Group #72216301 — subgroups

Subgroups — #72216301

Order 2048.

543 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 543 conjugacy classes of subgroups — too many to draw. Only its 272 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	#72216296	C₂	1	4	yes	maximal
2	#72216239	C₂	1	2	yes	maximal
3	C₁₆×M₆₄	C₂	1	1	yes	maximal, characteristic
4	#151055	—	2	8	no	—
5	C₈×M₆₄	C₂²	1	2	yes	—
6	#135796	—	2	2	no	—
7	#135796	C₂²	1	4	yes	—
8	#135781	—	2	2	no	—
9	C₂×C₁₆²	C₂²	1	1	yes	characteristic
10	#93664	C₄	1	2	yes	—
11	C₁₆×C₃₂	C₄	1	1	yes	characteristic
12	C₁₆×C₃₂	C₄	1	1	yes	characteristic
13	C₄×M₆₄	D₄	1	4	yes	—
14	#12909	—	4	4	no	—
15	#12023	—	2	8	no	—
16	#12023	—	4	4	no	—
17	C₂×C₈×C₁₆	D₄	1	2	yes	—
18	C₂×C₈×C₁₆	C₂³	1	1	yes	characteristic
19	#8823	—	2	4	no	—
20	#7463	—	4	16	no	—
21	#7329	Q₈	1	4	yes	—
22	C₈×C₃₂	C₂×C₄	1	2	yes	—
23	C₈×C₃₂	C₂×C₄	1	2	yes	—
24	C₁₆²	C₂×C₄	1	1	yes	characteristic
25	C₁₆²	C₂×C₄	1	1	yes	characteristic
26	C₁₆²	Q₈	1	2	yes	—
27	C₂×M₆₄	Q₁₆	1	8	yes	—
28	C₂×#368	—	8	8	no	—
29	C₂×C₄×C₁₆	D₈	1	4	yes	—
30	C₂×C₄×C₁₆	C₂×D₄	1	2	yes	—
31	#1848	—	4	4	no	—
32	#1567	—	4	4	no	—
33	#1567	—	4	4	no	—
34	C₂×C₈²	C₂×D₄	1	1	yes	characteristic
35	#1397	D₈	1	8	yes	—
36	C₄×C₃₂	C₄⋊C₄	1	4	yes	—
37	C₄×C₃₂	C₄⋊C₄	1	4	yes	—
38	#1372	—	4	16	no	—
39	#1372	—	8	8	no	—
40	C₈×C₁₆	C₄⋊C₄	1	2	yes	—
41	C₈×C₁₆	C₄⋊C₄	1	2	yes	—
42	C₈×C₁₆	Q₁₆	1	4	yes	—
43	C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
44	C₈×C₁₆	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₁₆	C₂×Q₈	1	2	yes	—
46	C₂²×C₁₆	SD₃₂	1	8	yes	—
47	C₂²×C₁₆	C₂×Q₁₆	1	4	yes	—
48	C₂×#108	—	8	8	no	—
49	#423	—	8	4	no	—
50	C₂×C₄×C₈	#127	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂×D₈	1	2	yes	—
52	M₆₄	—	2	16	no	—
53	C₂×C₃₂	#107	1	8	yes	—
54	C₂×C₃₂	#107	1	8	yes	—
55	#368	—	16	16	no	—
56	C₄×C₁₆	#107	1	4	yes	—
57	C₄×C₁₆	#107	1	4	yes	—
58	C₄×C₁₆	SD₃₂	1	8	yes	—
59	C₄×C₁₆	C₂×C₄⋊C₄	1	2	yes	—
60	C₄×C₁₆	C₄⋊Q₈	1	2	yes	—
61	C₄×C₁₆	C₂×D₈	1	4	yes	—
62	#335	—	8	8	no	—
63	#335	—	8	8	no	—
64	C₈²	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₈²	C₄⋊Q₈	1	1	yes	characteristic
66	C₈²	C₂×Q₁₆	1	2	yes	—
67	C₂×M₁₆	—	16	4	no	—
68	C₂²×C₈	#494	1	2	yes	—
69	C₂²×C₈	C₂×SD₃₂	1	4	yes	—
70	C₂×C₄²	#493	1	1	yes	characteristic
71	C₂×C₁₆	—	2	8	no	—
72	C₂×C₁₆	—	2	16	no	—
73	C₂×C₁₆	#367	1	8	yes	—
74	C₂×C₁₆	#367	1	8	yes	—
75	C₂×C₁₆	C₂×#107	1	4	yes	—
76	C₂×C₁₆	#500	1	4	yes	—
77	#108	—	16	16	no	—
78	C₄⋊C₈	—	16	4	no	—
79	C₄⋊C₈	—	16	4	no	—
80	C₄×C₈	#384	1	1	yes	characteristic
81	C₄×C₈	C₂×#107	1	2	yes	—
82	C₄×C₈	#493	1	2	yes	—
83	C₄×C₈	#494	1	1	yes	characteristic
84	C₄×C₈	#500	1	2	yes	—
85	C₄×C₈	C₂×SD₃₂	1	4	yes	—
86	C₃₂	—	2	8	no	—
87	C₃₂	#368	1	16	yes	complemented
88	C₂²×C₄	#1713	1	1	yes	characteristic
89	C₂²×C₄	#2248	1	2	yes	—
90	M₁₆	—	32	8	no	—
91	C₂×C₈	—	2	8	no	—
92	C₂×C₈	—	2	4	no	—
93	C₂×C₈	—	32	4	no	—
94	C₂×C₈	#1713	1	2	yes	—
95	C₂×C₈	#1845	1	2	yes	—
96	C₂×C₈	C₂×#367	1	4	yes	—
97	C₂×C₈	#2254	1	4	yes	—
98	C₄²	#1712	1	1	yes	characteristic
99	C₄²	#1845	1	1	yes	characteristic
100	C₄²	#2248	1	2	yes	—
101	C₁₆	—	2	8	no	—
102	C₁₆	—	2	4	no	—
103	C₁₆	#1426	1	16	yes	—
104	C₁₆	C₂×#368	1	8	yes	—
105	C₂³	#12523	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	#8818	1	1	yes	characteristic
109	C₂×C₄	#12517	1	2	yes	—
110	C₂×C₄	#12523	1	1	yes	characteristic
111	C₂×C₄	#12906	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	—	2	2	no	—
114	C₈	—	64	4	no	—
115	C₈	#12909	1	4	yes	—
116	C₈	C₂×#1426	1	8	yes	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	2	no	—
119	C₂²	#135778	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	2	2	no	—
122	C₄	#135781	1	2	yes	—
123	C₄	#152407	1	4	yes	—
124	C₂	—	4	1	no	—
125	C₂	#72216239	1	1	yes	characteristic
126	C₂	#72216298	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#72216296

C₂

yes

maximal

#72216239

C₂

yes

maximal

C₁₆×M₆₄

C₂

yes

maximal, characteristic

#151055

—

C₈×M₆₄

C₂²

yes

—

#135796

—

#135796

C₂²

yes

—

#135781

—

C₂×C₁₆²

C₂²

yes

characteristic

#93664

C₄

yes

—

C₁₆×C₃₂

C₄

yes

characteristic

C₁₆×C₃₂

C₄

yes

characteristic

C₄×M₆₄

D₄

yes

—

#12909

—

#12023

—

#12023

—

C₂×C₈×C₁₆

D₄

yes

—

C₂×C₈×C₁₆

C₂³

yes

—

—

Q₈

yes

—

C₈×C₃₂

C₂×C₄

yes

—

C₈×C₃₂

C₂×C₄

yes

—

C₁₆²

C₂×C₄

yes

characteristic

C₁₆²

C₂×C₄

yes

characteristic

C₁₆²

Q₈

yes

—

C₂×M₆₄

Q₁₆

yes

—

C₂×#368

—

C₂×C₄×C₁₆

D₈

yes

—

C₂×C₄×C₁₆

C₂×D₄

yes

—

#1848

—

#1567

—

#1567

—

C₂×C₈²

C₂×D₄

yes

characteristic

#1397

D₈

yes

—

C₄×C₃₂

C₄⋊C₄

yes

—

C₄×C₃₂

C₄⋊C₄

yes

—

#1372

—

#1372

—

C₈×C₁₆

C₄⋊C₄

yes

—

C₈×C₁₆

C₄⋊C₄

yes

—

C₈×C₁₆

Q₁₆

yes

—

C₈×C₁₆

C₂²×C₄

yes

characteristic

C₈×C₁₆

C₂×Q₈

yes

characteristic

C₈×C₁₆

C₂×Q₈

yes

—

C₂²×C₁₆

SD₃₂

yes

—

C₂²×C₁₆

C₂×Q₁₆

yes

—

C₂×#108

—

#423

—

C₂×C₄×C₈

#127

yes

characteristic

C₂×C₄×C₈

C₂×D₈

yes

—

M₆₄

—

C₂×C₃₂

#107

yes

—

C₂×C₃₂

#107

yes

—

#368

—

C₄×C₁₆

#107

yes

—

C₄×C₁₆

#107

yes

—

C₄×C₁₆

SD₃₂

yes

—

C₄×C₁₆

C₂×C₄⋊C₄

yes

—

C₄×C₁₆

C₄⋊Q₈

yes

—

C₄×C₁₆

C₂×D₈

yes

—

#335

—

#335

—

C₈²

C₂×C₄⋊C₄

yes

characteristic

C₈²

C₄⋊Q₈

yes

characteristic

C₈²

C₂×Q₁₆

yes

—

C₂×M₁₆

—

C₂²×C₈

#494

yes

—

C₂²×C₈

C₂×SD₃₂

yes

—

C₂×C₄²

#493

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

#367

yes

—

C₂×C₁₆

#367

yes

—

C₂×C₁₆

C₂×#107

yes

—

C₂×C₁₆

#500

yes

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

#384

yes

characteristic

C₄×C₈

C₂×#107

yes

—

C₄×C₈

#493

yes

—

C₄×C₈

#494

yes

characteristic

C₄×C₈

#500

yes

—

C₄×C₈

C₂×SD₃₂

yes

—

C₃₂

—

C₃₂

yes

C₂²×C₄

yes

C₂²×C₄

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#1713

yes

—

C₂×C₈

#1845

yes

—

C₂×C₈

C₂×#367

yes

—

C₂×C₈

#2254

yes

—

C₄²

yes

C₄²

yes

100

C₄²

#2248

yes

—

101

C₁₆

—

102

C₁₆

—

103

C₁₆

#1426

yes

—

104

C₁₆

C₂×#368

yes

—

105

C₂³

#12523

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

#8818

yes

characteristic

109

C₂×C₄

#12517

yes

—

110

C₂×C₄

#12523

yes

characteristic

111

C₂×C₄

#12906

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

#12909

yes

—

116

C₈

C₂×#1426

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

#135778

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

#135781

yes

—

123

C₄

#152407

yes

—

124

C₂

—

125

C₂

#72216239

yes

characteristic

126

C₂

#72216298

yes

—

	subgroup	G/H
center, socle	C₂²	#135778
commutator	C₈×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
prime radical	C₂³	#12523
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×C₁₆² ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#135778

commutator

C₈×C₁₆

C₂²×C₄

Frattini, cosocle

C₂×C₈×C₁₆

C₂³

prime radical

C₂³

#12523

LCS

G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×C₁₆² ◁ G

LECS

G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
87. C₃₂	#368	semidirect

H ≅ G/N

type

87. C₃₂

#368

semidirect

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

subgroup	index
#72216296	2
#72216239	2
C₁₆×M₆₄	2

subgroup

index

#72216296

#72216239

C₁₆×M₆₄

Subgroups — #72216301

Conjugacy classes of subgroups

Complements

Maximal subgroups