Finite Group Databasebeta

Subgroups — #38743499

Order 25216.

184 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 79 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	#27161127	C₂	1	1	yes	maximal, characteristic
2	C₂×#20263261	C₂	1	1	yes	maximal, characteristic
3	C₂×#20263261	C₂	1	1	yes	maximal, characteristic
4	#27161037	C₂	1	1	yes	maximal, characteristic
5	#27160851	C₂	1	1	yes	maximal, characteristic
6	#27160850	C₂	1	1	yes	maximal, characteristic
7	#430×C₁₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#15691342	C₂²	1	1	yes	characteristic
9	C₂×#15691342	C₂²	1	1	yes	characteristic
10	C₂²×#13096802	C₂²	1	1	yes	characteristic
11	#20263285	C₂²	1	1	yes	characteristic
12	#20263261	—	2	1	no	—
13	#20263261	—	2	1	no	—
14	#20263261	—	2	2	no	—
15	#20263261	C₄	1	4	yes	complemented
16	#20263236	—	2	1	no	—
17	C₈×Dic₁₉₇	—	2	1	no	—
18	C₂×M₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
21	#108×C₁₉₇	—	2	2	no	—
22	#15691342	—	2	1	no	—
23	#15691342	—	2	1	no	—
24	#15691342	—	2	1	no	—
25	#15691342	C₂×C₄	1	2	yes	—
26	C₂×#13096802	—	2	1	no	—
27	C₂×#13096802	—	2	1	no	—
28	C₂×#13096802	C₂×C₄	1	2	yes	—
29	C₂×#13096802	D₄	1	1	yes	characteristic
30	C₂×#13096802	D₄	1	1	yes	characteristic
31	C₂×#13096802	Q₈	1	2	yes	—
32	C₄⋊Dic₁₉₇	—	2	1	no	—
33	C₂²⋊Dic₁₉₇	Q₈	1	2	yes	—
34	C₄×Dic₁₉₇	—	2	1	no	—
35	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
36	M₁₆×C₁₉₇	—	2	1	no	—
37	M₁₆×C₁₉₇	—	2	1	no	—
38	M₁₆×C₁₉₇	—	2	1	no	—
39	M₁₆×C₁₉₇	—	2	1	no	—
40	M₁₆×C₁₉₇	C₂×C₄	1	2	yes	—
41	M₁₆×C₁₉₇	D₄	1	2	yes	—
42	C₂×C₈×C₁₉₇	—	2	1	no	—
43	C₂×C₈×C₁₉₇	—	2	1	no	—
44	C₂×C₈×C₁₉₇	—	2	1	no	—
45	C₂×Dic₁₉₇	—	2	2	no	—
46	#13096802	—	2	2	no	—
47	#13096802	—	4	1	no	—
48	#13096802	C₄⋊C₄	1	2	yes	—
49	#13096802	C₄⋊C₄	1	2	yes	—
50	C₂³×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
56	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
57	C₈×C₁₉₇	—	2	2	no	—
58	C₈×C₁₉₇	—	2	2	no	—
59	C₈×C₁₉₇	—	4	1	no	—
60	C₂²×C₁₉₇	—	4	1	no	—
61	C₂²×C₁₉₇	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₁₉₇	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₁₉₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₄×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₄×C₁₉₇	C₄×D₄	1	1	yes	characteristic
66	C₄×C₁₉₇	#121	1	1	yes	characteristic
67	C₄×C₁₉₇	#125	1	1	yes	characteristic
68	Dic₁₉₇	—	4	2	no	—
69	C₂×C₁₉₇	—	2	1	no	—
70	C₂×C₁₉₇	—	2	1	no	—
71	C₂×C₁₉₇	—	2	1	no	—
72	C₂×C₁₉₇	#389	1	1	yes	characteristic
73	C₁₉₇	#1953	1	1	yes	characteristic, complemented, Sylow
74	#1953	—	197	1	no	maximal, Sylow
75	#430	—	197	1	no	—
76	#430	D₁₉₇	1	1	yes	characteristic
77	C₂×#108	—	197	1	no	—
78	C₂×#108	—	197	1	no	—
79	#405	—	197	1	no	—
80	#344	—	197	1	no	—
81	#344	—	197	1	no	—
82	C₂×M₁₆	—	197	1	no	—
83	C₂×M₁₆	—	197	1	no	—
84	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
85	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
86	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
87	C₂²×C₈	—	197	1	no	—
88	#117	—	197	1	no	—
89	#108	—	2	2	no	—
90	#108	—	394	2	no	—
91	#108	—	394	1	no	—
92	#108	—	394	1	no	—
93	#108	—	197	4	no	—
94	#97	—	394	1	no	—
95	C₄×C₈	—	394	1	no	—
96	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
97	M₁₆	—	2	1	no	—
98	M₁₆	—	2	1	no	—
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	394	1	no	—
102	M₁₆	—	394	1	no	—
103	M₁₆	—	394	1	no	—
104	M₁₆	—	197	2	no	—
105	M₁₆	C₄×D₁₉₇	1	2	yes	—
106	M₁₆	C₂²⋊D₁₉₇	1	2	yes	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	394	1	no	—
111	C₂×C₈	—	394	1	no	—
112	C₂×C₈	—	197	2	no	—
113	C₂×C₈	—	197	1	no	—
114	C₂×C₈	—	197	1	no	—
115	C₂×C₈	—	197	2	no	—
116	C₄⋊C₄	—	394	1	no	—
117	C₂²⋊C₄	—	197	2	no	—
118	C₄²	—	394	1	no	—
119	C₂³	Q₈×D₁₉₇	1	1	yes	characteristic
120	C₂×C₄	—	394	2	no	—
121	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
122	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₁₉₇	1	1	yes	characteristic
125	C₂×C₄	#15691357	1	1	yes	characteristic
126	C₂×C₄	#15691358	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	2	2	no	—
129	C₈	—	4	1	no	—
130	C₈	—	788	1	no	—
131	C₈	—	394	2	no	—
132	C₈	—	197	2	no	—
133	C₈	—	197	2	no	—
134	C₂²	—	4	1	no	—
135	C₂²	#20263291	1	1	yes	characteristic
136	C₂²	#20263304	1	1	yes	characteristic
137	C₂²	#20263324	1	1	yes	characteristic
138	C₄	—	788	2	no	—
139	C₄	C₄⋊C₄×D₁₉₇	1	1	yes	characteristic
140	C₄	C₄×C₂²⋊D₁₉₇	1	1	yes	characteristic
141	C₄	#20263298	1	1	yes	characteristic
142	C₄	#20263316	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#27160974	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#20263298
commutator	C₂×C₄×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	#430×C₁₉₇	C₂
socle	C₂×C₁₉₇	#389
prime radical	C₂³×C₁₉₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	#430	D₁₉₇
hypocenter	C₁₉₇	#1953
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

Complements

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

N	H ≅ G/N	type
15. #20263261	C₄	semidirect
73. C₁₉₇	#1953	semidirect

Maximal subgroups

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

subgroup	index
#27161127	2
C₂×#20263261	2
#27161037	2
#27160851	2
#27160850	2
#430×C₁₉₇	2
#1953	197