Finite Group Databasebeta

Subgroups — #38742173

Order 25216.

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

normal subgroups only

This group has 144 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	#27161431	C₂	1	1	yes	maximal, characteristic
2	#27161371	C₂	1	1	yes	maximal, characteristic
3	#27161365	C₂	1	1	yes	maximal, characteristic
4	#27161203	C₂	1	1	yes	maximal, characteristic
5	#27160831	C₂	1	1	yes	maximal, characteristic
6	#27160826	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	#20263259	—	2	1	no	—
11	#20263258	—	2	1	no	—
12	C₄⋊#13096802	C₂²	1	1	yes	characteristic
13	#20263255	—	2	1	no	—
14	#20263251	—	2	1	no	—
15	#20263249	—	2	1	no	—
16	#20263249	—	2	1	no	—
17	#20263249	—	2	1	no	—
18	C₄×#13096802	C₂²	1	1	yes	characteristic
19	C₈×Dic₁₉₇	—	2	1	no	—
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	—	4	1	no	—
41	Dic₃₉₄	—	4	1	no	—
42	Dic₃₉₄	Q₁₆	1	2	yes	—
43	C₂×Dic₁₉₇	—	2	1	no	—
44	C₂×Dic₁₉₇	—	2	1	no	—
45	C₂×Dic₁₉₇	—	4	1	no	—
46	#13096802	—	2	1	no	—
47	#13096802	—	2	1	no	—
48	#13096802	—	4	1	no	—
49	Q₈×C₁₉₇	—	8	1	no	—
50	C₂×C₄×C₁₉₇	—	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	—	4	1	no	—
55	C₈×C₁₉₇	—	4	1	no	—
56	C₂²×C₁₉₇	#121	1	1	yes	characteristic
57	C₄×C₁₉₇	—	2	1	no	—
58	C₄×C₁₉₇	—	8	1	no	—
59	C₄×C₁₉₇	C₂×Q₁₆	1	1	yes	characteristic
60	C₄×C₁₉₇	#135	1	1	yes	characteristic
61	C₄×C₁₉₇	#135	1	1	yes	characteristic
62	C₄×C₁₉₇	#137	1	1	yes	characteristic
63	Dic₁₉₇	—	2	2	no	—
64	Dic₁₉₇	—	4	1	no	—
65	Dic₁₉₇	—	8	1	no	—
66	C₂×C₁₉₇	#464	1	1	yes	characteristic
67	C₂×C₁₉₇	#467	1	1	yes	characteristic
68	C₂×C₁₉₇	#471	1	1	yes	characteristic
69	C₁₉₇	#1684	1	1	yes	characteristic, complemented, Sylow
70	#1684	—	197	1	no	maximal, Sylow
71	#487	—	197	1	no	—
72	#477	—	197	1	no	—
73	#475	—	197	1	no	—
74	C₈×Q₈	—	197	1	no	—
75	#335	—	197	1	no	—
76	#333	D₁₉₇	1	1	yes	characteristic
77	#332	—	197	1	no	—
78	C₄⋊Q₈	—	197	1	no	—
79	C₄⋊Q₈	C₂×D₁₉₇	1	1	yes	characteristic
80	C₄×Q₈	—	197	1	no	—
81	#107	—	394	1	no	—
82	#106	—	394	1	no	—
83	C₄⋊C₈	—	394	1	no	—
84	C₄⋊C₈	—	197	1	no	—
85	C₄⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic
86	C₄⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic
87	#103	—	394	1	no	—
88	#103	—	394	1	no	—
89	#103	—	394	1	no	—
90	#103	—	394	1	no	—
91	C₄×C₈	—	394	1	no	—
92	C₄×C₈	—	197	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	394	1	no	—
95	C₂×Q₈	—	197	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	394	1	no	—
99	C₂×C₈	—	197	1	no	—
100	C₂×C₈	—	197	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	394	1	no	—
104	C₄⋊C₄	—	394	1	no	—
105	C₄⋊C₄	—	394	1	no	—
106	C₄⋊C₄	—	197	1	no	—
107	C₄²	—	394	1	no	—
108	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
109	Q₈	—	8	1	no	—
110	Q₈	—	788	1	no	—
111	Q₈	—	788	1	no	—
112	Q₈	—	394	1	no	—
113	Q₈	—	197	2	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	788	1	no	—
116	C₂×C₄	—	394	1	no	—
117	C₂×C₄	—	394	1	no	—
118	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
119	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
120	C₂×C₄	#15691358	1	1	yes	characteristic
121	C₈	—	4	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	788	1	no	—
124	C₈	—	394	1	no	—
125	C₈	—	394	1	no	—
126	C₂²	#20263299	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	8	1	no	—
129	C₄	—	788	1	no	—
130	C₄	—	1576	1	no	—
131	C₄	—	394	2	no	—
132	C₄	Q₁₆×D₁₉₇	1	1	yes	characteristic
133	C₄	#20263347	1	1	yes	characteristic
134	C₄	#20263347	1	1	yes	characteristic
135	C₄	#20263358	1	1	yes	characteristic
136	C₂	#27161304	1	1	yes	characteristic
137	C₂	#27161319	1	1	yes	characteristic
138	C₂	#27161341	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	#1684
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
69. C₁₉₇	#1684	semidirect

Maximal subgroups

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

subgroup	index
#27161431	2
#27161371	2
#27161365	2
#27161203	2
#27160831	2
#27160826	2
#333×C₁₉₇	2
#1684	197