Group #27140507 — subgroups

Subgroups — #27140507

Order 12416.

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

normal subgroups only

This group has 168 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	#20260759	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11715895	C₂	1	1	yes	maximal, characteristic
3	#20260486	C₂	1	4	yes	maximal, complemented
4	#431×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095495	—	2	1	no	—
6	C₂²×#11715895	—	2	1	no	—
7	C₄⋊#11715895	—	2	1	no	—
8	C₄⋊#11715895	—	2	1	no	—
9	C₄⋊#11715895	C₂²	1	2	yes	—
10	C₄⋊#11715895	C₂²	1	2	yes	—
11	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₉₇	C₄	1	2	yes	—
13	C₂²⋊C₈×C₉₇	C₄	1	2	yes	—
14	C₄×C₈×C₉₇	C₂²	1	2	yes	—
15	#13095495	—	4	2	no	—
16	C₂×#11715895	—	2	2	no	—
17	C₂×#11715895	—	2	2	no	—
18	C₂×#11715895	—	4	2	no	—
19	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₉₇	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	—
23	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₉₇	D₄	1	1	yes	characteristic
25	C₄²×C₉₇	Q₈	1	1	yes	characteristic
26	C₄²×C₉₇	C₂³	1	1	yes	characteristic
27	C₂³×C₉₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₉₇	—	2	2	no	—
29	C₂×C₄×C₉₇	—	4	1	no	—
30	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
34	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
35	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
38	C₈×C₉₇	—	2	4	no	—
39	#11715895	—	4	2	no	—
40	#11715895	—	4	2	no	—
41	C₂²×C₉₇	—	2	1	no	—
42	C₂²×C₉₇	—	2	1	no	—
43	C₂²×C₉₇	#106	1	2	yes	—
44	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₉₇	—	2	1	no	—
46	C₄×C₉₇	—	2	1	no	—
47	C₄×C₉₇	—	2	1	no	—
48	C₄×C₉₇	#106	1	2	yes	—
49	C₄×C₉₇	#131	1	2	yes	—
50	C₄×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
51	C₄×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
52	C₂×C₉₇	—	2	2	no	—
53	C₂×C₉₇	#424	1	1	yes	characteristic
54	C₂×C₉₇	C₂×#106	1	1	yes	characteristic
55	C₂×C₉₇	#430	1	1	yes	characteristic
56	#1570	—	97	1	no	maximal, Sylow
57	C₉₇	#1570	1	1	yes	characteristic, complemented, Sylow
58	#431	D₉₇	1	1	yes	characteristic
59	#423	—	97	1	no	—
60	C₂×C₄⋊C₈	—	97	1	no	—
61	#334	—	97	4	no	—
62	C₂×M₁₆	—	194	1	no	—
63	C₂²×C₈	—	194	1	no	—
64	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
65	C₄⋊C₈	—	97	2	no	—
66	C₄⋊C₈	—	97	2	no	—
67	C₄⋊C₈	—	194	1	no	—
68	C₄⋊C₈	—	194	1	no	—
69	C₄⋊C₈	Dic₉₇	1	2	yes	—
70	C₂²⋊C₈	Dic₉₇	1	2	yes	—
71	C₄×C₈	C₂×D₉₇	1	2	yes	—
72	C₂²×C₄	Dic₁₉₄	1	1	yes	characteristic
73	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
74	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
75	M₁₆	—	388	2	no	—
76	C₂×C₈	—	388	2	no	—
77	C₂×C₈	—	194	2	no	—
78	C₂×C₈	—	194	2	no	—
79	C₂×C₈	C₂×Dic₉₇	1	4	yes	—
80	C₄²	Dic₁₉₄	1	1	yes	characteristic
81	C₄²	D₃₈₈	1	1	yes	characteristic
82	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
83	C₄²	C₂²×D₉₇	1	1	yes	characteristic
84	C₂³	C₄⋊Dic₉₇	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	C₄⋊Dic₉₇	1	1	yes	characteristic
88	C₂×C₄	C₄⋊Dic₉₇	1	1	yes	characteristic
89	C₂×C₄	C₄⋊Dic₉₇	1	1	yes	characteristic
90	C₂×C₄	#13095500	1	2	yes	—
91	C₂×C₄	#13095500	1	2	yes	—
92	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
93	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
95	C₈	—	2	4	no	—
96	C₈	—	388	2	no	—
97	C₈	—	388	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#15690742	1	2	yes	—
101	C₂²	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#15690742	1	2	yes	—
106	C₄	#15690817	1	2	yes	—
107	C₄	C₂×#13095500	1	1	yes	characteristic
108	C₄	C₂×#13095500	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#20260762	1	1	yes	characteristic
111	C₂	C₂×#15690742	1	1	yes	characteristic
112	C₂	#20260784	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260759

C₂

yes

maximal, characteristic

C₂×C₄⋊#11715895

C₂

yes

maximal, characteristic

#20260486

C₂

yes

maximal, complemented

#431×C₉₇

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

C₄⋊#11715895

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

C₄

yes

—

C₂²⋊C₈×C₉₇

C₄

yes

—

C₄×C₈×C₉₇

C₂²

yes

—

—

—

—

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

Q₈

yes

characteristic

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

Q₈

yes

characteristic

C₄²×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₈×C₉₇

—

#11715895

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#106

yes

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#106

yes

—

C₄×C₉₇

#131

yes

—

C₄×C₉₇

C₂×SD₁₆

yes

characteristic

C₄×C₉₇

C₂×SD₁₆

yes

characteristic

C₂×C₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

#431

D₉₇

yes

characteristic

#423

—

C₂×C₄⋊C₈

—

#334

—

C₂×M₁₆

—

194

—

C₂²×C₈

—

194

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

194

—

C₄⋊C₈

—

194

—

C₄⋊C₈

Dic₉₇

yes

—

C₂²⋊C₈

Dic₉₇

yes

—

C₄×C₈

C₂×D₉₇

yes

—

C₂²×C₄

Dic₁₉₄

yes

characteristic

C₂²×C₄

D₃₈₈

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

M₁₆

—

388

—

C₂×C₈

—

388

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

C₂×Dic₉₇

yes

—

C₄²

Dic₁₉₄

yes

characteristic

C₄²

D₃₈₈

yes

characteristic

C₄²

C₂×Dic₉₇

yes

characteristic

C₄²

C₂²×D₉₇

yes

characteristic

C₂³

C₄⋊Dic₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₉₇

yes

characteristic

C₂×C₄

C₄⋊Dic₉₇

yes

characteristic

C₂×C₄

C₄⋊Dic₉₇

yes

characteristic

C₂×C₄

#13095500

yes

—

C₂×C₄

#13095500

yes

—

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

C₂×C₄

C₂×D₃₈₈

yes

characteristic

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

C₈

—

C₈

—

388

—

C₈

—

388

—

C₂²

—

C₂²

—

100

C₂²

#15690742

yes

—

101

C₂²

C₂×C₄⋊Dic₉₇

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#15690742

yes

—

106

C₄

#15690817

yes

—

107

C₄

C₂×#13095500

yes

characteristic

108

C₄

C₂×#13095500

yes

characteristic

109

C₂

—

110

C₂

#20260762

yes

characteristic

111

C₂

C₂×#15690742

yes

characteristic

112

C₂

#20260784

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₈₈
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₄²	C₂²×D₉₇
Fitting	#431×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₄⋊C₄
prime radical	C₂³×C₉₇	C₄⋊C₄
cosocle	C₄²×C₉₇	C₂³
hypercenter	#431	D₉₇
hypocenter	C₉₇	#1570
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₃₈₈

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₄²

C₂²×D₉₇

Fitting

#431×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₄⋊C₄

prime radical

C₂³×C₉₇

C₄⋊C₄

cosocle

C₄²×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #20260486	C₂	semidirect
57. C₉₇	#1570	semidirect

H ≅ G/N

type

3. #20260486

C₂

semidirect

57. C₉₇

#1570

semidirect

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

subgroup	index
#20260759	2
C₂×C₄⋊#11715895	2
#20260486	2
#431×C₉₇	2
#1570	97

subgroup

index

#431×C₉₇

Subgroups — #27140507

Conjugacy classes of subgroups

Complements

Maximal subgroups