Finite Group Databasebeta

Subgroups — #27156823

Order 12416.

166 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 38 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	#20261715	C₂	1	1	yes	maximal, characteristic
2	#20261708	C₂	1	1	yes	maximal, characteristic
3	#20260707	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095520	C₄	1	1	yes	characteristic, complemented
5	C₂²×#11715897	C₄	1	1	yes	characteristic, complemented
6	#15690889	—	2	1	no	—
7	#15690868	—	2	1	no	—
8	#15690868	C₄	1	2	yes	complemented
9	#15690867	—	2	1	no	—
10	C₂²⋊#11715897	—	2	1	no	—
11	C₂×#13095510	C₂²	1	1	yes	characteristic
12	#15690766	—	2	1	no	—
13	#15690725	C₄	1	1	yes	characteristic, complemented
14	#15690724	—	2	1	no	—
15	#99×C₉₇	C₄	1	1	yes	characteristic, complemented
16	#13095520	—	4	1	no	—
17	C₂×#11715897	—	2	1	no	—
18	C₂×#11715897	—	2	1	no	—
19	C₂×#11715897	—	4	1	no	—
20	C₄⋊#87604	—	4	1	no	—
21	C₂²⋊#87604	—	2	2	no	—
22	C₄×#87604	—	4	1	no	—
23	#13095510	—	4	1	no	—
24	#13095510	—	4	1	no	—
25	C₂×Dic₁₉₄	Q₈	1	1	yes	characteristic
26	C₂×C₂²⋊D₉₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₉₇	D₄	1	1	yes	characteristic
28	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₉₇	D₄	1	1	yes	characteristic
30	C₂×C₄×D₉₇	D₄	1	1	yes	characteristic
31	#13095490	—	4	1	no	—
32	C₂²⋊Dic₉₇	—	2	1	no	—
33	C₄×Dic₉₇	—	4	1	no	—
34	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
35	C₂²⋊C₄×C₉₇	—	2	1	no	—
36	C₂³×C₉₇	C₄²	1	1	yes	characteristic
37	C₂³×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×D₉₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×#87604	—	4	2	no	—
40	D₄×C₉₇	—	4	1	no	—
41	C₂×C₄×C₉₇	—	4	1	no	—
42	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂²⋊D₉₇	—	4	1	no	—
44	C₂²⋊D₉₇	—	4	1	no	—
45	C₂×Dic₉₇	—	2	1	no	—
46	C₂×Dic₉₇	—	2	1	no	—
47	C₂×Dic₉₇	—	2	1	no	—
48	C₂×Dic₉₇	—	2	1	no	—
49	C₂×Dic₉₇	—	4	1	no	—
50	C₂×Dic₉₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₉₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂×Dic₉₇	C₄⋊C₄	1	1	yes	characteristic
53	C₄×D₉₇	—	2	1	no	—
54	C₄×D₉₇	—	2	1	no	—
55	Dic₁₉₄	—	4	1	no	—
56	#11715897	—	4	1	no	—
57	#11715897	—	4	1	no	—
58	C₂²×C₉₇	—	2	1	no	—
59	C₂²×C₉₇	—	2	1	no	—
60	C₂²×C₉₇	—	4	1	no	—
61	C₂²×C₉₇	—	4	1	no	—
62	C₂²×C₉₇	#95	1	1	yes	characteristic
63	C₂×D₉₇	—	2	1	no	—
64	C₂×D₉₇	—	4	1	no	—
65	#87604	—	8	2	no	—
66	C₄×C₉₇	—	2	1	no	—
67	C₄×C₉₇	—	8	1	no	—
68	Dic₉₇	—	2	1	no	—
69	Dic₉₇	—	2	1	no	—
70	Dic₉₇	—	8	1	no	—
71	Dic₉₇	#99	1	1	yes	characteristic
72	Dic₉₇	#99	1	1	yes	characteristic
73	C₂×C₉₇	—	2	1	no	—
74	C₂×C₉₇	—	4	1	no	—
75	C₂×C₉₇	—	4	1	no	—
76	C₂×C₉₇	#342	1	1	yes	characteristic
77	D₉₇	—	4	1	no	—
78	#1391	—	97	1	no	maximal, Sylow
79	C₉₇	#1391	1	1	yes	characteristic, complemented, Sylow
80	#410	—	97	1	no	—
81	#410	—	97	1	no	—
82	#408	—	97	1	no	—
83	C₂×#41	—	97	1	no	—
84	C₂×M₁₆	—	97	1	no	—
85	C₂²×C₈	—	97	1	no	—
86	#117	—	194	1	no	—
87	#117	—	194	1	no	—
88	#99	—	97	2	no	—
89	#99	—	97	1	no	—
90	#99	—	194	1	no	—
91	#99	—	194	1	no	—
92	#99	#87604	1	1	yes	characteristic, complemented
93	C₂²⋊C₈	—	194	1	no	—
94	C₂²⋊C₈	—	194	1	no	—
95	#41	—	388	1	no	—
96	#41	—	388	1	no	—
97	C₂×Q₈	—	97	1	no	—
98	C₂×D₄	—	97	1	no	—
99	C₂×D₄	—	97	1	no	—
100	C₂×D₄	C₂×#87604	1	1	yes	characteristic
101	C₂²×C₄	—	97	1	no	—
102	C₂²×C₄	—	97	1	no	—
103	C₂²×C₄	—	97	1	no	—
104	M₁₆	—	388	1	no	—
105	C₂×C₈	—	388	1	no	—
106	C₂×C₈	—	194	1	no	—
107	C₂×C₈	—	194	1	no	—
108	C₄⋊C₄	—	388	1	no	—
109	C₄⋊C₄	—	388	1	no	—
110	C₂²⋊C₄	—	2	1	no	—
111	C₂²⋊C₄	—	194	2	no	—
112	C₂²⋊C₄	—	194	1	no	—
113	C₄²	—	388	1	no	—
114	C₄²	—	388	1	no	—
115	C₂³	—	97	1	no	—
116	C₂³	C₄×#87604	1	1	yes	characteristic
117	C₂³	C₂²⋊#87604	1	1	yes	characteristic
118	Q₈	—	388	1	no	—
119	D₄	—	4	1	no	—
120	D₄	—	388	1	no	—
121	D₄	—	388	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	97	1	no	—
124	C₂×C₄	—	97	1	no	—
125	C₂×C₄	—	97	1	no	—
126	C₂×C₄	—	388	1	no	—
127	C₂×C₄	—	388	2	no	—
128	C₂×C₄	—	194	1	no	—
129	C₂×C₄	—	194	1	no	—
130	C₂×C₄	—	194	1	no	—
131	C₂×C₄	—	194	1	no	—
132	C₂×C₄	—	194	1	no	—
133	C₂×C₄	—	194	1	no	—
134	C₂×C₄	C₄⋊#87604	1	1	yes	characteristic
135	C₈	—	388	1	no	—
136	C₈	—	388	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	4	1	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	388	1	no	—
142	C₂²	—	194	1	no	—
143	C₂²	(C₂×#87604)⋊C₄	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	8	1	no	—
146	C₄	—	776	1	no	—
147	C₄	—	776	2	no	—
148	C₄	—	97	1	no	—
149	C₄	—	97	1	no	—
150	C₄	—	194	1	no	—
151	C₄	—	194	1	no	—
152	C₂	—	2	1	no	—
153	C₂	—	4	1	no	—
154	C₂	—	4	1	no	—
155	C₂	—	388	1	no	—
156	C₂	#20261612	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261612
commutator	C₂³×C₉₇	C₄²
Frattini	C₂×D₄	C₂×#87604
Fitting	#99×C₉₇	C₄
socle	C₂×C₉₇	#342
prime radical, cosocle	C₂×#13095510	C₂²
hypercenter	#99	#87604
hypocenter	C₉₇	#1391
LCS	G ▷ C₂³×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #99

Complements

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

N	H ≅ G/N	type
4. C₂×#13095520	C₄	semidirect
5. C₂²×#11715897	C₄	semidirect
8. #15690868	C₄	semidirect
13. #15690725	C₄	semidirect
15. #99×C₉₇	C₄	semidirect
79. C₉₇	#1391	semidirect
92. #99	#87604	semidirect

Maximal subgroups

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

subgroup	index
#20261715	2
#20261708	2
#20260707	2
#1391	97