Group #27137681 — subgroups

Subgroups — C₉₇×#1723 #27137681

Order 12416.

176 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 80 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	#500×C₉₇	C₂	1	1	yes	maximal, characteristic
2	#500×C₉₇	C₂	1	1	yes	maximal, characteristic
3	#498×C₉₇	C₂	1	1	yes	maximal, characteristic
4	#494×C₉₇	C₂	1	1	yes	maximal, characteristic
5	#487×C₉₇	C₂	1	1	yes	maximal, characteristic
6	#487×C₉₇	C₂	1	1	yes	maximal, characteristic
7	#322×C₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆×C₉₇	—	2	2	no	—
9	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
13	#107×C₉₇	—	2	2	no	—
14	#107×C₉₇	—	2	2	no	—
15	#106×C₉₇	—	2	2	no	—
16	#103×C₉₇	—	2	2	no	—
17	#103×C₉₇	—	2	2	no	—
18	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₉₇	—	2	1	no	—
22	C₂×Q₈×C₉₇	—	2	1	no	—
23	C₂×Q₈×C₉₇	—	2	1	no	—
24	C₂×Q₈×C₉₇	—	2	1	no	—
25	Q₁₆×C₉₇	—	4	4	no	—
26	C₂×C₈×C₉₇	D₄	1	2	yes	—
27	C₂×C₈×C₉₇	D₄	1	2	yes	—
28	C₂×C₈×C₉₇	D₄	1	2	yes	—
29	C₄⋊C₄×C₉₇	—	2	1	no	—
30	C₄⋊C₄×C₉₇	—	2	1	no	—
31	C₄⋊C₄×C₉₇	—	2	1	no	—
32	C₄⋊C₄×C₉₇	—	2	1	no	—
33	C₄⋊C₄×C₉₇	—	2	1	no	—
34	C₄⋊C₄×C₉₇	—	2	1	no	—
35	C₄⋊C₄×C₉₇	—	2	1	no	—
36	C₄⋊C₄×C₉₇	—	2	1	no	—
37	C₄²×C₉₇	C₂³	1	1	yes	characteristic
38	Q₈×C₉₇	—	4	2	no	—
39	Q₈×C₉₇	—	4	2	no	—
40	Q₈×C₉₇	—	4	2	no	—
41	Q₈×C₉₇	—	4	2	no	—
42	C₂×C₄×C₉₇	—	4	1	no	—
43	C₂×C₄×C₉₇	—	4	1	no	—
44	C₂×C₄×C₉₇	—	4	1	no	—
45	C₂×C₄×C₉₇	—	4	1	no	—
46	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
49	C₈×C₉₇	—	2	2	no	—
50	C₈×C₉₇	—	2	2	no	—
51	C₈×C₉₇	Q₁₆	1	4	yes	complemented
52	C₂²×C₉₇	#127	1	1	yes	characteristic
53	C₄×C₉₇	—	8	1	no	—
54	C₄×C₉₇	—	8	1	no	—
55	C₄×C₉₇	—	8	1	no	—
56	C₄×C₉₇	—	8	1	no	—
57	C₄×C₉₇	C₂×Q₁₆	1	2	yes	—
58	C₄×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
59	C₄×C₉₇	#137	1	2	yes	—
60	C₂×C₉₇	#494	1	1	yes	characteristic
61	C₂×C₉₇	#496	1	1	yes	characteristic
62	C₂×C₉₇	#497	1	1	yes	characteristic
63	#1723	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₉₇	#1723	1	1	yes	characteristic, direct product factor, Sylow
65	#500	C₂×C₉₇	1	1	yes	characteristic
66	#500	C₂×C₉₇	1	1	yes	characteristic
67	#498	C₂×C₉₇	1	1	yes	characteristic
68	#494	C₂×C₉₇	1	1	yes	characteristic
69	#487	C₂×C₉₇	1	1	yes	characteristic
70	#487	C₂×C₉₇	1	1	yes	characteristic
71	#322	C₂×C₉₇	1	1	yes	characteristic
72	C₂×Q₁₆	—	2	2	no	—
73	C₄⋊Q₈	C₂²×C₉₇	1	1	yes	characteristic
74	C₄⋊Q₈	C₂²×C₉₇	1	1	yes	characteristic
75	C₄⋊Q₈	C₂²×C₉₇	1	1	yes	characteristic
76	C₄⋊Q₈	C₂²×C₉₇	1	1	yes	characteristic
77	#107	—	2	2	no	—
78	#107	—	2	2	no	—
79	#106	—	2	2	no	—
80	#103	—	2	2	no	—
81	#103	—	2	2	no	—
82	C₄×C₈	C₂²×C₉₇	1	1	yes	characteristic
83	C₄×C₈	C₂²×C₉₇	1	1	yes	characteristic
84	C₄×C₈	C₂²×C₉₇	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	2	1	no	—
88	C₂×Q₈	—	2	1	no	—
89	Q₁₆	—	4	4	no	—
90	C₂×C₈	D₄×C₉₇	1	2	yes	—
91	C₂×C₈	D₄×C₉₇	1	2	yes	—
92	C₂×C₈	D₄×C₉₇	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄²	C₂³×C₉₇	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	4	2	no	—
105	Q₈	—	4	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
111	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
112	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	Q₁₆×C₉₇	1	4	yes	complemented
116	C₂²	#127×C₉₇	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	8	1	no	—
120	C₄	—	8	1	no	—
121	C₄	C₂×Q₁₆×C₉₇	1	2	yes	—
122	C₄	AGL(1,ℤ/8)×C₉₇	1	2	yes	—
123	C₄	#137×C₉₇	1	2	yes	—
124	C₂	#494×C₉₇	1	1	yes	characteristic
125	C₂	#496×C₉₇	1	1	yes	characteristic
126	C₂	#497×C₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#500×C₉₇

C₂

yes

maximal, characteristic

#500×C₉₇

C₂

yes

maximal, characteristic

#498×C₉₇

C₂

yes

maximal, characteristic

#494×C₉₇

C₂

yes

maximal, characteristic

#487×C₉₇

C₂

yes

maximal, characteristic

#487×C₉₇

C₂

yes

maximal, characteristic

#322×C₉₇

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₉₇

—

C₄⋊Q₈×C₉₇

C₂²

yes

characteristic

C₄⋊Q₈×C₉₇

C₂²

yes

characteristic

C₄⋊Q₈×C₉₇

C₂²

yes

characteristic

C₄⋊Q₈×C₉₇

C₂²

yes

characteristic

#107×C₉₇

—

#107×C₉₇

—

#106×C₉₇

—

#103×C₉₇

—

#103×C₉₇

—

C₄×C₈×C₉₇

C₂²

yes

characteristic

C₄×C₈×C₉₇

C₂²

yes

characteristic

C₄×C₈×C₉₇

C₂²

yes

characteristic

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

Q₁₆×C₉₇

—

C₂×C₈×C₉₇

D₄

yes

—

C₂×C₈×C₉₇

D₄

yes

—

C₂×C₈×C₉₇

D₄

yes

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

C₂³

yes

characteristic

Q₈×C₉₇

—

Q₈×C₉₇

—

Q₈×C₉₇

—

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₈×C₉₇

—

C₈×C₉₇

—

C₈×C₉₇

Q₁₆

yes

complemented

C₂²×C₉₇

#127

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×Q₁₆

yes

—

C₄×C₉₇

AGL(1,ℤ/8)

yes

—

C₄×C₉₇

#137

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₉₇

#1723

yes

characteristic, direct product factor, Sylow

#500

C₂×C₉₇

yes

characteristic

#500

C₂×C₉₇

yes

characteristic

#498

C₂×C₉₇

yes

characteristic

#494

C₂×C₉₇

yes

characteristic

#487

C₂×C₉₇

yes

characteristic

#487

C₂×C₉₇

yes

characteristic

#322

C₂×C₉₇

yes

characteristic

C₂×Q₁₆

—

C₄⋊Q₈

C₂²×C₉₇

yes

characteristic

C₄⋊Q₈

C₂²×C₉₇

yes

characteristic

C₄⋊Q₈

C₂²×C₉₇

yes

characteristic

C₄⋊Q₈

C₂²×C₉₇

yes

—

—

—

—

—

C₄×C₈

C₂²×C₉₇

yes

characteristic

C₄×C₈

C₂²×C₉₇

yes

characteristic

C₄×C₈

C₂²×C₉₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

C₂×C₈

D₄×C₉₇

yes

—

C₂×C₈

D₄×C₉₇

yes

—

C₂×C₈

D₄×C₉₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

C₂³×C₉₇

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂×D₄×C₉₇

yes

characteristic

111

C₂×C₄

C₂×D₄×C₉₇

yes

characteristic

112

C₂×C₄

C₂×D₄×C₉₇

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

Q₁₆×C₉₇

yes

complemented

116

C₂²

#127×C₉₇

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂×Q₁₆×C₉₇

yes

—

122

C₄

AGL(1,ℤ/8)×C₉₇

yes

—

123

C₄

#137×C₉₇

yes

—

124

C₂

#494×C₉₇

yes

characteristic

125

C₂

#496×C₉₇

yes

characteristic

126

C₂

#497×C₉₇

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₉₇	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₉₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₄²×C₉₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₉₇

#127

commutator, Frattini, cosocle

C₄²

C₂³×C₉₇

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₉₇ ◁ C₄²×C₉₇ ◁ G

N	H ≅ G/N	type
51. C₈×C₉₇	Q₁₆	semidirect
63. #1723	C₉₇	direct (N × H)
64. C₉₇	#1723	direct (N × H)
115. C₈	Q₁₆×C₉₇	semidirect

H ≅ G/N

type

51. C₈×C₉₇

Q₁₆

semidirect

63. #1723

C₉₇

direct (N × H)

64. C₉₇

#1723

direct (N × H)

115. C₈

Q₁₆×C₉₇

semidirect

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

subgroup	index
#500×C₉₇	2
#498×C₉₇	2
#494×C₉₇	2
#487×C₉₇	2
#322×C₉₇	2
#1723	97

subgroup

index

#500×C₉₇

#498×C₉₇

#494×C₉₇

#487×C₉₇

#322×C₉₇

#1723

Subgroups — C₉₇×#1723 #27137681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C97×#1723 #27137681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#1723 #27137681