Subgroups — M₃₂×#88324 #28307751

Order 13568.

354 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290921	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#88324	C₂	1	1	yes	maximal, characteristic
3	#20289626	C₂	1	2	yes	maximal, complemented
4	C₁₆×#88324	C₂	1	2	yes	maximal, complemented
5	M₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133291	C₄	1	2	yes	—
7	C₂×C₄×#88324	C₄	1	2	yes	—
8	#16811512	C₄	1	2	yes	—
9	#16811509	C₂²	1	2	yes	—
10	C₈×#88324	C₄	1	2	yes	—
11	C₈×#88324	C₂²	1	1	yes	characteristic
12	C₈×#88324	C₂²	1	1	yes	characteristic
13	C₂×#13133157	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	#16810398	C₄	1	2	yes	—
16	C₁₆×Dic₅₃	C₂²	1	2	yes	—
17	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
18	C₂²×#88324	C₈	1	4	yes	—
19	#13133291	C₈	1	8	yes	complemented
20	C₂×#11719910	C₂×C₄	1	4	yes	—
21	C₄×#88324	C₈	1	4	yes	—
22	C₄×#88324	C₂×C₄	1	2	yes	—
23	C₄×#88324	C₂×C₄	1	2	yes	—
24	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	#13133157	C₈	1	4	yes	complemented
27	C₂×#11719908	C₂×C₄	1	2	yes	—
28	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₅₃	C₈	1	4	yes	complemented
34	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
36	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
37	C₂×#88324	—	2	8	no	—
38	C₂×#88324	C₂×C₈	1	4	yes	—
39	C₂×#88324	C₂×C₈	1	4	yes	—
40	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
45	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
46	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×#88322	C₄²	1	1	yes	characteristic
48	C₂×#88322	C₂×C₈	1	4	yes	—
49	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
50	#11719910	C₂×C₈	1	8	yes	complemented
51	C₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
52	#11719908	C₂×C₈	1	4	yes	complemented
53	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
56	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
58	C₂×Dic₅₃	—	2	4	no	—
59	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
60	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
61	#88324	—	2	4	no	—
62	#88324	M₃₂	1	8	yes	direct product factor
63	C₈×C₅₃	C₄×C₈	1	2	yes	—
64	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
66	#88322	C₄×C₈	1	2	yes	—
67	#88322	C₂²×C₈	1	2	yes	—
68	C₈×M₃₂	—	53	1	no	maximal, Sylow
69	C₂²×C₅₃	—	2	2	no	—
70	C₂²×C₅₃	C₈²	1	2	yes	—
71	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
72	C₄×C₅₃	C₈²	1	2	yes	—
73	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
74	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
75	Dic₅₃	—	2	2	no	—
76	Dic₅₃	C₂×M₃₂	1	4	yes	—
77	C₄×M₃₂	—	53	1	no	—
78	C₄×M₃₂	—	53	1	no	—
79	C₂×C₈²	—	53	1	no	—
80	#1311	—	53	2	no	—
81	C₈×C₁₆	—	53	2	no	—
82	C₂×C₅₃	—	2	2	no	—
83	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
84	C₂×C₅₃	C₄×M₃₂	1	2	yes	—
85	C₂×M₃₂	—	53	2	no	—
86	C₂×M₃₂	—	53	1	no	—
87	C₂×M₃₂	#6157	1	1	yes	characteristic
88	C₂×C₄×C₈	—	53	2	no	—
89	C₂×C₄×C₈	—	53	1	no	—
90	#346	—	53	2	no	—
91	#346	—	53	2	no	—
92	C₄×C₁₆	—	53	2	no	—
93	C₄×C₁₆	—	53	2	no	—
94	C₈²	—	53	1	no	—
95	C₈²	—	53	1	no	—
96	C₈²	—	53	2	no	—
97	C₅₃	C₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₈	—	53	1	no	—
99	C₂²×C₈	—	53	4	no	—
100	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
101	C₂×C₄²	—	53	1	no	—
102	M₃₂	—	53	8	no	—
103	M₃₂	—	53	4	no	—
104	M₃₂	#88324	1	4	yes	direct product factor
105	C₂×C₁₆	—	53	4	no	—
106	C₂×C₁₆	—	53	2	no	—
107	C₂×C₁₆	C₂×#6157	1	2	yes	—
108	C₄×C₈	—	53	2	no	—
109	C₄×C₈	—	53	2	no	—
110	C₄×C₈	—	53	4	no	—
111	C₄×C₈	—	53	1	no	—
112	C₄×C₈	—	53	1	no	—
113	C₄×C₈	—	53	1	no	—
114	C₄×C₈	—	53	1	no	—
115	C₂²×C₄	—	53	2	no	—
116	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
117	C₂×C₈	—	53	4	no	—
118	C₂×C₈	—	53	4	no	—
119	C₂×C₈	—	53	1	no	—
120	C₂×C₈	—	53	1	no	—
121	C₂×C₈	—	53	4	no	—
122	C₂×C₈	—	106	8	no	—
123	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
124	C₂×C₈	C₄×#6157	1	1	yes	characteristic
125	C₂×C₈	C₂×#88324	1	2	yes	—
126	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
127	C₄²	—	53	2	no	—
128	C₄²	—	53	1	no	—
129	C₄²	—	53	1	no	—
130	C₁₆	—	53	8	no	—
131	C₁₆	—	53	4	no	—
132	C₁₆	C₂×#88324	1	4	yes	complemented
133	C₂³	C₈×#6157	1	1	yes	characteristic
134	C₂×C₄	—	53	2	no	—
135	C₂×C₄	—	53	2	no	—
136	C₂×C₄	—	106	4	no	—
137	C₂×C₄	C₈×#6157	1	2	yes	—
138	C₂×C₄	C₈×#6157	1	1	yes	characteristic
139	C₂×C₄	C₄×#88324	1	2	yes	—
140	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
141	C₈	—	53	8	no	—
142	C₈	—	53	2	no	—
143	C₈	—	53	2	no	—
144	C₈	—	106	4	no	—
145	C₈	C₄×#88324	1	2	yes	—
146	C₈	C₂²×#88324	1	1	yes	characteristic
147	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
148	C₂²	—	2	2	no	—
149	C₂²	C₈×#88324	1	2	yes	—
150	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
151	C₄	—	53	4	no	—
152	C₄	—	106	2	no	—
153	C₄	C₈×#88324	1	2	yes	—
154	C₄	C₂×C₈×#6157	1	1	yes	characteristic
155	C₄	C₂×C₄×#88324	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	C₂×C₈×#88324	1	1	yes	characteristic
158	C₂	M₃₂×#6157	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6157
commutator	C₂×C₅₃	C₂×C₈²
Fitting	C₂×M₃₂×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₈×Dic₅₃	C₂³
hypercenter	C₂×M₃₂	#6157
hypocenter	C₅₃	C₈×M₃₂
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20289626	C₂	semidirect
4. C₁₆×#88324	C₂	semidirect
19. #13133291	C₈	semidirect
26. #13133157	C₈	semidirect
33. M₃₂×C₅₃	C₈	semidirect
50. #11719910	C₂×C₈	semidirect
51. C₁₆×C₅₃	C₂×C₈	semidirect
52. #11719908	C₂×C₈	semidirect
62. #88324	M₃₂	direct (N × H)
97. C₅₃	C₈×M₃₂	semidirect
104. M₃₂	#88324	direct (N × H)
132. C₁₆	C₂×#88324	semidirect

Maximal subgroups

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

subgroup	index
#20290921	2
C₂×C₈×#88324	2
#20289626	2
C₁₆×#88324	2
M₃₂×Dic₅₃	2
C₈×M₃₂	53

Subgroups — M32×#88324 #28307751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₃₂×#88324 #28307751