Finite Group Databasebeta

Subgroups — #38741315

Order 25216.

212 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 73 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×#20263318	C₂	1	1	yes	maximal, characteristic
2	C₄×#15691342	C₂	1	1	yes	maximal, characteristic
3	#27160820	C₂	1	4	yes	maximal, complemented
4	#432×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#15691342	—	2	1	no	—
6	#20263318	—	2	1	no	—
7	#20263318	—	2	1	no	—
8	#20263318	C₄	1	2	yes	—
9	#20263318	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₉₇	C₄	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₉₇	C₄	1	1	yes	characteristic
12	C₂×#15691337	—	2	1	no	—
13	C₂×#15691337	—	2	1	no	—
14	#20263237	C₂²	1	2	yes	—
15	C₄×#13096802	—	2	1	no	—
16	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₁₉₇	—	2	1	no	—
18	C₂²⋊C₈×C₁₉₇	—	2	1	no	—
19	#97×C₁₉₇	C₂²	1	2	yes	—
20	C₂²×Dic₁₉₇	—	2	1	no	—
21	C₂²×Dic₁₉₇	—	2	1	no	—
22	#15691342	—	2	4	no	—
23	C₂×#13096802	—	2	2	no	—
24	C₄⋊Dic₁₉₇	—	2	1	no	—
25	C₄⋊Dic₁₉₇	—	2	1	no	—
26	C₄⋊Dic₁₉₇	C₂×C₄	1	2	yes	—
27	C₄⋊Dic₁₉₇	C₂×C₄	1	2	yes	—
28	#15691337	—	2	2	no	—
29	#15691337	—	2	2	no	—
30	#15691337	—	2	2	no	—
31	#15691337	—	2	2	no	—
32	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₉₇	—	2	2	no	—
36	C₄²×C₁₉₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
38	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
39	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
40	C₂×Dic₁₉₇	—	2	2	no	—
41	C₂×Dic₁₉₇	—	2	2	no	—
42	C₂×Dic₁₉₇	—	4	2	no	—
43	C₂×Dic₁₉₇	—	4	2	no	—
44	#13096802	—	2	4	no	—
45	C₂³×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₉₇	—	2	2	no	—
47	C₂×C₄×C₁₉₇	—	2	2	no	—
48	C₂×C₄×C₁₉₇	—	4	1	no	—
49	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
55	C₈×C₁₉₇	—	4	2	no	—
56	C₂²×C₁₉₇	—	2	1	no	—
57	C₂²×C₁₉₇	—	2	1	no	—
58	C₂²×C₁₉₇	#104	1	2	yes	—
59	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₁₉₇	—	2	1	no	—
61	C₄×C₁₉₇	—	2	1	no	—
62	C₄×C₁₉₇	—	2	1	no	—
63	C₄×C₁₉₇	—	2	1	no	—
64	C₄×C₁₉₇	—	2	1	no	—
65	C₄×C₁₉₇	#101	1	2	yes	—
66	Dic₁₉₇	—	4	2	no	—
67	Dic₁₉₇	—	4	2	no	—
68	C₂×C₁₉₇	—	2	2	no	—
69	C₂×C₁₉₇	C₂×#101	1	1	yes	characteristic
70	C₂×C₁₉₇	C₂×#104	1	1	yes	characteristic
71	C₂×C₁₉₇	#421	1	1	yes	characteristic
72	C₁₉₇	#1528	1	1	yes	characteristic, complemented, Sylow
73	#1528	—	197	1	no	maximal, Sylow
74	C₂×#125	—	197	1	no	—
75	#432	D₁₉₇	1	1	yes	characteristic
76	C₄×M₁₆	—	197	1	no	—
77	#330	—	197	4	no	—
78	C₂×M₁₆	—	394	1	no	—
79	#125	—	394	1	no	—
80	#125	—	394	1	no	—
81	#125	—	197	2	no	—
82	#125	—	197	2	no	—
83	C₂×C₄⋊C₄	—	394	1	no	—
84	C₂×C₄⋊C₄	—	394	1	no	—
85	C₂×C₄⋊C₄	—	197	1	no	—
86	C₂×C₄⋊C₄	—	197	1	no	—
87	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
88	C₄⋊C₈	—	2	1	no	—
89	C₂²⋊C₈	—	2	1	no	—
90	#97	—	197	2	no	—
91	#97	C₂×D₁₉₇	1	2	yes	—
92	C₄×C₈	—	394	1	no	—
93	C₂²×C₄	—	394	1	no	—
94	C₂²×C₄	—	394	1	no	—
95	C₂²×C₄	C₄×D₁₉₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
97	C₂²×C₄	D₇₈₈	1	1	yes	characteristic
98	M₁₆	—	394	4	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	394	2	no	—
101	C₄⋊C₄	—	394	2	no	—
102	C₄⋊C₄	—	394	2	no	—
103	C₄⋊C₄	—	394	1	no	—
104	C₄⋊C₄	—	394	1	no	—
105	C₄⋊C₄	—	394	2	no	—
106	C₄⋊C₄	—	394	2	no	—
107	C₄⋊C₄	—	197	2	no	—
108	C₄⋊C₄	—	197	2	no	—
109	C₄²	C₄×D₁₉₇	1	1	yes	characteristic
110	C₄²	C₂²⋊D₁₉₇	1	1	yes	characteristic
111	C₄²	D₇₈₈	1	1	yes	characteristic
112	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
113	C₂³	#15691335	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	788	2	no	—
118	C₂×C₄	—	788	2	no	—
119	C₂×C₄	—	394	2	no	—
120	C₂×C₄	—	394	2	no	—
121	C₂×C₄	#15691335	1	1	yes	characteristic
122	C₂×C₄	#15691335	1	1	yes	characteristic
123	C₂×C₄	#15691335	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
126	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₈	—	394	4	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#20263252	1	2	yes	—
132	C₂²	C₂×#15691335	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	788	2	no	—
139	C₄	—	788	2	no	—
140	C₄	#20263244	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	C₂×#20263244	1	1	yes	characteristic
143	C₂	C₂×#20263252	1	1	yes	characteristic
144	C₂	#27161097	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15691335
commutator	C₂×C₄×C₁₉₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₉₇
Fitting	#432×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₉₇	C₂²⋊C₄
cosocle	C₄²×C₁₉₇	C₂³
hypercenter	#432	D₁₉₇
hypocenter	C₁₉₇	#1528
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

Complements

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

N	H ≅ G/N	type
3. #27160820	C₂	semidirect
72. C₁₉₇	#1528	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263318	2
C₄×#15691342	2
#27160820	2
#432×C₁₉₇	2
#1528	197