Subgroups — C₇×D₄×C₃₂ #13190860

Order 1792.

214 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₃₂	C₂	1	2	yes	maximal, complemented
2	C₇×D₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₇×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×C₂²⋊C₃₂	C₂	1	2	yes	maximal, complemented
5	C₄×C₇×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
7	C₇×C₈×D₄	C₄	1	1	yes	characteristic
8	C₂×C₇×C₃₂	—	2	4	no	—
9	C₂×C₇×C₃₂	C₂²	1	2	yes	—
10	C₂×C₇×C₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₃₂	C₂²	1	1	yes	characteristic
12	C₇×C₄⋊C₁₆	C₄	1	1	yes	characteristic
13	C₇×C₂²⋊C₁₆	C₄	1	2	yes	—
14	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
15	D₄×C₃₂	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
17	C₄×C₇×D₄	C₈	1	1	yes	characteristic
18	C₂×C₇×C₁₆	—	2	2	no	—
19	C₂×C₇×C₁₆	—	2	2	no	—
20	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
23	C₇×C₄⋊C₈	C₈	1	1	yes	characteristic
24	C₇×C₂²⋊C₈	C₈	1	2	yes	—
25	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₇×C₃₂	—	2	2	no	—
27	C₇×C₃₂	—	2	1	no	—
28	C₇×C₃₂	D₄	1	2	yes	direct product factor
29	C₂²×C₃₂	C₂×C₇	1	2	yes	complemented
30	D₄×C₁₆	C₂×C₇	1	1	yes	characteristic
31	C₄⋊C₃₂	C₂×C₇	1	1	yes	characteristic, complemented
32	C₂²⋊C₃₂	C₂×C₇	1	2	yes	complemented
33	C₄×C₃₂	C₂×C₇	1	1	yes	characteristic, complemented
34	C₂×C₇×D₄	C₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
36	C₂×C₇×C₈	—	2	2	no	—
37	C₂×C₇×C₈	—	2	2	no	—
38	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
39	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
40	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
41	C₇×C₄⋊C₄	C₁₆	1	1	yes	characteristic
42	C₇×C₂²⋊C₄	C₁₆	1	2	yes	—
43	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
44	C₇×C₁₆	—	2	2	no	—
45	C₇×C₁₆	—	2	1	no	—
46	C₇×C₁₆	C₂×D₄	1	1	yes	characteristic
47	C₇×C₁₆	#41	1	1	yes	characteristic
48	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
49	C₈×D₄	C₄×C₇	1	1	yes	characteristic
50	C₂×C₃₂	—	2	4	no	—
51	C₂×C₃₂	C₂²×C₇	1	1	yes	characteristic
52	C₂×C₃₂	C₂²×C₇	1	2	yes	—
53	C₂×C₃₂	C₂²×C₇	1	1	yes	characteristic
54	C₄⋊C₁₆	C₄×C₇	1	1	yes	characteristic
55	C₂²⋊C₁₆	C₄×C₇	1	2	yes	—
56	C₄×C₁₆	C₂²×C₇	1	1	yes	characteristic
57	C₂³×C₇	C₂×C₁₆	1	2	yes	—
58	C₇×D₄	C₃₂	1	4	yes	direct product factor
59	C₂×C₄×C₇	—	2	2	no	—
60	C₂×C₄×C₇	—	2	2	no	—
61	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
62	C₂×C₄×C₇	C₂×C₁₆	1	2	yes	—
63	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
64	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
65	C₇×C₈	—	2	2	no	—
66	C₇×C₈	—	2	1	no	—
67	C₇×C₈	C₄×D₄	1	1	yes	characteristic
68	C₇×C₈	#131	1	1	yes	characteristic
69	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
70	C₄×D₄	C₇×C₈	1	1	yes	characteristic
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
74	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
75	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
76	C₄⋊C₈	C₇×C₈	1	1	yes	characteristic
77	C₂²⋊C₈	C₇×C₈	1	2	yes	—
78	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
79	C₃₂	—	2	2	no	—
80	C₃₂	—	2	1	no	—
81	C₃₂	C₇×D₄	1	2	yes	direct product factor
82	C₂²×C₇	—	2	2	no	—
83	C₂²×C₇	—	2	2	no	—
84	C₂²×C₇	C₂×C₃₂	1	4	yes	complemented
85	C₂²×C₇	C₂²×C₁₆	1	1	yes	characteristic
86	C₄×C₇	—	2	2	no	—
87	C₄×C₇	—	2	1	no	—
88	C₄×C₇	C₂×C₃₂	1	2	yes	complemented
89	C₄×C₇	C₈×D₄	1	1	yes	characteristic
90	C₄×C₇	#504	1	1	yes	characteristic
91	C₂×D₄	C₇×C₁₆	1	1	yes	characteristic
92	C₂²×C₄	C₂×C₇×C₈	1	2	yes	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	C₂×C₇×C₈	1	1	yes	characteristic
96	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
97	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
98	C₄⋊C₄	C₇×C₁₆	1	1	yes	characteristic
99	C₂²⋊C₄	C₇×C₁₆	1	2	yes	—
100	C₄²	C₂×C₇×C₈	1	1	yes	characteristic
101	C₁₆	—	2	2	no	—
102	C₁₆	—	2	1	no	—
103	C₁₆	C₂×C₇×D₄	1	1	yes	characteristic
104	C₁₆	C₇×#41	1	1	yes	characteristic
105	C₂×C₇	—	2	4	no	—
106	C₂×C₇	D₄×C₁₆	1	1	yes	characteristic
107	C₂×C₇	C₂²×C₃₂	1	1	yes	characteristic
108	C₂×C₇	#2258	1	1	yes	characteristic
109	C₂³	C₂×C₇×C₁₆	1	2	yes	—
110	D₄	C₇×C₃₂	1	4	yes	direct product factor
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	C₂×C₇×C₁₆	1	1	yes	characteristic
114	C₂×C₄	C₂×C₇×C₁₆	1	2	yes	—
115	C₂×C₄	C₂×C₇×C₁₆	1	1	yes	characteristic
116	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
117	C₈	—	2	1	no	—
118	C₈	—	2	2	no	—
119	C₈	C₄×C₇×D₄	1	1	yes	characteristic
120	C₈	C₇×#131	1	1	yes	characteristic
121	C₇	D₄×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	C₂×C₇×C₃₂	1	4	yes	complemented
125	C₂²	C₂²×C₇×C₁₆	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	2	2	no	—
128	C₄	C₂×C₇×C₃₂	1	2	yes	complemented
129	C₄	C₇×C₈×D₄	1	1	yes	characteristic
130	C₄	C₇×#504	1	1	yes	characteristic
131	C₂	—	2	4	no	—
132	C₂	C₇×D₄×C₁₆	1	1	yes	characteristic
133	C₂	C₂²×C₇×C₃₂	1	1	yes	characteristic
134	C₂	C₇×#2258	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇×C₃₂	C₂²
commutator	C₂	C₂²×C₇×C₃₂
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂²×C₇	C₂²×C₁₆
prime radical	C₂×C₇×D₄	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×C₃₂	C₂	semidirect
3. C₇×C₄⋊C₃₂	C₂	semidirect
4. C₇×C₂²⋊C₃₂	C₂	semidirect
5. C₄×C₇×C₃₂	C₂	semidirect
15. D₄×C₃₂	C₇	direct (N × H)
28. C₇×C₃₂	D₄	direct (N × H)
29. C₂²×C₃₂	C₂×C₇	semidirect
31. C₄⋊C₃₂	C₂×C₇	semidirect
32. C₂²⋊C₃₂	C₂×C₇	semidirect
33. C₄×C₃₂	C₂×C₇	semidirect
58. C₇×D₄	C₃₂	direct (N × H)
81. C₃₂	C₇×D₄	direct (N × H)
84. C₂²×C₇	C₂×C₃₂	semidirect
88. C₄×C₇	C₂×C₃₂	semidirect
110. D₄	C₇×C₃₂	direct (N × H)
121. C₇	D₄×C₃₂	direct (N × H)
124. C₂²	C₂×C₇×C₃₂	semidirect
128. C₄	C₂×C₇×C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₃₂	2
C₇×D₄×C₁₆	2
C₇×C₄⋊C₃₂	2
C₇×C₂²⋊C₃₂	2
C₄×C₇×C₃₂	2
D₄×C₃₂	7

Subgroups — C7×D4×C32 #13190860

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×D₄×C₃₂ #13190860