Group #27138032 — subgroups

Subgroups — C₉₇×#2074 #27138032

Order 12416.

222 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 92 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×#103×C₉₇	C₂	1	2	yes	maximal
2	#391×C₉₇	C₂	1	1	yes	maximal, characteristic
3	#382×C₉₇	C₂	1	1	yes	maximal, characteristic
4	#340×C₉₇	C₂	1	2	yes	maximal
5	#336×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
11	#103×C₉₇	—	2	4	no	—
12	#95×C₉₇	—	2	2	no	—
13	#95×C₉₇	—	2	2	no	—
14	C₂×Q₈×C₉₇	—	2	1	no	—
15	C₂×Q₈×C₉₇	—	2	2	no	—
16	C₂×Q₈×C₉₇	—	2	2	no	—
17	C₂×Q₈×C₉₇	D₄	1	2	yes	—
18	C₂²×C₄×C₉₇	—	2	1	no	—
19	C₂²×C₄×C₉₇	—	2	2	no	—
20	C₂²×C₄×C₉₇	—	2	1	no	—
21	C₂²×C₄×C₉₇	D₄	1	2	yes	—
22	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₉₇	—	2	2	no	—
24	C₂×C₈×C₉₇	—	2	2	no	—
25	C₂×C₈×C₉₇	—	2	2	no	—
26	C₄⋊C₄×C₉₇	—	2	2	no	—
27	C₄⋊C₄×C₉₇	—	2	2	no	—
28	C₄⋊C₄×C₉₇	—	2	2	no	—
29	C₄²×C₉₇	—	2	2	no	—
30	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₉₇	—	2	4	no	—
32	Q₈×C₉₇	—	4	2	no	—
33	C₂×C₄×C₉₇	—	2	2	no	—
34	C₂×C₄×C₉₇	—	2	2	no	—
35	C₂×C₄×C₉₇	—	2	2	no	—
36	C₂×C₄×C₉₇	—	2	2	no	—
37	C₂×C₄×C₉₇	—	4	1	no	—
38	C₂×C₄×C₉₇	—	4	1	no	—
39	C₂×C₄×C₉₇	—	4	2	no	—
40	C₂×C₄×C₉₇	—	4	1	no	—
41	C₂×C₄×C₉₇	—	4	2	no	—
42	C₂×C₄×C₉₇	—	4	2	no	—
43	C₂×C₄×C₉₇	—	4	2	no	—
44	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₉₇	#41	1	2	yes	—
46	C₂×C₄×C₉₇	#41	1	2	yes	—
47	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
48	C₈×C₉₇	—	4	2	no	—
49	C₂²×C₉₇	#121	1	1	yes	characteristic
50	C₂²×C₉₇	#123	1	1	yes	characteristic
51	C₂²×C₉₇	#124	1	1	yes	characteristic
52	C₂²×C₉₇	#135	1	2	yes	—
53	C₂²×C₉₇	#137	1	2	yes	—
54	C₄×C₉₇	—	4	2	no	—
55	C₄×C₉₇	—	4	2	no	—
56	C₄×C₉₇	—	8	2	no	—
57	C₄×C₉₇	—	8	1	no	—
58	C₄×C₉₇	#123	1	2	yes	—
59	C₄×C₉₇	#126	1	2	yes	—
60	C₂×C₉₇	#397	1	1	yes	characteristic
61	C₂×C₉₇	#464	1	2	yes	—
62	C₂×C₉₇	#485	1	2	yes	—
63	C₂×C₉₇	#488	1	1	yes	characteristic
64	C₂×C₉₇	#491	1	1	yes	characteristic
65	#2074	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₉₇	#2074	1	1	yes	characteristic, direct product factor, Sylow
67	C₂×#103	C₂×C₉₇	1	2	yes	—
68	#391	C₂×C₉₇	1	1	yes	characteristic
69	#382	C₂×C₉₇	1	1	yes	characteristic
70	#340	C₂×C₉₇	1	2	yes	—
71	#336	C₂×C₉₇	1	1	yes	characteristic
72	C₂²×Q₈	C₂²×C₉₇	1	1	yes	characteristic
73	C₂²×C₈	C₂²×C₉₇	1	2	yes	—
74	C₂×C₄⋊C₄	C₂²×C₉₇	1	1	yes	characteristic
75	C₂×C₄⋊C₄	C₂²×C₉₇	1	2	yes	—
76	C₂×C₄²	C₂²×C₉₇	1	1	yes	characteristic
77	#103	—	2	4	no	—
78	#95	—	2	2	no	—
79	#95	—	2	2	no	—
80	C₂×Q₈	—	2	1	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	D₄×C₉₇	1	2	yes	—
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	D₄×C₉₇	1	2	yes	—
88	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄²	—	2	2	no	—
96	C₂³	C₂×D₄×C₉₇	1	1	yes	characteristic
97	Q₈	—	2	4	no	—
98	Q₈	—	4	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
111	C₂×C₄	#41×C₉₇	1	2	yes	—
112	C₂×C₄	#41×C₉₇	1	2	yes	—
113	C₂×C₄	#41×C₉₇	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₂²	#121×C₉₇	1	1	yes	characteristic
116	C₂²	#123×C₉₇	1	1	yes	characteristic
117	C₂²	#124×C₉₇	1	1	yes	characteristic
118	C₂²	#135×C₉₇	1	2	yes	—
119	C₂²	#137×C₉₇	1	2	yes	—
120	C₄	—	4	2	no	—
121	C₄	—	4	2	no	—
122	C₄	—	8	1	no	—
123	C₄	—	8	2	no	—
124	C₄	#123×C₉₇	1	2	yes	—
125	C₄	#126×C₉₇	1	2	yes	—
126	C₂	#397×C₉₇	1	1	yes	characteristic
127	C₂	#464×C₉₇	1	2	yes	—
128	C₂	#485×C₉₇	1	2	yes	—
129	C₂	#488×C₉₇	1	1	yes	characteristic
130	C₂	#491×C₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#103×C₉₇

C₂

yes

maximal

#391×C₉₇

C₂

yes

maximal, characteristic

#382×C₉₇

C₂

yes

maximal, characteristic

#340×C₉₇

C₂

yes

maximal

#336×C₉₇

C₂

yes

maximal, characteristic

C₂²×Q₈×C₉₇

C₂²

yes

characteristic

C₂²×C₈×C₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#103×C₉₇

—

#95×C₉₇

—

#95×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

Q₈×C₉₇

—

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

#41

yes

characteristic

C₈×C₉₇

—

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

—

C₂²×C₉₇

#137

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#123

yes

—

C₄×C₉₇

#126

yes

—

C₂×C₉₇

#397

yes

characteristic

C₂×C₉₇

#464

yes

—

C₂×C₉₇

#485

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₉₇

#2074

yes

characteristic, direct product factor, Sylow

C₂×#103

C₂×C₉₇

yes

—

#391

C₂×C₉₇

yes

characteristic

#382

C₂×C₉₇

yes

characteristic

#340

C₂×C₉₇

yes

—

#336

C₂×C₉₇

yes

characteristic

C₂²×Q₈

C₂²×C₉₇

yes

characteristic

C₂²×C₈

C₂²×C₉₇

yes

—

C₂×C₄⋊C₄

C₂²×C₉₇

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₉₇

yes

—

C₂×C₄²

C₂²×C₉₇

yes

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₄×C₉₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₉₇

yes

—

C₂²×C₄

C₂³×C₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₂³

C₂×D₄×C₉₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂×D₄×C₉₇

yes

characteristic

111

C₂×C₄

#41×C₉₇

yes

—

112

C₂×C₄

#41×C₉₇

yes

—

113

C₂×C₄

#41×C₉₇

yes

characteristic

114

C₈

—

115

C₂²

#121×C₉₇

yes

characteristic

116

C₂²

#123×C₉₇

yes

characteristic

117

C₂²

#124×C₉₇

yes

characteristic

118

C₂²

#135×C₉₇

yes

—

119

C₂²

#137×C₉₇

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#123×C₉₇

yes

—

125

C₄

#126×C₉₇

yes

—

126

C₂

#397×C₉₇

yes

characteristic

127

C₂

#464×C₉₇

yes

—

128

C₂

#485×C₉₇

yes

—

129

C₂

#488×C₉₇

yes

characteristic

130

C₂

#491×C₉₇

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₉₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₉₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉₇ ◁ C₂×C₄²×C₉₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₉₇

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₉₇

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₉₇ ◁ C₂×C₄²×C₉₇ ◁ G

N	H ≅ G/N	type
65. #2074	C₉₇	direct (N × H)
66. C₉₇	#2074	direct (N × H)

H ≅ G/N

type

65. #2074

C₉₇

direct (N × H)

66. C₉₇

#2074

direct (N × H)

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

subgroup	index
C₂×#103×C₉₇	2
#391×C₉₇	2
#382×C₉₇	2
#340×C₉₇	2
#336×C₉₇	2
#2074	97

subgroup

index

C₂×#103×C₉₇

#391×C₉₇

#382×C₉₇

#340×C₉₇

#336×C₉₇

#2074

Subgroups — C₉₇×#2074 #27138032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C97×#2074 #27138032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#2074 #27138032