Group #14544178 — subgroups

Subgroups — #14544178

Order 2176.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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	#11763831	C₂	1	1	yes	maximal, characteristic
2	#11763745	C₂	1	1	yes	maximal, characteristic
3	#11763663	C₂	1	4	yes	maximal
4	#11763364	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#595	—	2	1	no	—
6	C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
7	#10587632	C₂²	1	2	yes	—
8	Q₈×Dic₁₇	C₄	1	2	yes	complemented
9	#10587583	—	2	1	no	—
10	#10587578	—	2	1	no	—
11	#10587576	—	2	1	no	—
12	#10587573	—	2	1	no	—
13	C₄²×D₁₇	C₂²	1	1	yes	characteristic
14	#10587555	C₄	1	1	yes	characteristic, complemented
15	(C₂×#595)⋊C₄	—	2	1	no	—
16	(C₂×#595)⋊C₄	C₂²	1	2	yes	—
17	#10587543	C₂²	1	2	yes	—
18	#10587542	—	2	1	no	—
19	C₄⋊#3624	—	2	1	no	—
20	C₂²×#595	—	2	2	no	—
21	C₂×Q₈×C₁₇	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
23	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
24	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
25	C₂×Dic₃₄	—	2	1	no	—
26	C₂×#3624	—	2	2	no	—
27	C₄×#595	—	2	4	no	—
28	C₄⋊C₄×C₁₇	—	2	1	no	—
29	C₄⋊C₄×C₁₇	—	2	1	no	—
30	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
31	#63227	—	2	2	no	—
32	C₄⋊Dic₁₇	—	2	1	no	—
33	C₄⋊Dic₁₇	—	2	2	no	—
34	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
35	#63225	—	2	2	no	—
36	C₄×Dic₁₇	—	2	2	no	—
37	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₁₇	D₄	1	1	yes	characteristic
39	C₄×Dic₁₇	D₄	1	1	yes	characteristic
40	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂×#595	—	2	4	no	—
42	C₂×#595	—	4	2	no	—
43	C₂×#595	—	4	2	no	—
44	Q₈×C₁₇	—	4	2	no	—
45	C₂×C₄×C₁₇	—	2	2	no	—
46	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₁₇	—	2	2	no	—
50	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
53	C₄×D₁₇	—	2	2	no	—
54	C₄×D₁₇	—	2	2	no	—
55	C₄×D₁₇	—	4	1	no	—
56	Dic₃₄	—	4	2	no	—
57	#3624	—	4	2	no	—
58	#1506	—	17	1	no	maximal, Sylow
59	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×D₁₇	—	2	1	no	—
61	C₂×D₁₇	—	2	1	no	—
62	C₂×D₁₇	#137	1	2	yes	—
63	#595	—	4	4	no	—
64	C₄×C₁₇	—	2	1	no	—
65	C₄×C₁₇	—	2	1	no	—
66	C₄×C₁₇	—	4	2	no	—
67	C₄×C₁₇	#104	1	2	yes	—
68	Dic₁₇	—	2	1	no	—
69	Dic₁₇	—	2	1	no	—
70	Dic₁₇	—	2	1	no	—
71	Dic₁₇	—	4	2	no	—
72	#533	—	17	1	no	—
73	#432	—	17	1	no	—
74	#376	—	17	1	no	—
75	#328	—	17	4	no	—
76	C₂×C₁₇	#410	1	1	yes	characteristic
77	C₂×C₁₇	#419	1	1	yes	characteristic
78	C₂×C₁₇	C₂×#104	1	1	yes	characteristic
79	D₁₇	—	2	2	no	—
80	C₄⋊Q₈	—	17	1	no	—
81	C₄⋊Q₈	#595	1	1	yes	characteristic, complemented
82	#125	—	34	1	no	—
83	C₂²⋊Q₈	—	17	2	no	—
84	C₂²⋊Q₈	—	34	1	no	—
85	C₄×Q₈	—	17	2	no	—
86	C₄×Q₈	—	34	1	no	—
87	#117	—	34	1	no	—
88	C₂×C₄²	—	17	1	no	—
89	C₂×C₄²	—	34	1	no	—
90	C₄⋊C₈	—	34	1	no	—
91	C₂²⋊C₈	—	17	2	no	—
92	#97	—	34	1	no	—
93	#95	—	17	2	no	—
94	#95	—	34	1	no	—
95	C₁₇	#1506	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₈	—	34	1	no	—
97	C₂×Q₈	C₂×#595	1	2	yes	—
98	C₂²×C₄	—	17	1	no	—
99	C₂²×C₄	—	17	1	no	—
100	C₂²×C₄	—	17	1	no	—
101	C₂²×C₄	—	34	2	no	—
102	C₂×C₈	—	34	2	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	17	2	no	—
106	C₄⋊C₄	—	34	2	no	—
107	C₄⋊C₄	—	34	2	no	—
108	C₄⋊C₄	—	34	1	no	—
109	C₂²⋊C₄	—	34	2	no	—
110	C₄²	—	17	1	no	—
111	C₄²	—	17	1	no	—
112	C₄²	—	17	1	no	—
113	C₄²	—	34	2	no	—
114	C₄²	—	34	4	no	—
115	C₄²	C₂×#595	1	1	yes	characteristic
116	C₂³	—	17	1	no	—
117	Q₈	—	4	2	no	—
118	Q₈	—	68	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	17	1	no	—
121	C₂×C₄	—	17	1	no	—
122	C₂×C₄	—	17	1	no	—
123	C₂×C₄	—	34	2	no	—
124	C₂×C₄	—	34	2	no	—
125	C₂×C₄	—	34	4	no	—
126	C₂×C₄	—	34	2	no	—
127	C₂×C₄	—	68	1	no	—
128	C₂×C₄	—	68	2	no	—
129	C₂×C₄	—	68	2	no	—
130	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
131	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
132	C₂×C₄	C₂²×#595	1	1	yes	characteristic
133	C₈	—	68	2	no	—
134	C₂²	—	17	2	no	—
135	C₂²	—	34	1	no	—
136	C₂²	—	34	1	no	—
137	C₂²	C₂×C₂²⋊#595	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	2	no	—
141	C₄	—	34	1	no	—
142	C₄	—	34	1	no	—
143	C₄	—	34	1	no	—
144	C₄	—	68	4	no	—
145	C₄	—	68	2	no	—
146	C₄	#10587550	1	2	yes	—
147	C₂	—	34	2	no	—
148	C₂	#11763789	1	1	yes	characteristic
149	C₂	#11763802	1	1	yes	characteristic
150	C₂	C₂×#10587550	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763831

C₂

yes

maximal, characteristic

#11763745

C₂

yes

maximal, characteristic

#11763663

C₂

yes

maximal

#11763364

C₂

yes

maximal, characteristic

C₂×C₄×#595

—

C₁₇×C₄⋊Q₈

C₄

yes

characteristic, complemented

#10587632

C₂²

yes

—

Q₈×Dic₁₇

C₄

yes

—

—

—

—

C₄²×D₁₇

C₂²

yes

characteristic

#10587555

C₄

yes

characteristic, complemented

(C₂×#595)⋊C₄

—

(C₂×#595)⋊C₄

C₂²

yes

—

#10587543

C₂²

yes

—

#10587542

—

C₄⋊#3624

—

C₂²×#595

—

C₂×Q₈×C₁₇

C₂×C₄

yes

—

C₂×C₄×D₁₇

D₄

yes

characteristic

C₂×C₄×D₁₇

D₄

yes

characteristic

C₂×C₄×D₁₇

C₂³

yes

characteristic

C₂×Dic₃₄

—

C₂×#3624

—

C₄×#595

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

#63227

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

#63225

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

D₄

yes

characteristic

C₄×Dic₁₇

D₄

yes

characteristic

C₂²×D₁₇

C₂×D₄

yes

—

—

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₇

C₂×D₄

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

#3624

—

#1506

—

maximal, Sylow

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

#137

yes

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#104

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

—

—

—

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₄⋊Q₈

—

C₄⋊Q₈

#595

yes

characteristic, complemented

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#95

—

#95

—

C₁₇

#1506

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂×#595

yes

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

C₂×#595

yes

characteristic

116

C₂³

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂²⋊#595

yes

characteristic

131

C₂×C₄

C₂²⋊#595

yes

characteristic

132

C₂×C₄

C₂²×#595

yes

characteristic

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×C₂²⋊#595

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#10587550

yes

—

147

C₂

—

148

C₂

#11763789

yes

characteristic

149

C₂

#11763802

yes

characteristic

150

C₂

C₂×#10587550

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#595
Fitting	C₁₇×C₄⋊Q₈	C₄
socle	C₂²×C₁₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₇	C₂×D₄
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	C₄⋊Q₈	#595
hypocenter	C₁₇	#1506
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#595

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#595

Fitting

C₁₇×C₄⋊Q₈

C₄

socle

C₂²×C₁₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₇

C₂×D₄

cosocle

C₂×C₄×D₁₇

C₂³

C₄⋊Q₈

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
6. C₁₇×C₄⋊Q₈	C₄	semidirect
8. Q₈×Dic₁₇	C₄	semidirect
14. #10587555	C₄	semidirect
81. C₄⋊Q₈	#595	semidirect
95. C₁₇	#1506	semidirect

H ≅ G/N

type

6. C₁₇×C₄⋊Q₈

C₄

semidirect

8. Q₈×Dic₁₇

C₄

semidirect

14. #10587555

C₄

semidirect

81. C₄⋊Q₈

#595

semidirect

95. C₁₇

#1506

semidirect

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

subgroup	index
#11763831	2
#11763745	2
#11763663	2
#11763364	2
#1506	17

subgroup

index

Subgroups — #14544178

Conjugacy classes of subgroups

Complements

Maximal subgroups