Group #44480488 — subgroups

Subgroups — #1718×C₂₃₉ #44480488

Order 30592.

164 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 72 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	#500×C₂₃₉	C₂	1	1	yes	maximal, characteristic
2	#498×C₂₃₉	C₂	1	1	yes	maximal, characteristic
3	#487×C₂₃₉	C₂	1	1	yes	maximal, characteristic
4	#487×C₂₃₉	C₂	1	1	yes	maximal, characteristic
5	#487×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	#487×C₂₃₉	C₂	1	1	yes	maximal, characteristic
7	#322×C₂₃₉	C₂	1	1	yes	maximal, characteristic
8	C₄⋊Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
9	C₄⋊Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
12	#107×C₂₃₉	—	2	2	no	—
13	#106×C₂₃₉	—	2	2	no	—
14	#103×C₂₃₉	—	2	2	no	—
15	#103×C₂₃₉	—	2	2	no	—
16	#103×C₂₃₉	—	2	2	no	—
17	#103×C₂₃₉	—	2	2	no	—
18	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₂₃₉	—	2	1	no	—
22	C₂×Q₈×C₂₃₉	—	2	1	no	—
23	C₂×Q₈×C₂₃₉	—	2	1	no	—
24	C₂×Q₈×C₂₃₉	—	2	1	no	—
25	C₂×C₈×C₂₃₉	D₄	1	2	yes	—
26	C₂×C₈×C₂₃₉	D₄	1	2	yes	—
27	C₂×C₈×C₂₃₉	D₄	1	2	yes	—
28	C₄⋊C₄×C₂₃₉	—	2	1	no	—
29	C₄⋊C₄×C₂₃₉	—	2	1	no	—
30	C₄⋊C₄×C₂₃₉	—	2	1	no	—
31	C₄⋊C₄×C₂₃₉	—	2	1	no	—
32	C₄⋊C₄×C₂₃₉	—	2	1	no	—
33	C₄⋊C₄×C₂₃₉	—	2	1	no	—
34	C₄⋊C₄×C₂₃₉	—	2	1	no	—
35	C₄⋊C₄×C₂₃₉	—	2	1	no	—
36	C₄²×C₂₃₉	C₂³	1	1	yes	characteristic
37	Q₈×C₂₃₉	—	4	2	no	—
38	Q₈×C₂₃₉	—	4	2	no	—
39	Q₈×C₂₃₉	—	4	2	no	—
40	Q₈×C₂₃₉	—	4	2	no	—
41	C₂×C₄×C₂₃₉	—	4	1	no	—
42	C₂×C₄×C₂₃₉	—	4	1	no	—
43	C₂×C₄×C₂₃₉	—	4	1	no	—
44	C₂×C₄×C₂₃₉	—	4	1	no	—
45	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
48	C₈×C₂₃₉	—	2	2	no	—
49	C₈×C₂₃₉	—	2	2	no	—
50	C₈×C₂₃₉	—	2	2	no	—
51	C₂²×C₂₃₉	#127	1	1	yes	characteristic
52	C₄×C₂₃₉	—	8	1	no	—
53	C₄×C₂₃₉	—	8	1	no	—
54	C₄×C₂₃₉	—	8	1	no	—
55	C₄×C₂₃₉	—	8	1	no	—
56	C₄×C₂₃₉	#135	1	2	yes	—
57	C₄×C₂₃₉	#137	1	2	yes	—
58	C₄×C₂₃₉	#137	1	2	yes	—
59	C₂×C₂₃₉	#495	1	1	yes	characteristic
60	C₂×C₂₃₉	#497	1	1	yes	characteristic
61	C₂×C₂₃₉	#497	1	1	yes	characteristic
62	C₂₃₉	#1718	1	1	yes	characteristic, direct product factor, Sylow
63	#1718	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	#500	C₂×C₂₃₉	1	1	yes	characteristic
65	#498	C₂×C₂₃₉	1	1	yes	characteristic
66	#487	C₂×C₂₃₉	1	1	yes	characteristic
67	#487	C₂×C₂₃₉	1	1	yes	characteristic
68	#487	C₂×C₂₃₉	1	1	yes	characteristic
69	#487	C₂×C₂₃₉	1	1	yes	characteristic
70	#322	C₂×C₂₃₉	1	1	yes	characteristic
71	C₄⋊Q₈	C₂²×C₂₃₉	1	1	yes	characteristic
72	C₄⋊Q₈	C₂²×C₂₃₉	1	1	yes	characteristic
73	C₄⋊Q₈	C₂²×C₂₃₉	1	1	yes	characteristic
74	C₄⋊Q₈	C₂²×C₂₃₉	1	1	yes	characteristic
75	#107	—	2	2	no	—
76	#106	—	2	2	no	—
77	#103	—	2	2	no	—
78	#103	—	2	2	no	—
79	#103	—	2	2	no	—
80	#103	—	2	2	no	—
81	C₄×C₈	C₂²×C₂₃₉	1	1	yes	characteristic
82	C₄×C₈	C₂²×C₂₃₉	1	1	yes	characteristic
83	C₄×C₈	C₂²×C₂₃₉	1	1	yes	characteristic
84	C₂×Q₈	—	2	1	no	—
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	2	1	no	—
88	C₂×C₈	D₄×C₂₃₉	1	2	yes	—
89	C₂×C₈	D₄×C₂₃₉	1	2	yes	—
90	C₂×C₈	D₄×C₂₃₉	1	2	yes	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄²	C₂³×C₂₃₉	1	1	yes	characteristic
100	Q₈	—	4	2	no	—
101	Q₈	—	4	2	no	—
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	C₂×D₄×C₂₃₉	1	1	yes	characteristic
109	C₂×C₄	C₂×D₄×C₂₃₉	1	1	yes	characteristic
110	C₂×C₄	C₂×D₄×C₂₃₉	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	2	no	—
113	C₈	—	2	2	no	—
114	C₂²	#127×C₂₃₉	1	1	yes	characteristic
115	C₄	—	8	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	#135×C₂₃₉	1	2	yes	—
120	C₄	#137×C₂₃₉	1	2	yes	—
121	C₄	#137×C₂₃₉	1	2	yes	—
122	C₂	#495×C₂₃₉	1	1	yes	characteristic
123	C₂	#497×C₂₃₉	1	1	yes	characteristic
124	C₂	#497×C₂₃₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#500×C₂₃₉

C₂

yes

maximal, characteristic

#498×C₂₃₉

C₂

yes

maximal, characteristic

#487×C₂₃₉

C₂

yes

maximal, characteristic

#487×C₂₃₉

C₂

yes

maximal, characteristic

#487×C₂₃₉

C₂

yes

maximal, characteristic

#487×C₂₃₉

C₂

yes

maximal, characteristic

#322×C₂₃₉

C₂

yes

maximal, characteristic

C₄⋊Q₈×C₂₃₉

C₂²

yes

characteristic

C₄⋊Q₈×C₂₃₉

C₂²

yes

characteristic

C₄⋊Q₈×C₂₃₉

C₂²

yes

characteristic

C₄⋊Q₈×C₂₃₉

C₂²

yes

characteristic

#107×C₂₃₉

—

#106×C₂₃₉

—

#103×C₂₃₉

—

#103×C₂₃₉

—

#103×C₂₃₉

—

#103×C₂₃₉

—

C₄×C₈×C₂₃₉

C₂²

yes

characteristic

C₄×C₈×C₂₃₉

C₂²

yes

characteristic

C₄×C₈×C₂₃₉

C₂²

yes

characteristic

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×C₈×C₂₃₉

D₄

yes

—

C₂×C₈×C₂₃₉

D₄

yes

—

C₂×C₈×C₂₃₉

D₄

yes

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄²×C₂₃₉

C₂³

yes

characteristic

Q₈×C₂₃₉

—

Q₈×C₂₃₉

—

Q₈×C₂₃₉

—

Q₈×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₈×C₂₃₉

—

C₈×C₂₃₉

—

C₈×C₂₃₉

—

C₂²×C₂₃₉

#127

yes

characteristic

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

#135

yes

—

C₄×C₂₃₉

#137

yes

—

C₄×C₂₃₉

#137

yes

—

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂₃₉

yes

characteristic, direct product factor, Sylow

#1718

C₂₃₉

yes

maximal, characteristic, direct product factor, Sylow

#500

C₂×C₂₃₉

yes

characteristic

#498

C₂×C₂₃₉

yes

characteristic

#487

C₂×C₂₃₉

yes

characteristic

#487

C₂×C₂₃₉

yes

characteristic

#487

C₂×C₂₃₉

yes

characteristic

#487

C₂×C₂₃₉

yes

characteristic

#322

C₂×C₂₃₉

yes

characteristic

C₄⋊Q₈

C₂²×C₂₃₉

yes

characteristic

C₄⋊Q₈

C₂²×C₂₃₉

yes

characteristic

C₄⋊Q₈

C₂²×C₂₃₉

yes

characteristic

C₄⋊Q₈

C₂²×C₂₃₉

yes

—

—

—

—

—

—

C₄×C₈

C₂²×C₂₃₉

yes

characteristic

C₄×C₈

C₂²×C₂₃₉

yes

characteristic

C₄×C₈

C₂²×C₂₃₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

D₄×C₂₃₉

yes

—

C₂×C₈

D₄×C₂₃₉

yes

—

C₂×C₈

D₄×C₂₃₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂³×C₂₃₉

yes

characteristic

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×D₄×C₂₃₉

yes

characteristic

109

C₂×C₄

C₂×D₄×C₂₃₉

yes

characteristic

110

C₂×C₄

C₂×D₄×C₂₃₉

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₂²

#127×C₂₃₉

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#135×C₂₃₉

yes

—

120

C₄

#137×C₂₃₉

yes

—

121

C₄

#137×C₂₃₉

yes

—

122

C₂

#495×C₂₃₉

yes

characteristic

123

C₂

#497×C₂₃₉

yes

characteristic

124

C₂

#497×C₂₃₉

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₃₉	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₂₃₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃₉ ◁ C₄²×C₂₃₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₂₃₉

#127

commutator, Frattini, cosocle

C₄²

C₂³×C₂₃₉

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃₉ ◁ C₄²×C₂₃₉ ◁ G

N	H ≅ G/N	type
62. C₂₃₉	#1718	direct (N × H)
63. #1718	C₂₃₉	direct (N × H)

H ≅ G/N

type

62. C₂₃₉

#1718

direct (N × H)

63. #1718

C₂₃₉

direct (N × H)

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

subgroup	index
#500×C₂₃₉	2
#498×C₂₃₉	2
#487×C₂₃₉	2
#322×C₂₃₉	2
#1718	239

subgroup

index

#500×C₂₃₉

#498×C₂₃₉

#487×C₂₃₉

#322×C₂₃₉

#1718

239

Subgroups — #1718×C₂₃₉ #44480488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1718×C239 #44480488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1718×C₂₃₉ #44480488