Group #44396296 — subgroups

Subgroups — Q₈×C₂²⋊C₄×C₂₂₇ #44396296

Order 29056.

820 conjugacy classes of subgroups; only the 392 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 392 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	C₂³×Q₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₂₂₇	C₂	1	2	yes	maximal, complemented
3	#391×C₂₂₇	C₂	1	6	yes	maximal, complemented
4	#380×C₂₂₇	C₂	1	3	yes	maximal
5	C₄×C₂²⋊C₄×C₂₂₇	C₂	1	3	yes	maximal
6	C₂²×Q₈×C₂₂₇	—	2	8	no	—
7	C₂²×Q₈×C₂₂₇	C₄	1	8	yes	complemented
8	C₂²×Q₈×C₂₂₇	C₂²	1	3	yes	—
9	C₂²×Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₂₇	C₂²	1	3	yes	—
11	C₄×Q₈×C₂₂₇	—	2	8	no	—
12	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	6	yes	—
13	C₂×C₂²⋊C₄×C₂₂₇	C₂²	1	3	yes	—
14	C₂×C₂²⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₂₂₇	C₂²	1	6	yes	—
16	#95×C₂₂₇	C₂²	1	6	yes	—
17	#95×C₂₂₇	C₂²	1	6	yes	—
18	C₂⁴×C₂₂₇	C₂³	1	1	yes	characteristic
19	C₂×Q₈×C₂₂₇	—	2	32	no	—
20	C₂×Q₈×C₂₂₇	—	2	12	no	—
21	C₂×Q₈×C₂₂₇	C₂×C₄	1	12	yes	—
22	C₂×Q₈×C₂₂₇	C₂×C₄	1	4	yes	—
23	C₂×Q₈×C₂₂₇	D₄	1	8	yes	—
24	C₂²×C₄×C₂₂₇	—	2	12	no	—
25	C₂²×C₄×C₂₂₇	C₂×C₄	1	12	yes	—
26	C₂²×C₄×C₂₂₇	C₂³	1	6	yes	—
27	C₂²×C₄×C₂₂₇	C₂³	1	3	yes	—
28	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
29	C₂²×C₄×C₂₂₇	C₂³	1	3	yes	—
30	C₄⋊C₄×C₂₂₇	—	2	12	no	—
31	C₂²⋊C₄×C₂₂₇	—	2	6	no	—
32	C₂²⋊C₄×C₂₂₇	Q₈	1	4	yes	direct product factor
33	C₄²×C₂₂₇	—	2	12	no	—
34	C₂³×C₂₂₇	—	2	2	no	—
35	C₂³×C₂₂₇	—	2	2	no	—
36	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
37	C₂³×C₂₂₇	C₂×Q₈	1	2	yes	—
38	C₂³×C₂₂₇	#41	1	2	yes	—
39	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
40	Q₈×C₂₂₇	—	2	24	no	—
41	Q₈×C₂₂₇	C₂²⋊C₄	1	16	yes	direct product factor
42	C₂×C₄×C₂₂₇	—	2	2	no	—
43	C₂×C₄×C₂₂₇	—	2	6	no	—
44	C₂×C₄×C₂₂₇	—	2	2	no	—
45	C₂×C₄×C₂₂₇	—	2	24	no	—
46	C₂×C₄×C₂₂₇	—	2	6	no	—
47	C₂×C₄×C₂₂₇	—	2	6	no	—
48	C₂×C₄×C₂₂₇	—	2	6	no	—
49	C₂×C₄×C₂₂₇	C₂²×C₄	1	6	yes	—
50	C₂×C₄×C₂₂₇	C₂²×C₄	1	6	yes	—
51	C₂×C₄×C₂₂₇	C₂×D₄	1	12	yes	—
52	C₂×C₄×C₂₂₇	C₂×Q₈	1	4	yes	complemented
53	C₂²×C₂₂₇	—	2	2	no	—
54	C₂²×C₂₂₇	—	2	2	no	—
55	C₂²×C₂₂₇	—	2	4	no	—
56	C₂²×C₂₂₇	—	2	4	no	—
57	C₂²×C₂₂₇	C₄×Q₈	1	4	yes	complemented
58	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
59	C₂²×C₂₂₇	C₂²×D₄	1	2	yes	—
60	C₂²×C₂₂₇	C₂²×Q₈	1	1	yes	characteristic
61	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
62	C₂²×C₂₂₇	2₋¹⁺⁴	1	2	yes	—
63	C₄×C₂₂₇	—	2	6	no	—
64	C₄×C₂₂₇	—	2	4	no	—
65	C₄×C₂₂₇	—	4	6	no	—
66	C₄×C₂₂₇	C₂×C₂²⋊C₄	1	12	yes	—
67	C₂×C₂₂₇	—	2	4	no	—
68	C₂×C₂₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₂₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
70	C₂×C₂₂₇	#519	1	1	yes	characteristic
71	C₂×C₂₂₇	#548	1	2	yes	—
72	C₂×C₂₂₇	D₄×Q₈	1	2	yes	—
73	C₂₂₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
74	Q₈×C₂²⋊C₄	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×Q₈	C₂×C₂₂₇	1	1	yes	characteristic
76	C₂×C₄×Q₈	C₂×C₂₂₇	1	2	yes	complemented
77	#391	C₂×C₂₂₇	1	6	yes	complemented
78	#380	C₂×C₂₂₇	1	3	yes	—
79	C₄×C₂²⋊C₄	C₂×C₂₂₇	1	3	yes	—
80	C₂²×Q₈	—	2	8	no	—
81	C₂²×Q₈	C₄×C₂₂₇	1	8	yes	complemented
82	C₂²×Q₈	C₂²×C₂₂₇	1	3	yes	—
83	C₂²×Q₈	C₂²×C₂₂₇	1	1	yes	characteristic
84	C₂³×C₄	C₂²×C₂₂₇	1	3	yes	—
85	C₄×Q₈	—	2	8	no	—
86	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	6	yes	—
87	C₂×C₂²⋊C₄	C₂²×C₂₂₇	1	3	yes	—
88	C₂×C₂²⋊C₄	C₂²×C₂₂₇	1	1	yes	characteristic
89	C₂×C₄²	C₂²×C₂₂₇	1	6	yes	—
90	#95	C₂²×C₂₂₇	1	6	yes	—
91	#95	C₂²×C₂₂₇	1	6	yes	—
92	C₂⁴	C₂³×C₂₂₇	1	1	yes	characteristic
93	C₂×Q₈	—	2	32	no	—
94	C₂×Q₈	—	2	12	no	—
95	C₂×Q₈	C₂×C₄×C₂₂₇	1	12	yes	—
96	C₂×Q₈	C₂×C₄×C₂₂₇	1	4	yes	—
97	C₂×Q₈	D₄×C₂₂₇	1	8	yes	—
98	C₂²×C₄	—	2	12	no	—
99	C₂²×C₄	C₂×C₄×C₂₂₇	1	12	yes	—
100	C₂²×C₄	C₂³×C₂₂₇	1	6	yes	—
101	C₂²×C₄	C₂³×C₂₂₇	1	3	yes	—
102	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
103	C₂²×C₄	C₂³×C₂₂₇	1	3	yes	—
104	C₄⋊C₄	—	2	12	no	—
105	C₂²⋊C₄	—	2	6	no	—
106	C₂²⋊C₄	Q₈×C₂₂₇	1	4	yes	direct product factor
107	C₄²	—	2	12	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	C₂²×C₄×C₂₂₇	1	2	yes	—
111	C₂³	C₂×Q₈×C₂₂₇	1	2	yes	—
112	C₂³	#41×C₂₂₇	1	2	yes	—
113	C₂³	C₂⁴×C₂₂₇	1	1	yes	characteristic
114	Q₈	—	2	24	no	—
115	Q₈	C₂²⋊C₄×C₂₂₇	1	16	yes	direct product factor
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	6	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	24	no	—
120	C₂×C₄	—	2	6	no	—
121	C₂×C₄	—	2	6	no	—
122	C₂×C₄	—	2	6	no	—
123	C₂×C₄	C₂²×C₄×C₂₂₇	1	6	yes	—
124	C₂×C₄	C₂²×C₄×C₂₂₇	1	6	yes	—
125	C₂×C₄	C₂×D₄×C₂₂₇	1	12	yes	—
126	C₂×C₄	C₂×Q₈×C₂₂₇	1	4	yes	complemented
127	C₂²	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	4	no	—
131	C₂²	C₄×Q₈×C₂₂₇	1	4	yes	complemented
132	C₂²	C₂³×C₄×C₂₂₇	1	1	yes	characteristic
133	C₂²	C₂²×D₄×C₂₂₇	1	2	yes	—
134	C₂²	C₂²×Q₈×C₂₂₇	1	1	yes	characteristic
135	C₂²	C₂×#41×C₂₂₇	1	1	yes	characteristic
136	C₂²	2₋¹⁺⁴×C₂₂₇	1	2	yes	—
137	C₄	—	2	6	no	—
138	C₄	—	2	4	no	—
139	C₄	—	4	6	no	—
140	C₄	C₂×C₂²⋊C₄×C₂₂₇	1	12	yes	—
141	C₂	—	2	4	no	—
142	C₂	C₂²×C₂²⋊C₄×C₂₂₇	1	1	yes	characteristic
143	C₂	C₂×C₄×Q₈×C₂₂₇	1	1	yes	characteristic
144	C₂	#519×C₂₂₇	1	1	yes	characteristic
145	C₂	#548×C₂₂₇	1	2	yes	—
146	C₂	D₄×Q₈×C₂₂₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Q₈×C₂₂₇

C₂

yes

maximal, characteristic

C₂×C₄×Q₈×C₂₂₇

C₂

yes

maximal, complemented

#391×C₂₂₇

C₂

yes

maximal, complemented

#380×C₂₂₇

C₂

yes

maximal

C₄×C₂²⋊C₄×C₂₂₇

C₂

yes

maximal

C₂²×Q₈×C₂₂₇

—

C₂²×Q₈×C₂₂₇

C₄

yes

complemented

C₂²×Q₈×C₂₂₇

C₂²

yes

—

C₂²×Q₈×C₂₂₇

C₂²

yes

characteristic

C₂³×C₄×C₂₂₇

C₂²

yes

—

C₄×Q₈×C₂₂₇

—

C₂×C₄⋊C₄×C₂₂₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₂₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₂₇

C₂²

yes

characteristic

C₂×C₄²×C₂₂₇

C₂²

yes

—

#95×C₂₂₇

C₂²

yes

—

#95×C₂₂₇

C₂²

yes

—

C₂⁴×C₂₂₇

C₂³

yes

characteristic

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

C₂×C₄

yes

—

C₂×Q₈×C₂₂₇

C₂×C₄

yes

—

C₂×Q₈×C₂₂₇

D₄

yes

—

C₂²×C₄×C₂₂₇

—

C₂²×C₄×C₂₂₇

C₂×C₄

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

—

C₄⋊C₄×C₂₂₇

—

C₂²⋊C₄×C₂₂₇

—

C₂²⋊C₄×C₂₂₇

Q₈

yes

direct product factor

C₄²×C₂₂₇

—

C₂³×C₂₂₇

—

C₂³×C₂₂₇

—

C₂³×C₂₂₇

C₂²×C₄

yes

—

C₂³×C₂₂₇

C₂×Q₈

yes

—

C₂³×C₂₂₇

#41

yes

—

C₂³×C₂₂₇

C₂⁴

yes

characteristic

Q₈×C₂₂₇

—

Q₈×C₂₂₇

C₂²⋊C₄

yes

direct product factor

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

C₂×C₄×C₂₂₇

C₂×Q₈

yes

complemented

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

C₄×Q₈

yes

complemented

C₂²×C₂₂₇

C₂³×C₄

yes

characteristic

C₂²×C₂₂₇

C₂²×D₄

yes

—

C₂²×C₂₂₇

C₂²×Q₈

yes

characteristic

C₂²×C₂₂₇

C₂×#41

yes

characteristic

C₂²×C₂₂₇

2₋¹⁺⁴

yes

—

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

C₂×C₂²⋊C₄

yes

—

C₂×C₂₂₇

—

C₂×C₂₂₇

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₂₂₇

C₂×C₄×Q₈

yes

C₂×C₂₂₇

yes

C₂×C₂₂₇

yes

—

C₂×C₂₂₇

D₄×Q₈

yes

—

C₂₂₇

Q₈×C₂²⋊C₄

yes

characteristic, direct product factor, Sylow

Q₈×C₂²⋊C₄

C₂₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₂³×Q₈

C₂×C₂₂₇

yes

characteristic

C₂×C₄×Q₈

C₂×C₂₂₇

yes

complemented

#391

C₂×C₂₂₇

yes

complemented

#380

C₂×C₂₂₇

yes

—

C₄×C₂²⋊C₄

C₂×C₂₂₇

yes

—

C₂²×Q₈

—

C₂²×Q₈

C₄×C₂₂₇

yes

complemented

C₂²×Q₈

C₂²×C₂₂₇

yes

—

C₂²×Q₈

C₂²×C₂₂₇

yes

characteristic

C₂³×C₄

C₂²×C₂₂₇

yes

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₂²×C₂₂₇

yes

—

C₂×C₂²⋊C₄

C₂²×C₂₂₇

yes

—

C₂×C₂²⋊C₄

C₂²×C₂₂₇

yes

characteristic

C₂×C₄²

C₂²×C₂₂₇

yes

—

#95

C₂²×C₂₂₇

yes

—

#95

C₂²×C₂₂₇

yes

—

C₂⁴

C₂³×C₂₂₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₂₂₇

yes

—

C₂×Q₈

C₂×C₄×C₂₂₇

yes

—

C₂×Q₈

D₄×C₂₂₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₂₂₇

yes

—

100

C₂²×C₄

C₂³×C₂₂₇

yes

—

101

C₂²×C₄

C₂³×C₂₂₇

yes

—

102

C₂²×C₄

C₂³×C₂₂₇

yes

—

103

C₂²×C₄

C₂³×C₂₂₇

yes

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

Q₈×C₂₂₇

yes

direct product factor

107

C₄²

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₂²×C₄×C₂₂₇

yes

—

111

C₂³

C₂×Q₈×C₂₂₇

yes

—

112

C₂³

#41×C₂₂₇

yes

—

113

C₂³

C₂⁴×C₂₂₇

yes

characteristic

114

Q₈

—

115

Q₈

C₂²⋊C₄×C₂₂₇

yes

direct product factor

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂²×C₄×C₂₂₇

yes

—

124

C₂×C₄

C₂²×C₄×C₂₂₇

yes

—

125

C₂×C₄

C₂×D₄×C₂₂₇

yes

—

126

C₂×C₄

C₂×Q₈×C₂₂₇

yes

complemented

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₄×Q₈×C₂₂₇

yes

complemented

132

C₂²

C₂³×C₄×C₂₂₇

yes

characteristic

133

C₂²

C₂²×D₄×C₂₂₇

yes

—

134

C₂²

C₂²×Q₈×C₂₂₇

yes

characteristic

135

C₂²

C₂×#41×C₂₂₇

yes

characteristic

136

C₂²

2₋¹⁺⁴×C₂₂₇

yes

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

C₂×C₂²⋊C₄×C₂₂₇

yes

—

141

C₂

—

142

C₂

C₂²×C₂²⋊C₄×C₂₂₇

yes

characteristic

143

C₂

C₂×C₄×Q₈×C₂₂₇

yes

characteristic

144

C₂

#519×C₂₂₇

yes

characteristic

145

C₂

#548×C₂₂₇

yes

—

146

C₂

D₄×Q₈×C₂₂₇

yes

—

	subgroup	G/H
center, socle	C₂³×C₂₂₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₂₇
Frattini, cosocle	C₂³	C₂⁴×C₂₂₇
prime radical	C₂⁴×C₂₂₇	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₂₂₇

C₂⁴

commutator

C₂²

C₂³×C₄×C₂₂₇

Frattini, cosocle

C₂³

C₂⁴×C₂₂₇

prime radical

C₂⁴×C₂₂₇

C₂³

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₂₂₇ ◁ G

N	H ≅ G/N	type
2. C₂×C₄×Q₈×C₂₂₇	C₂	semidirect
3. #391×C₂₂₇	C₂	semidirect
7. C₂²×Q₈×C₂₂₇	C₄	semidirect
32. C₂²⋊C₄×C₂₂₇	Q₈	direct (N × H)
41. Q₈×C₂₂₇	C₂²⋊C₄	direct (N × H)
52. C₂×C₄×C₂₂₇	C₂×Q₈	semidirect
57. C₂²×C₂₂₇	C₄×Q₈	semidirect
73. C₂₂₇	Q₈×C₂²⋊C₄	direct (N × H)
74. Q₈×C₂²⋊C₄	C₂₂₇	direct (N × H)
76. C₂×C₄×Q₈	C₂×C₂₂₇	semidirect
77. #391	C₂×C₂₂₇	semidirect
81. C₂²×Q₈	C₄×C₂₂₇	semidirect
106. C₂²⋊C₄	Q₈×C₂₂₇	direct (N × H)
115. Q₈	C₂²⋊C₄×C₂₂₇	direct (N × H)
126. C₂×C₄	C₂×Q₈×C₂₂₇	semidirect
131. C₂²	C₄×Q₈×C₂₂₇	semidirect

H ≅ G/N

type

2. C₂×C₄×Q₈×C₂₂₇

C₂

semidirect

3. #391×C₂₂₇

C₂

semidirect

7. C₂²×Q₈×C₂₂₇

C₄

semidirect

32. C₂²⋊C₄×C₂₂₇

Q₈

direct (N × H)

41. Q₈×C₂₂₇

C₂²⋊C₄

direct (N × H)

52. C₂×C₄×C₂₂₇

C₂×Q₈

semidirect

57. C₂²×C₂₂₇

C₄×Q₈

semidirect

73. C₂₂₇

Q₈×C₂²⋊C₄

direct (N × H)

74. Q₈×C₂²⋊C₄

C₂₂₇

direct (N × H)

76. C₂×C₄×Q₈

C₂×C₂₂₇

semidirect

77. #391

C₂×C₂₂₇

semidirect

81. C₂²×Q₈

C₄×C₂₂₇

semidirect

106. C₂²⋊C₄

Q₈×C₂₂₇

direct (N × H)

115. Q₈

C₂²⋊C₄×C₂₂₇

direct (N × H)

126. C₂×C₄

C₂×Q₈×C₂₂₇

semidirect

131. C₂²

C₄×Q₈×C₂₂₇

semidirect

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

subgroup	index
C₂³×Q₈×C₂₂₇	2
C₂×C₄×Q₈×C₂₂₇	2
#391×C₂₂₇	2
#380×C₂₂₇	2
C₄×C₂²⋊C₄×C₂₂₇	2
Q₈×C₂²⋊C₄	227

subgroup

index

C₂³×Q₈×C₂₂₇

C₂×C₄×Q₈×C₂₂₇

#391×C₂₂₇

#380×C₂₂₇

C₄×C₂²⋊C₄×C₂₂₇

Q₈×C₂²⋊C₄

227

Subgroups — Q₈×C₂²⋊C₄×C₂₂₇ #44396296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×C22⋊C4×C227 #44396296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₂²⋊C₄×C₂₂₇ #44396296