Group #27137987 — subgroups

Subgroups — C₉₇×#2029 #27137987

Order 12416.

466 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 84 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	#543×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#379×C₉₇	C₂	1	3	yes	maximal
3	#342×C₉₇	C₂	1	3	yes	maximal, complemented
4	C₂⁵×C₉₇	C₂²	1	1	yes	characteristic
5	C₂²⋊Q₈×C₉₇	—	2	6	no	—
6	C₂×C₂²⋊C₄×C₉₇	—	2	6	no	—
7	C₂×C₂²⋊C₄×C₉₇	C₂²	1	3	yes	—
8	C₂×C₂²⋊C₄×C₉₇	C₂²	1	3	yes	—
9	C₂⁴×C₉₇	—	2	3	no	—
10	C₂⁴×C₉₇	—	2	3	no	—
11	C₂⁴×C₉₇	—	4	4	no	—
12	C₂⁴×C₉₇	Q₈	1	2	yes	complemented
13	C₂⁴×C₉₇	C₂³	1	1	yes	characteristic
14	C₂×Q₈×C₉₇	—	2	2	no	—
15	C₂²×C₄×C₉₇	—	2	3	no	—
16	C₂²×C₄×C₉₇	D₄	1	6	yes	—
17	C₄⋊C₄×C₉₇	—	2	6	no	—
18	C₂²⋊C₄×C₉₇	—	2	6	no	—
19	C₂²⋊C₄×C₉₇	—	2	6	no	—
20	C₂²⋊C₄×C₉₇	—	4	12	no	—
21	C₂³×C₉₇	—	2	12	no	—
22	C₂³×C₉₇	—	2	6	no	—
23	C₂³×C₉₇	—	4	4	no	—
24	C₂³×C₉₇	—	4	2	no	—
25	C₂³×C₉₇	—	4	6	no	—
26	C₂³×C₉₇	—	4	12	no	—
27	C₂³×C₉₇	—	4	1	no	—
28	C₂³×C₉₇	—	4	3	no	—
29	C₂³×C₉₇	C₂×D₄	1	3	yes	—
30	C₂³×C₉₇	C₂×Q₈	1	1	yes	characteristic
31	C₂³×C₉₇	#41	1	3	yes	—
32	Q₈×C₉₇	—	8	2	no	—
33	C₂×C₄×C₉₇	—	2	6	no	—
34	C₂×C₄×C₉₇	—	4	6	no	—
35	C₂×C₄×C₉₇	—	4	6	no	—
36	C₂×C₄×C₉₇	—	4	3	no	—
37	C₂²×C₉₇	—	2	6	no	—
38	C₂²×C₉₇	—	2	12	no	—
39	C₂²×C₉₇	—	4	1	no	—
40	C₂²×C₉₇	—	4	3	no	—
41	C₂²×C₉₇	—	4	12	no	—
42	C₂²×C₉₇	—	4	6	no	—
43	C₂²×C₉₇	—	4	2	no	—
44	C₂²×C₉₇	—	4	4	no	—
45	C₂²×C₉₇	#120	1	1	yes	characteristic
46	C₂²×C₉₇	C₂²⋊Q₈	1	3	yes	—
47	C₂²×C₉₇	#124	1	3	yes	—
48	C₄×C₉₇	—	4	6	no	—
49	C₄×C₉₇	—	8	3	no	—
50	C₂×C₉₇	—	2	3	no	—
51	C₂×C₉₇	—	2	3	no	—
52	C₂×C₉₇	—	4	4	no	—
53	C₂×C₉₇	#393	1	1	yes	characteristic
54	C₂×C₉₇	B(4,𝔽₂)	1	2	yes	—
55	#2029	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₉₇	#2029	1	1	yes	characteristic, direct product factor, Sylow
57	#543	C₂×C₉₇	1	1	yes	characteristic, complemented
58	#379	C₂×C₉₇	1	3	yes	—
59	#342	C₂×C₉₇	1	3	yes	complemented
60	C₂⁵	C₂²×C₉₇	1	1	yes	characteristic
61	C₂²⋊Q₈	—	2	6	no	—
62	C₂×C₂²⋊C₄	—	2	6	no	—
63	C₂×C₂²⋊C₄	C₂²×C₉₇	1	3	yes	—
64	C₂×C₂²⋊C₄	C₂²×C₉₇	1	3	yes	—
65	C₂⁴	—	2	3	no	—
66	C₂⁴	—	2	3	no	—
67	C₂⁴	—	4	4	no	—
68	C₂⁴	Q₈×C₉₇	1	2	yes	complemented
69	C₂⁴	C₂³×C₉₇	1	1	yes	characteristic
70	C₂×Q₈	—	2	2	no	—
71	C₂²×C₄	—	2	3	no	—
72	C₂²×C₄	D₄×C₉₇	1	6	yes	—
73	C₄⋊C₄	—	2	6	no	—
74	C₂²⋊C₄	—	2	6	no	—
75	C₂²⋊C₄	—	2	6	no	—
76	C₂²⋊C₄	—	4	12	no	—
77	C₂³	—	2	12	no	—
78	C₂³	—	2	6	no	—
79	C₂³	—	4	4	no	—
80	C₂³	—	4	2	no	—
81	C₂³	—	4	6	no	—
82	C₂³	—	4	12	no	—
83	C₂³	—	4	1	no	—
84	C₂³	—	4	3	no	—
85	C₂³	C₂×D₄×C₉₇	1	3	yes	—
86	C₂³	C₂×Q₈×C₉₇	1	1	yes	characteristic
87	C₂³	#41×C₉₇	1	3	yes	—
88	Q₈	—	8	2	no	—
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	4	6	no	—
91	C₂×C₄	—	4	6	no	—
92	C₂×C₄	—	4	3	no	—
93	C₂²	—	2	6	no	—
94	C₂²	—	2	12	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	4	3	no	—
97	C₂²	—	4	12	no	—
98	C₂²	—	4	6	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	4	no	—
101	C₂²	#120×C₉₇	1	1	yes	characteristic
102	C₂²	C₂²⋊Q₈×C₉₇	1	3	yes	—
103	C₂²	#124×C₉₇	1	3	yes	—
104	C₄	—	4	6	no	—
105	C₄	—	8	3	no	—
106	C₂	—	2	3	no	—
107	C₂	—	2	3	no	—
108	C₂	—	4	4	no	—
109	C₂	#393×C₉₇	1	1	yes	characteristic
110	C₂	B(4,𝔽₂)×C₉₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#543×C₉₇

C₂

yes

maximal, characteristic, complemented

#379×C₉₇

C₂

yes

maximal

#342×C₉₇

C₂

yes

maximal, complemented

C₂⁵×C₉₇

C₂²

yes

characteristic

C₂²⋊Q₈×C₉₇

—

C₂×C₂²⋊C₄×C₉₇

—

C₂×C₂²⋊C₄×C₉₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₉₇

C₂²

yes

—

C₂⁴×C₉₇

—

C₂⁴×C₉₇

—

C₂⁴×C₉₇

—

C₂⁴×C₉₇

Q₈

yes

complemented

C₂⁴×C₉₇

C₂³

yes

characteristic

C₂×Q₈×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₄⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

—

C₂³×C₉₇

C₂×Q₈

yes

characteristic

C₂³×C₉₇

#41

yes

—

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#120

yes

characteristic

C₂²×C₉₇

C₂²⋊Q₈

yes

—

C₂²×C₉₇

#124

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#393

yes

characteristic

C₂×C₉₇

B(4,𝔽₂)

yes

—

#2029

C₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₉₇

#2029

yes

characteristic, direct product factor, Sylow

#543

C₂×C₉₇

yes

characteristic, complemented

#379

C₂×C₉₇

yes

—

#342

C₂×C₉₇

yes

complemented

C₂⁵

C₂²×C₉₇

yes

characteristic

C₂²⋊Q₈

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂²×C₉₇

yes

—

C₂×C₂²⋊C₄

C₂²×C₉₇

yes

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

Q₈×C₉₇

yes

complemented

C₂⁴

C₂³×C₉₇

yes

characteristic

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₄×C₉₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×D₄×C₉₇

yes

—

C₂³

C₂×Q₈×C₉₇

yes

characteristic

C₂³

#41×C₉₇

yes

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

#120×C₉₇

yes

characteristic

102

C₂²

C₂²⋊Q₈×C₉₇

yes

—

103

C₂²

#124×C₉₇

yes

—

104

C₄

—

105

C₄

—

106

C₂

—

107

C₂

—

108

C₂

—

109

C₂

#393×C₉₇

yes

characteristic

110

C₂

B(4,𝔽₂)×C₉₇

yes

—

	subgroup	G/H
center, socle	C₂²×C₉₇	#120
commutator, Frattini, cosocle	C₂⁴	C₂³×C₉₇
prime radical	C₂⁵×C₉₇	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₂⁴×C₉₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₉₇

#120

commutator, Frattini, cosocle

C₂⁴

C₂³×C₉₇

prime radical

C₂⁵×C₉₇

C₂²

LCS

G ▷ C₂⁴ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₉₇ ◁ C₂⁴×C₉₇ ◁ G

N	H ≅ G/N	type
1. #543×C₉₇	C₂	semidirect
3. #342×C₉₇	C₂	semidirect
12. C₂⁴×C₉₇	Q₈	semidirect
55. #2029	C₉₇	direct (N × H)
56. C₉₇	#2029	direct (N × H)
57. #543	C₂×C₉₇	semidirect
59. #342	C₂×C₉₇	semidirect
68. C₂⁴	Q₈×C₉₇	semidirect

H ≅ G/N

type

1. #543×C₉₇

C₂

semidirect

3. #342×C₉₇

C₂

semidirect

12. C₂⁴×C₉₇

Q₈

semidirect

55. #2029

C₉₇

direct (N × H)

56. C₉₇

#2029

direct (N × H)

57. #543

C₂×C₉₇

semidirect

59. #342

C₂×C₉₇

semidirect

68. C₂⁴

Q₈×C₉₇

semidirect

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

subgroup	index
#543×C₉₇	2
#379×C₉₇	2
#342×C₉₇	2
#2029	97

subgroup

index

#543×C₉₇

#379×C₉₇

#342×C₉₇

#2029

Subgroups — C₉₇×#2029 #27137987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C97×#2029 #27137987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#2029 #27137987