Group #72225123 — subgroups

Subgroups — #72225123

Order 4096.

933 conjugacy classes of subgroups; only the 427 normal subgroups are shown.

normal subgroups only

This group has 933 conjugacy classes of subgroups — too many to draw. Only its 427 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	#72225067	C₂	1	1	yes	maximal, characteristic
2	#72225038	C₄	1	3	yes	complemented
3	#72216379	C₂	1	3	yes	maximal
4	#72216301	C₂	1	3	yes	maximal
5	#72216298	—	2	6	no	—
6	#72216296	—	2	6	no	—
7	#72216239	C₂²	1	3	yes	—
8	C₁₆×M₆₄	C₂²	1	3	yes	—
9	#72216197	C₂²	1	1	yes	characteristic
10	C₃₂²	C₄	1	1	yes	characteristic, complemented
11	#152407	—	4	12	no	—
12	#151055	—	4	12	no	—
13	C₈×M₆₄	D₄	1	6	yes	—
14	#135796	—	2	6	no	—
15	#135781	—	2	3	no	—
16	#135778	—	2	3	no	—
17	C₂×C₁₆²	C₂³	1	1	yes	characteristic
18	#93664	Q₈	1	6	yes	—
19	C₁₆×C₃₂	C₂×C₄	1	3	yes	—
20	C₁₆×C₃₂	C₂×C₄	1	3	yes	—
21	C₂×#1426	—	8	24	no	—
22	C₄×M₆₄	D₈	1	12	yes	—
23	#12909	—	4	6	no	—
24	#12906	—	4	6	no	—
25	#12023	—	4	12	no	—
26	C₂×C₈×C₁₆	C₂×D₄	1	3	yes	—
27	#8823	—	4	3	no	—
28	#8818	—	4	1	no	—
29	#7463	—	8	24	no	—
30	#7329	Q₁₆	1	12	yes	—
31	C₈×C₃₂	C₄⋊C₄	1	6	yes	—
32	C₈×C₃₂	C₄⋊C₄	1	6	yes	—
33	C₁₆²	C₂²×C₄	1	1	yes	characteristic
34	C₁₆²	C₂×Q₈	1	3	yes	—
35	C₂×M₆₄	SD₃₂	1	24	yes	—
36	C₂×#368	—	8	12	no	—
37	C₂×#367	—	8	12	no	—
38	C₂×C₄×C₁₆	C₂×D₈	1	6	yes	—
39	#1848	—	8	3	no	—
40	#1845	—	8	3	no	—
41	#1567	—	8	6	no	—
42	C₂×C₈²	#127	1	1	yes	characteristic
43	#1426	—	16	48	no	—
44	#1397	SD₃₂	1	24	yes	—
45	C₄×C₃₂	#107	1	12	yes	—
46	C₄×C₃₂	#107	1	12	yes	—
47	#1372	—	8	24	no	—
48	C₈×C₁₆	C₂×C₄⋊C₄	1	3	yes	—
49	C₈×C₁₆	C₄⋊Q₈	1	3	yes	—
50	C₈×C₁₆	C₂×Q₁₆	1	6	yes	—
51	C₂²×C₁₆	C₂×SD₃₂	1	12	yes	—
52	C₂×#108	—	16	6	no	—
53	C₂×#107	—	16	6	no	—
54	#423	—	16	3	no	—
55	C₂×C₄×C₈	#493	1	3	yes	—
56	#384	—	16	1	no	—
57	M₆₄	—	2	48	no	—
58	C₂×C₃₂	#367	1	24	yes	—
59	C₂×C₃₂	#367	1	24	yes	—
60	#368	—	16	24	no	—
61	#367	—	16	24	no	—
62	C₄×C₁₆	C₂×#107	1	6	yes	—
63	C₄×C₁₆	#500	1	6	yes	—
64	C₄×C₁₆	C₂×SD₃₂	1	12	yes	—
65	#335	—	16	12	no	—
66	C₈²	#384	1	1	yes	characteristic
67	C₈²	#494	1	3	yes	—
68	C₂×M₁₆	—	32	3	no	—
69	C₂²×C₈	#2248	1	6	yes	—
70	C₂×C₄⋊C₄	—	32	3	no	—
71	C₂×C₄²	#1712	1	1	yes	characteristic
72	C₂×C₁₆	—	2	24	no	—
73	C₂×C₁₆	C₂×#367	1	12	yes	—
74	C₂×C₁₆	#2254	1	12	yes	—
75	#108	—	32	12	no	—
76	#107	—	32	12	no	—
77	C₄⋊C₈	—	32	6	no	—
78	C₄×C₈	#1713	1	3	yes	—
79	C₄×C₈	#1845	1	3	yes	—
80	C₄×C₈	#2248	1	6	yes	—
81	C₃₂	—	2	24	no	—
82	C₃₂	#1426	1	48	yes	complemented
83	C₂²×C₄	—	64	1	no	—
84	C₂²×C₄	#12517	1	3	yes	—
85	M₁₆	—	64	6	no	—
86	C₂×C₈	—	2	12	no	—
87	C₂×C₈	—	64	3	no	—
88	C₂×C₈	#12523	1	6	yes	—
89	C₂×C₈	#12906	1	6	yes	—
90	C₄⋊C₄	—	64	6	no	—
91	C₄²	#8818	1	1	yes	characteristic
92	C₄²	#12517	1	3	yes	—
93	C₁₆	—	2	12	no	—
94	C₁₆	C₂×#1426	1	24	yes	—
95	C₂³	#123411	1	1	yes	characteristic
96	C₂×C₄	—	2	6	no	—
97	C₂×C₄	—	128	3	no	—
98	C₂×C₄	#123411	1	3	yes	—
99	C₂×C₄	#135778	1	3	yes	—
100	C₈	—	2	6	no	—
101	C₈	—	128	3	no	—
102	C₈	#152407	1	12	yes	—
103	C₂²	—	2	3	no	—
104	C₂²	#72216197	1	1	yes	characteristic
105	C₄	—	256	1	no	—
106	C₄	—	2	3	no	—
107	C₄	#72216298	1	6	yes	—
108	C₂	—	4	1	no	—
109	C₂	#72216379	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#72225067

C₂

yes

maximal, characteristic

#72225038

C₄

yes

complemented

#72216379

C₂

yes

maximal

#72216301

C₂

yes

—

—

C₂²

yes

—

C₁₆×M₆₄

C₂²

yes

—

#72216197

C₂²

yes

characteristic

C₃₂²

C₄

yes

characteristic, complemented

#152407

—

#151055

—

C₈×M₆₄

D₄

yes

—

#135796

—

#135781

—

#135778

—

C₂×C₁₆²

C₂³

yes

characteristic

#93664

Q₈

yes

—

C₁₆×C₃₂

C₂×C₄

yes

—

C₁₆×C₃₂

C₂×C₄

yes

—

C₂×#1426

—

C₄×M₆₄

D₈

yes

—

#12909

—

#12906

—

#12023

—

C₂×C₈×C₁₆

C₂×D₄

yes

—

—

—

—

Q₁₆

yes

—

C₈×C₃₂

C₄⋊C₄

yes

—

C₈×C₃₂

C₄⋊C₄

yes

—

C₁₆²

C₂²×C₄

yes

characteristic

C₁₆²

C₂×Q₈

yes

—

C₂×M₆₄

SD₃₂

yes

—

C₂×#368

—

C₂×#367

—

C₂×C₄×C₁₆

C₂×D₈

yes

—

#1848

—

#1845

—

#1567

—

C₂×C₈²

yes

—

SD₃₂

yes

—

C₄×C₃₂

#107

yes

—

C₄×C₃₂

#107

yes

—

#1372

—

C₈×C₁₆

C₂×C₄⋊C₄

yes

—

C₈×C₁₆

C₄⋊Q₈

yes

—

C₈×C₁₆

C₂×Q₁₆

yes

—

C₂²×C₁₆

C₂×SD₃₂

yes

—

C₂×#108

—

C₂×#107

—

#423

—

C₂×C₄×C₈

#493

yes

—

#384

—

M₆₄

—

C₂×C₃₂

#367

yes

—

C₂×C₃₂

#367

yes

—

#368

—

#367

—

C₄×C₁₆

C₂×#107

yes

—

C₄×C₁₆

#500

yes

—

C₄×C₁₆

C₂×SD₃₂

yes

—

#335

—

C₈²

#384

yes

characteristic

C₈²

#494

yes

—

C₂×M₁₆

—

C₂²×C₈

#2248

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

#1712

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×#367

yes

—

C₂×C₁₆

#2254

yes

—

#108

—

#107

—

C₄⋊C₈

—

C₄×C₈

#1713

yes

—

C₄×C₈

#1845

yes

—

C₄×C₈

#2248

yes

—

C₃₂

—

C₃₂

#1426

yes

complemented

C₂²×C₄

—

C₂²×C₄

#12517

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#12523

yes

—

C₂×C₈

#12906

yes

—

C₄⋊C₄

—

C₄²

#8818

yes

characteristic

C₄²

#12517

yes

—

C₁₆

—

C₁₆

C₂×#1426

yes

—

C₂³

#123411

yes

characteristic

C₂×C₄

—

C₂×C₄

—

128

—

C₂×C₄

#123411

yes

—

C₂×C₄

#135778

yes

—

100

C₈

—

101

C₈

—

128

—

102

C₈

#152407

yes

—

103

C₂²

—

104

C₂²

#72216197

yes

characteristic

105

C₄

—

256

—

106

C₄

—

107

C₄

#72216298

yes

—

108

C₂

—

109

C₂

#72216379

yes

—

	subgroup	G/H
center, socle	C₂²	#72216197
commutator	C₁₆²	C₂²×C₄
Frattini, cosocle	C₂×C₁₆²	C₂³
prime radical	C₂³	#123411
LCS	G ▷ C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×C₁₆² ◁ G
LECS	G ▷ C₂×C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#72216197

commutator

C₁₆²

C₂²×C₄

Frattini, cosocle

C₂×C₁₆²

C₂³

prime radical

C₂³

#123411

LCS

G ▷ C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

UCS

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

LECS

G ▷ C₂×C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. #72225038	C₄	semidirect
10. C₃₂²	C₄	semidirect
82. C₃₂	#1426	semidirect

H ≅ G/N

type

2. #72225038

C₄

semidirect

10. C₃₂²

C₄

semidirect

82. C₃₂

#1426

semidirect

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

subgroup	index
#72225067	2
#72216379	2
#72216301	2

subgroup

index

#72225067

#72216379

#72216301

Subgroups — #72225123

Conjugacy classes of subgroups

Complements

Maximal subgroups