Group #27141720 — subgroups

Subgroups — #27141720

Order 12416.

312 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 125 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, complemented
2	C₂×C₂²⋊#11715895	C₂	1	2	yes	maximal
3	#20260490	C₂	1	2	yes	maximal, complemented
4	C₂×#117×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11715895	C₂²	1	4	yes	—
6	C₄⋊#11715895	—	2	4	no	—
7	C₂²⋊#11715895	—	2	4	no	—
8	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
9	#117×C₉₇	—	2	4	no	—
10	C₂×C₄⋊C₄×C₉₇	C₄	1	2	yes	—
11	C₂×C₂²⋊C₄×C₉₇	C₄	1	2	yes	—
12	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
14	C₂×#11715895	—	2	4	no	—
15	C₂×#11715895	—	2	4	no	—
16	C₂×#11715895	—	2	4	no	—
17	C₂⁴×C₉₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₉₇	—	2	2	no	—
19	C₂²×C₄×C₉₇	—	2	2	no	—
20	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₉₇	D₄	1	2	yes	—
24	C₂²×C₄×C₉₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₉₇	—	2	4	no	—
27	C₂²⋊C₄×C₉₇	—	2	4	no	—
28	C₄²×C₉₇	—	2	2	no	—
29	C₄²×C₉₇	D₄	1	4	yes	—
30	C₂³×C₉₇	—	2	1	no	—
31	C₂³×C₉₇	—	2	1	no	—
32	C₂³×C₉₇	—	2	4	no	—
33	C₂³×C₉₇	C₄⋊C₄	1	2	yes	—
34	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₉₇	—	2	1	no	—
36	C₂×C₄×C₉₇	—	2	1	no	—
37	C₂×C₄×C₉₇	—	2	4	no	—
38	C₂×C₄×C₉₇	—	2	4	no	—
39	C₂×C₄×C₉₇	—	2	4	no	—
40	C₂×C₄×C₉₇	—	2	2	no	—
41	C₂×C₄×C₉₇	—	2	4	no	—
42	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₉₇	#41	1	2	yes	—
48	#11715895	—	4	4	no	—
49	C₂²×C₉₇	—	2	2	no	—
50	C₂²×C₉₇	—	2	4	no	—
51	C₂²×C₉₇	—	4	2	no	—
52	C₂²×C₉₇	—	4	2	no	—
53	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₉₇	#117	1	1	yes	characteristic
56	C₂²×C₉₇	#131	1	4	yes	—
57	C₄×C₉₇	—	2	2	no	—
58	C₄×C₉₇	—	2	4	no	—
59	C₄×C₉₇	—	4	2	no	—
60	C₄×C₉₇	#121	1	2	yes	—
61	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
62	C₂×C₉₇	—	4	2	no	—
63	C₂×C₉₇	#380	1	1	yes	characteristic
64	C₂×C₉₇	#408	1	2	yes	—
65	C₂×C₉₇	#424	1	2	yes	—
66	C₂×C₉₇	#433	1	2	yes	—
67	#1798	—	97	1	no	maximal, Sylow
68	C₉₇	#1798	1	1	yes	characteristic, complemented, Sylow
69	C₂×#117	D₉₇	1	1	yes	characteristic
70	C₂×C₄⋊C₈	—	97	2	no	—
71	C₂×C₂²⋊C₈	—	97	2	no	—
72	#336	—	97	2	no	—
73	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
74	C₂²×C₈	—	97	4	no	—
75	#117	—	2	4	no	—
76	C₂×C₄⋊C₄	Dic₉₇	1	2	yes	—
77	C₂×C₂²⋊C₄	Dic₉₇	1	2	yes	—
78	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
79	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
80	C₄⋊C₈	—	194	4	no	—
81	C₂²⋊C₈	—	194	4	no	—
82	C₂⁴	C₂×Dic₉₇	1	1	yes	characteristic
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	Dic₁₉₄	1	2	yes	—
86	C₂²×C₄	D₃₈₈	1	2	yes	—
87	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
88	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
89	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
91	C₂×C₈	—	194	4	no	—
92	C₂×C₈	—	194	4	no	—
93	C₂×C₈	—	194	4	no	—
94	C₄⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	4	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²⋊D₉₇	1	4	yes	—
98	C₂³	—	2	1	no	—
99	C₂³	—	2	1	no	—
100	C₂³	—	2	4	no	—
101	C₂³	C₄⋊Dic₉₇	1	2	yes	—
102	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	C₂²⋊Dic₉₇	1	4	yes	—
111	C₂×C₄	C₄⋊Dic₉₇	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
113	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
114	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
115	C₂×C₄	#13095511	1	2	yes	—
116	C₈	—	388	4	no	—
117	C₂²	—	2	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	2	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
122	C₂²	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
123	C₂²	#15690769	1	1	yes	characteristic
124	C₂²	#15690817	1	4	yes	—
125	C₄	—	2	4	no	—
126	C₄	—	2	2	no	—
127	C₄	—	4	2	no	—
128	C₄	#15690784	1	2	yes	—
129	C₄	#15690789	1	2	yes	—
130	C₂	—	4	2	no	—
131	C₂	#20260595	1	1	yes	characteristic
132	C₂	#20260702	1	2	yes	—
133	C₂	#20260762	1	2	yes	—
134	C₂	#20260796	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11715895

C₂

yes

maximal, complemented

C₂×C₂²⋊#11715895

C₂

yes

maximal

#20260490

C₂

yes

maximal, complemented

C₂×#117×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₄⋊#11715895

—

C₂²⋊#11715895

—

C₂³×C₄×C₉₇

C₂²

yes

characteristic

#117×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₄

yes

—

C₂×C₂²⋊C₄×C₉₇

C₄

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

—

—

—

C₂⁴×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

Q₈

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

C₄⋊C₄

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

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

C₂×C₄⋊C₄

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#121

yes

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

C₂×C₉₇

—

C₂×C₉₇

#380

yes

characteristic

C₂×C₉₇

#408

yes

—

C₂×C₉₇

#424

yes

—

C₂×C₉₇

#433

yes

—

#1798

—

maximal, Sylow

C₉₇

#1798

yes

characteristic, complemented, Sylow

C₂×#117

D₉₇

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

#336

—

C₂³×C₄

C₂×D₉₇

yes

characteristic

C₂²×C₈

—

#117

—

C₂×C₄⋊C₄

Dic₉₇

yes

—

C₂×C₂²⋊C₄

Dic₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

194

—

C₂²⋊C₈

—

194

—

C₂⁴

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₉₄

yes

—

C₂²×C₄

D₃₈₈

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²⋊D₉₇

yes

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₄⋊Dic₉₇

yes

—

102

C₂³

C₂²×Dic₉₇

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂²⋊Dic₉₇

yes

—

111

C₂×C₄

C₄⋊Dic₉₇

yes

—

112

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

113

C₂×C₄

C₂×D₃₈₈

yes

characteristic

114

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

115

C₂×C₄

#13095511

yes

—

116

C₈

—

388

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×C₂²⋊Dic₉₇

yes

characteristic

122

C₂²

C₂×C₄⋊Dic₉₇

yes

characteristic

123

C₂²

#15690769

yes

characteristic

124

C₂²

#15690817

yes

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#15690784

yes

—

129

C₄

#15690789

yes

—

130

C₂

—

131

C₂

#20260595

yes

characteristic

132

C₂

#20260702

yes

—

133

C₂

#20260762

yes

—

134

C₂

#20260796

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator, socle	C₂³×C₉₇	C₂²×C₄
Fitting	C₂×#117×C₉₇	C₂
prime radical	C₂⁴×C₉₇	C₂×C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×#117	D₉₇
hypocenter	C₉₇	#1798
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₉₇

commutator, socle

C₂³×C₉₇

C₂²×C₄

Fitting

C₂×#117×C₉₇

C₂

prime radical

C₂⁴×C₉₇

C₂×C₄

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂³×C₉₇ ▷ C₉₇

C₁ ◁ C₂²×C₄ ◁ C₂×#117

N	H ≅ G/N	type
1. C₂×C₄⋊#11715895	C₂	semidirect
3. #20260490	C₂	semidirect
68. C₉₇	#1798	semidirect

H ≅ G/N

type

1. C₂×C₄⋊#11715895

C₂

semidirect

3. #20260490

C₂

semidirect

68. C₉₇

#1798

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×C₂²⋊#11715895	2
#20260490	2
C₂×#117×C₉₇	2
#1798	97

subgroup

index

C₂×C₄⋊#11715895

C₂×C₂²⋊#11715895

#20260490

C₂×#117×C₉₇

#1798

Subgroups — #27141720

Conjugacy classes of subgroups

Complements

Maximal subgroups