Group #10610 — subgroups

Subgroups — #10610

Order 256.

153 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 153 conjugacy classes of subgroups — too many to draw. Only its 57 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	#1997	C₂	1	1	yes	maximal, characteristic
2	#1950	C₂	1	1	yes	maximal, characteristic
3	#1570	C₂	1	1	yes	maximal, characteristic
4	#1565	C₂	1	1	yes	maximal, characteristic
5	#1549	C₂	1	1	yes	maximal, characteristic
6	C₂×#332	C₂	1	1	yes	maximal, characteristic
7	#1292	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	#431	C₂²	1	1	yes	characteristic
10	C₂×#108	—	2	1	no	—
11	#423	C₂²	1	1	yes	characteristic
12	#423	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×M₁₆	C₂²	1	1	yes	characteristic
16	#343	—	2	1	no	—
17	#343	—	2	1	no	—
18	#341	—	2	1	no	—
19	#335	—	2	1	no	—
20	#334	—	2	1	no	—
21	#334	—	2	2	no	—
22	#332	—	2	2	no	—
23	#332	—	2	1	no	—
24	#332	—	2	1	no	—
25	C₂²×Q₈	—	2	1	no	—
26	C₂×M₁₆	—	2	1	no	—
27	C₂×M₁₆	—	2	1	no	—
28	C₂×M₁₆	D₄	1	1	yes	characteristic
29	C₂×M₁₆	D₄	1	1	yes	characteristic
30	C₂²×C₈	—	2	1	no	—
31	C₂²×C₈	—	2	1	no	—
32	C₄⋊Q₈	—	2	1	no	—
33	C₄⋊Q₈	—	2	1	no	—
34	C₄⋊Q₈	—	2	1	no	—
35	C₄⋊Q₈	—	2	1	no	—
36	C₂×C₄⋊C₄	—	2	1	no	—
37	C₂×C₄⋊C₄	Q₈	1	2	yes	—
38	C₂×C₄²	C₂³	1	1	yes	characteristic
39	#108	—	4	2	no	—
40	C₄⋊C₈	—	2	1	no	—
41	C₄⋊C₈	—	2	1	no	—
42	C₄⋊C₈	—	2	1	no	—
43	C₄⋊C₈	—	2	1	no	—
44	C₄⋊C₈	—	2	1	no	—
45	C₄⋊C₈	—	2	1	no	—
46	C₄⋊C₈	—	2	1	no	—
47	C₄⋊C₈	D₄	1	2	yes	—
48	C₄⋊C₈	D₄	1	1	yes	characteristic
49	C₄⋊C₈	D₄	1	1	yes	characteristic
50	C₂²⋊C₈	—	2	1	no	—
51	#97	—	2	1	no	—
52	C₄×C₈	—	2	1	no	—
53	C₄×C₈	—	2	1	no	—
54	C₂×Q₈	—	4	1	no	—
55	C₂×Q₈	—	4	1	no	—
56	C₂×Q₈	—	8	1	no	—
57	C₂×Q₈	—	8	1	no	—
58	C₂²×C₄	—	2	2	no	—
59	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄	#41	1	1	yes	characteristic
62	M₁₆	—	2	1	no	—
63	M₁₆	—	2	1	no	—
64	M₁₆	—	2	1	no	—
65	M₁₆	—	4	1	no	—
66	M₁₆	—	4	1	no	—
67	M₁₆	—	4	1	no	—
68	M₁₆	—	4	1	no	—
69	M₁₆	SD₁₆	1	2	yes	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	4	1	no	—
76	C₂×C₈	—	4	1	no	—
77	C₂×C₈	—	4	1	no	—
78	C₂×C₈	—	4	1	no	—
79	C₂×C₈	—	4	1	no	—
80	C₂×C₈	—	8	1	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	4	1	no	—
84	C₄⋊C₄	—	4	1	no	—
85	C₄²	C₂×D₄	1	1	yes	characteristic
86	C₄²	C₂×D₄	1	1	yes	characteristic
87	C₄²	#41	1	1	yes	characteristic
88	C₄²	#41	1	1	yes	characteristic
89	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
90	Q₈	—	8	2	no	—
91	Q₈	—	8	1	no	—
92	Q₈	—	8	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	8	2	no	—
98	C₂×C₄	#121	1	1	yes	characteristic
99	C₂×C₄	#121	1	1	yes	characteristic
100	C₂×C₄	#121	1	1	yes	characteristic
101	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄	#126	1	1	yes	characteristic
104	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄	#135	1	1	yes	characteristic
106	C₂×C₄	#137	1	1	yes	characteristic
107	C₂×C₄	#137	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	—	4	2	no	—
110	C₈	—	4	1	no	—
111	C₈	—	4	1	no	—
112	C₈	—	4	1	no	—
113	C₈	—	8	1	no	—
114	C₈	—	8	1	no	—
115	C₈	—	8	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	#396	1	1	yes	characteristic
119	C₂²	#475	1	1	yes	characteristic
120	C₂²	#479	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	8	2	no	—
125	C₄	#461	1	1	yes	characteristic
126	C₄	#464	1	1	yes	characteristic
127	C₄	#465	1	1	yes	characteristic
128	C₄	#470	1	1	yes	characteristic
129	C₄	#471	1	1	yes	characteristic
130	C₄	#472	1	1	yes	characteristic
131	C₂	—	2	1	no	—
132	C₂	—	2	1	no	—
133	C₂	#2063	1	1	yes	characteristic
134	C₂	#2066	1	1	yes	characteristic
135	C₂	#2070	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1997

C₂

yes

maximal, characteristic

#1950

C₂

yes

maximal, characteristic

#1570

C₂

yes

maximal, characteristic

#1565

C₂

yes

maximal, characteristic

#1549

C₂

yes

maximal, characteristic

C₂×#332

C₂

yes

maximal, characteristic

#1292

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

C₂×#108

—

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₄×M₁₆

C₂²

yes

—

—

—

—

—

—

—

—

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

characteristic

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Q₈

yes

—

C₂×C₄²

C₂³

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

C₄⋊C₈

D₄

yes

characteristic

C₄⋊C₈

D₄

yes

characteristic

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

SD₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

C₄²

yes

C₄²

yes

C₂³

C₂²⋊Q₈

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

#121

yes

characteristic

101

C₂×C₄

C₂²⋊Q₈

yes

characteristic

102

C₂×C₄

C₂²⋊Q₈

yes

characteristic

103

C₂×C₄

#126

yes

characteristic

104

C₂×C₄

C₂×SD₁₆

yes

characteristic

105

C₂×C₄

#135

yes

characteristic

106

C₂×C₄

#137

yes

characteristic

107

C₂×C₄

#137

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₂²

—

117

C₂²

—

118

C₂²

#396

yes

characteristic

119

C₂²

#475

yes

characteristic

120

C₂²

#479

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#461

yes

characteristic

126

C₄

#464

yes

characteristic

127

C₄

#465

yes

characteristic

128

C₄

#470

yes

characteristic

129

C₄

#471

yes

characteristic

130

C₄

#472

yes

characteristic

131

C₂

—

132

C₂

—

133

C₂

#2063

yes

characteristic

134

C₂

#2066

yes

characteristic

135

C₂

#2070

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#396

commutator, Frattini, cosocle

C₂×C₄²

C₂³

prime radical

C₂³

C₂²⋊Q₈

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#1997	2
#1950	2
#1570	2
#1565	2
#1549	2
C₂×#332	2
#1292	2

subgroup

index

Subgroups — #10610

Conjugacy classes of subgroups

Maximal subgroups