Finite Group Databasebeta

Subgroups — #38742267

Order 25216.

146 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161430	C₂	1	1	yes	maximal, characteristic
2	#27161365	C₂	1	1	yes	maximal, characteristic
3	#27161290	C₂	1	1	yes	maximal, characteristic
4	#27161209	C₂	1	1	yes	maximal, characteristic
5	#27160826	C₂	1	1	yes	maximal, characteristic
6	#27160824	C₂	1	1	yes	maximal, characteristic
7	#335×C₁₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₈₈	—	2	1	no	—
9	#20263326	C₂²	1	1	yes	characteristic
10	#20263326	C₂²	1	1	yes	characteristic
11	C₄×Dic₃₉₄	C₂²	1	1	yes	characteristic
12	#20263258	—	2	1	no	—
13	C₄⋊#13096802	C₂²	1	1	yes	characteristic
14	#20263255	—	2	1	no	—
15	#20263251	—	2	1	no	—
16	#20263251	—	2	1	no	—
17	#20263251	—	2	1	no	—
18	#20263251	—	2	1	no	—
19	#20263236	—	2	1	no	—
20	C₄⋊C₈×C₁₉₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₉₇	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₃₉₄	—	2	1	no	—
24	C₂×Dic₃₉₄	—	2	1	no	—
25	C₂×Dic₃₉₄	D₄	1	1	yes	characteristic
26	Dic₇₈₈	—	4	2	no	—
27	C₂×#13096802	—	2	1	no	—
28	C₄⋊Dic₁₉₇	—	2	1	no	—
29	C₄⋊Dic₁₉₇	—	2	1	no	—
30	C₄⋊Dic₁₉₇	—	2	1	no	—
31	C₄⋊Dic₁₉₇	—	2	1	no	—
32	C₄⋊Dic₁₉₇	D₄	1	1	yes	characteristic
33	#15691337	—	2	1	no	—
34	C₄×Dic₁₉₇	—	2	1	no	—
35	C₂×C₈×C₁₉₇	—	2	1	no	—
36	C₂×C₈×C₁₉₇	—	2	1	no	—
37	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
39	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
40	Dic₃₉₄	—	2	1	no	—
41	Dic₃₉₄	—	2	1	no	—
42	Dic₃₉₄	—	4	1	no	—
43	Dic₃₉₄	—	4	1	no	—
44	Dic₃₉₄	—	4	1	no	—
45	Dic₃₉₄	—	4	1	no	—
46	C₂×Dic₁₉₇	—	2	1	no	—
47	C₂×Dic₁₉₇	—	2	1	no	—
48	C₂×Dic₁₉₇	—	4	1	no	—
49	C₂×Dic₁₉₇	—	4	1	no	—
50	#13096802	—	4	1	no	—
51	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
54	C₈×C₁₉₇	—	2	1	no	—
55	C₈×C₁₉₇	—	2	1	no	—
56	C₈×C₁₉₇	—	4	1	no	—
57	C₈×C₁₉₇	—	4	1	no	—
58	C₂²×C₁₉₇	#121	1	1	yes	characteristic
59	C₄×C₁₉₇	—	2	1	no	—
60	C₄×C₁₉₇	#135	1	1	yes	characteristic
61	C₄×C₁₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₄×C₁₉₇	#137	1	1	yes	characteristic
63	C₄×C₁₉₇	#137	1	1	yes	characteristic
64	Dic₁₉₇	—	4	1	no	—
65	Dic₁₉₇	—	4	1	no	—
66	Dic₁₉₇	—	8	1	no	—
67	Dic₁₉₇	—	8	1	no	—
68	C₂×C₁₉₇	#464	1	1	yes	characteristic
69	C₂×C₁₉₇	#468	1	1	yes	characteristic
70	C₂×C₁₉₇	#473	1	1	yes	characteristic
71	C₁₉₇	#1700	1	1	yes	characteristic, complemented, Sylow
72	#1700	—	197	1	no	maximal, Sylow
73	#487	—	197	1	no	—
74	#475	—	197	1	no	—
75	#462	—	197	1	no	—
76	#446	—	197	1	no	—
77	#335	D₁₉₇	1	1	yes	characteristic
78	#332	—	197	1	no	—
79	#332	—	197	1	no	—
80	C₂×Q₁₆	—	394	1	no	—
81	C₄⋊Q₈	—	197	1	no	—
82	C₄⋊Q₈	—	197	1	no	—
83	C₄×Q₈	—	197	1	no	—
84	#106	—	394	1	no	—
85	C₄⋊C₈	—	394	1	no	—
86	C₄⋊C₈	—	197	1	no	—
87	C₄⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic
88	C₄⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic
89	#103	—	394	1	no	—
90	#103	—	394	1	no	—
91	#103	—	394	1	no	—
92	#103	—	394	1	no	—
93	#97	—	394	1	no	—
94	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
95	C₂×Q₈	—	394	1	no	—
96	C₂×Q₈	—	394	1	no	—
97	C₂×Q₈	—	197	1	no	—
98	Q₁₆	—	788	2	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	394	1	no	—
102	C₂×C₈	D₇₈₈	1	1	yes	characteristic
103	C₂×C₈	D₇₈₈	1	1	yes	characteristic
104	C₄⋊C₄	—	394	1	no	—
105	C₄⋊C₄	—	394	1	no	—
106	C₄⋊C₄	—	394	1	no	—
107	C₄⋊C₄	—	394	1	no	—
108	C₄⋊C₄	—	394	1	no	—
109	C₄⋊C₄	—	197	1	no	—
110	C₄²	—	394	1	no	—
111	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
112	Q₈	—	788	1	no	—
113	Q₈	—	788	1	no	—
114	Q₈	—	788	1	no	—
115	Q₈	—	788	1	no	—
116	Q₈	—	394	1	no	—
117	Q₈	—	394	1	no	—
118	C₂×C₄	—	788	1	no	—
119	C₂×C₄	—	788	1	no	—
120	C₂×C₄	—	394	1	no	—
121	C₂×C₄	—	394	1	no	—
122	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
123	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
124	C₂×C₄	#15691359	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	2	1	no	—
127	C₈	—	4	1	no	—
128	C₈	—	4	1	no	—
129	C₈	—	788	1	no	—
130	C₂²	#20263301	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	788	1	no	—
133	C₄	—	788	1	no	—
134	C₄	—	1576	1	no	—
135	C₄	—	1576	1	no	—
136	C₄	#20263348	1	1	yes	characteristic
137	C₄	#20263350	1	1	yes	characteristic
138	C₄	#20263358	1	1	yes	characteristic
139	C₄	#20263359	1	1	yes	characteristic
140	C₂	#27161302	1	1	yes	characteristic
141	C₂	#27161326	1	1	yes	characteristic
142	C₂	#27161355	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20263301
commutator, cosocle	C₄²×C₁₉₇	C₂³
Frattini	C₄²	C₂²×D₁₉₇
Fitting	#335×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	#121
hypercenter	#335	D₁₉₇
hypocenter	C₁₉₇	#1700
LCS	G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

Complements

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

N	H ≅ G/N	type
71. C₁₉₇	#1700	semidirect

Maximal subgroups

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

subgroup	index
#27161430	2
#27161365	2
#27161290	2
#27161209	2
#27160826	2
#27160824	2
#335×C₁₉₇	2
#1700	197