Group #27141589 — subgroups

Subgroups — #27141589

Order 12416.

292 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 165 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	2	yes	maximal
3	#20260490	C₂	1	2	yes	maximal
4	C₄³×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11715895	C₂²	1	4	yes	—
6	C₄⋊#11715895	C₄	1	8	yes	complemented
7	#15690721	—	2	4	no	—
8	C₂×C₄²×C₉₇	C₄	1	2	yes	—
9	C₂×C₄²×C₉₇	C₄	1	2	yes	—
10	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
13	C₂×#11715895	—	2	4	no	—
14	C₂×#11715895	—	2	4	no	—
15	C₂×#11715895	C₂×C₄	1	8	yes	—
16	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
21	C₄²×C₉₇	—	2	4	no	—
22	C₄²×C₉₇	—	2	4	no	—
23	C₄²×C₉₇	—	2	2	no	—
24	C₄²×C₉₇	C₂×C₄	1	4	yes	—
25	C₄²×C₉₇	D₄	1	2	yes	—
26	C₄²×C₉₇	Q₈	1	2	yes	—
27	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₉₇	—	2	2	no	—
29	C₂×C₄×C₉₇	—	2	2	no	—
30	C₂×C₄×C₉₇	—	2	2	no	—
31	C₂×C₄×C₉₇	—	2	2	no	—
32	C₂×C₄×C₉₇	—	2	2	no	—
33	C₂×C₄×C₉₇	C₄²	1	2	yes	—
34	C₂×C₄×C₉₇	C₄²	1	2	yes	—
35	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
36	C₂×C₄×C₉₇	M₁₆	1	4	yes	—
37	C₂×C₄×C₉₇	M₁₆	1	4	yes	—
38	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₉₇	#41	1	2	yes	—
42	#11715895	—	2	8	no	—
43	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
44	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₉₇	#117	1	1	yes	characteristic
46	C₂²×C₉₇	C₂×M₁₆	1	2	yes	—
47	C₂²×C₉₇	C₂×M₁₆	1	2	yes	—
48	C₄×C₉₇	—	2	2	no	—
49	C₄×C₉₇	—	2	4	no	—
50	C₄×C₉₇	—	2	4	no	—
51	C₄×C₉₇	#97	1	4	yes	complemented
52	C₄×C₉₇	C₄×D₄	1	2	yes	—
53	C₄×C₉₇	C₄×Q₈	1	2	yes	—
54	C₂×C₉₇	C₄×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₉₇	C₂×#97	1	1	yes	characteristic
56	C₂×C₉₇	C₄×M₁₆	1	1	yes	characteristic
57	C₂×C₉₇	#423	1	2	yes	—
58	C₂×C₉₇	#432	1	2	yes	—
59	#1767	—	97	1	no	maximal, Sylow
60	C₉₇	#1767	1	1	yes	characteristic, complemented, Sylow
61	C₂×C₄⋊C₈	—	97	2	no	—
62	C₂×#97	—	97	2	no	—
63	C₄³	D₉₇	1	1	yes	characteristic
64	#336	—	97	2	no	—
65	C₂²×C₈	—	97	4	no	—
66	C₂×C₄²	Dic₉₇	1	2	yes	—
67	C₂×C₄²	Dic₉₇	1	2	yes	—
68	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
69	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
70	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
71	C₄⋊C₈	—	97	8	no	—
72	#97	—	194	4	no	—
73	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
74	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
75	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
76	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
77	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
78	C₂×C₈	—	97	8	no	—
79	C₂×C₈	—	194	4	no	—
80	C₂×C₈	—	194	4	no	—
81	C₄²	—	2	4	no	—
82	C₄²	—	2	4	no	—
83	C₄²	—	2	2	no	—
84	C₄²	Dic₁₉₄	1	2	yes	—
85	C₄²	C₄×D₉₇	1	4	yes	—
86	C₄²	D₃₈₈	1	2	yes	—
87	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	C₄×Dic₉₇	1	2	yes	—
94	C₂×C₄	C₄×Dic₉₇	1	2	yes	—
95	C₂×C₄	C₄⋊Dic₉₇	1	4	yes	—
96	C₂×C₄	#13095495	1	4	yes	—
97	C₂×C₄	#13095495	1	4	yes	—
98	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
99	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
101	C₂×C₄	#13095511	1	2	yes	—
102	C₈	—	194	8	no	—
103	C₂²	C₂×C₄×Dic₉₇	1	1	yes	characteristic
104	C₂²	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
105	C₂²	#15690769	1	1	yes	characteristic
106	C₂²	C₂×#13095495	1	2	yes	—
107	C₂²	C₂×#13095495	1	2	yes	—
108	C₄	—	2	2	no	—
109	C₄	—	2	4	no	—
110	C₄	—	2	4	no	—
111	C₄	#15690721	1	4	yes	complemented
112	C₄	C₄×D₃₈₈	1	2	yes	—
113	C₄	C₄×Dic₁₉₄	1	2	yes	—
114	C₂	C₄×C₄⋊Dic₉₇	1	1	yes	characteristic
115	C₂	C₂×#15690721	1	1	yes	characteristic
116	C₂	C₄×#13095495	1	1	yes	characteristic
117	C₂	#20260759	1	2	yes	—
118	C₂	#20260792	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11715895

C₂

yes

maximal

C₂×#15690721

C₂

yes

maximal

#20260490

C₂

yes

maximal

C₄³×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₄⋊#11715895

C₄

yes

complemented

#15690721

—

C₂×C₄²×C₉₇

C₄

yes

—

C₂×C₄²×C₉₇

C₄

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

—

—

C₂×C₄

yes

—

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

C₄²×C₉₇

—

C₄²×C₉₇

—

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

—

C₂×C₄×C₉₇

C₄²

yes

—

C₂×C₄×C₉₇

C₄²

yes

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

M₁₆

yes

—

C₂×C₄×C₉₇

M₁₆

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

—

C₂²×C₉₇

C₂×C₄²

yes

characteristic

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

#117

yes

characteristic

C₂²×C₉₇

C₂×M₁₆

yes

—

C₂²×C₉₇

C₂×M₁₆

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#97

yes

complemented

C₄×C₉₇

C₄×D₄

yes

—

C₄×C₉₇

C₄×Q₈

yes

—

C₂×C₉₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₉₇

C₂×#97

yes

characteristic

C₂×C₉₇

C₄×M₁₆

yes

characteristic

C₂×C₉₇

#423

yes

—

C₂×C₉₇

#432

yes

—

#1767

—

maximal, Sylow

C₉₇

#1767

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×#97

—

C₄³

D₉₇

yes

characteristic

#336

—

C₂²×C₈

—

C₂×C₄²

Dic₉₇

yes

—

C₂×C₄²

Dic₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

#97

—

194

—

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄²

—

C₄²

—

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

—

C₂×C₄

C₄×Dic₉₇

yes

—

C₂×C₄

C₄×Dic₉₇

yes

—

C₂×C₄

C₄⋊Dic₉₇

yes

—

C₂×C₄

#13095495

yes

—

C₂×C₄

#13095495

yes

—

C₂×C₄

C₂×C₄×D₉₇

yes

—

C₂×C₄

C₂×D₃₈₈

yes

characteristic

100

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

101

C₂×C₄

#13095511

yes

—

102

C₈

—

194

—

103

C₂²

C₂×C₄×Dic₉₇

yes

characteristic

104

C₂²

C₂×C₄⋊Dic₉₇

yes

characteristic

105

C₂²

#15690769

yes

characteristic

106

C₂²

C₂×#13095495

yes

—

107

C₂²

C₂×#13095495

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#15690721

yes

complemented

112

C₄

C₄×D₃₈₈

yes

—

113

C₄

C₄×Dic₁₉₄

yes

—

114

C₂

C₄×C₄⋊Dic₉₇

yes

characteristic

115

C₂

C₂×#15690721

yes

characteristic

116

C₂

C₄×#13095495

yes

characteristic

117

C₂

#20260759

yes

—

118

C₂

#20260792

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator	C₂²×C₉₇	C₂×C₄²
Fitting	C₄³×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₄³	D₉₇
hypocenter	C₉₇	#1767
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₉₇

commutator

C₂²×C₉₇

C₂×C₄²

Fitting

C₄³×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂²×C₄

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₄³

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂²×C₄ ◁ C₄³

N	H ≅ G/N	type
6. C₄⋊#11715895	C₄	semidirect
51. C₄×C₉₇	#97	semidirect
60. C₉₇	#1767	semidirect
111. C₄	#15690721	semidirect

H ≅ G/N

type

6. C₄⋊#11715895

C₄

semidirect

51. C₄×C₉₇

#97

semidirect

60. C₉₇

#1767

semidirect

111. C₄

#15690721

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
#20260490	2
C₄³×C₉₇	2
#1767	97

subgroup

index

C₄³×C₉₇

Subgroups — #27141589

Conjugacy classes of subgroups

Complements

Maximal subgroups