Finite Group Databasebeta

Subgroups — #38742239

Order 25216.

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

normal subgroups only

This group has 140 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	#27161482	C₂	1	1	yes	maximal, characteristic
2	#27161375	C₂	1	1	yes	maximal, characteristic
3	#27161371	C₂	1	1	yes	maximal, characteristic
4	#27161206	C₂	1	1	yes	maximal, characteristic
5	#27160829	C₂	1	1	yes	maximal, characteristic
6	#27160827	C₂	1	1	yes	maximal, characteristic
7	#333×C₁₉₇	C₂	1	1	yes	maximal, characteristic
8	#20263326	C₂²	1	1	yes	characteristic
9	C₄×Dic₃₉₄	C₂²	1	1	yes	characteristic
10	#20263260	—	2	1	no	—
11	#20263259	—	2	1	no	—
12	#20263259	—	2	1	no	—
13	#20263259	—	2	1	no	—
14	C₄⋊#13096802	C₂²	1	1	yes	characteristic
15	#20263255	—	2	1	no	—
16	#20263251	—	2	1	no	—
17	#20263249	—	2	1	no	—
18	#20263236	—	2	1	no	—
19	C₄×#13096802	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×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	D₄	1	1	yes	characteristic
25	C₂×#13096802	—	2	1	no	—
26	C₂×#13096802	D₄	1	1	yes	characteristic
27	C₂×#13096802	D₄	1	1	yes	characteristic
28	C₄⋊Dic₁₉₇	—	2	1	no	—
29	C₄⋊Dic₁₉₇	—	2	1	no	—
30	C₄⋊Dic₁₉₇	D₄	1	1	yes	characteristic
31	#15691337	—	2	1	no	—
32	C₄×Dic₁₉₇	—	2	1	no	—
33	C₂×Q₈×C₁₉₇	—	2	1	no	—
34	C₂×C₈×C₁₉₇	—	2	1	no	—
35	C₂×C₈×C₁₉₇	—	2	1	no	—
36	C₄⋊C₄×C₁₉₇	—	2	1	no	—
37	C₄⋊C₄×C₁₉₇	—	2	1	no	—
38	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
39	Dic₃₉₄	—	2	1	no	—
40	Dic₃₉₄	—	2	1	no	—
41	Dic₃₉₄	—	8	1	no	—
42	C₂×Dic₁₉₇	—	2	1	no	—
43	C₂×Dic₁₉₇	—	2	1	no	—
44	C₂×Dic₁₉₇	—	4	1	no	—
45	#13096802	—	2	1	no	—
46	#13096802	—	4	1	no	—
47	#13096802	Q₁₆	1	2	yes	—
48	Q₈×C₁₉₇	—	8	1	no	—
49	C₂×C₄×C₁₉₇	—	4	1	no	—
50	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
53	C₈×C₁₉₇	—	4	1	no	—
54	C₈×C₁₉₇	—	4	1	no	—
55	C₂²×C₁₉₇	#121	1	1	yes	characteristic
56	C₄×C₁₉₇	—	2	1	no	—
57	C₄×C₁₉₇	—	8	1	no	—
58	C₄×C₁₉₇	C₂×Q₁₆	1	1	yes	characteristic
59	C₄×C₁₉₇	#137	1	1	yes	characteristic
60	C₄×C₁₉₇	#137	1	1	yes	characteristic
61	C₄×C₁₉₇	#137	1	1	yes	characteristic
62	Dic₁₉₇	—	4	1	no	—
63	Dic₁₉₇	—	4	1	no	—
64	Dic₁₉₇	—	8	1	no	—
65	C₂×C₁₉₇	#462	1	1	yes	characteristic
66	C₂×C₁₉₇	#470	1	1	yes	characteristic
67	C₂×C₁₉₇	#472	1	1	yes	characteristic
68	C₁₉₇	#1696	1	1	yes	characteristic, complemented, Sylow
69	#1696	—	197	1	no	maximal, Sylow
70	#500	—	197	1	no	—
71	#477	—	197	1	no	—
72	#477	—	197	1	no	—
73	#446	—	197	1	no	—
74	#334	—	197	1	no	—
75	#333	—	197	1	no	—
76	#333	D₁₉₇	1	1	yes	characteristic
77	C₄⋊Q₈	—	197	1	no	—
78	C₄⋊Q₈	C₂×D₁₉₇	1	1	yes	characteristic
79	C₄×Q₈	—	197	1	no	—
80	#107	—	394	1	no	—
81	#107	—	394	1	no	—
82	#107	—	394	1	no	—
83	#107	—	394	1	no	—
84	C₄⋊C₈	—	394	1	no	—
85	C₄⋊C₈	—	197	1	no	—
86	C₄⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic
87	C₄⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic
88	#103	—	394	1	no	—
89	#103	—	394	1	no	—
90	#97	—	394	1	no	—
91	C₄×C₈	—	197	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	394	1	no	—
94	C₂×Q₈	—	197	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	394	1	no	—
98	C₂×C₈	—	197	1	no	—
99	C₂×C₈	—	197	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	394	1	no	—
103	C₄⋊C₄	—	394	1	no	—
104	C₄⋊C₄	—	394	1	no	—
105	C₄⋊C₄	—	197	1	no	—
106	C₄²	—	394	1	no	—
107	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
108	Q₈	—	8	1	no	—
109	Q₈	—	1576	1	no	—
110	Q₈	—	394	1	no	—
111	Q₈	—	394	1	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	788	1	no	—
114	C₂×C₄	—	394	1	no	—
115	C₂×C₄	—	394	1	no	—
116	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
117	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
118	C₂×C₄	#15691358	1	1	yes	characteristic
119	C₈	—	4	1	no	—
120	C₈	—	4	1	no	—
121	C₈	—	788	1	no	—
122	C₈	—	394	1	no	—
123	C₈	—	197	2	no	—
124	C₂²	#20263299	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	8	1	no	—
127	C₄	—	788	1	no	—
128	C₄	—	788	1	no	—
129	C₄	—	1576	1	no	—
130	C₄	Q₁₆×D₁₉₇	1	1	yes	characteristic
131	C₄	#20263355	1	1	yes	characteristic
132	C₄	#20263355	1	1	yes	characteristic
133	C₄	#20263358	1	1	yes	characteristic
134	C₂	#27161288	1	1	yes	characteristic
135	C₂	#27161336	1	1	yes	characteristic
136	C₂	#27161348	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20263299
commutator, cosocle	C₄²×C₁₉₇	C₂³
Frattini	C₄²	C₂²×D₁₉₇
Fitting	#333×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	#121
hypercenter	#333	D₁₉₇
hypocenter	C₁₉₇	#1696
LCS	G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

Complements

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

N	H ≅ G/N	type
68. C₁₉₇	#1696	semidirect

Maximal subgroups

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

subgroup	index
#27161482	2
#27161375	2
#27161371	2
#27161206	2
#27160829	2
#27160827	2
#333×C₁₉₇	2
#1696	197