Group #27167380 — subgroups

Subgroups — C₁₀₁×#2026 #27167380

Order 12928.

258 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 100 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×#125×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#103×C₁₀₁	C₂	1	2	yes	maximal
3	#391×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#340×C₁₀₁	C₂	1	2	yes	maximal
5	#336×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
8	#125×C₁₀₁	—	2	4	no	—
9	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
10	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
13	#103×C₁₀₁	—	2	4	no	—
14	#95×C₁₀₁	—	2	2	no	—
15	C₂×Q₈×C₁₀₁	—	2	1	no	—
16	C₂×Q₈×C₁₀₁	—	2	2	no	—
17	C₂×Q₈×C₁₀₁	—	2	2	no	—
18	C₂×Q₈×C₁₀₁	Q₈	1	2	yes	—
19	C₂²×C₄×C₁₀₁	—	2	1	no	—
20	C₂²×C₄×C₁₀₁	—	2	2	no	—
21	C₂²×C₄×C₁₀₁	—	2	1	no	—
22	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
23	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₁₀₁	—	2	2	no	—
25	C₂×C₈×C₁₀₁	—	2	2	no	—
26	C₂×C₈×C₁₀₁	—	2	2	no	—
27	C₄⋊C₄×C₁₀₁	—	2	8	no	—
28	C₄⋊C₄×C₁₀₁	—	2	2	no	—
29	C₄⋊C₄×C₁₀₁	—	2	2	no	—
30	C₄⋊C₄×C₁₀₁	D₄	1	4	yes	—
31	C₄²×C₁₀₁	—	2	2	no	—
32	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₁₀₁	—	2	4	no	—
34	Q₈×C₁₀₁	—	4	2	no	—
35	C₂×C₄×C₁₀₁	—	2	2	no	—
36	C₂×C₄×C₁₀₁	—	2	4	no	—
37	C₂×C₄×C₁₀₁	—	2	2	no	—
38	C₂×C₄×C₁₀₁	—	2	2	no	—
39	C₂×C₄×C₁₀₁	—	2	2	no	—
40	C₂×C₄×C₁₀₁	—	4	1	no	—
41	C₂×C₄×C₁₀₁	—	4	1	no	—
42	C₂×C₄×C₁₀₁	—	4	2	no	—
43	C₂×C₄×C₁₀₁	—	4	2	no	—
44	C₂×C₄×C₁₀₁	—	4	1	no	—
45	C₂×C₄×C₁₀₁	—	4	2	no	—
46	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₀₁	#41	1	2	yes	—
49	C₂×C₄×C₁₀₁	#41	1	1	yes	characteristic
50	C₈×C₁₀₁	—	4	2	no	—
51	C₂²×C₁₀₁	#120	1	1	yes	characteristic
52	C₂²×C₁₀₁	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₁₀₁	#124	1	1	yes	characteristic
54	C₂²×C₁₀₁	#135	1	2	yes	—
55	C₂²×C₁₀₁	#137	1	2	yes	—
56	C₄×C₁₀₁	—	4	2	no	—
57	C₄×C₁₀₁	—	4	2	no	—
58	C₄×C₁₀₁	—	4	4	no	—
59	C₄×C₁₀₁	—	8	1	no	—
60	C₄×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₁₀₁	#124	1	2	yes	—
62	C₂×C₁₀₁	#393	1	1	yes	characteristic
63	C₂×C₁₀₁	#452	1	2	yes	—
64	C₂×C₁₀₁	#479	1	2	yes	—
65	C₂×C₁₀₁	#488	1	1	yes	characteristic
66	C₂×C₁₀₁	#491	1	1	yes	characteristic
67	#2026	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₁₀₁	#2026	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×#125	C₂×C₁₀₁	1	1	yes	characteristic
70	C₂×#103	C₂×C₁₀₁	1	2	yes	—
71	#391	C₂×C₁₀₁	1	1	yes	characteristic
72	#340	C₂×C₁₀₁	1	2	yes	—
73	#336	C₂×C₁₀₁	1	1	yes	characteristic
74	C₂²×Q₈	C₂²×C₁₀₁	1	1	yes	characteristic
75	C₂²×C₈	C₂²×C₁₀₁	1	2	yes	—
76	#125	—	2	4	no	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	C₂²×C₁₀₁	1	1	yes	characteristic
79	C₂×C₄⋊C₄	C₂²×C₁₀₁	1	2	yes	—
80	C₂×C₄²	C₂²×C₁₀₁	1	1	yes	characteristic
81	#103	—	2	4	no	—
82	#95	—	2	2	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	—	2	2	no	—
86	C₂×Q₈	Q₈×C₁₀₁	1	2	yes	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	D₄×C₁₀₁	1	2	yes	—
91	C₂²×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	D₄×C₁₀₁	1	4	yes	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×D₄×C₁₀₁	1	1	yes	characteristic
101	Q₈	—	2	4	no	—
102	Q₈	—	4	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
115	C₂×C₄	C₂×Q₈×C₁₀₁	1	1	yes	characteristic
116	C₂×C₄	#41×C₁₀₁	1	2	yes	—
117	C₂×C₄	#41×C₁₀₁	1	1	yes	characteristic
118	C₈	—	4	2	no	—
119	C₂²	#120×C₁₀₁	1	1	yes	characteristic
120	C₂²	C₂²⋊Q₈×C₁₀₁	1	1	yes	characteristic
121	C₂²	#124×C₁₀₁	1	1	yes	characteristic
122	C₂²	#135×C₁₀₁	1	2	yes	—
123	C₂²	#137×C₁₀₁	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	4	2	no	—
127	C₄	—	8	1	no	—
128	C₄	C₂²⋊Q₈×C₁₀₁	1	2	yes	—
129	C₄	#124×C₁₀₁	1	2	yes	—
130	C₂	#393×C₁₀₁	1	1	yes	characteristic
131	C₂	#452×C₁₀₁	1	2	yes	—
132	C₂	#479×C₁₀₁	1	2	yes	—
133	C₂	#488×C₁₀₁	1	1	yes	characteristic
134	C₂	#491×C₁₀₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#125×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#103×C₁₀₁

C₂

yes

maximal

#391×C₁₀₁

C₂

yes

maximal, characteristic

#340×C₁₀₁

C₂

yes

maximal

#336×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₀₁

C₂²

yes

characteristic

C₂²×C₈×C₁₀₁

C₂²

yes

—

#125×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

#103×C₁₀₁

—

#95×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

Q₈

yes

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

#41

yes

characteristic

C₈×C₁₀₁

—

C₂²×C₁₀₁

#120

yes

characteristic

C₂²×C₁₀₁

C₂²⋊Q₈

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

—

C₂²×C₁₀₁

#137

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂²⋊Q₈

yes

—

C₄×C₁₀₁

#124

yes

—

C₂×C₁₀₁

#393

yes

characteristic

C₂×C₁₀₁

#452

yes

—

C₂×C₁₀₁

#479

yes

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₁₀₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₁

#2026

yes

characteristic, direct product factor, Sylow

C₂×#125

C₂×C₁₀₁

yes

characteristic

C₂×#103

C₂×C₁₀₁

yes

—

#391

C₂×C₁₀₁

yes

characteristic

#340

C₂×C₁₀₁

yes

—

#336

C₂×C₁₀₁

yes

characteristic

C₂²×Q₈

C₂²×C₁₀₁

yes

characteristic

C₂²×C₈

C₂²×C₁₀₁

yes

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₀₁

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₀₁

yes

—

C₂×C₄²

C₂²×C₁₀₁

yes

characteristic

#103

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Q₈×C₁₀₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₀₁

yes

—

C₂²×C₄

C₂³×C₁₀₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₁₀₁

yes

—

C₄²

—

100

C₂³

C₂×D₄×C₁₀₁

yes

characteristic

101

Q₈

—

102

Q₈

—

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

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×D₄×C₁₀₁

yes

—

115

C₂×C₄

C₂×Q₈×C₁₀₁

yes

characteristic

116

C₂×C₄

#41×C₁₀₁

yes

—

117

C₂×C₄

#41×C₁₀₁

yes

characteristic

118

C₈

—

119

C₂²

#120×C₁₀₁

yes

characteristic

120

C₂²

C₂²⋊Q₈×C₁₀₁

yes

characteristic

121

C₂²

#124×C₁₀₁

yes

characteristic

122

C₂²

#135×C₁₀₁

yes

—

123

C₂²

#137×C₁₀₁

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

C₂²⋊Q₈×C₁₀₁

yes

—

129

C₄

#124×C₁₀₁

yes

—

130

C₂

#393×C₁₀₁

yes

characteristic

131

C₂

#452×C₁₀₁

yes

—

132

C₂

#479×C₁₀₁

yes

—

133

C₂

#488×C₁₀₁

yes

characteristic

134

C₂

#491×C₁₀₁

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₁ ◁ C₂×C₄²×C₁₀₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₀₁

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₁₀₁

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₁ ◁ C₂×C₄²×C₁₀₁ ◁ G

N	H ≅ G/N	type
67. #2026	C₁₀₁	direct (N × H)
68. C₁₀₁	#2026	direct (N × H)

H ≅ G/N

type

67. #2026

C₁₀₁

direct (N × H)

68. C₁₀₁

#2026

direct (N × H)

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

subgroup	index
C₂×#125×C₁₀₁	2
C₂×#103×C₁₀₁	2
#391×C₁₀₁	2
#340×C₁₀₁	2
#336×C₁₀₁	2
#2026	101

subgroup

index

C₂×#125×C₁₀₁

C₂×#103×C₁₀₁

#391×C₁₀₁

#340×C₁₀₁

#336×C₁₀₁

#2026

101

Subgroups — C₁₀₁×#2026 #27167380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C101×#2026 #27167380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#2026 #27167380