Finite Group Databasebeta

Subgroups — #38744561

Order 25216.

162 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 59 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	C₂×#20263269	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#27161161	C₂	1	1	yes	maximal, characteristic
4	#27160906	C₂	1	1	yes	maximal, characteristic, complemented
5	#27160881	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#358×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₅₇₆	C₂²	1	1	yes	characteristic
9	Q₈×Dic₁₉₇	—	2	1	no	—
10	#20263288	—	2	1	no	—
11	#20263269	C₄	1	4	yes	complemented
12	C₂×#15691333	C₂²	1	1	yes	characteristic
13	#20263260	C₂²	1	1	yes	characteristic
14	#20263250	—	2	1	no	—
15	#20263246	—	2	1	no	—
16	C₈×Dic₁₉₇	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
19	#107×C₁₉₇	C₂²	1	1	yes	characteristic
20	C₂×D₇₈₈	—	2	1	no	—
21	C₂×C₄×D₁₉₇	—	4	1	no	—
22	D₁₅₇₆	—	2	1	no	—
23	D₁₅₇₆	C₂×C₄	1	2	yes	—
24	C₂×#13096802	D₄	1	1	yes	characteristic
25	C₄⋊Dic₁₉₇	—	2	1	no	—
26	C₄⋊Dic₁₉₇	—	4	1	no	—
27	#15691335	—	4	1	no	—
28	C₄×Dic₁₉₇	—	4	1	no	—
29	C₄×Dic₁₉₇	D₄	1	1	yes	characteristic
30	#15691333	C₂×C₄	1	2	yes	complemented
31	C₂×Q₈×C₁₉₇	—	2	1	no	—
32	Q₁₆×C₁₉₇	—	2	1	no	—
33	Q₁₆×C₁₉₇	C₂×C₄	1	2	yes	complemented
34	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₉₇	—	2	1	no	—
36	C₁₆×C₁₉₇	—	2	1	no	—
37	C₂²×D₁₉₇	—	4	1	no	—
38	D₇₈₈	—	2	2	no	—
39	D₇₈₈	—	4	1	no	—
40	C₂×Dic₁₉₇	—	4	1	no	—
41	C₂×Dic₁₉₇	D₈	1	1	yes	characteristic
42	C₂×Dic₁₉₇	D₈	1	1	yes	characteristic
43	C₄×D₁₉₇	—	8	2	no	—
44	#13096802	—	2	1	no	—
45	Q₈×C₁₉₇	—	2	2	no	—
46	Q₈×C₁₉₇	—	4	1	no	—
47	C₂×C₄×C₁₉₇	—	4	1	no	—
48	C₂×C₄×C₁₉₇	—	4	1	no	—
49	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
50	C₈×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
51	C₈×C₁₉₇	#41	1	1	yes	characteristic
52	C₂²×C₁₉₇	C₂×D₈	1	1	yes	characteristic
53	C₂×D₁₉₇	—	4	2	no	—
54	C₂×D₁₉₇	—	8	1	no	—
55	C₂×D₁₉₇	—	8	1	no	—
56	C₄×C₁₉₇	—	4	2	no	—
57	C₄×C₁₉₇	—	8	1	no	—
58	C₄×C₁₉₇	C₄×D₄	1	1	yes	characteristic
59	C₄×C₁₉₇	#135	1	1	yes	characteristic
60	Dic₁₉₇	—	2	1	no	—
61	Dic₁₉₇	—	8	1	no	—
62	Dic₁₉₇	SD₃₂	1	2	yes	complemented
63	C₂×C₁₉₇	C₄×D₈	1	1	yes	characteristic
64	C₂×C₁₉₇	C₂×SD₃₂	1	1	yes	characteristic
65	C₂×C₁₉₇	#508	1	1	yes	characteristic
66	D₁₉₇	—	8	2	no	—
67	C₁₉₇	C₄×SD₃₂	1	1	yes	characteristic, complemented, Sylow
68	C₄×SD₃₂	—	197	1	no	maximal, Sylow
69	C₂×SD₃₂	—	197	1	no	—
70	C₄×Q₁₆	—	197	1	no	—
71	C₄×D₈	—	197	1	no	—
72	#367	—	197	1	no	—
73	#358	D₁₉₇	1	1	yes	characteristic, complemented
74	#357	—	197	1	no	—
75	C₄×C₁₆	—	197	1	no	—
76	C₂×Q₁₆	C₂×D₁₉₇	1	1	yes	characteristic
77	C₂×D₈	—	197	1	no	—
78	C₄×Q₈	—	394	1	no	—
79	C₄×D₄	—	394	1	no	—
80	SD₃₂	—	197	4	no	—
81	C₂×C₁₆	—	197	1	no	—
82	C₂×C₁₆	C₂×D₁₉₇	1	1	yes	characteristic
83	#107	—	197	1	no	—
84	#107	C₂×D₁₉₇	1	1	yes	characteristic
85	#103	—	394	1	no	—
86	#102	—	394	1	no	—
87	C₄×C₈	—	197	1	no	—
88	C₂×Q₈	—	2	1	no	—
89	C₂×D₄	—	394	1	no	—
90	C₂²×C₄	—	788	1	no	—
91	Q₁₆	—	2	1	no	—
92	Q₁₆	C₄×D₁₉₇	1	2	yes	complemented
93	D₈	—	394	1	no	—
94	D₈	—	197	2	no	—
95	C₂×C₈	—	197	1	no	—
96	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	788	1	no	—
99	C₄⋊C₄	—	394	1	no	—
100	C₂²⋊C₄	—	788	1	no	—
101	C₄²	—	788	1	no	—
102	C₄²	—	197	1	no	—
103	C₁₆	—	2	1	no	—
104	C₁₆	—	197	2	no	—
105	C₂³	—	788	1	no	—
106	Q₈	—	2	2	no	—
107	Q₈	—	4	1	no	—
108	D₄	—	788	1	no	—
109	D₄	—	394	2	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	788	1	no	—
113	C₂×C₄	—	1576	2	no	—
114	C₂×C₄	—	197	1	no	—
115	C₂×C₄	—	197	1	no	—
116	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
117	C₈	—	394	1	no	—
118	C₈	C₂×C₄×D₁₉₇	1	1	yes	characteristic
119	C₈	#15691357	1	1	yes	characteristic
120	C₂²	—	788	2	no	—
121	C₂²	—	1576	1	no	—
122	C₂²	—	1576	1	no	—
123	C₂²	D₈×D₁₉₇	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	—	8	1	no	—
126	C₄	—	1576	1	no	—
127	C₄	—	394	1	no	—
128	C₄	—	197	2	no	—
129	C₄	#20263286	1	1	yes	characteristic
130	C₄	#20263347	1	1	yes	characteristic
131	C₂	—	1576	2	no	—
132	C₂	#27161159	1	1	yes	characteristic
133	C₂	SD₃₂×D₁₉₇	1	1	yes	characteristic
134	C₂	#27161512	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₉₇
commutator	C₈×C₁₉₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₉₇
Fitting	#358×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×D₈
prime radical	C₂×D₁₅₇₆	C₂²
cosocle	C₂×C₈×C₁₉₇	C₂³
hypercenter	#358	D₁₉₇
hypocenter	C₁₉₇	C₄×SD₃₂
LCS	G ▷ C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

Complements

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

N	H ≅ G/N	type
2. Q₁₆×Dic₁₉₇	C₂	semidirect
4. #27160906	C₂	semidirect
6. C₁₆×Dic₁₉₇	C₂	semidirect
7. #358×C₁₉₇	C₂	semidirect
11. #20263269	C₄	semidirect
30. #15691333	C₂×C₄	semidirect
33. Q₁₆×C₁₉₇	C₂×C₄	semidirect
62. Dic₁₉₇	SD₃₂	semidirect
67. C₁₉₇	C₄×SD₃₂	semidirect
73. #358	D₁₉₇	semidirect
92. Q₁₆	C₄×D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263269	2
Q₁₆×Dic₁₉₇	2
#27161161	2
#27160906	2
#27160881	2
C₁₆×Dic₁₉₇	2
#358×C₁₉₇	2
C₄×SD₃₂	197