Group #73304 — subgroups

Subgroups — C₄×#870 #73304

Order 384.

484 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 229 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×#870	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄³	C₂	1	1	yes	maximal, characteristic
3	#4936	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊Dic₃	C₂	1	2	yes	maximal
5	#4933	C₂	1	4	yes	maximal
6	C₄×Dic₃⋊C₄	C₂	1	4	yes	maximal
7	#4928	C₂	1	2	yes	maximal
8	C₄×#125	—	3	1	no	maximal, Sylow
9	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄²	C₂²	1	2	yes	—
11	C₂×C₃×C₄²	C₂²	1	2	yes	—
12	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
17	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
18	C₂×C₄×Dic₃	C₂²	1	4	yes	—
19	#870	C₄	1	8	yes	direct product factor
20	C₃⋊#95	C₂²	1	4	yes	—
21	C₃⋊#95	C₂²	1	4	yes	—
22	C₃⋊#95	C₂²	1	4	yes	—
23	C₂×#125	—	3	1	no	—
24	#389	—	3	2	no	—
25	#387	—	3	4	no	—
26	#382	—	3	1	no	—
27	C₄×C₄⋊C₄	—	3	4	no	—
28	C₄×C₄⋊C₄	—	3	2	no	—
29	C₄³	D₃	1	1	yes	characteristic
30	C₂²×C₃×C₄	C₂³	1	2	yes	—
31	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃×C₄	C₂³	1	2	yes	—
33	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
34	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
35	C₂²×Dic₃	C₂³	1	4	yes	—
36	C₂²×Dic₃	C₂³	1	4	yes	—
37	C₃×C₄²	—	2	4	no	—
38	C₃×C₄²	—	2	2	no	—
39	C₃×C₄²	—	2	4	no	—
40	C₃×C₄²	C₂×C₄	1	4	yes	—
41	C₃×C₄²	Q₈	1	4	yes	—
42	C₄⋊Dic₃	—	2	4	no	—
43	C₄⋊Dic₃	C₂×C₄	1	8	yes	—
44	Dic₃⋊C₄	—	2	8	no	—
45	Dic₃⋊C₄	C₂×C₄	1	16	yes	—
46	C₄×Dic₃	—	2	8	no	—
47	#125	—	3	8	no	—
48	C₂×C₄⋊C₄	—	3	4	no	—
49	C₂×C₄⋊C₄	—	3	2	no	—
50	C₂×C₄⋊C₄	—	3	2	no	—
51	C₂×C₄⋊C₄	—	3	4	no	—
52	C₂×C₄²	—	3	4	no	—
53	C₂×C₄²	C₂×D₃	1	2	yes	—
54	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
55	C₂×C₄²	C₂×D₃	1	2	yes	—
56	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
58	#95	—	3	4	no	—
59	#95	—	3	4	no	—
60	#95	—	3	4	no	—
61	C₂³×C₃	C₂⁴	1	1	yes	characteristic
62	C₂×C₃×C₄	—	2	2	no	—
63	C₂×C₃×C₄	—	2	2	no	—
64	C₂×C₃×C₄	—	2	2	no	—
65	C₂×C₃×C₄	—	2	2	no	—
66	C₂×C₃×C₄	—	2	2	no	—
67	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
68	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
69	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
70	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
71	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
72	C₂×C₃×C₄	#41	1	4	yes	—
73	C₂×C₃×C₄	#41	1	2	yes	—
74	C₂×C₃×C₄	#41	1	4	yes	—
75	C₂×Dic₃	—	2	4	no	—
76	C₂×Dic₃	—	2	4	no	—
77	C₂×Dic₃	—	2	4	no	—
78	C₂×Dic₃	—	2	4	no	—
79	C₂×Dic₃	—	2	4	no	—
80	C₂×Dic₃	C₂²×C₄	1	8	yes	—
81	C₂²×C₄	—	3	4	no	—
82	C₂²×C₄	—	3	4	no	—
83	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
84	C₂²×C₄	C₂²×D₃	1	2	yes	—
85	C₂²×C₄	C₂²×D₃	1	2	yes	—
86	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
88	C₄⋊C₄	—	3	16	no	—
89	C₄⋊C₄	—	3	8	no	—
90	C₄⋊C₄	—	6	8	no	—
91	C₄⋊C₄	—	6	4	no	—
92	C₄²	—	2	4	no	—
93	C₄²	—	2	2	no	—
94	C₄²	—	2	4	no	—
95	C₄²	—	6	8	no	—
96	C₄²	Dic₆	1	4	yes	—
97	C₄²	C₄×D₃	1	4	yes	—
98	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
99	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
100	C₂²×C₃	C₂×#41	1	1	yes	characteristic
101	C₂²×C₃	C₂×#41	1	2	yes	—
102	C₂²×C₃	C₂×#41	1	2	yes	—
103	C₃×C₄	—	2	4	no	—
104	C₃×C₄	—	2	4	no	—
105	C₃×C₄	—	2	2	no	—
106	C₃×C₄	C₄×Q₈	1	4	yes	—
107	C₃×C₄	#125	1	4	yes	complemented
108	Dic₃	—	2	8	no	—
109	Dic₃	—	4	4	no	—
110	C₂³	C₂³×D₃	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	3	8	no	—
117	C₂×C₄	—	6	4	no	—
118	C₂×C₄	—	6	4	no	—
119	C₂×C₄	—	6	4	no	—
120	C₂×C₄	—	6	4	no	—
121	C₂×C₄	—	6	4	no	—
122	C₂×C₄	C₂×Dic₆	1	2	yes	—
123	C₂×C₄	C₂×Dic₆	1	2	yes	—
124	C₂×C₄	C₂×Dic₆	1	2	yes	—
125	C₂×C₄	C₂×C₄×D₃	1	4	yes	—
126	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
127	C₂×C₄	#235	1	4	yes	—
128	C₂×C₄	#235	1	2	yes	—
129	C₂×C₄	#235	1	4	yes	—
130	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
131	C₂×C₃	C₄×#41	1	2	yes	—
132	C₂×C₃	C₂×#125	1	1	yes	characteristic
133	C₂×C₃	#529	1	2	yes	—
134	C₂×C₃	#533	1	1	yes	characteristic
135	C₂²	C₂²×Dic₆	1	1	yes	characteristic
136	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
137	C₂²	C₂×#235	1	1	yes	characteristic
138	C₂²	C₂×#235	1	2	yes	—
139	C₂²	C₂×#235	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	2	4	no	—
142	C₄	—	2	4	no	—
143	C₄	—	6	8	no	—
144	C₄	—	12	4	no	—
145	C₄	C₄×Dic₆	1	4	yes	—
146	C₄	#870	1	4	yes	direct product factor
147	C₃	C₄×#125	1	1	yes	characteristic, complemented, Sylow
148	C₂	C₂×C₄×Dic₆	1	1	yes	characteristic
149	C₂	C₂×#870	1	1	yes	characteristic
150	C₂	#5470	1	1	yes	characteristic
151	C₂	C₄×#235	1	2	yes	—
152	C₂	#5479	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#870

C₂

yes

maximal, characteristic

C₃×C₄³

C₂

yes

maximal, characteristic

#4936

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₃

C₂

yes

maximal

#4933

C₂

yes

maximal

C₄×Dic₃⋊C₄

C₂

yes

maximal

#4928

C₂

yes

maximal

C₄×#125

—

maximal, Sylow

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₃×C₄²

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

—

#870

C₄

yes

direct product factor

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

—

—

—

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

D₃

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×C₄

yes

—

C₃×C₄²

Q₈

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂×C₄

yes

—

Dic₃⋊C₄

—

Dic₃⋊C₄

C₂×C₄

yes

—

C₄×Dic₃

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₂×C₄²

C₂×D₃

yes

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₂×C₄²

C₂×D₃

yes

—

—

—

C₂³×C₃

C₂⁴

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

C₂²×C₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₆

yes

—

C₄²

C₄×D₃

yes

—

C₂²×C₃

C₂³×C₄

yes

characteristic

C₂²×C₃

C₂²×Q₈

yes

characteristic

100

C₂²×C₃

C₂×#41

yes

characteristic

101

C₂²×C₃

C₂×#41

yes

—

102

C₂²×C₃

C₂×#41

yes

—

103

C₃×C₄

—

104

C₃×C₄

—

105

C₃×C₄

—

106

C₃×C₄

C₄×Q₈

yes

—

107

C₃×C₄

#125

yes

complemented

108

Dic₃

—

109

Dic₃

—

110

C₂³

C₂³×D₃

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂×Dic₆

yes

—

123

C₂×C₄

C₂×Dic₆

yes

—

124

C₂×C₄

C₂×Dic₆

yes

—

125

C₂×C₄

C₂×C₄×D₃

yes

—

126

C₂×C₄

C₂×C₄×D₃

yes

—

127

C₂×C₄

#235

yes

—

128

C₂×C₄

#235

yes

—

129

C₂×C₄

#235

yes

—

130

C₂×C₃

C₂×C₄×Q₈

yes

characteristic

131

C₂×C₃

C₄×#41

yes

—

132

C₂×C₃

C₂×#125

yes

characteristic

133

C₂×C₃

#529

yes

—

134

C₂×C₃

#533

yes

characteristic

135

C₂²

C₂²×Dic₆

yes

characteristic

136

C₂²

C₂²×C₄×D₃

yes

characteristic

137

C₂²

C₂×#235

yes

characteristic

138

C₂²

C₂×#235

yes

—

139

C₂²

C₂×#235

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

C₄×Dic₆

yes

—

146

C₄

#870

yes

direct product factor

147

C₃

C₄×#125

yes

characteristic, complemented, Sylow

148

C₂

C₂×C₄×Dic₆

yes

characteristic

149

C₂

C₂×#870

yes

characteristic

150

C₂

#5470

yes

characteristic

151

C₂

C₄×#235

yes

—

152

C₂

#5479

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂³×C₄
Frattini	C₂³	C₂³×D₃
Fitting	C₃×C₄³	C₂
socle, prime radical, cosocle	C₂³×C₃	C₂⁴
hypercenter	C₄³	D₃
hypocenter	C₃	C₄×#125
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

subgroup

G/H

center

C₂²×C₄

C₂²×D₃

commutator

C₂²×C₃

C₂³×C₄

Frattini

C₂³

C₂³×D₃

Fitting

C₃×C₄³

C₂

socle, prime radical, cosocle

C₂³×C₃

C₂⁴

hypercenter

C₄³

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₂²×C₄ ◁ C₄³

N	H ≅ G/N	type
19. #870	C₄	direct (N × H)
107. C₃×C₄	#125	semidirect
146. C₄	#870	direct (N × H)
147. C₃	C₄×#125	semidirect

H ≅ G/N

type

19. #870

C₄

direct (N × H)

107. C₃×C₄

#125

semidirect

146. C₄

#870

direct (N × H)

147. C₃

C₄×#125

semidirect

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

subgroup	index
C₂×#870	2
C₃×C₄³	2
#4936	2
C₄×C₄⋊Dic₃	2
#4933	2
C₄×Dic₃⋊C₄	2
#4928	2
C₄×#125	3

subgroup

index

C₃×C₄³

C₄×C₄⋊Dic₃

C₄×Dic₃⋊C₄

Subgroups — C₄×#870 #73304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#870 #73304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#870 #73304