Group #27142712 — subgroups

Subgroups — #27142712

Order 12416.

284 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261292	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₉₇	C₂	1	1	yes	maximal, characteristic
3	#20260496	C₂	1	4	yes	maximal
4	#423×C₉₇	C₂	1	1	yes	maximal, characteristic
5	#15690810	C₄	1	2	yes	complemented
6	#15690802	C₄	1	2	yes	complemented
7	#15690789	—	2	2	no	—
8	C₄×Dic₁₉₄	—	2	2	no	—
9	#15690769	C₂²	1	2	yes	—
10	C₂×C₄×Dic₉₇	—	2	2	no	—
11	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
12	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₉₇	—	2	2	no	—
15	C₂×Dic₁₉₄	—	2	2	no	—
16	C₂²×Dic₉₇	—	2	2	no	—
17	C₄⋊Dic₉₇	—	2	2	no	—
18	C₄⋊Dic₉₇	C₂×C₄	1	4	yes	—
19	#13095490	—	2	4	no	—
20	C₂²⋊Dic₉₇	—	2	2	no	—
21	C₄×Dic₉₇	—	2	8	no	—
22	C₄×Dic₉₇	—	2	2	no	—
23	C₄×Dic₉₇	—	2	2	no	—
24	C₄×Dic₉₇	Q₈	1	4	yes	complemented
25	C₂²×C₄×C₉₇	D₄	1	2	yes	—
26	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₉₇	—	2	4	no	—
28	C₂×C₈×C₉₇	—	2	2	no	—
29	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₉₇	D₄	1	2	yes	—
32	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₉₇	—	2	4	no	—
34	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₉₇	#41	1	2	yes	—
40	C₂×Dic₉₇	—	2	8	no	—
41	C₂×Dic₉₇	—	2	4	no	—
42	C₂×Dic₉₇	—	2	4	no	—
43	Dic₁₉₄	—	4	4	no	—
44	C₈×C₉₇	—	4	2	no	—
45	C₂²×C₉₇	—	2	2	no	—
46	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₉₇	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₉₇	—	2	2	no	—
49	C₄×C₉₇	#104	1	4	yes	—
50	C₄×C₉₇	C₄×Q₈	1	2	yes	—
51	C₄×C₉₇	#124	1	1	yes	characteristic
52	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₉₇	—	2	8	no	—
54	Dic₉₇	—	4	4	no	—
55	C₂×C₉₇	—	2	2	no	—
56	C₂×C₉₇	#391	1	1	yes	characteristic
57	C₂×C₉₇	C₂×#104	1	2	yes	—
58	#1994	—	97	1	no	maximal, Sylow
59	C₉₇	#1994	1	1	yes	characteristic, complemented, Sylow
60	#533	—	97	1	no	—
61	#423	D₉₇	1	1	yes	characteristic
62	C₄³	—	97	1	no	—
63	#339	—	97	4	no	—
64	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
65	C₄⋊Q₈	—	97	2	no	—
66	#125	—	97	2	no	—
67	C₂²⋊Q₈	—	194	2	no	—
68	C₄×Q₈	—	194	2	no	—
69	#117	—	97	2	no	—
70	C₂×C₄²	—	97	2	no	—
71	C₂×C₄²	—	194	2	no	—
72	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
73	C₄⋊C₈	—	2	2	no	—
74	C₂×Q₈	—	194	2	no	—
75	C₂²×C₄	—	194	2	no	—
76	C₂²×C₄	D₃₈₈	1	2	yes	—
77	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
78	M₁₆	—	2	4	no	—
79	C₂×C₈	—	2	2	no	—
80	C₄⋊C₄	—	97	4	no	—
81	C₄⋊C₄	—	194	4	no	—
82	C₄⋊C₄	—	194	2	no	—
83	C₂²⋊C₄	—	194	2	no	—
84	C₄²	—	97	4	no	—
85	C₄²	—	194	8	no	—
86	C₄²	—	194	2	no	—
87	C₄²	—	194	2	no	—
88	C₄²	C₄×D₉₇	1	1	yes	characteristic
89	C₄²	C₄×D₉₇	1	1	yes	characteristic
90	C₄²	C₂²⋊D₉₇	1	2	yes	—
91	C₂³	C₂×D₃₈₈	1	1	yes	characteristic
92	Q₈	—	388	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	194	4	no	—
95	C₂×C₄	—	194	8	no	—
96	C₂×C₄	—	194	4	no	—
97	C₂×C₄	#13095488	1	2	yes	—
98	C₂×C₄	#13095488	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
101	C₂×C₄	Q₈×D₉₇	1	2	yes	—
102	C₂×C₄	#13095510	1	2	yes	—
103	C₈	—	4	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×#13095488	1	1	yes	characteristic
106	C₂²	#15690790	1	2	yes	—
107	C₄	—	2	2	no	—
108	C₄	—	388	4	no	—
109	C₄	—	194	8	no	—
110	C₄	#15690736	1	4	yes	—
111	C₄	#15690775	1	2	yes	—
112	C₄	#15690797	1	1	yes	characteristic
113	C₄	#15690808	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#20260640	1	1	yes	characteristic
116	C₂	C₂×#15690736	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20261292

C₂

yes

maximal, characteristic

C₄²×Dic₉₇

C₂

yes

maximal, characteristic

#20260496

C₂

yes

maximal

#423×C₉₇

C₂

yes

maximal, characteristic

#15690810

C₄

yes

complemented

#15690802

C₄

yes

complemented

#15690789

—

C₄×Dic₁₉₄

—

#15690769

C₂²

yes

—

C₂×C₄×Dic₉₇

—

C₂×C₄×Dic₉₇

C₂²

yes

—

C₂×M₁₆×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

—

C₂×Dic₁₉₄

—

C₂²×Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

C₂×C₄

yes

—

#13095490

—

C₂²⋊Dic₉₇

—

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₄×Dic₉₇

Q₈

yes

complemented

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

C₈×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

—

C₄×C₉₇

#104

yes

—

C₄×C₉₇

C₄×Q₈

yes

—

C₄×C₉₇

#124

yes

characteristic

C₄×C₉₇

C₄⋊Q₈

yes

characteristic

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#391

yes

characteristic

C₂×C₉₇

C₂×#104

yes

—

#1994

—

maximal, Sylow

C₉₇

#1994

yes

characteristic, complemented, Sylow

#533

—

#423

D₉₇

yes

characteristic

C₄³

—

#339

—

C₂×M₁₆

C₂×D₉₇

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

194

—

C₄×Q₈

—

194

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

194

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

C₂×Q₈

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

D₃₈₈

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₂²⋊C₄

—

194

—

C₄²

—

C₄²

—

194

—

C₄²

—

194

—

C₄²

—

194

—

C₄²

C₄×D₉₇

yes

characteristic

C₄²

C₄×D₉₇

yes

characteristic

C₄²

C₂²⋊D₉₇

yes

—

C₂³

C₂×D₃₈₈

yes

characteristic

Q₈

—

388

—

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

#13095488

yes

—

C₂×C₄

#13095488

yes

—

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

101

C₂×C₄

Q₈×D₉₇

yes

—

102

C₂×C₄

#13095510

yes

—

103

C₈

—

104

C₂²

—

105

C₂²

C₂×#13095488

yes

characteristic

106

C₂²

#15690790

yes

—

107

C₄

—

108

C₄

—

388

—

109

C₄

—

194

—

110

C₄

#15690736

yes

—

111

C₄

#15690775

yes

—

112

C₄

#15690797

yes

characteristic

113

C₄

#15690808

yes

characteristic

114

C₂

—

115

C₂

#20260640

yes

characteristic

116

C₂

C₂×#15690736

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#423×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#423	D₉₇
hypocenter	C₉₇	#1994
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#423×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
5. #15690810	C₄	semidirect
6. #15690802	C₄	semidirect
24. C₄×Dic₉₇	Q₈	semidirect
59. C₉₇	#1994	semidirect

H ≅ G/N

type

5. #15690810

C₄

semidirect

6. #15690802

C₄

semidirect

24. C₄×Dic₉₇

Q₈

semidirect

59. C₉₇

#1994

semidirect

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

subgroup	index
#20261292	2
C₄²×Dic₉₇	2
#20260496	2
#423×C₉₇	2
#1994	97

subgroup

index

#20261292

C₄²×Dic₉₇

#20260496

#423×C₉₇

#1994

Subgroups — #27142712

Conjugacy classes of subgroups

Complements

Maximal subgroups