Subgroups — M₃₂×#10596833 #34009643

Order 18688.

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	#22626137	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#10596833	C₂	1	1	yes	maximal, characteristic
3	#22624842	C₂	1	2	yes	maximal, complemented
4	C₁₆×#10596833	C₂	1	2	yes	maximal, complemented
5	M₃₂×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556665	C₄	1	2	yes	—
7	C₂×C₄×#10596833	C₄	1	2	yes	—
8	#17989476	C₄	1	2	yes	—
9	#17989473	C₂²	1	2	yes	—
10	C₈×#10596833	C₄	1	2	yes	—
11	C₈×#10596833	C₂²	1	1	yes	characteristic
12	C₈×#10596833	C₂²	1	1	yes	characteristic
13	C₂×#14556531	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	#17988362	C₄	1	2	yes	—
16	C₁₆×Dic₇₃	C₂²	1	2	yes	—
17	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic
18	C₂²×#10596833	C₈	1	4	yes	—
19	#14556665	C₈	1	8	yes	complemented
20	C₂×#11924031	C₂×C₄	1	4	yes	—
21	C₄×#10596833	C₈	1	4	yes	—
22	C₄×#10596833	C₂×C₄	1	2	yes	—
23	C₄×#10596833	C₂×C₄	1	2	yes	—
24	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
26	#14556531	C₈	1	4	yes	complemented
27	C₂×#11924003	C₂×C₄	1	2	yes	—
28	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₇₃	C₈	1	4	yes	complemented
34	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
35	C₂×#10596833	—	2	8	no	—
36	C₂×#10596833	C₂×C₈	1	4	yes	—
37	C₂×#10596833	C₂×C₈	1	4	yes	—
38	#11924031	C₂×C₈	1	8	yes	complemented
39	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
40	C₂×#10596831	C₄²	1	1	yes	characteristic
41	C₂×#10596831	C₂×C₈	1	4	yes	—
42	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
43	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
44	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
45	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
46	#11924003	C₂×C₈	1	4	yes	complemented
47	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₁₆×C₇₃	C₂×C₈	1	4	yes	complemented
53	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
56	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
57	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
58	C₂×Dic₇₃	—	2	4	no	—
59	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
60	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
61	#10596833	—	2	4	no	—
62	#10596833	M₃₂	1	8	yes	direct product factor
63	C₈×C₇₃	C₄×C₈	1	2	yes	—
64	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
65	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
66	#10596831	C₄×C₈	1	2	yes	—
67	#10596831	C₂²×C₈	1	2	yes	—
68	C₂²×C₇₃	—	2	2	no	—
69	C₂²×C₇₃	C₈²	1	2	yes	—
70	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
71	C₄×C₇₃	C₈²	1	2	yes	—
72	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
73	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
74	Dic₇₃	—	2	2	no	—
75	Dic₇₃	C₂×M₃₂	1	4	yes	—
76	C₈×M₃₂	—	73	1	no	maximal, Sylow
77	C₂×C₇₃	—	2	2	no	—
78	C₂×C₇₃	C₂×C₈²	1	1	yes	characteristic
79	C₂×C₇₃	C₄×M₃₂	1	2	yes	—
80	C₄×M₃₂	—	73	1	no	—
81	C₄×M₃₂	—	73	1	no	—
82	C₂×C₈²	—	73	1	no	—
83	#1311	—	73	2	no	—
84	C₈×C₁₆	—	73	2	no	—
85	C₇₃	C₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₃₂	—	73	2	no	—
87	C₂×M₃₂	—	73	1	no	—
88	C₂×M₃₂	#64408	1	1	yes	characteristic
89	C₂×C₄×C₈	—	73	2	no	—
90	C₂×C₄×C₈	—	73	1	no	—
91	#346	—	73	2	no	—
92	#346	—	73	2	no	—
93	C₄×C₁₆	—	73	2	no	—
94	C₄×C₁₆	—	73	2	no	—
95	C₈²	—	73	1	no	—
96	C₈²	—	73	1	no	—
97	C₈²	—	73	2	no	—
98	C₂²×C₈	—	73	1	no	—
99	C₂²×C₈	—	73	4	no	—
100	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
101	C₂×C₄²	—	73	1	no	—
102	M₃₂	—	73	8	no	—
103	M₃₂	—	73	4	no	—
104	M₃₂	#10596833	1	4	yes	direct product factor
105	C₂×C₁₆	—	73	4	no	—
106	C₂×C₁₆	—	73	2	no	—
107	C₂×C₁₆	C₂×#64408	1	2	yes	—
108	C₄×C₈	—	73	2	no	—
109	C₄×C₈	—	73	2	no	—
110	C₄×C₈	—	73	4	no	—
111	C₄×C₈	—	73	1	no	—
112	C₄×C₈	—	73	1	no	—
113	C₄×C₈	—	73	1	no	—
114	C₄×C₈	—	73	1	no	—
115	C₂²×C₄	—	73	2	no	—
116	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
117	C₂×C₈	—	73	4	no	—
118	C₂×C₈	—	73	4	no	—
119	C₂×C₈	—	73	1	no	—
120	C₂×C₈	—	73	1	no	—
121	C₂×C₈	—	73	4	no	—
122	C₂×C₈	—	146	8	no	—
123	C₂×C₈	C₄×#64408	1	1	yes	characteristic
124	C₂×C₈	C₂×#10596833	1	2	yes	—
125	C₂×C₈	#11924036	1	2	yes	—
126	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
127	C₄²	—	73	2	no	—
128	C₄²	—	73	1	no	—
129	C₄²	—	73	1	no	—
130	C₁₆	—	73	8	no	—
131	C₁₆	—	73	4	no	—
132	C₁₆	C₂×#10596833	1	4	yes	complemented
133	C₂³	C₈×#64408	1	1	yes	characteristic
134	C₂×C₄	—	73	2	no	—
135	C₂×C₄	—	73	2	no	—
136	C₂×C₄	—	146	4	no	—
137	C₂×C₄	C₈×#64408	1	1	yes	characteristic
138	C₂×C₄	C₈×#64408	1	2	yes	—
139	C₂×C₄	C₄×#10596833	1	2	yes	—
140	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
141	C₈	—	73	8	no	—
142	C₈	—	73	2	no	—
143	C₈	—	73	2	no	—
144	C₈	—	146	4	no	—
145	C₈	C₄×#10596833	1	2	yes	—
146	C₈	C₂²×#10596833	1	1	yes	characteristic
147	C₈	C₂×#11924036	1	1	yes	characteristic
148	C₂²	—	2	2	no	—
149	C₂²	C₈×#10596833	1	2	yes	—
150	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
151	C₄	—	73	4	no	—
152	C₄	—	146	2	no	—
153	C₄	C₈×#10596833	1	2	yes	—
154	C₄	C₂×C₈×#64408	1	1	yes	characteristic
155	C₄	C₂×C₄×#10596833	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	C₂×C₈×#10596833	1	1	yes	characteristic
158	C₂	M₃₂×#64408	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#64408
commutator	C₂×C₇₃	C₂×C₈²
Fitting	C₂×M₃₂×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂³×C₇₃	C₄×C₈
cosocle	C₈×Dic₇₃	C₂³
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	C₈×M₃₂
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22624842	C₂	semidirect
4. C₁₆×#10596833	C₂	semidirect
19. #14556665	C₈	semidirect
26. #14556531	C₈	semidirect
33. M₃₂×C₇₃	C₈	semidirect
38. #11924031	C₂×C₈	semidirect
46. #11924003	C₂×C₈	semidirect
52. C₁₆×C₇₃	C₂×C₈	semidirect
62. #10596833	M₃₂	direct (N × H)
85. C₇₃	C₈×M₃₂	semidirect
104. M₃₂	#10596833	direct (N × H)
132. C₁₆	C₂×#10596833	semidirect

Maximal subgroups

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

subgroup	index
#22626137	2
C₂×C₈×#10596833	2
#22624842	2
C₁₆×#10596833	2
M₃₂×Dic₇₃	2
C₈×M₃₂	73

Subgroups — M32×#10596833 #34009643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₃₂×#10596833 #34009643