Subgroups — C₂²×C₄×M₃₂ #7622413

Order 512.

2794 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₃₂	C₂	1	2	yes	maximal
2	C₂³×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₃₂	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#346	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₈	C₄	1	1	yes	characteristic
7	C₂⁴×C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₄²	C₄	1	1	yes	characteristic
9	C₂²×M₃₂	C₄	1	32	yes	direct product factor
10	C₂²×M₃₂	C₂²	1	24	yes	—
11	C₂³×C₁₆	C₂²	1	4	yes	—
12	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
13	C₂²×C₄×C₈	C₄	1	12	yes	—
14	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
15	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₈	C₂²	1	12	yes	—
17	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₄×M₃₂	C₂²	1	64	yes	direct product factor
19	C₂×#346	C₂²	1	24	yes	complemented
20	C₂×C₄×C₁₆	C₂²	1	24	yes	complemented
21	C₂⁴×C₄	C₈	1	2	yes	—
22	C₂⁴×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₈	C₈	1	16	yes	—
24	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₈	C₂×C₄	1	6	yes	—
26	C₂³×C₈	C₂×C₄	1	16	yes	—
27	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₈	C₂×C₄	1	12	yes	—
29	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₈	C₂³	1	6	yes	—
31	C₂³×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₄²	C₈	1	12	yes	—
33	C₂²×C₄²	C₈	1	2	yes	—
34	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄²	C₂×C₄	1	12	yes	—
36	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₂×M₃₂	C₂×C₄	1	192	yes	direct product factor
38	C₂×M₃₂	C₂³	1	32	yes	—
39	C₂²×C₁₆	C₂×C₄	1	32	yes	complemented
40	C₂²×C₁₆	C₂³	1	24	yes	—
41	C₂×C₄×C₈	C₂×C₄	1	6	yes	—
42	C₂×C₄×C₈	C₂×C₄	1	48	yes	—
43	C₂×C₄×C₈	C₂×C₄	1	12	yes	—
44	C₂×C₄×C₈	C₂×C₄	1	12	yes	—
45	C₂×C₄×C₈	C₂×C₄	1	6	yes	—
46	C₂×C₄×C₈	C₂³	1	6	yes	—
47	C₂×C₄×C₈	C₂³	1	16	yes	—
48	C₂×C₄×C₈	C₂³	1	6	yes	—
49	#346	C₂³	1	32	yes	complemented
50	C₄×C₁₆	C₂³	1	32	yes	complemented
51	C₂⁵	C₂×C₈	1	1	yes	characteristic
52	C₂³×C₄	—	2	16	no	—
53	C₂³×C₄	C₄²	1	8	yes	—
54	C₂³×C₄	C₂×C₈	1	2	yes	—
55	C₂³×C₄	C₂×C₈	1	24	yes	—
56	C₂³×C₄	C₂×C₈	1	6	yes	—
57	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
58	C₂³×C₄	C₂×C₈	1	8	yes	—
59	C₂³×C₄	C₂×C₈	1	2	yes	—
60	C₂³×C₄	C₂²×C₄	1	6	yes	—
61	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₈	C₄²	1	48	yes	—
63	C₂²×C₈	C₄²	1	8	yes	—
64	C₂²×C₈	C₂×C₈	1	96	yes	—
65	C₂²×C₈	C₂×C₈	1	8	yes	—
66	C₂²×C₈	C₂×C₈	1	8	yes	—
67	C₂²×C₈	C₂²×C₄	1	12	yes	—
68	C₂²×C₈	C₂²×C₄	1	6	yes	—
69	C₂²×C₈	C₂²×C₄	1	3	yes	—
70	C₂²×C₈	C₂²×C₄	1	8	yes	—
71	C₂²×C₈	C₂²×C₄	1	48	yes	—
72	C₂²×C₈	C₂²×C₄	1	6	yes	—
73	C₂²×C₈	C₂²×C₄	1	16	yes	—
74	C₂²×C₈	C₂²×C₄	1	6	yes	—
75	C₂²×C₈	C₂⁴	1	3	yes	—
76	C₂²×C₈	C₂⁴	1	4	yes	—
77	C₂×C₄²	C₂×C₈	1	6	yes	—
78	C₂×C₄²	C₂×C₈	1	48	yes	—
79	C₂×C₄²	C₂×C₈	1	24	yes	—
80	C₂×C₄²	C₂×C₈	1	6	yes	—
81	C₂×C₄²	C₂²×C₄	1	6	yes	—
82	C₂×C₄²	C₂²×C₄	1	16	yes	—
83	C₂×C₄²	C₂²×C₄	1	6	yes	—
84	M₃₂	C₂²×C₄	1	128	yes	direct product factor
85	C₂×C₁₆	C₂²×C₄	1	96	yes	complemented
86	C₂×C₁₆	C₂⁴	1	16	yes	—
87	C₄×C₈	C₂²×C₄	1	12	yes	—
88	C₄×C₈	C₂²×C₄	1	16	yes	—
89	C₄×C₈	C₂²×C₄	1	16	yes	—
90	C₄×C₈	C₂²×C₄	1	12	yes	—
91	C₄×C₈	C₂⁴	1	4	yes	—
92	C₄×C₈	C₂⁴	1	4	yes	—
93	C₂⁴	—	2	8	no	—
94	C₂⁴	C₄×C₈	1	8	yes	—
95	C₂⁴	C₂²×C₈	1	6	yes	—
96	C₂⁴	C₂²×C₈	1	1	yes	characteristic
97	C₂²×C₄	—	2	8	no	—
98	C₂²×C₄	—	2	96	no	—
99	C₂²×C₄	C₄×C₈	1	48	yes	—
100	C₂²×C₄	C₄×C₈	1	8	yes	—
101	C₂²×C₄	M₃₂	1	16	yes	direct product factor
102	C₂²×C₄	C₂×C₄²	1	24	yes	—
103	C₂²×C₄	C₂×C₄²	1	4	yes	—
104	C₂²×C₄	C₂²×C₈	1	12	yes	—
105	C₂²×C₄	C₂²×C₈	1	32	yes	—
106	C₂²×C₄	C₂²×C₈	1	24	yes	—
107	C₂²×C₄	C₂²×C₈	1	4	yes	—
108	C₂²×C₄	C₂²×C₈	1	12	yes	—
109	C₂²×C₄	C₂²×C₈	1	3	yes	—
110	C₂²×C₄	C₂²×C₈	1	6	yes	—
111	C₂²×C₄	C₂²×C₈	1	12	yes	—
112	C₂²×C₄	C₂³×C₄	1	4	yes	—
113	C₂²×C₄	C₂³×C₄	1	3	yes	—
114	C₂×C₈	C₂×C₄²	1	12	yes	—
115	C₂×C₈	C₂×C₄²	1	48	yes	—
116	C₂×C₈	C₂×C₄²	1	24	yes	—
117	C₂×C₈	C₂²×C₈	1	64	yes	—
118	C₂×C₈	C₂²×C₈	1	24	yes	—
119	C₂×C₈	C₂²×C₈	1	24	yes	—
120	C₂×C₈	C₂³×C₄	1	4	yes	—
121	C₂×C₈	C₂³×C₄	1	4	yes	—
122	C₂×C₈	C₂³×C₄	1	12	yes	—
123	C₂×C₈	C₂³×C₄	1	16	yes	—
124	C₂×C₈	C₂³×C₄	1	3	yes	—
125	C₂×C₈	C₂³×C₄	1	4	yes	—
126	C₂×C₈	C₂⁵	1	1	yes	characteristic
127	C₄²	C₂²×C₈	1	12	yes	—
128	C₄²	C₂²×C₈	1	32	yes	—
129	C₄²	C₂²×C₈	1	12	yes	—
130	C₄²	C₂³×C₄	1	4	yes	—
131	C₄²	C₂³×C₄	1	4	yes	—
132	C₁₆	C₂³×C₄	1	32	yes	complemented
133	C₂³	—	2	32	no	—
134	C₂³	—	2	24	no	—
135	C₂³	C₂×C₄×C₈	1	24	yes	—
136	C₂³	C₂×C₄×C₈	1	4	yes	—
137	C₂³	C₂×M₃₂	1	8	yes	—
138	C₂³	C₂³×C₈	1	4	yes	—
139	C₂³	C₂³×C₈	1	3	yes	—
140	C₂×C₄	—	2	24	no	—
141	C₂×C₄	—	2	64	no	—
142	C₂×C₄	C₂×C₄×C₈	1	48	yes	—
143	C₂×C₄	C₂×C₄×C₈	1	12	yes	—
144	C₂×C₄	C₂×C₄×C₈	1	24	yes	—
145	C₂×C₄	C₂×M₃₂	1	48	yes	direct product factor
146	C₂×C₄	C₂²×C₄²	1	8	yes	—
147	C₂×C₄	C₂²×C₄²	1	6	yes	—
148	C₂×C₄	C₂³×C₈	1	8	yes	—
149	C₂×C₄	C₂³×C₈	1	3	yes	—
150	C₂×C₄	C₂³×C₈	1	4	yes	—
151	C₂×C₄	C₂³×C₈	1	8	yes	—
152	C₂×C₄	C₂³×C₈	1	6	yes	—
153	C₂×C₄	C₂³×C₈	1	8	yes	—
154	C₂×C₄	C₂⁴×C₄	1	1	yes	characteristic
155	C₈	C₂²×C₄²	1	6	yes	—
156	C₈	C₂²×C₄²	1	8	yes	—
157	C₈	C₂³×C₈	1	8	yes	—
158	C₈	C₂³×C₈	1	8	yes	—
159	C₈	C₂⁴×C₄	1	2	yes	—
160	C₂²	—	2	48	no	—
161	C₂²	—	2	8	no	—
162	C₂²	C₄×M₃₂	1	16	yes	direct product factor
163	C₂²	C₂²×C₄×C₈	1	8	yes	—
164	C₂²	C₂²×C₄×C₈	1	6	yes	—
165	C₂²	C₂²×M₃₂	1	12	yes	—
166	C₂²	C₂⁴×C₈	1	1	yes	characteristic
167	C₄	—	2	8	no	—
168	C₄	C₂²×C₄×C₈	1	8	yes	—
169	C₄	C₂²×C₄×C₈	1	6	yes	—
170	C₄	C₂²×M₃₂	1	16	yes	direct product factor
171	C₄	C₂³×C₄²	1	1	yes	characteristic
172	C₄	C₂⁴×C₈	1	1	yes	characteristic
173	C₂	—	2	8	no	—
174	C₂	C₂×C₄×M₃₂	1	12	yes	direct product factor
175	C₂	C₂³×C₄×C₈	1	1	yes	characteristic
176	C₂	C₂³×M₃₂	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₈	C₂²
commutator	C₂	C₂³×C₄×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂⁴	C₂²×C₈
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×M₃₂	C₂	direct (N × H)
4. C₂²×#346	C₂	semidirect
5. C₂²×C₄×C₁₆	C₂	semidirect
9. C₂²×M₃₂	C₄	direct (N × H)
18. C₄×M₃₂	C₂²	direct (N × H)
19. C₂×#346	C₂²	semidirect
20. C₂×C₄×C₁₆	C₂²	semidirect
37. C₂×M₃₂	C₂×C₄	direct (N × H)
39. C₂²×C₁₆	C₂×C₄	semidirect
49. #346	C₂³	semidirect
50. C₄×C₁₆	C₂³	semidirect
84. M₃₂	C₂²×C₄	direct (N × H)
85. C₂×C₁₆	C₂²×C₄	semidirect
101. C₂²×C₄	M₃₂	direct (N × H)
132. C₁₆	C₂³×C₄	semidirect
145. C₂×C₄	C₂×M₃₂	direct (N × H)
162. C₂²	C₄×M₃₂	direct (N × H)
170. C₄	C₂²×M₃₂	direct (N × H)
174. C₂	C₂×C₄×M₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×M₃₂	2
C₂³×C₄×C₈	2
C₂×C₄×M₃₂	2
C₂²×#346	2
C₂²×C₄×C₁₆	2

Subgroups — C22×C4×M32 #7622413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×M₃₂ #7622413