Subgroups — #38751720

Order 25216.

406 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15691357	C₂	1	1	yes	maximal, characteristic
2	#27161209	C₂	1	1	yes	maximal, characteristic, complemented
3	#27161206	C₂	1	1	yes	maximal, characteristic, complemented
4	#27161150	C₂	1	2	yes	maximal, complemented
5	#27161147	C₂	1	2	yes	maximal, complemented
6	D₄×#13096802	C₂	1	1	yes	maximal, characteristic, complemented
7	#27161135	C₂	1	1	yes	maximal, characteristic, complemented
8	#27161133	C₂	1	2	yes	maximal, complemented
9	C₄×#15691341	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#20263236	C₂	1	2	yes	maximal, complemented
11	C₈×D₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15691357	C₄	1	1	yes	characteristic
13	C₂×#15691341	C₂²	1	1	yes	characteristic
14	C₂×#15691341	C₂²	1	1	yes	characteristic
15	C₂²×#13096802	C₂²	1	2	yes	—
16	C₄×Dic₃₉₄	C₂²	1	1	yes	characteristic
17	#20263291	C₄	1	1	yes	characteristic
18	D₄×Dic₁₉₇	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊D₁₉₇	C₂²	1	2	yes	—
20	#20263286	C₄	1	2	yes	—
21	#20263284	C₄	1	1	yes	characteristic
22	#20263282	C₄	1	2	yes	—
23	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
24	C₄²×D₁₉₇	C₂²	1	1	yes	characteristic
25	C₄⋊#13096802	C₂²	1	1	yes	characteristic
26	#20263255	C₂²	1	2	yes	—
27	#20263255	C₂²	1	2	yes	—
28	C₂²⋊#13096802	C₂²	1	2	yes	—
29	#20263238	C₂²	1	2	yes	—
30	#20263236	C₂²	1	1	yes	characteristic
31	#20263236	C₂²	1	2	yes	—
32	#20263236	C₂²	1	2	yes	—
33	#20263236	C₂²	1	2	yes	—
34	#20263236	C₂²	1	1	yes	characteristic
35	C₄×#13096802	C₂²	1	1	yes	characteristic
36	C₂²×C₈×C₁₉₇	C₂²	1	2	yes	—
37	C₄×D₄×C₁₉₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₉₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₉₇	C₂²	1	2	yes	—
40	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
41	#15691357	—	2	4	no	—
42	C₂×Dic₃₉₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₉₇	C₂×C₄	1	2	yes	—
44	C₂²×Dic₁₉₇	C₂×C₄	1	2	yes	—
45	C₂²×Dic₁₉₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₉₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₉₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₉₇	C₂³	1	1	yes	characteristic
49	#15691341	—	2	2	no	—
50	#15691341	—	2	2	no	—
51	C₂×#13096802	—	2	4	no	—
52	C₂×#13096802	C₂³	1	1	yes	characteristic
53	C₂×#13096802	C₂³	1	2	yes	—
54	C₂×#13096802	C₂³	1	1	yes	characteristic
55	C₄⋊Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
56	#15691337	C₂×C₄	1	2	yes	—
57	#15691337	C₂×C₄	1	2	yes	—
58	C₂²⋊Dic₁₉₇	C₂×C₄	1	2	yes	—
59	#15691335	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₉₇	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₉₇	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₉₇	C₂³	1	1	yes	characteristic
65	C₄×Dic₁₉₇	C₂³	1	2	yes	—
66	C₂×D₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
68	C₂×C₈×C₁₉₇	—	2	4	no	—
69	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₉₇	C₂³	1	2	yes	—
71	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
73	C₂²⋊C₄×C₁₉₇	C₂×C₄	1	2	yes	—
74	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
75	C₂²×D₁₉₇	C₂²×C₄	1	1	yes	characteristic
76	Dic₃₉₄	—	2	2	no	—
77	C₂²⋊D₁₉₇	—	2	4	no	—
78	C₂×Dic₁₉₇	—	2	4	no	—
79	C₂×Dic₁₉₇	—	4	2	no	—
80	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₉₇	C₂²×C₄	1	2	yes	—
84	C₂×Dic₁₉₇	C₂²×C₄	1	2	yes	—
85	C₄×D₁₉₇	—	2	2	no	—
86	C₄×D₁₉₇	—	2	1	no	—
87	C₄×D₁₉₇	—	2	1	no	—
88	C₄×D₁₉₇	—	4	1	no	—
89	#13096802	—	2	2	no	—
90	#13096802	—	2	1	no	—
91	#13096802	#41	1	2	yes	complemented
92	C₂³×C₁₉₇	C₂²×C₄	1	2	yes	—
93	D₄×C₁₉₇	M₁₆	1	4	yes	complemented
94	C₂×C₄×C₁₉₇	—	2	2	no	—
95	C₂×C₄×C₁₉₇	—	2	2	no	—
96	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
100	C₈×C₁₉₇	—	2	2	no	—
101	C₈×C₁₉₇	—	2	1	no	—
102	C₈×C₁₉₇	#41	1	2	yes	complemented
103	C₂²×C₁₉₇	—	2	2	no	—
104	C₂²×C₁₉₇	—	2	2	no	—
105	C₂²×C₁₉₇	C₂×M₁₆	1	4	yes	—
106	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×D₁₉₇	—	2	1	no	—
108	C₂×D₁₉₇	—	2	1	no	—
109	C₂×D₁₉₇	—	2	1	no	—
110	C₄×C₁₉₇	—	2	2	no	—
111	C₄×C₁₉₇	—	2	1	no	—
112	C₄×C₁₉₇	C₂×M₁₆	1	2	yes	—
113	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
114	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
115	Dic₁₉₇	—	2	1	no	—
116	Dic₁₉₇	—	2	1	no	—
117	Dic₁₉₇	—	2	2	no	—
118	Dic₁₉₇	—	2	1	no	—
119	Dic₁₉₇	—	2	2	no	—
120	C₂×C₁₉₇	—	2	4	no	—
121	C₂×C₁₉₇	C₄×#41	1	1	yes	characteristic
122	C₂×C₁₉₇	C₂²×M₁₆	1	1	yes	characteristic
123	C₂×C₁₉₇	#568	1	1	yes	characteristic
124	D₁₉₇	—	4	1	no	—
125	C₁₉₇	#2970	1	1	yes	characteristic, complemented, Sylow
126	#2970	—	197	1	no	maximal, Sylow
127	C₄×#41	—	197	1	no	—
128	#446	—	197	1	no	—
129	#446	—	197	1	no	—
130	#435	—	197	2	no	—
131	#435	—	197	2	no	—
132	C₈×D₄	—	197	1	no	—
133	C₈×D₄	D₁₉₇	1	1	yes	characteristic, complemented
134	#432	—	197	1	no	—
135	#432	—	197	2	no	—
136	C₄×M₁₆	—	197	1	no	—
137	C₂×#97	—	197	2	no	—
138	C₂×#41	—	197	1	no	—
139	C₂×M₁₆	—	197	1	no	—
140	C₂×M₁₆	—	197	1	no	—
141	C₂²×C₈	—	197	2	no	—
142	C₂²×C₈	C₂×D₁₉₇	1	2	yes	—
143	C₄×Q₈	—	197	1	no	—
144	C₄×Q₈	—	197	1	no	—
145	C₄×D₄	—	197	1	no	—
146	C₄×D₄	—	197	2	no	—
147	C₄×D₄	—	197	2	no	—
148	C₄×D₄	C₂×D₁₉₇	1	1	yes	characteristic
149	#117	—	197	1	no	—
150	#117	—	197	2	no	—
151	C₂×C₄²	—	197	2	no	—
152	C₂×C₄²	—	197	1	no	—
153	C₄⋊C₈	—	197	2	no	—
154	C₄⋊C₈	—	197	2	no	—
155	C₄⋊C₈	—	197	1	no	—
156	C₄⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic
157	C₂²⋊C₈	—	197	2	no	—
158	C₂²⋊C₈	—	197	2	no	—
159	C₂²⋊C₈	C₂×D₁₉₇	1	2	yes	—
160	#97	—	197	1	no	—
161	#97	—	197	2	no	—
162	#97	—	197	2	no	—
163	#97	—	197	2	no	—
164	#97	—	197	1	no	—
165	C₄×C₈	—	197	1	no	—
166	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
167	#41	—	394	4	no	—
168	C₂×Q₈	—	197	1	no	—
169	C₂×D₄	—	197	2	no	—
170	C₂×D₄	C₄×D₁₉₇	1	1	yes	characteristic
171	C₂²×C₄	—	197	2	no	—
172	C₂²×C₄	—	197	1	no	—
173	C₂²×C₄	—	197	2	no	—
174	C₂²×C₄	—	197	1	no	—
175	C₂²×C₄	—	197	1	no	—
176	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
177	M₁₆	—	394	2	no	—
178	M₁₆	—	394	2	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	394	4	no	—
181	C₂×C₈	—	197	1	no	—
182	C₂×C₈	—	197	2	no	—
183	C₂×C₈	—	197	1	no	—
184	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
185	C₂×C₈	C₂²×D₁₉₇	1	2	yes	—
186	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
187	C₄⋊C₄	—	197	2	no	—
188	C₄⋊C₄	—	197	2	no	—
189	C₄⋊C₄	—	197	1	no	—
190	C₄⋊C₄	C₄×D₁₉₇	1	1	yes	characteristic
191	C₂²⋊C₄	—	197	2	no	—
192	C₂²⋊C₄	—	197	2	no	—
193	C₂²⋊C₄	C₄×D₁₉₇	1	2	yes	—
194	C₄²	—	197	2	no	—
195	C₄²	—	197	1	no	—
196	C₄²	—	197	1	no	—
197	C₄²	—	197	2	no	—
198	C₄²	—	197	1	no	—
199	C₄²	—	197	2	no	—
200	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
201	C₂³	—	197	1	no	—
202	C₂³	C₂×C₄×D₁₉₇	1	2	yes	—
203	Q₈	—	394	2	no	—
204	D₄	—	394	4	no	—
205	D₄	#15691341	1	4	yes	complemented
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	788	1	no	—
209	C₂×C₄	—	788	2	no	—
210	C₂×C₄	—	394	4	no	—
211	C₂×C₄	—	394	2	no	—
212	C₂×C₄	—	394	1	no	—
213	C₂×C₄	—	394	1	no	—
214	C₂×C₄	—	197	1	no	—
215	C₂×C₄	—	197	1	no	—
216	C₂×C₄	—	197	1	no	—
217	C₂×C₄	—	197	2	no	—
218	C₂×C₄	—	197	2	no	—
219	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
222	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	2	no	—
225	C₈	—	394	2	no	—
226	C₈	—	394	1	no	—
227	C₈	—	197	2	no	—
228	C₈	#15691357	1	2	yes	complemented
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	394	1	no	—
232	C₂²	—	394	1	no	—
233	C₂²	—	394	1	no	—
234	C₂²	C₂×#15691341	1	4	yes	—
235	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	394	1	no	—
239	C₄	—	394	1	no	—
240	C₄	—	394	2	no	—
241	C₄	—	394	1	no	—
242	C₄	—	394	2	no	—
243	C₄	C₂×#15691341	1	2	yes	—
244	C₄	#41×D₁₉₇	1	1	yes	characteristic
245	C₄	C₂×#15691357	1	1	yes	characteristic
246	C₂	—	2	4	no	—
247	C₂	—	788	1	no	—
248	C₂	C₄×#15691357	1	1	yes	characteristic
249	C₂	C₂²×#15691341	1	1	yes	characteristic
250	C₂	#27161844	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉₇
commutator, socle	C₂²×C₁₉₇	C₂³×C₄
Fitting	C₈×D₄×C₁₉₇	C₂
prime radical	C₂×#15691357	C₄
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	C₈×D₄	D₁₉₇
hypocenter	C₁₉₇	#2970
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #27161209	C₂	semidirect
3. #27161206	C₂	semidirect
4. #27161150	C₂	semidirect
5. #27161147	C₂	semidirect
6. D₄×#13096802	C₂	semidirect
7. #27161135	C₂	semidirect
8. #27161133	C₂	semidirect
9. C₄×#15691341	C₂	semidirect
10. C₂×#20263236	C₂	semidirect
11. C₈×D₄×C₁₉₇	C₂	semidirect
91. #13096802	#41	semidirect
93. D₄×C₁₉₇	M₁₆	semidirect
102. C₈×C₁₉₇	#41	semidirect
125. C₁₉₇	#2970	semidirect
133. C₈×D₄	D₁₉₇	semidirect
205. D₄	#15691341	semidirect
228. C₈	#15691357	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15691357	2
#27161209	2
#27161206	2
#27161150	2
#27161147	2
D₄×#13096802	2
#27161135	2
#27161133	2
C₄×#15691341	2
C₂×#20263236	2
C₈×D₄×C₁₉₇	2
#2970	197