Group #27141593 — subgroups

Subgroups — #27141593

Order 12416.

260 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 141 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⋊#11715895	C₂	1	2	yes	maximal
2	C₂×#15690721	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11715895	C₂	1	1	yes	maximal, characteristic
4	#20260490	C₂	1	2	yes	maximal
5	#376×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11715895	C₂²	1	2	yes	—
7	C₂²×#11715895	C₂²	1	2	yes	—
8	C₄⋊#11715895	C₄	1	8	yes	complemented
9	#15690721	—	2	2	no	—
10	C₄×#11715895	—	2	2	no	—
11	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
14	#95×C₉₇	C₄	1	4	yes	—
15	C₂×#11715895	—	2	2	no	—
16	C₂×#11715895	—	2	2	no	—
17	C₂×#11715895	—	2	2	no	—
18	C₂×#11715895	—	2	2	no	—
19	C₂×#11715895	C₂×C₄	1	4	yes	—
20	C₂×#11715895	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
25	C₄²×C₉₇	—	2	2	no	—
26	C₄²×C₉₇	C₂×C₄	1	4	yes	—
27	C₄²×C₉₇	D₄	1	2	yes	—
28	C₄²×C₉₇	Q₈	1	2	yes	—
29	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₉₇	—	2	4	no	—
31	C₂×C₄×C₉₇	—	2	4	no	—
32	C₂×C₄×C₉₇	—	2	2	no	—
33	C₂×C₄×C₉₇	—	2	4	no	—
34	C₂×C₄×C₉₇	C₄²	1	4	yes	—
35	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
36	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₉₇	#41	1	2	yes	—
40	#11715895	—	2	4	no	—
41	#11715895	—	2	4	no	—
42	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
43	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₉₇	#117	1	1	yes	characteristic
45	C₂²×C₉₇	#131	1	4	yes	—
46	C₄×C₉₇	—	2	4	no	—
47	C₄×C₉₇	—	2	4	no	—
48	C₄×C₉₇	—	4	2	no	—
49	C₄×C₉₇	C₄×D₄	1	2	yes	—
50	C₄×C₉₇	C₄×Q₈	1	2	yes	—
51	C₂×C₉₇	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₉₇	#405	1	2	yes	—
53	C₂×C₉₇	#424	1	2	yes	—
54	C₂×C₉₇	#433	1	2	yes	—
55	#1768	—	97	1	no	maximal, Sylow
56	C₉₇	#1768	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₄⋊C₈	—	97	2	no	—
58	C₂×#97	—	97	1	no	—
59	C₂×C₄×C₈	—	97	1	no	—
60	#376	D₉₇	1	1	yes	characteristic
61	#336	—	97	2	no	—
62	C₂²×C₈	—	97	2	no	—
63	C₂²×C₈	—	97	2	no	—
64	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
66	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
67	C₄⋊C₈	—	97	8	no	—
68	#97	—	194	2	no	—
69	C₄×C₈	—	194	2	no	—
70	#95	Dic₉₇	1	4	yes	—
71	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
72	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
73	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
74	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
75	C₂×C₈	—	97	4	no	—
76	C₂×C₈	—	97	4	no	—
77	C₂×C₈	—	194	2	no	—
78	C₂×C₈	—	194	2	no	—
79	C₂×C₈	—	194	2	no	—
80	C₂×C₈	—	194	2	no	—
81	C₄²	—	2	2	no	—
82	C₄²	Dic₁₉₄	1	2	yes	—
83	C₄²	C₄×D₉₇	1	4	yes	—
84	C₄²	D₃₈₈	1	2	yes	—
85	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	C₄×Dic₉₇	1	4	yes	—
91	C₂×C₄	C₄⋊Dic₉₇	1	4	yes	—
92	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
93	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
95	C₂×C₄	#13095511	1	2	yes	—
96	C₈	—	194	4	no	—
97	C₈	—	194	4	no	—
98	C₂²	C₂×C₄×Dic₉₇	1	1	yes	characteristic
99	C₂²	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
100	C₂²	#15690769	1	1	yes	characteristic
101	C₂²	#15690817	1	4	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	2	4	no	—
104	C₄	—	4	2	no	—
105	C₄	C₄×D₃₈₈	1	2	yes	—
106	C₄	C₄×Dic₁₉₄	1	2	yes	—
107	C₂	C₄×C₄⋊Dic₉₇	1	1	yes	characteristic
108	C₂	#20260691	1	2	yes	—
109	C₂	#20260762	1	2	yes	—
110	C₂	#20260796	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11715895

C₂

yes

maximal

C₂×#15690721

C₂

yes

maximal, characteristic

C₂×C₄×#11715895

C₂

yes

maximal, characteristic

#20260490

C₂

yes

maximal

#376×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₂²×#11715895

C₂²

yes

—

C₄⋊#11715895

C₄

yes

complemented

#15690721

—

C₄×#11715895

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#95×C₉₇

C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×#11715895

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄²×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

—

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

C₄²

yes

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

#11715895

—

#11715895

—

C₂²×C₉₇

C₂×C₄²

yes

characteristic

C₂²×C₉₇

C₂×C₄⋊C₄

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄×D₄

yes

—

C₄×C₉₇

C₄×Q₈

yes

—

C₂×C₉₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₉₇

#405

yes

—

C₂×C₉₇

#424

yes

—

C₂×C₉₇

#433

yes

—

#1768

—

maximal, Sylow

C₉₇

#1768

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#376

D₉₇

yes

characteristic

#336

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

#97

—

194

—

C₄×C₈

—

194

—

#95

Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄²

—

C₄²

Dic₁₉₄

yes

—

C₄²

C₄×D₉₇

yes

—

C₄²

D₃₈₈

yes

—

C₂³

C₂²×Dic₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₉₇

yes

—

C₂×C₄

C₄⋊Dic₉₇

yes

—

C₂×C₄

C₂×C₄×D₉₇

yes

—

C₂×C₄

C₂×D₃₈₈

yes

characteristic

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

C₂×C₄

#13095511

yes

—

C₈

—

194

—

C₈

—

194

—

C₂²

C₂×C₄×Dic₉₇

yes

characteristic

C₂²

C₂×C₄⋊Dic₉₇

yes

characteristic

100

C₂²

#15690769

yes

characteristic

101

C₂²

#15690817

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄×D₃₈₈

yes

—

106

C₄

C₄×Dic₁₉₄

yes

—

107

C₂

C₄×C₄⋊Dic₉₇

yes

characteristic

108

C₂

#20260691

yes

—

109

C₂

#20260762

yes

—

110

C₂

#20260796

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator	C₂²×C₉₇	C₂×C₄²
Fitting	#376×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#376	D₉₇
hypocenter	C₉₇	#1768
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ #376

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₉₇

commutator

C₂²×C₉₇

C₂×C₄²

Fitting

#376×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂²×C₄

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂²×C₄ ◁ #376

N	H ≅ G/N	type
8. C₄⋊#11715895	C₄	semidirect
56. C₉₇	#1768	semidirect

H ≅ G/N

type

8. C₄⋊#11715895

C₄

semidirect

56. C₉₇

#1768

semidirect

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

subgroup	index
C₂×C₄⋊#11715895	2
C₂×#15690721	2
C₂×C₄×#11715895	2
#20260490	2
#376×C₉₇	2
#1768	97

subgroup

index

#376×C₉₇

Subgroups — #27141593

Conjugacy classes of subgroups

Complements

Maximal subgroups