊#95	semidirect
135. C₂×C₄⋊C₄	Dic₁₉₇	semidirect
229. C₄	#20263233	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₁₉₇	2
C₂²×C₄×Dic₁₉₇	2
C₂×#20263233	2
C₂²×C₄⋊C₄×C₁₉₇	2
C₄⋊#95	197