Group #27137789 — subgroups

Subgroups — C₉₇×#1831 #27137789

Order 12416.

260 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 160 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	C₂×C₄⋊C₈×C₉₇	C₂	1	2	yes	maximal
2	C₄³×C₉₇	C₂	1	1	yes	maximal, characteristic
3	#336×C₉₇	C₂	1	4	yes	maximal
4	C₂²×C₈×C₉₇	C₂²	1	4	yes	—
5	C₂×C₄²×C₉₇	C₄	1	2	yes	—
6	C₂×C₄²×C₉₇	C₄	1	1	yes	characteristic
7	C₂×C₄²×C₉₇	C₄	1	1	yes	characteristic
8	C₂×C₄²×C₉₇	C₂²	1	2	yes	—
9	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
10	C₄⋊C₈×C₉₇	—	2	4	no	—
11	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
16	C₂×C₈×C₉₇	—	2	4	no	—
17	C₂×C₈×C₉₇	—	2	8	no	—
18	C₄²×C₉₇	—	2	4	no	—
19	C₄²×C₉₇	—	2	2	no	—
20	C₄²×C₉₇	—	2	4	no	—
21	C₄²×C₉₇	C₈	1	4	yes	complemented
22	C₄²×C₉₇	D₄	1	2	yes	—
23	C₄²×C₉₇	Q₈	1	2	yes	—
24	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₄×C₉₇	—	2	2	no	—
26	C₂×C₄×C₉₇	—	2	2	no	—
27	C₂×C₄×C₉₇	—	2	2	no	—
28	C₂×C₄×C₉₇	—	2	2	no	—
29	C₂×C₄×C₉₇	—	2	2	no	—
30	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
32	C₂×C₄×C₉₇	C₂×C₈	1	4	yes	—
33	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
34	C₂×C₄×C₉₇	M₁₆	1	4	yes	—
35	C₂×C₄×C₉₇	M₁₆	1	2	yes	—
36	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₉₇	#41	1	4	yes	—
39	C₈×C₉₇	—	4	4	no	—
40	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₉₇	#117	1	2	yes	—
42	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
43	C₂²×C₉₇	C₂×M₁₆	1	1	yes	characteristic
44	C₂²×C₉₇	C₂×M₁₆	1	2	yes	—
45	C₄×C₉₇	—	2	2	no	—
46	C₄×C₉₇	—	2	4	no	—
47	C₄×C₉₇	—	2	4	no	—
48	C₄×C₉₇	C₄⋊C₈	1	4	yes	—
49	C₄×C₉₇	#124	1	2	yes	—
50	C₄×C₉₇	#125	1	2	yes	—
51	C₂×C₉₇	#382	1	1	yes	characteristic
52	C₂×C₉₇	C₂×C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₉₇	#423	1	1	yes	characteristic
54	C₂×C₉₇	#431	1	2	yes	—
55	C₂×C₉₇	#432	1	2	yes	—
56	#1831	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₉₇	#1831	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×C₄⋊C₈	C₂×C₉₇	1	2	yes	—
59	C₄³	C₂×C₉₇	1	1	yes	characteristic
60	#336	C₂×C₉₇	1	4	yes	—
61	C₂²×C₈	C₂²×C₉₇	1	4	yes	—
62	C₂×C₄²	C₄×C₉₇	1	2	yes	—
63	C₂×C₄²	C₄×C₉₇	1	1	yes	characteristic
64	C₂×C₄²	C₄×C₉₇	1	1	yes	characteristic
65	C₂×C₄²	C₂²×C₉₇	1	2	yes	—
66	C₂×C₄²	C₂²×C₉₇	1	1	yes	characteristic
67	C₄⋊C₈	—	2	4	no	—
68	C₂²×C₄	C₂×C₄×C₉₇	1	1	yes	characteristic
69	C₂²×C₄	C₂×C₄×C₉₇	1	2	yes	—
70	C₂²×C₄	C₂×C₄×C₉₇	1	2	yes	—
71	C₂²×C₄	C₂×C₄×C₉₇	1	1	yes	characteristic
72	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	8	no	—
75	C₄²	—	2	4	no	—
76	C₄²	—	2	2	no	—
77	C₄²	—	2	4	no	—
78	C₄²	C₈×C₉₇	1	4	yes	complemented
79	C₄²	D₄×C₉₇	1	2	yes	—
80	C₄²	Q₈×C₉₇	1	2	yes	—
81	C₂³	C₂²×C₄×C₉₇	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₄⋊C₄×C₉₇	1	2	yes	—
88	C₂×C₄	C₄⋊C₄×C₉₇	1	2	yes	—
89	C₂×C₄	C₂×C₈×C₉₇	1	4	yes	—
90	C₂×C₄	C₂×C₈×C₉₇	1	2	yes	—
91	C₂×C₄	M₁₆×C₉₇	1	4	yes	—
92	C₂×C₄	M₁₆×C₉₇	1	2	yes	—
93	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
94	C₂×C₄	C₂×Q₈×C₉₇	1	1	yes	characteristic
95	C₂×C₄	#41×C₉₇	1	4	yes	—
96	C₈	—	4	4	no	—
97	C₂²	C₂×C₄⋊C₄×C₉₇	1	1	yes	characteristic
98	C₂²	#117×C₉₇	1	2	yes	—
99	C₂²	C₂²×C₈×C₉₇	1	1	yes	characteristic
100	C₂²	C₂×M₁₆×C₉₇	1	1	yes	characteristic
101	C₂²	C₂×M₁₆×C₉₇	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	4	no	—
104	C₄	—	2	4	no	—
105	C₄	C₄⋊C₈×C₉₇	1	4	yes	—
106	C₄	#124×C₉₇	1	2	yes	—
107	C₄	#125×C₉₇	1	2	yes	—
108	C₂	#382×C₉₇	1	1	yes	characteristic
109	C₂	C₂×C₄⋊C₈×C₉₇	1	1	yes	characteristic
110	C₂	#423×C₉₇	1	1	yes	characteristic
111	C₂	#431×C₉₇	1	2	yes	—
112	C₂	#432×C₉₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₈×C₉₇

C₂

yes

maximal

C₄³×C₉₇

C₂

yes

maximal, characteristic

#336×C₉₇

C₂

yes

maximal

C₂²×C₈×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₄

yes

—

C₂×C₄²×C₉₇

C₄

yes

characteristic

C₂×C₄²×C₉₇

C₄

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

C₈

yes

complemented

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

Q₈

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂×C₈

yes

—

C₂×C₄×C₉₇

C₂×C₈

yes

—

C₂×C₄×C₉₇

M₁₆

yes

—

C₂×C₄×C₉₇

M₁₆

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₈×C₉₇

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

#117

yes

—

C₂²×C₉₇

C₂²×C₈

yes

characteristic

C₂²×C₉₇

C₂×M₁₆

yes

characteristic

C₂²×C₉₇

C₂×M₁₆

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄⋊C₈

yes

—

C₄×C₉₇

#124

yes

—

C₄×C₉₇

#125

yes

—

C₂×C₉₇

#382

yes

characteristic

C₂×C₉₇

C₂×C₄⋊C₈

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₂×C₉₇

#432

yes

—

#1831

C₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₉₇

#1831

yes

characteristic, direct product factor, Sylow

C₂×C₄⋊C₈

C₂×C₉₇

yes

—

C₄³

C₂×C₉₇

yes

characteristic

#336

C₂×C₉₇

yes

—

C₂²×C₈

C₂²×C₉₇

yes

—

C₂×C₄²

C₄×C₉₇

yes

—

C₂×C₄²

C₄×C₉₇

yes

characteristic

C₂×C₄²

C₄×C₉₇

yes

characteristic

C₂×C₄²

C₂²×C₉₇

yes

—

C₂×C₄²

C₂²×C₉₇

yes

characteristic

C₄⋊C₈

—

C₂²×C₄

C₂×C₄×C₉₇

yes

characteristic

C₂²×C₄

C₂×C₄×C₉₇

yes

—

C₂²×C₄

C₂×C₄×C₉₇

yes

—

C₂²×C₄

C₂×C₄×C₉₇

yes

characteristic

C₂²×C₄

C₂³×C₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₈×C₉₇

yes

complemented

C₄²

D₄×C₉₇

yes

—

C₄²

Q₈×C₉₇

yes

—

C₂³

C₂²×C₄×C₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₄×C₉₇

yes

—

C₂×C₄

C₄⋊C₄×C₉₇

yes

—

C₂×C₄

C₂×C₈×C₉₇

yes

—

C₂×C₄

C₂×C₈×C₉₇

yes

—

C₂×C₄

M₁₆×C₉₇

yes

—

C₂×C₄

M₁₆×C₉₇

yes

—

C₂×C₄

C₂×D₄×C₉₇

yes

characteristic

C₂×C₄

C₂×Q₈×C₉₇

yes

characteristic

C₂×C₄

#41×C₉₇

yes

—

C₈

—

C₂²

C₂×C₄⋊C₄×C₉₇

yes

characteristic

C₂²

#117×C₉₇

yes

—

C₂²

C₂²×C₈×C₉₇

yes

characteristic

100

C₂²

C₂×M₁₆×C₉₇

yes

characteristic

101

C₂²

C₂×M₁₆×C₉₇

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄⋊C₈×C₉₇

yes

—

106

C₄

#124×C₉₇

yes

—

107

C₄

#125×C₉₇

yes

—

108

C₂

#382×C₉₇

yes

characteristic

109

C₂

C₂×C₄⋊C₈×C₉₇

yes

characteristic

110

C₂

#423×C₉₇

yes

characteristic

111

C₂

#431×C₉₇

yes

—

112

C₂

#432×C₉₇

yes

—

	subgroup	G/H
center	C₂²×C₄×C₉₇	C₂³
commutator	C₂²	C₂²×C₈×C₉₇
Frattini, cosocle	C₂²×C₄	C₂³×C₉₇
socle, prime radical	C₂³×C₉₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉₇ ◁ G

subgroup

G/H

center

C₂²×C₄×C₉₇

C₂³

commutator

C₂²

C₂²×C₈×C₉₇

Frattini, cosocle

C₂²×C₄

C₂³×C₉₇

socle, prime radical

C₂³×C₉₇

C₂²×C₄

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₉₇ ◁ G

N	H ≅ G/N	type
21. C₄²×C₉₇	C₈	semidirect
56. #1831	C₉₇	direct (N × H)
57. C₉₇	#1831	direct (N × H)
78. C₄²	C₈×C₉₇	semidirect

H ≅ G/N

type

21. C₄²×C₉₇

C₈

semidirect

56. #1831

C₉₇

direct (N × H)

57. C₉₇

#1831

direct (N × H)

78. C₄²

C₈×C₉₇

semidirect

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

subgroup	index
C₂×C₄⋊C₈×C₉₇	2
C₄³×C₉₇	2
#336×C₉₇	2
#1831	97

subgroup

index

C₂×C₄⋊C₈×C₉₇

C₄³×C₉₇

#336×C₉₇

#1831

Subgroups — C₉₇×#1831 #27137789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C97×#1831 #27137789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#1831 #27137789