Subgroups — M₃₂×#92418 #29588990

Order 15616.

354 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14519717	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
3	#21443735	C₂	1	2	yes	maximal, complemented
4	C₁₆×#92418	C₂	1	2	yes	maximal, complemented
5	C₄×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519717	C₄	1	2	yes	—
7	C₂×C₄×#92418	C₄	1	2	yes	—
8	#16839538	C₄	1	2	yes	—
9	C₄×#11758185	C₂²	1	2	yes	—
10	C₈×#92418	C₄	1	2	yes	—
11	C₈×#92418	C₂²	1	1	yes	characteristic
12	C₈×#92418	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₆₁	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#346	C₄	1	2	yes	—
16	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
17	C₂²×#92418	C₈	1	4	yes	—
18	#14519717	C₈	1	8	yes	complemented
19	C₂×#11758185	C₂×C₄	1	4	yes	—
20	C₄×#92418	C₈	1	4	yes	—
21	C₄×#92418	C₂×C₄	1	2	yes	—
22	C₄×#92418	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₆₁	C₈	1	8	yes	complemented
27	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
37	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
38	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₆₁	C₄²	1	1	yes	characteristic
41	C₄²×C₆₁	C₂×C₈	1	2	yes	—
42	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×#92418	—	2	8	no	—
44	C₂×#92418	C₂×C₈	1	4	yes	—
45	C₂×#92418	C₂×C₈	1	4	yes	—
46	C₁₆×C₆₁	C₂×C₈	1	8	yes	complemented
47	#11758185	C₂×C₈	1	8	yes	complemented
48	C₂³×C₆₁	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₆₁	—	2	4	no	—
50	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
51	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
53	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
54	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
56	C₈×C₆₁	C₄×C₈	1	2	yes	—
57	C₈×C₆₁	C₄×C₈	1	2	yes	—
58	C₈×C₆₁	C₂²×C₈	1	2	yes	—
59	C₈×C₆₁	C₂²×C₈	1	2	yes	—
60	#92418	—	2	4	no	—
61	#92418	M₃₂	1	8	yes	direct product factor
62	C₈×M₃₂	—	61	1	no	maximal, Sylow
63	C₂²×C₆₁	—	2	2	no	—
64	C₂²×C₆₁	C₈²	1	2	yes	—
65	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
66	C₄×C₆₁	—	2	2	no	—
67	C₄×C₆₁	C₈²	1	2	yes	—
68	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
69	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
70	C₄×C₆₁	C₂×M₃₂	1	4	yes	—
71	C₄×M₃₂	—	61	1	no	—
72	C₄×M₃₂	D₆₁	1	1	yes	characteristic
73	C₂×C₈²	—	61	1	no	—
74	#1311	—	61	2	no	—
75	C₈×C₁₆	—	61	2	no	—
76	C₂×C₆₁	—	2	2	no	—
77	C₂×C₆₁	C₂×C₈²	1	1	yes	characteristic
78	C₂×C₆₁	C₄×M₃₂	1	2	yes	—
79	C₂×M₃₂	—	61	2	no	—
80	C₂×M₃₂	Dic₆₁	1	2	yes	—
81	C₂×C₄×C₈	—	61	2	no	—
82	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
83	#346	—	61	2	no	—
84	#346	Dic₆₁	1	2	yes	—
85	C₄×C₁₆	—	61	2	no	—
86	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
87	C₈²	—	61	2	no	—
88	C₈²	—	61	1	no	—
89	C₈²	—	61	1	no	—
90	C₆₁	C₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	—	61	4	no	—
92	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
93	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
94	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
95	M₃₂	—	61	8	no	—
96	M₃₂	#92418	1	8	yes	direct product factor
97	C₂×C₁₆	—	61	4	no	—
98	C₂×C₁₆	C₂×Dic₆₁	1	4	yes	—
99	C₄×C₈	—	61	4	no	—
100	C₄×C₈	—	61	2	no	—
101	C₄×C₈	—	61	2	no	—
102	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
103	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
104	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
105	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
106	C₂²×C₄	C₈×D₆₁	1	2	yes	—
107	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
108	C₂×C₈	—	61	4	no	—
109	C₂×C₈	—	61	4	no	—
110	C₂×C₈	—	122	8	no	—
111	C₂×C₈	C₈×D₆₁	1	4	yes	—
112	C₂×C₈	C₂×#92418	1	4	yes	—
113	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
114	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
115	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
116	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
117	C₄²	C₈×D₆₁	1	2	yes	—
118	C₄²	C₄×Dic₆₁	1	1	yes	characteristic
119	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
120	C₁₆	—	61	8	no	—
121	C₁₆	C₂×#92418	1	8	yes	complemented
122	C₂³	C₈×Dic₆₁	1	1	yes	characteristic
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	C₈×Dic₆₁	1	2	yes	—
125	C₂×C₄	C₈×Dic₆₁	1	2	yes	—
126	C₂×C₄	C₈×Dic₆₁	1	1	yes	characteristic
127	C₂×C₄	C₄×#92418	1	2	yes	—
128	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
129	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
130	C₈	—	61	8	no	—
131	C₈	—	122	4	no	—
132	C₈	C₈×Dic₆₁	1	2	yes	—
133	C₈	C₄×#92418	1	2	yes	—
134	C₈	C₂×C₈×D₆₁	1	2	yes	—
135	C₈	C₂²×#92418	1	2	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	C₈×#92418	1	2	yes	—
138	C₂²	C₂×C₈×Dic₆₁	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	C₈×#92418	1	2	yes	—
141	C₄	C₂×C₈×Dic₆₁	1	1	yes	characteristic
142	C₄	C₂×C₄×#92418	1	1	yes	characteristic
143	C₄	M₃₂×D₆₁	1	4	yes	—
144	C₂	—	2	2	no	—
145	C₂	C₂×C₈×#92418	1	1	yes	characteristic
146	C₂	M₃₂×Dic₆₁	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₆₁
commutator	C₂×C₆₁	C₂×C₈²
Fitting	C₄×M₃₂×C₆₁	C₂
socle	C₂²×C₆₁	C₂×C₄×C₈
prime radical	C₂³×C₆₁	C₄×C₈
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	C₄×M₃₂	D₆₁
hypocenter	C₆₁	C₈×M₃₂
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄×C₈ ◁ C₄×M₃₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #21443735	C₂	semidirect
4. C₁₆×#92418	C₂	semidirect
18. #14519717	C₈	semidirect
26. M₃₂×C₆₁	C₈	semidirect
46. C₁₆×C₆₁	C₂×C₈	semidirect
47. #11758185	C₂×C₈	semidirect
61. #92418	M₃₂	direct (N × H)
90. C₆₁	C₈×M₃₂	semidirect
96. M₃₂	#92418	direct (N × H)
121. C₁₆	C₂×#92418	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519717	2
C₂×C₈×#92418	2
#21443735	2
C₁₆×#92418	2
C₄×M₃₂×C₆₁	2
C₈×M₃₂	61

Subgroups — M32×#92418 #29588990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₃₂×#92418 #29588990