Group #27141863 — subgroups

Subgroups — #27141863

Order 12416.

248 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 133 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	1	yes	maximal, characteristic
2	#20260605	C₂	1	1	yes	maximal, characteristic
3	#20260497	C₂	1	4	yes	maximal
4	#384×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11715895	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₉₇	—	2	1	no	—
7	C₂×C₄⋊Dic₉₇	—	2	1	no	—
8	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
9	C₄×#11715895	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₉₇	—	2	1	no	—
11	C₂×C₄⋊C₄×C₉₇	—	2	1	no	—
12	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₉₇	—	2	2	no	—
15	C₂×#11715895	—	2	2	no	—
16	C₂×#11715895	—	2	2	no	—
17	C₂×#11715895	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₉₇	—	2	4	no	—
19	C₄⋊Dic₉₇	—	4	2	no	—
20	C₄⋊Dic₉₇	—	4	2	no	—
21	C₂²×C₄×C₉₇	—	2	2	no	—
22	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₉₇	—	2	4	no	—
26	C₄⋊C₄×C₉₇	—	4	2	no	—
27	C₄⋊C₄×C₉₇	—	4	2	no	—
28	C₄²×C₉₇	C₂×C₄	1	2	yes	—
29	C₄²×C₉₇	Q₈	1	2	yes	—
30	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₉₇	—	4	2	no	—
32	C₂×C₄×C₉₇	—	4	2	no	—
33	C₂×C₄×C₉₇	—	4	2	no	—
34	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₉₇	D₈	1	2	yes	—
37	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
38	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
39	C₂×C₄×C₉₇	Q₁₆	1	2	yes	—
40	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₉₇	#41	1	2	yes	—
43	C₂×C₄×C₉₇	#41	1	2	yes	—
44	C₂×Dic₉₇	—	4	2	no	—
45	C₂×Dic₉₇	—	4	2	no	—
46	C₂×Dic₉₇	—	4	2	no	—
47	#11715895	—	2	4	no	—
48	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₉₇	#124	1	1	yes	characteristic
50	C₂²×C₉₇	#124	1	1	yes	characteristic
51	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
55	C₄×C₉₇	—	8	2	no	—
56	C₄×C₉₇	#106	1	4	yes	—
57	C₄×C₉₇	#107	1	4	yes	—
58	C₄×C₉₇	#117	1	2	yes	—
59	C₄×C₉₇	#125	1	2	yes	—
60	Dic₉₇	—	8	2	no	—
61	C₂×C₉₇	#382	1	1	yes	characteristic
62	C₂×C₉₇	C₂×#106	1	1	yes	characteristic
63	C₂×C₉₇	C₂×#107	1	1	yes	characteristic
64	C₂×C₉₇	#486	1	1	yes	characteristic
65	C₂×C₉₇	#486	1	1	yes	characteristic
66	C₂×C₉₇	#487	1	1	yes	characteristic
67	C₂×C₉₇	#487	1	1	yes	characteristic
68	#1834	—	97	1	no	maximal, Sylow
69	C₉₇	#1834	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₄×C₈	—	97	1	no	—
71	#384	—	97	1	no	—
72	#384	D₉₇	1	1	yes	characteristic
73	#340	—	97	4	no	—
74	C₂²×C₈	—	97	2	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	97	2	no	—
78	C₂×C₄⋊C₄	—	194	1	no	—
79	C₂×C₄⋊C₄	—	194	1	no	—
80	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
81	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
82	C₄×C₈	—	97	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	194	2	no	—
85	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
88	C₂×C₈	—	97	4	no	—
89	C₂×C₈	—	194	2	no	—
90	C₂×C₈	—	194	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	388	2	no	—
95	C₄⋊C₄	—	388	2	no	—
96	C₄⋊C₄	—	194	4	no	—
97	C₄²	Dic₁₉₄	1	2	yes	—
98	C₄²	C₄×D₉₇	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	388	2	no	—
104	C₂×C₄	—	388	2	no	—
105	C₂×C₄	—	388	2	no	—
106	C₂×C₄	#13095490	1	2	yes	—
107	C₂×C₄	#13095490	1	2	yes	—
108	C₂×C₄	#13095496	1	2	yes	—
109	C₂×C₄	#13095498	1	2	yes	—
110	C₂×C₄	#13095499	1	2	yes	—
111	C₂×C₄	#13095501	1	2	yes	—
112	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
114	C₂×C₄	#13095510	1	2	yes	—
115	C₂×C₄	#13095512	1	2	yes	—
116	C₈	—	194	4	no	—
117	C₂²	C₂×#13095490	1	1	yes	characteristic
118	C₂²	#15690797	1	1	yes	characteristic
119	C₂²	#15690799	1	1	yes	characteristic
120	C₂²	C₂×#13095496	1	1	yes	characteristic
121	C₂²	C₂×#13095498	1	1	yes	characteristic
122	C₂²	C₂×#13095499	1	1	yes	characteristic
123	C₂²	C₂×#13095501	1	1	yes	characteristic
124	C₄	—	8	2	no	—
125	C₄	—	776	2	no	—
126	C₄	#15690741	1	4	yes	—
127	C₄	#15690743	1	4	yes	—
128	C₄	#15690768	1	2	yes	—
129	C₄	#15690801	1	2	yes	—
130	C₂	#20260600	1	1	yes	characteristic
131	C₂	C₂×#15690741	1	1	yes	characteristic
132	C₂	C₂×#15690743	1	1	yes	characteristic
133	C₂	#20261081	1	1	yes	characteristic
134	C₂	#20261083	1	1	yes	characteristic
135	C₂	#20261085	1	1	yes	characteristic
136	C₂	#20261087	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#11715895

C₂

yes

maximal, characteristic

#20260605

C₂

yes

maximal, characteristic

#20260497

C₂

yes

maximal

#384×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₄×#11715895

C₄

yes

complemented

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

—

C₂×#11715895

C₂×C₄

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

Q₈

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

D₈

yes

—

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

Q₁₆

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

#11715895

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

C₂×D₈

yes

characteristic

C₂²×C₉₇

C₂×SD₁₆

yes

characteristic

C₂²×C₉₇

C₂×SD₁₆

yes

characteristic

C₂²×C₉₇

C₂×Q₁₆

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

#106

yes

—

C₄×C₉₇

#107

yes

—

C₄×C₉₇

#117

yes

—

C₄×C₉₇

#125

yes

—

Dic₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

#384

—

#384

D₉₇

yes

characteristic

#340

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

388

—

C₄⋊C₄

—

388

—

C₄⋊C₄

—

194

—

C₄²

Dic₁₉₄

yes

—

C₄²

C₄×D₉₇

yes

—

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

388

—

104

C₂×C₄

—

388

—

105

C₂×C₄

—

388

—

106

C₂×C₄

#13095490

yes

—

107

C₂×C₄

#13095490

yes

—

108

C₂×C₄

#13095496

yes

—

109

C₂×C₄

#13095498

yes

—

110

C₂×C₄

#13095499

yes

—

111

C₂×C₄

#13095501

yes

—

112

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

113

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

114

C₂×C₄

#13095510

yes

—

115

C₂×C₄

#13095512

yes

—

116

C₈

—

194

—

117

C₂²

C₂×#13095490

yes

characteristic

118

C₂²

#15690797

yes

characteristic

119

C₂²

#15690799

yes

characteristic

120

C₂²

C₂×#13095496

yes

characteristic

121

C₂²

C₂×#13095498

yes

characteristic

122

C₂²

C₂×#13095499

yes

characteristic

123

C₂²

C₂×#13095501

yes

characteristic

124

C₄

—

125

C₄

—

776

—

126

C₄

#15690741

yes

—

127

C₄

#15690743

yes

—

128

C₄

#15690768

yes

—

129

C₄

#15690801

yes

—

130

C₂

#20260600

yes

characteristic

131

C₂

C₂×#15690741

yes

characteristic

132

C₂

C₂×#15690743

yes

characteristic

133

C₂

#20261081

yes

characteristic

134

C₂

#20261083

yes

characteristic

135

C₂

#20261085

yes

characteristic

136

C₂

#20261087

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#384×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#384	D₉₇
hypocenter	C₉₇	#1834
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#384×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×#11715895	C₄	semidirect
69. C₉₇	#1834	semidirect

H ≅ G/N

type

9. C₄×#11715895

C₄

semidirect

69. C₉₇

#1834

semidirect

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

subgroup	index
C₂×C₄×#11715895	2
#20260605	2
#20260497	2
#384×C₉₇	2
#1834	97

subgroup

index

#384×C₉₇

Subgroups — #27141863

Conjugacy classes of subgroups

Complements

Maximal subgroups