Group #27153540 — subgroups

Subgroups — #27153540

Order 12416.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#20261548	C₂	1	4	yes	maximal, complemented
2	#20261007	C₂	1	4	yes	maximal, complemented
3	#20260917	C₂	1	4	yes	maximal, complemented
4	#20260853	C₂	1	2	yes	maximal, complemented
5	#494×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#15690858	C₂²	1	4	yes	—
7	#15690842	—	2	8	no	—
8	#15690841	C₂²	1	8	yes	—
9	#15690833	—	2	4	no	—
10	#15690832	C₂²	1	2	yes	—
11	Q₁₆×D₉₇	—	2	4	no	—
12	#15690816	C₂²	1	4	yes	—
13	#15690746	C₂²	1	2	yes	—
14	#15690738	C₂²	1	4	yes	—
15	#15690729	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₉₇	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₉₇	C₂²	1	2	yes	—
18	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
19	#13095512	—	2	8	no	—
20	#13095511	—	4	4	no	—
21	#13095511	C₂³	1	4	yes	—
22	Q₈×D₉₇	—	2	8	no	—
23	Dic₃₈₈	D₄	1	2	yes	—
24	#13095501	—	2	8	no	—
25	#13095500	D₄	1	4	yes	—
26	#13095498	—	2	8	no	—
27	D₇₇₆	D₄	1	2	yes	—
28	#13095495	C₂³	1	4	yes	—
29	#13095494	—	2	4	no	—
30	C₈×D₉₇	—	2	4	no	—
31	C₂×Q₈×C₉₇	C₂³	1	4	yes	—
32	Q₁₆×C₉₇	—	2	4	no	—
33	Q₁₆×C₉₇	—	2	4	no	—
34	C₂×C₈×C₉₇	C₂³	1	2	yes	—
35	C₄⋊C₄×C₉₇	—	4	2	no	—
36	C₄²×C₉₇	C₂³	1	1	yes	characteristic
37	Q₈×C₉₇	—	2	8	no	—
38	C₂×C₄×C₉₇	—	2	1	no	—
39	C₂×C₄×C₉₇	—	2	4	no	—
40	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
41	C₂²⋊D₉₇	—	2	4	no	—
42	D₃₈₈	—	4	4	no	—
43	D₃₈₈	C₂×D₄	1	4	yes	—
44	C₄×D₉₇	—	2	4	no	—
45	C₄×D₉₇	—	4	8	no	—
46	Dic₁₉₄	—	4	4	no	—
47	Dic₁₉₄	C₂×D₄	1	4	yes	—
48	C₈×C₉₇	C₂×D₄	1	2	yes	—
49	C₈×C₉₇	C₂×D₄	1	2	yes	—
50	#11715895	—	2	4	no	—
51	C₂²×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₉₇	—	2	4	no	—
53	C₄×C₉₇	—	2	2	no	—
54	C₄×C₉₇	—	4	4	no	—
55	C₄×C₉₇	C₂²×D₄	1	2	yes	—
56	Dic₉₇	—	2	4	no	—
57	C₂×C₉₇	—	2	1	no	—
58	C₂×C₉₇	D₄²	1	1	yes	characteristic
59	D₉₇	—	4	4	no	—
60	#3293	—	97	1	no	maximal, Sylow
61	C₉₇	#3293	1	1	yes	characteristic, complemented, Sylow
62	#578	—	97	4	no	—
63	#494	D₉₇	1	1	yes	characteristic, complemented
64	#473	—	97	4	no	—
65	#456	—	97	4	no	—
66	#443	—	97	2	no	—
67	2₋¹⁺⁴	—	97	4	no	—
68	#137	—	97	8	no	—
69	#137	—	194	8	no	—
70	#135	—	97	2	no	—
71	#135	—	194	4	no	—
72	C₂×Q₁₆	—	194	4	no	—
73	C₂×Q₁₆	C₂×D₉₇	1	4	yes	—
74	#131	—	97	4	no	—
75	C₄⋊Q₈	C₂×D₉₇	1	2	yes	—
76	#108	—	97	2	no	—
77	#104	—	97	4	no	—
78	#101	—	97	4	no	—
79	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
80	#41	—	97	4	no	—
81	#41	—	388	4	no	—
82	#41	—	194	8	no	—
83	C₂×Q₈	—	194	8	no	—
84	C₂×Q₈	C₂²×D₉₇	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	97	2	no	—
88	Q₁₆	—	194	8	no	—
89	SD₁₆	—	97	4	no	—
90	SD₁₆	—	194	8	no	—
91	D₈	—	97	2	no	—
92	M₁₆	—	97	4	no	—
93	M₁₆	—	194	4	no	—
94	C₂×C₈	—	194	4	no	—
95	C₂×C₈	C₂²×D₉₇	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₉₇	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	97	4	no	—
100	Q₈	—	388	4	no	—
101	D₄	—	97	4	no	—
102	D₄	—	388	4	no	—
103	D₄	—	194	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	388	8	no	—
107	C₂×C₄	—	194	4	no	—
108	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
109	C₈	—	194	4	no	—
110	C₈	D₄×D₉₇	1	2	yes	—
111	C₈	D₄×D₉₇	1	2	yes	—
112	C₂²	—	194	4	no	—
113	C₂²	#15690855	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	194	4	no	—
117	C₄	C₂×D₄×D₉₇	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	388	4	no	—
120	C₂	#20261370	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20261548

C₂

yes

maximal, complemented

#20261007

C₂

yes

maximal, complemented

#20260917

C₂

yes

maximal, complemented

#20260853

C₂

yes

maximal, complemented

#494×C₉₇

C₂

yes

maximal, characteristic, complemented

#15690858

C₂²

yes

—

#15690842

—

#15690841

C₂²

yes

—

#15690833

—

#15690832

C₂²

yes

—

Q₁₆×D₉₇

—

#15690816

C₂²

yes

—

#15690746

C₂²

yes

—

#15690738

C₂²

yes

—

#15690729

C₂²

yes

—

C₂×Q₁₆×C₉₇

C₂²

yes

—

C₄⋊Q₈×C₉₇

C₂²

yes

—

C₄×C₈×C₉₇

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₉₇

—

Dic₃₈₈

D₄

yes

—

#13095501

—

#13095500

D₄

yes

—

#13095498

—

D₇₇₆

D₄

yes

—

#13095495

C₂³

yes

—

#13095494

—

C₈×D₉₇

—

C₂×Q₈×C₉₇

C₂³

yes

—

Q₁₆×C₉₇

—

Q₁₆×C₉₇

—

C₂×C₈×C₉₇

C₂³

yes

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

C₂³

yes

characteristic

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂⁴

yes

characteristic

C₂²⋊D₉₇

—

D₃₈₈

—

D₃₈₈

C₂×D₄

yes

—

C₄×D₉₇

—

C₄×D₉₇

—

Dic₁₉₄

—

Dic₁₉₄

C₂×D₄

yes

—

C₈×C₉₇

C₂×D₄

yes

—

C₈×C₉₇

C₂×D₄

yes

—

#11715895

—

C₂²×C₉₇

H(5,𝔽₂)

yes

characteristic

C₂×D₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂²×D₄

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

D₄²

yes

characteristic

D₉₇

—

#3293

—

maximal, Sylow

C₉₇

#3293

yes

characteristic, complemented, Sylow

#578

—

#494

D₉₇

yes

characteristic, complemented

#473

—

#456

—

#443

—

2₋¹⁺⁴

—

#137

—

#137

—

194

—

#135

—

#135

—

194

—

C₂×Q₁₆

—

194

—

C₂×Q₁₆

C₂×D₉₇

yes

—

#131

—

C₄⋊Q₈

C₂×D₉₇

yes

—

#108

—

#104

—

#101

—

C₄×C₈

C₂×D₉₇

yes

characteristic

#41

—

#41

—

388

—

#41

—

194

—

C₂×Q₈

—

194

—

C₂×Q₈

C₂²×D₉₇

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

194

—

SD₁₆

—

SD₁₆

—

194

—

D₈

—

M₁₆

—

M₁₆

—

194

—

C₂×C₈

—

194

—

C₂×C₈

C₂²×D₉₇

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₉₇

yes

characteristic

Q₈

—

Q₈

—

100

Q₈

—

388

—

101

D₄

—

102

D₄

—

388

—

103

D₄

—

194

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

388

—

107

C₂×C₄

—

194

—

108

C₂×C₄

C₂³×D₉₇

yes

characteristic

109

C₈

—

194

—

110

C₈

D₄×D₉₇

yes

—

111

C₈

D₄×D₉₇

yes

—

112

C₂²

—

194

—

113

C₂²

#15690855

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

194

—

117

C₄

C₂×D₄×D₉₇

yes

—

118

C₂

—

119

C₂

—

388

—

120

C₂

#20261370

yes

characteristic

	subgroup	G/H
center	C₂	#20261370
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	#494×C₉₇	C₂
socle	C₂×C₉₇	D₄²
hypercenter	#494	D₉₇
hypocenter	C₉₇	#3293
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#20261370

commutator, cosocle

C₂×C₄×C₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₉₇

Fitting

#494×C₉₇

C₂

socle

C₂×C₉₇

D₄²

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #20261548	C₂	semidirect
2. #20261007	C₂	semidirect
3. #20260917	C₂	semidirect
4. #20260853	C₂	semidirect
5. #494×C₉₇	C₂	semidirect
61. C₉₇	#3293	semidirect
63. #494	D₉₇	semidirect

H ≅ G/N

type

1. #20261548

C₂

semidirect

2. #20261007

C₂

semidirect

3. #20260917

C₂

semidirect

4. #20260853

C₂

semidirect

5. #494×C₉₇

C₂

semidirect

61. C₉₇

#3293

semidirect

63. #494

D₉₇

semidirect

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

subgroup	index
#20261548	2
#20261007	2
#20260917	2
#20260853	2
#494×C₉₇	2
#3293	97

subgroup

index

#494×C₉₇

Subgroups — #27153540

Conjugacy classes of subgroups

Complements

Maximal subgroups