Subgroups — C₂×C₄×C₇²×C₃₂ #72190891

Order 12544.

1350 conjugacy classes of subgroups.

The subgroup lattice has 171 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇²×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇²×C₃₂	C₂	1	2	yes	maximal
3	C₄×C₇²×C₃₂	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₇²×C₈	C₄	1	1	yes	characteristic
5	C₂²×C₇²×C₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₇²×C₁₆	C₂²	1	1	yes	characteristic
7	C₄×C₇²×C₁₆	C₄	1	2	yes	—
8	C₄×C₇²×C₁₆	C₂²	1	2	yes	—
9	C₂×C₇²×C₃₂	C₄	1	8	yes	direct product factor
10	C₂×C₇²×C₃₂	C₂²	1	4	yes	—
11	C₂×C₄×C₇×C₃₂	C₇	1	8	yes	maximal, direct product factor
12	C₂²×C₇²×C₈	C₈	1	1	yes	characteristic
13	C₂²×C₇²×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₇²	C₈	1	1	yes	characteristic
15	C₂×C₇²×C₁₆	C₈	1	4	yes	—
16	C₂×C₇²×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₂×C₇²×C₁₆	C₂×C₄	1	2	yes	—
18	C₂×C₇²×C₁₆	C₂×C₄	1	4	yes	—
19	C₂×C₇²×C₁₆	C₂³	1	1	yes	characteristic
20	C₄×C₇²×C₈	C₈	1	2	yes	—
21	C₄×C₇²×C₈	C₂×C₄	1	2	yes	—
22	C₇²×C₃₂	C₂×C₄	1	8	yes	direct product factor
23	C₂²×C₇×C₃₂	C₂×C₇	1	16	yes	—
24	C₂×C₄×C₇×C₁₆	C₂×C₇	1	8	yes	—
25	C₄×C₇×C₃₂	C₂×C₇	1	32	yes	direct product factor
26	C₂²×C₄×C₇²	C₁₆	1	2	yes	—
27	C₂²×C₄×C₇²	C₂×C₈	1	1	yes	characteristic
28	C₂×C₇²×C₈	C₁₆	1	4	yes	—
29	C₂×C₇²×C₈	C₄²	1	2	yes	—
30	C₂×C₇²×C₈	C₂×C₈	1	1	yes	characteristic
31	C₂×C₇²×C₈	C₂×C₈	1	2	yes	—
32	C₂×C₇²×C₈	C₂×C₈	1	2	yes	—
33	C₂×C₇²×C₈	C₂²×C₄	1	1	yes	characteristic
34	C₄²×C₇²	C₁₆	1	2	yes	—
35	C₄²×C₇²	C₂×C₈	1	2	yes	—
36	C₇²×C₁₆	C₄²	1	2	yes	—
37	C₇²×C₁₆	C₂×C₈	1	4	yes	—
38	C₇²×C₁₆	C₂²×C₄	1	2	yes	—
39	C₂²×C₇×C₁₆	C₄×C₇	1	8	yes	—
40	C₂²×C₇×C₁₆	C₂²×C₇	1	8	yes	—
41	C₂×C₄×C₇×C₈	C₄×C₇	1	8	yes	—
42	C₂×C₇×C₃₂	C₄×C₇	1	64	yes	direct product factor
43	C₂×C₇×C₃₂	C₂²×C₇	1	32	yes	—
44	C₄×C₇×C₁₆	C₄×C₇	1	16	yes	—
45	C₄×C₇×C₁₆	C₂²×C₇	1	16	yes	—
46	C₂³×C₇²	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₇²	C₃₂	1	8	yes	direct product factor
48	C₂×C₄×C₇²	C₄×C₈	1	2	yes	—
49	C₂×C₄×C₇²	C₂×C₁₆	1	1	yes	characteristic
50	C₂×C₄×C₇²	C₂×C₁₆	1	2	yes	—
51	C₂×C₄×C₇²	C₂×C₁₆	1	4	yes	—
52	C₂×C₄×C₇²	C₂²×C₈	1	1	yes	characteristic
53	C₇²×C₈	C₄×C₈	1	2	yes	—
54	C₇²×C₈	C₂×C₁₆	1	4	yes	—
55	C₇²×C₈	C₂×C₄²	1	1	yes	characteristic
56	C₇²×C₈	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄×C₃₂	C₇²	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²×C₇×C₈	C₇×C₈	1	8	yes	—
59	C₂²×C₇×C₈	C₂×C₄×C₇	1	8	yes	—
60	C₂×C₄²×C₇	C₇×C₈	1	8	yes	—
61	C₂×C₇×C₁₆	C₇×C₈	1	32	yes	—
62	C₂×C₇×C₁₆	C₂×C₄×C₇	1	8	yes	—
63	C₂×C₇×C₁₆	C₂×C₄×C₇	1	16	yes	—
64	C₂×C₇×C₁₆	C₂×C₄×C₇	1	32	yes	—
65	C₂×C₇×C₁₆	C₂³×C₇	1	8	yes	—
66	C₄×C₇×C₈	C₇×C₈	1	16	yes	—
67	C₄×C₇×C₈	C₂×C₄×C₇	1	16	yes	—
68	C₇×C₃₂	C₂×C₄×C₇	1	64	yes	direct product factor
69	C₂²×C₇²	C₄×C₁₆	1	2	yes	—
70	C₂²×C₇²	C₂×C₃₂	1	4	yes	—
71	C₂²×C₇²	C₂²×C₁₆	1	1	yes	characteristic
72	C₄×C₇²	C₄×C₁₆	1	2	yes	—
73	C₄×C₇²	C₂×C₃₂	1	8	yes	direct product factor
74	C₄×C₇²	C₂×C₄×C₈	1	1	yes	characteristic
75	C₄×C₇²	C₂²×C₁₆	1	1	yes	characteristic
76	C₂²×C₃₂	C₂×C₇²	1	2	yes	—
77	C₂×C₄×C₁₆	C₂×C₇²	1	1	yes	characteristic
78	C₄×C₃₂	C₂×C₇²	1	4	yes	direct product factor
79	C₂²×C₄×C₇	C₇×C₁₆	1	16	yes	—
80	C₂²×C₄×C₇	C₂×C₇×C₈	1	8	yes	—
81	C₂×C₇×C₈	C₇×C₁₆	1	32	yes	—
82	C₂×C₇×C₈	C₄²×C₇	1	16	yes	—
83	C₂×C₇×C₈	C₂×C₇×C₈	1	8	yes	—
84	C₂×C₇×C₈	C₂×C₇×C₈	1	16	yes	—
85	C₂×C₇×C₈	C₂×C₇×C₈	1	16	yes	—
86	C₂×C₇×C₈	C₂²×C₄×C₇	1	8	yes	—
87	C₄²×C₇	C₇×C₁₆	1	16	yes	—
88	C₄²×C₇	C₂×C₇×C₈	1	16	yes	—
89	C₇×C₁₆	C₄²×C₇	1	16	yes	—
90	C₇×C₁₆	C₂×C₇×C₈	1	32	yes	—
91	C₇×C₁₆	C₂²×C₄×C₇	1	16	yes	—
92	C₂×C₇²	C₄×C₃₂	1	4	yes	direct product factor
93	C₂×C₇²	C₂×C₄×C₁₆	1	1	yes	characteristic
94	C₂×C₇²	C₂²×C₃₂	1	2	yes	—
95	C₂²×C₁₆	C₄×C₇²	1	1	yes	characteristic
96	C₂²×C₁₆	C₂²×C₇²	1	1	yes	characteristic
97	C₂×C₄×C₈	C₄×C₇²	1	1	yes	characteristic
98	C₂×C₃₂	C₄×C₇²	1	8	yes	direct product factor
99	C₂×C₃₂	C₂²×C₇²	1	4	yes	—
100	C₄×C₁₆	C₄×C₇²	1	2	yes	—
101	C₄×C₁₆	C₂²×C₇²	1	2	yes	—
102	C₂³×C₇	C₂×C₇×C₁₆	1	8	yes	—
103	C₂×C₄×C₇	C₇×C₃₂	1	64	yes	direct product factor
104	C₂×C₄×C₇	C₄×C₇×C₈	1	16	yes	—
105	C₂×C₄×C₇	C₂×C₇×C₁₆	1	8	yes	—
106	C₂×C₄×C₇	C₂×C₇×C₁₆	1	16	yes	—
107	C₂×C₄×C₇	C₂×C₇×C₁₆	1	32	yes	—
108	C₂×C₄×C₇	C₂²×C₇×C₈	1	8	yes	—
109	C₇×C₈	C₄×C₇×C₈	1	16	yes	—
110	C₇×C₈	C₂×C₇×C₁₆	1	32	yes	—
111	C₇×C₈	C₂×C₄²×C₇	1	8	yes	—
112	C₇×C₈	C₂²×C₇×C₈	1	8	yes	—
113	C₇²	C₂×C₄×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²×C₈	C₇²×C₈	1	1	yes	characteristic
115	C₂²×C₈	C₂×C₄×C₇²	1	1	yes	characteristic
116	C₂×C₄²	C₇²×C₈	1	1	yes	characteristic
117	C₂×C₁₆	C₇²×C₈	1	4	yes	—
118	C₂×C₁₆	C₂×C₄×C₇²	1	1	yes	characteristic
119	C₂×C₁₆	C₂×C₄×C₇²	1	2	yes	—
120	C₂×C₁₆	C₂×C₄×C₇²	1	4	yes	—
121	C₂×C₁₆	C₂³×C₇²	1	1	yes	characteristic
122	C₄×C₈	C₇²×C₈	1	2	yes	—
123	C₄×C₈	C₂×C₄×C₇²	1	2	yes	—
124	C₃₂	C₂×C₄×C₇²	1	8	yes	direct product factor
125	C₂²×C₇	C₄×C₇×C₁₆	1	16	yes	—
126	C₂²×C₇	C₂×C₇×C₃₂	1	32	yes	—
127	C₂²×C₇	C₂²×C₇×C₁₆	1	8	yes	—
128	C₄×C₇	C₄×C₇×C₁₆	1	16	yes	—
129	C₄×C₇	C₂×C₇×C₃₂	1	64	yes	direct product factor
130	C₄×C₇	C₂×C₄×C₇×C₈	1	8	yes	—
131	C₄×C₇	C₂²×C₇×C₁₆	1	8	yes	—
132	C₂²×C₄	C₇²×C₁₆	1	2	yes	—
133	C₂²×C₄	C₂×C₇²×C₈	1	1	yes	characteristic
134	C₂×C₈	C₇²×C₁₆	1	4	yes	—
135	C₂×C₈	C₄²×C₇²	1	2	yes	—
136	C₂×C₈	C₂×C₇²×C₈	1	1	yes	characteristic
137	C₂×C₈	C₂×C₇²×C₈	1	2	yes	—
138	C₂×C₈	C₂×C₇²×C₈	1	2	yes	—
139	C₂×C₈	C₂²×C₄×C₇²	1	1	yes	characteristic
140	C₄²	C₇²×C₁₆	1	2	yes	—
141	C₄²	C₂×C₇²×C₈	1	2	yes	—
142	C₁₆	C₄²×C₇²	1	2	yes	—
143	C₁₆	C₂×C₇²×C₈	1	4	yes	—
144	C₁₆	C₂²×C₄×C₇²	1	2	yes	—
145	C₂×C₇	C₄×C₇×C₃₂	1	32	yes	direct product factor
146	C₂×C₇	C₂×C₄×C₇×C₁₆	1	8	yes	—
147	C₂×C₇	C₂²×C₇×C₃₂	1	16	yes	—
148	C₂³	C₂×C₇²×C₁₆	1	1	yes	characteristic
149	C₂×C₄	C₇²×C₃₂	1	8	yes	direct product factor
150	C₂×C₄	C₄×C₇²×C₈	1	2	yes	—
151	C₂×C₄	C₂×C₇²×C₁₆	1	1	yes	characteristic
152	C₂×C₄	C₂×C₇²×C₁₆	1	2	yes	—
153	C₂×C₄	C₂×C₇²×C₁₆	1	4	yes	—
154	C₂×C₄	C₂²×C₇²×C₈	1	1	yes	characteristic
155	C₈	C₄×C₇²×C₈	1	2	yes	—
156	C₈	C₂×C₇²×C₁₆	1	4	yes	—
157	C₈	C₂×C₄²×C₇²	1	1	yes	characteristic
158	C₈	C₂²×C₇²×C₈	1	1	yes	characteristic
159	C₇	C₂×C₄×C₇×C₃₂	1	8	yes	direct product factor
160	C₂²	C₂×C₇²×C₃₂	1	4	yes	—
161	C₂²	C₄×C₇²×C₁₆	1	2	yes	—
162	C₂²	C₂²×C₇²×C₁₆	1	1	yes	characteristic
163	C₄	C₂×C₇²×C₃₂	1	8	yes	direct product factor
164	C₄	C₄×C₇²×C₁₆	1	2	yes	—
165	C₄	C₂²×C₇²×C₁₆	1	1	yes	characteristic
166	C₄	C₂×C₄×C₇²×C₈	1	1	yes	characteristic
167	C₂	C₄×C₇²×C₃₂	1	4	yes	direct product factor
168	C₂	C₂²×C₇²×C₃₂	1	2	yes	—
169	C₂	C₂×C₄×C₇²×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₁₆	C₂³×C₇²
socle, prime radical	C₂³×C₇²	C₂×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₇²×C₃₂	C₂	direct (N × H)
9. C₂×C₇²×C₃₂	C₄	direct (N × H)
11. C₂×C₄×C₇×C₃₂	C₇	direct (N × H)
22. C₇²×C₃₂	C₂×C₄	direct (N × H)
25. C₄×C₇×C₃₂	C₂×C₇	direct (N × H)
42. C₂×C₇×C₃₂	C₄×C₇	direct (N × H)
47. C₂×C₄×C₇²	C₃₂	direct (N × H)
57. C₂×C₄×C₃₂	C₇²	direct (N × H)
68. C₇×C₃₂	C₂×C₄×C₇	direct (N × H)
73. C₄×C₇²	C₂×C₃₂	direct (N × H)
78. C₄×C₃₂	C₂×C₇²	direct (N × H)
92. C₂×C₇²	C₄×C₃₂	direct (N × H)
98. C₂×C₃₂	C₄×C₇²	direct (N × H)
103. C₂×C₄×C₇	C₇×C₃₂	direct (N × H)
113. C₇²	C₂×C₄×C₃₂	direct (N × H)
124. C₃₂	C₂×C₄×C₇²	direct (N × H)
129. C₄×C₇	C₂×C₇×C₃₂	direct (N × H)
145. C₂×C₇	C₄×C₇×C₃₂	direct (N × H)
149. C₂×C₄	C₇²×C₃₂	direct (N × H)
159. C₇	C₂×C₄×C₇×C₃₂	direct (N × H)
163. C₄	C₂×C₇²×C₃₂	direct (N × H)
167. C₂	C₄×C₇²×C₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇²×C₁₆	2
C₂²×C₇²×C₃₂	2
C₄×C₇²×C₃₂	2
C₂×C₄×C₇×C₃₂	7

Subgroups — C2×C4×C72×C32 #72190891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₇²×C₃₂ #72190891