Group #21425777 — subgroups

Subgroups — C₂×#16837201 #21425777

Order 7552.

240 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 77 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×#14273646	C₂	1	2	yes	maximal
2	#16837201	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#11745664	—	2	2	no	—
5	#14273646	—	2	2	no	—
6	#14273646	C₂²	1	4	yes	—
7	C₂²×Q₈×C₅₉	C₄	1	1	yes	characteristic
8	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
9	C₄⋊Q₈×C₅₉	—	2	2	no	—
10	C₄⋊Q₈×C₅₉	C₄	1	2	yes	—
11	C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
13	C₂×C₄²×C₅₉	C₄	1	1	yes	characteristic
14	C₂×Q₈×C₅₉	—	2	1	no	—
15	C₂×Q₈×C₅₉	—	4	1	no	—
16	C₂×Q₈×C₅₉	—	4	1	no	—
17	C₂×Q₈×C₅₉	—	4	1	no	—
18	C₂×Q₈×C₅₉	—	4	1	no	—
19	C₂×Q₈×C₅₉	C₂×C₄	1	2	yes	—
20	C₂×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂×Q₈×C₅₉	D₄	1	2	yes	—
22	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₅₉	—	2	1	no	—
24	C₂²×C₄×C₅₉	—	2	1	no	—
25	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₉	D₄	1	2	yes	—
27	C₄⋊C₄×C₅₉	—	2	4	no	—
28	C₄⋊C₄×C₅₉	—	2	4	no	—
29	C₄²×C₅₉	—	2	1	no	—
30	C₄²×C₅₉	C₂×C₄	1	2	yes	—
31	#11745664	—	2	4	no	—
32	#11745664	—	4	2	no	—
33	C₂×C₅₉⋊C₈	—	4	2	no	—
34	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
35	Q₈×C₅₉	—	4	1	no	—
36	Q₈×C₅₉	—	4	2	no	—
37	Q₈×C₅₉	—	4	1	no	—
38	Q₈×C₅₉	—	4	2	no	—
39	Q₈×C₅₉	—	4	2	no	—
40	C₂×C₄×C₅₉	—	2	4	no	—
41	C₂×C₄×C₅₉	—	2	2	no	—
42	C₂×C₄×C₅₉	—	2	2	no	—
43	C₂×C₄×C₅₉	—	2	1	no	—
44	C₂×C₄×C₅₉	—	2	1	no	—
45	C₂×C₄×C₅₉	—	2	2	no	—
46	C₂×C₄×C₅₉	—	2	2	no	—
47	C₂×C₄×C₅₉	—	4	1	no	—
48	C₂×C₄×C₅₉	—	4	1	no	—
49	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
53	C₅₉⋊C₈	—	4	4	no	—
54	C₂²×C₅₉	—	2	2	no	—
55	C₂²×C₅₉	#99	1	1	yes	characteristic
56	C₂²×C₅₉	#99	1	1	yes	characteristic
57	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₅₉	—	2	4	no	—
59	C₄×C₅₉	—	2	2	no	—
60	C₄×C₅₉	—	2	2	no	—
61	C₄×C₅₉	—	2	1	no	—
62	C₄×C₅₉	—	2	1	no	—
63	C₄×C₅₉	—	4	2	no	—
64	C₂×#356	—	59	1	no	maximal, Sylow
65	C₂×C₅₉	—	2	1	no	—
66	C₂×C₅₉	—	2	1	no	—
67	C₂×C₅₉	#356	1	2	yes	complemented
68	C₂×C₅₉	C₂×#99	1	1	yes	characteristic
69	C₂×C₄⋊Q₈	D₅₉	1	1	yes	characteristic
70	C₂×#101	—	59	2	no	—
71	#356	—	59	4	no	—
72	C₅₉	C₂×#356	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	Dic₅₉	1	1	yes	characteristic
74	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
75	C₂×M₁₆	—	118	2	no	—
76	C₄⋊Q₈	—	2	2	no	—
77	C₄⋊Q₈	Dic₅₉	1	2	yes	—
78	C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
79	C₂×C₄⋊C₄	—	2	2	no	—
80	C₂×C₄²	Dic₅₉	1	1	yes	characteristic
81	#101	—	59	4	no	—
82	#101	—	118	2	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	4	1	no	—
85	C₂×Q₈	—	4	1	no	—
86	C₂×Q₈	—	4	1	no	—
87	C₂×Q₈	—	4	1	no	—
88	C₂×Q₈	C₂×Dic₅₉	1	2	yes	—
89	C₂×Q₈	C₂×Dic₅₉	1	1	yes	characteristic
90	C₂×Q₈	C₂²⋊D₅₉	1	2	yes	—
91	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
95	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
96	M₁₆	—	118	4	no	—
97	M₁₆	—	236	2	no	—
98	C₂×C₈	—	236	2	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄²	—	2	1	no	—
102	C₄²	C₂×Dic₅₉	1	2	yes	—
103	C₂³	C₂²⋊Dic₅₉	1	1	yes	characteristic
104	Q₈	—	4	1	no	—
105	Q₈	—	4	2	no	—
106	Q₈	—	4	1	no	—
107	Q₈	—	4	2	no	—
108	Q₈	—	4	2	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
119	C₂×C₄	C₂²⋊Dic₅₉	1	2	yes	—
120	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
122	C₈	—	236	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#14273642	1	1	yes	characteristic
125	C₂²	#14273642	1	1	yes	characteristic
126	C₂²	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	1	no	—
130	C₄	—	2	2	no	—
131	C₄	—	2	1	no	—
132	C₄	—	4	2	no	—
133	C₂	—	2	1	no	—
134	C₂	—	2	1	no	—
135	C₂	#16837201	1	2	yes	direct product factor
136	C₂	C₂×#14273642	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273646

C₂

yes

maximal

#16837201

C₂

yes

maximal, direct product factor

C₂×C₄⋊Q₈×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

—

#14273646

—

#14273646

C₂²

yes

—

C₂²×Q₈×C₅₉

C₄

yes

characteristic

C₂²×Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

C₄

yes

—

C₄⋊Q₈×C₅₉

C₂²

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₄

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

C₂×C₄

yes

—

C₂×Q₈×C₅₉

C₂×C₄

yes

characteristic

C₂×Q₈×C₅₉

D₄

yes

—

C₂×Q₈×C₅₉

C₂³

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂²⋊C₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₂×#356

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#356

yes

complemented

C₂×C₅₉

C₂×#99

yes

characteristic

C₂×C₄⋊Q₈

D₅₉

yes

characteristic

C₂×#101

—

#356

—

C₅₉

C₂×#356

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₅₉

yes

characteristic

C₂²×Q₈

C₂×D₅₉

yes

characteristic

C₂×M₁₆

—

118

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₅₉

yes

—

C₄⋊Q₈

C₂×D₅₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

Dic₅₉

yes

characteristic

#101

—

#101

—

118

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₉

yes

—

C₂×Q₈

C₂×Dic₅₉

yes

characteristic

C₂×Q₈

C₂²⋊D₅₉

yes

—

C₂×Q₈

C₂²×D₅₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

—

M₁₆

—

118

—

M₁₆

—

236

—

C₂×C₈

—

236

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

C₂×Dic₅₉

yes

—

103

C₂³

C₂²⋊Dic₅₉

yes

characteristic

104

Q₈

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

119

C₂×C₄

C₂²⋊Dic₅₉

yes

—

120

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

122

C₈

—

236

—

123

C₂²

—

124

C₂²

#14273642

yes

characteristic

125

C₂²

#14273642

yes

characteristic

126

C₂²

C₂×C₂²⋊Dic₅₉

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₂

—

134

C₂

—

135

C₂

#16837201

yes

direct product factor

136

C₂

C₂×#14273642

yes

characteristic

	subgroup	G/H
center	C₂²	#14273642
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂×Q₈	C₂²×D₅₉
Fitting	C₂×C₄⋊Q₈×C₅₉	C₂
socle	C₂²×C₅₉	#99
hypercenter	C₂×C₄⋊Q₈	D₅₉
hypocenter	C₅₉	C₂×#356
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

#14273642

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂×Q₈

C₂²×D₅₉

Fitting

C₂×C₄⋊Q₈×C₅₉

C₂

socle

C₂²×C₅₉

#99

hypercenter

C₂×C₄⋊Q₈

D₅₉

hypocenter

C₅₉

C₂×#356

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #16837201	C₂	direct (N × H)
67. C₂×C₅₉	#356	semidirect
72. C₅₉	C₂×#356	semidirect
135. C₂	#16837201	direct (N × H)

H ≅ G/N

type

2. #16837201

C₂

direct (N × H)

67. C₂×C₅₉

#356

semidirect

72. C₅₉

C₂×#356

semidirect

135. C₂

#16837201

direct (N × H)

Subgroups — C₂×#16837201 #21425777

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16837201 #21425777

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16837201 #21425777