Group #27138054 — subgroups

Subgroups — C₉₇×#2096 #27138054

Order 12416.

254 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 108 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×#106×C₉₇	C₂	1	2	yes	maximal
2	C₂×#103×C₉₇	C₂	1	2	yes	maximal
3	#391×C₉₇	C₂	1	1	yes	maximal, characteristic
4	#384×C₉₇	C₂	1	1	yes	maximal, characteristic
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	—	2	2	no	—
9	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
12	#106×C₉₇	—	2	4	no	—
13	#103×C₉₇	—	2	4	no	—
14	#95×C₉₇	—	2	2	no	—
15	C₂×Q₈×C₉₇	—	2	1	no	—
16	C₂×Q₈×C₉₇	—	2	2	no	—
17	C₂×Q₈×C₉₇	—	2	2	no	—
18	C₂×Q₈×C₉₇	Q₈	1	2	yes	—
19	C₂²×C₄×C₉₇	—	2	1	no	—
20	C₂²×C₄×C₉₇	—	2	1	no	—
21	C₂²×C₄×C₉₇	—	2	2	no	—
22	C₂²×C₄×C₉₇	D₄	1	2	yes	—
23	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₉₇	—	2	2	no	—
25	C₂×C₈×C₉₇	—	2	2	no	—
26	C₂×C₈×C₉₇	D₄	1	4	yes	—
27	C₄⋊C₄×C₉₇	—	2	2	no	—
28	C₄⋊C₄×C₉₇	—	2	2	no	—
29	C₄⋊C₄×C₉₇	—	2	2	no	—
30	C₄⋊C₄×C₉₇	—	4	4	no	—
31	C₄²×C₉₇	—	2	2	no	—
32	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₉₇	—	2	4	no	—
34	Q₈×C₉₇	—	4	2	no	—
35	C₂×C₄×C₉₇	—	2	2	no	—
36	C₂×C₄×C₉₇	—	2	2	no	—
37	C₂×C₄×C₉₇	—	2	2	no	—
38	C₂×C₄×C₉₇	—	4	1	no	—
39	C₂×C₄×C₉₇	—	4	1	no	—
40	C₂×C₄×C₉₇	—	4	2	no	—
41	C₂×C₄×C₉₇	—	4	1	no	—
42	C₂×C₄×C₉₇	—	4	4	no	—
43	C₂×C₄×C₉₇	—	4	2	no	—
44	C₂×C₄×C₉₇	SD₁₆	1	4	yes	—
45	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₉₇	#41	1	2	yes	—
48	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
49	C₈×C₉₇	—	2	4	no	—
50	C₂²×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₉₇	#123	1	1	yes	characteristic
52	C₂²×C₉₇	#127	1	1	yes	characteristic
53	C₂²×C₉₇	C₂×SD₁₆	1	2	yes	—
54	C₂²×C₉₇	#137	1	2	yes	—
55	C₄×C₉₇	—	2	4	no	—
56	C₄×C₉₇	—	4	2	no	—
57	C₄×C₉₇	—	8	2	no	—
58	C₄×C₉₇	—	8	1	no	—
59	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₉₇	#123	1	2	yes	—
61	C₂×C₉₇	#399	1	1	yes	characteristic
62	C₂×C₉₇	#475	1	2	yes	—
63	C₂×C₉₇	#483	1	2	yes	—
64	C₂×C₉₇	#492	1	1	yes	characteristic
65	C₂×C₉₇	#497	1	1	yes	characteristic
66	#2096	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₉₇	#2096	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×#106	C₂×C₉₇	1	2	yes	—
69	C₂×#103	C₂×C₉₇	1	2	yes	—
70	#391	C₂×C₉₇	1	1	yes	characteristic
71	#384	C₂×C₉₇	1	1	yes	characteristic
72	#336	C₂×C₉₇	1	1	yes	characteristic
73	C₂²×Q₈	C₂²×C₉₇	1	1	yes	characteristic
74	C₂²×C₈	C₂²×C₉₇	1	2	yes	—
75	C₂×C₄⋊C₄	—	2	2	no	—
76	C₂×C₄⋊C₄	C₂²×C₉₇	1	2	yes	—
77	C₂×C₄⋊C₄	C₂²×C₉₇	1	1	yes	characteristic
78	C₂×C₄²	C₂²×C₉₇	1	1	yes	characteristic
79	#106	—	2	4	no	—
80	#103	—	2	4	no	—
81	#95	—	2	2	no	—
82	C₂×Q₈	—	2	1	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	Q₈×C₉₇	1	2	yes	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	D₄×C₉₇	1	2	yes	—
90	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	D₄×C₉₇	1	4	yes	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	4	4	no	—
98	C₄²	—	2	2	no	—
99	C₂³	C₂×D₄×C₉₇	1	1	yes	characteristic
100	Q₈	—	2	4	no	—
101	Q₈	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	4	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	SD₁₆×C₉₇	1	4	yes	—
112	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
113	C₂×C₄	C₂×Q₈×C₉₇	1	1	yes	characteristic
114	C₂×C₄	#41×C₉₇	1	2	yes	—
115	C₂×C₄	#41×C₉₇	1	1	yes	characteristic
116	C₈	—	2	4	no	—
117	C₂²	C₂²⋊Q₈×C₉₇	1	1	yes	characteristic
118	C₂²	#123×C₉₇	1	1	yes	characteristic
119	C₂²	#127×C₉₇	1	1	yes	characteristic
120	C₂²	C₂×SD₁₆×C₉₇	1	2	yes	—
121	C₂²	#137×C₉₇	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	4	2	no	—
124	C₄	—	8	2	no	—
125	C₄	—	8	1	no	—
126	C₄	C₂²⋊Q₈×C₉₇	1	2	yes	—
127	C₄	#123×C₉₇	1	2	yes	—
128	C₂	#399×C₉₇	1	1	yes	characteristic
129	C₂	#475×C₉₇	1	2	yes	—
130	C₂	#483×C₉₇	1	2	yes	—
131	C₂	#492×C₉₇	1	1	yes	characteristic
132	C₂	#497×C₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#106×C₉₇

C₂

yes

maximal

C₂×#103×C₉₇

C₂

yes

maximal

#391×C₉₇

C₂

yes

maximal, characteristic

#384×C₉₇

C₂

yes

maximal, characteristic

#336×C₉₇

C₂

yes

maximal, characteristic

C₂²×Q₈×C₉₇

C₂²

yes

characteristic

C₂²×C₈×C₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#106×C₉₇

—

#103×C₉₇

—

#95×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

Q₈

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

D₄

yes

—

C₄⋊C₄×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

SD₁₆

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

#41

yes

characteristic

C₈×C₉₇

—

C₂²×C₉₇

C₂²⋊Q₈

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

C₂×SD₁₆

yes

—

C₂²×C₉₇

#137

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

#123

yes

—

C₂×C₉₇

#399

yes

characteristic

C₂×C₉₇

#475

yes

—

C₂×C₉₇

#483

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₉₇

#2096

yes

characteristic, direct product factor, Sylow

C₂×#106

C₂×C₉₇

yes

—

C₂×#103

C₂×C₉₇

yes

—

#391

C₂×C₉₇

yes

characteristic

#384

C₂×C₉₇

yes

characteristic

#336

C₂×C₉₇

yes

characteristic

C₂²×Q₈

C₂²×C₉₇

yes

characteristic

C₂²×C₈

C₂²×C₉₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₉₇

yes

—

C₂×C₄⋊C₄

C₂²×C₉₇

yes

characteristic

C₂×C₄²

C₂²×C₉₇

yes

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Q₈×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

D₄×C₉₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₂³

C₂×D₄×C₉₇

yes

characteristic

100

Q₈

—

101

Q₈

—

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

—

111

C₂×C₄

SD₁₆×C₉₇

yes

—

112

C₂×C₄

C₂×D₄×C₉₇

yes

—

113

C₂×C₄

C₂×Q₈×C₉₇

yes

characteristic

114

C₂×C₄

#41×C₉₇

yes

—

115

C₂×C₄

#41×C₉₇

yes

characteristic

116

C₈

—

117

C₂²

C₂²⋊Q₈×C₉₇

yes

characteristic

118

C₂²

#123×C₉₇

yes

characteristic

119

C₂²

#127×C₉₇

yes

characteristic

120

C₂²

C₂×SD₁₆×C₉₇

yes

—

121

C₂²

#137×C₉₇

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₂²⋊Q₈×C₉₇

yes

—

127

C₄

#123×C₉₇

yes

—

128

C₂

#399×C₉₇

yes

characteristic

129

C₂

#475×C₉₇

yes

—

130

C₂

#483×C₉₇

yes

—

131

C₂

#492×C₉₇

yes

characteristic

132

C₂

#497×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
66. #2096	C₉₇	direct (N × H)
67. C₉₇	#2096	direct (N × H)

H ≅ G/N

type

66. #2096

C₉₇

direct (N × H)

67. C₉₇

#2096

direct (N × H)

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

subgroup	index
C₂×#106×C₉₇	2
C₂×#103×C₉₇	2
#391×C₉₇	2
#384×C₉₇	2
#336×C₉₇	2
#2096	97

subgroup

index

C₂×#106×C₉₇

C₂×#103×C₉₇

#391×C₉₇

#384×C₉₇

#336×C₉₇

#2096

Subgroups — C₉₇×#2096 #27138054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C97×#2096 #27138054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#2096 #27138054