Subgroups — M₃₂×#1055 #15771896

Order 3328.

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×#88052	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
3	#13112752	C₂	1	2	yes	maximal, complemented
4	C₁₆×#1055	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88052	C₄	1	2	yes	—
9	C₂×C₄×#1055	C₄	1	2	yes	—
10	C₁₃×#346	C₄	1	2	yes	—
11	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
12	#11718053	C₄	1	2	yes	—
13	C₄×#6090	C₂²	1	2	yes	—
14	C₈×#1055	C₄	1	2	yes	—
15	C₈×#1055	C₂²	1	1	yes	characteristic
16	C₈×#1055	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×#1055	C₈	1	4	yes	—
21	C₁₃×M₃₂	C₈	1	8	yes	complemented
22	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	#88052	C₈	1	8	yes	complemented
28	C₂×#6090	C₂×C₄	1	4	yes	—
29	C₄×#1055	C₈	1	4	yes	—
30	C₄×#1055	C₂×C₄	1	2	yes	—
31	C₄×#1055	C₂×C₄	1	2	yes	—
32	C₈×M₃₂	—	13	1	no	maximal, Sylow
33	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
38	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
39	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₁₃	C₄²	1	1	yes	characteristic
42	C₄²×C₁₃	C₂×C₈	1	2	yes	—
43	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×#1055	—	2	8	no	—
45	C₂×#1055	C₂×C₈	1	4	yes	—
46	C₂×#1055	C₂×C₈	1	4	yes	—
47	C₁₃×C₁₆	C₂×C₈	1	8	yes	complemented
48	#6090	C₂×C₈	1	8	yes	complemented
49	C₄×M₃₂	—	13	1	no	—
50	C₄×M₃₂	D₁₃	1	1	yes	characteristic
51	C₂×C₈²	—	13	1	no	—
52	#1311	—	13	2	no	—
53	C₈×C₁₆	—	13	2	no	—
54	C₂³×C₁₃	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	—	2	4	no	—
56	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
57	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
59	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
62	C₈×C₁₃	C₄×C₈	1	2	yes	—
63	C₈×C₁₃	C₄×C₈	1	2	yes	—
64	C₈×C₁₃	C₂²×C₈	1	2	yes	—
65	C₈×C₁₃	C₂²×C₈	1	2	yes	—
66	#1055	—	2	4	no	—
67	#1055	M₃₂	1	8	yes	direct product factor
68	C₂×M₃₂	—	13	2	no	—
69	C₂×M₃₂	Dic₁₃	1	2	yes	—
70	C₂×C₄×C₈	—	13	2	no	—
71	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
72	#346	—	13	2	no	—
73	#346	Dic₁₃	1	2	yes	—
74	C₄×C₁₆	—	13	2	no	—
75	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
76	C₈²	—	13	2	no	—
77	C₈²	—	13	1	no	—
78	C₈²	—	13	1	no	—
79	C₂²×C₁₃	—	2	2	no	—
80	C₂²×C₁₃	C₈²	1	2	yes	—
81	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
82	C₄×C₁₃	—	2	2	no	—
83	C₄×C₁₃	C₈²	1	2	yes	—
84	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
85	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
86	C₄×C₁₃	C₂×M₃₂	1	4	yes	—
87	C₂²×C₈	—	13	4	no	—
88	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
89	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
90	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
91	M₃₂	—	13	8	no	—
92	M₃₂	#1055	1	8	yes	direct product factor
93	C₂×C₁₆	—	13	4	no	—
94	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
95	C₄×C₈	—	13	4	no	—
96	C₄×C₈	—	13	2	no	—
97	C₄×C₈	—	13	2	no	—
98	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
99	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
100	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
102	C₂×C₁₃	—	2	2	no	—
103	C₂×C₁₃	C₂×C₈²	1	1	yes	characteristic
104	C₂×C₁₃	C₄×M₃₂	1	2	yes	—
105	C₂²×C₄	C₈×D₁₃	1	2	yes	—
106	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
107	C₂×C₈	—	13	4	no	—
108	C₂×C₈	—	13	4	no	—
109	C₂×C₈	—	26	8	no	—
110	C₂×C₈	C₈×D₁₃	1	4	yes	—
111	C₂×C₈	C₂×#1055	1	4	yes	—
112	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
113	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
114	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
115	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
116	C₄²	C₈×D₁₃	1	2	yes	—
117	C₄²	C₄×Dic₁₃	1	1	yes	characteristic
118	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
119	C₁₆	—	13	8	no	—
120	C₁₆	C₂×#1055	1	8	yes	complemented
121	C₁₃	C₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
122	C₂³	C₈×Dic₁₃	1	1	yes	characteristic
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	C₄×#1055	1	2	yes	—
125	C₂×C₄	C₈×Dic₁₃	1	2	yes	—
126	C₂×C₄	C₈×Dic₁₃	1	2	yes	—
127	C₂×C₄	C₈×Dic₁₃	1	1	yes	characteristic
128	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
129	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
130	C₈	—	13	8	no	—
131	C₈	—	26	4	no	—
132	C₈	C₄×#1055	1	2	yes	—
133	C₈	C₈×Dic₁₃	1	2	yes	—
134	C₈	C₂×C₈×D₁₃	1	2	yes	—
135	C₈	C₂²×#1055	1	2	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	C₈×#1055	1	2	yes	—
138	C₂²	C₂×C₈×Dic₁₃	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	C₈×#1055	1	2	yes	—
141	C₄	D₁₃×M₃₂	1	4	yes	—
142	C₄	C₂×C₄×#1055	1	1	yes	characteristic
143	C₄	C₂×C₈×Dic₁₃	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	C₂×C₈×#1055	1	1	yes	characteristic
146	C₂	M₃₂×Dic₁₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₁₃
commutator	C₂×C₁₃	C₂×C₈²
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂×C₄×C₈
prime radical	C₂³×C₁₃	C₄×C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	C₈×M₃₂
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #13112752	C₂	semidirect
4. C₁₆×#1055	C₂	semidirect
21. C₁₃×M₃₂	C₈	semidirect
27. #88052	C₈	semidirect
47. C₁₃×C₁₆	C₂×C₈	semidirect
48. #6090	C₂×C₈	semidirect
67. #1055	M₃₂	direct (N × H)
92. M₃₂	#1055	direct (N × H)
120. C₁₆	C₂×#1055	semidirect
121. C₁₃	C₈×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88052	2
C₂×C₈×#1055	2
#13112752	2
C₁₆×#1055	2
C₄×C₁₃×M₃₂	2
C₈×M₃₂	13

Subgroups — M32×#1055 #15771896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₃₂×#1055 #15771896