Subgroups — #13274322

Order 1792.

202 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₇⋊C₃₂)	C₂	1	2	yes	maximal
3	C₄⋊(C₇⋊C₃₂)	C₂	1	2	yes	maximal, complemented
4	C₄×C₇⋊C₃₂	C₂	1	2	yes	maximal, complemented
5	C₂²×C₇×C₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
8	C₄×C₇×C₁₆	C₄	1	1	yes	characteristic
9	C₄×C₇×C₁₆	C₄	1	1	yes	characteristic
10	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
11	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₇⋊C₃₂	C₂²	1	4	yes	—
13	#13639	—	7	1	no	maximal, Sylow
14	C₂²×C₇×C₈	C₈	1	1	yes	characteristic
15	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₇	C₈	1	1	yes	characteristic
17	C₂×C₇×C₁₆	—	2	2	no	—
18	C₂×C₇×C₁₆	—	2	2	no	—
19	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
23	C₄×C₇×C₈	C₈	1	1	yes	characteristic
24	C₄×C₇×C₈	C₈	1	1	yes	characteristic
25	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₇⋊C₃₂	—	2	4	no	—
28	C₂×C₄×C₁₆	D₇	1	1	yes	characteristic
29	C₄⋊C₃₂	—	7	2	no	—
30	C₂²⋊C₃₂	—	7	2	no	—
31	C₄×C₃₂	—	7	2	no	—
32	C₂²×C₄×C₇	C₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₇	C₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₇×C₈	—	2	2	no	—
36	C₂×C₇×C₈	—	2	1	no	—
37	C₂×C₇×C₈	—	2	1	no	—
38	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
39	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
40	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
41	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₇	C₁₆	1	1	yes	characteristic
43	C₄²×C₇	C₁₆	1	1	yes	characteristic
44	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
46	C₇×C₁₆	—	2	1	no	—
47	C₇×C₁₆	—	2	1	no	—
48	C₇×C₁₆	M₁₆	1	2	yes	—
49	C₇×C₁₆	#41	1	2	yes	—
50	C₂²×C₁₆	Dic₇	1	1	yes	characteristic
51	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
52	C₂×C₄×C₈	Dic₇	1	1	yes	characteristic
53	C₂×C₃₂	—	7	4	no	—
54	C₄×C₁₆	Dic₇	1	1	yes	characteristic
55	C₄×C₁₆	Dic₇	1	1	yes	characteristic
56	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
57	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
58	C₂³×C₇	C₂×C₁₆	1	1	yes	characteristic
59	C₂×C₄×C₇	—	2	2	no	—
60	C₂×C₄×C₇	—	2	1	no	—
61	C₂×C₄×C₇	—	2	1	no	—
62	C₂×C₄×C₇	C₃₂	1	4	yes	complemented
63	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
64	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
65	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
67	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
68	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
69	C₇×C₈	—	2	1	no	—
70	C₇×C₈	—	2	1	no	—
71	C₇×C₈	M₃₂	1	2	yes	—
72	C₇×C₈	#117	1	1	yes	characteristic
73	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
74	C₂²×C₈	C₇⋊C₈	1	1	yes	characteristic
75	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
76	C₂×C₄²	C₇⋊C₈	1	1	yes	characteristic
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	C₂×Dic₇	1	1	yes	characteristic
80	C₂×C₁₆	C₂×Dic₇	1	1	yes	characteristic
81	C₂×C₁₆	C₂×Dic₇	1	1	yes	characteristic
82	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
83	C₄×C₈	C₇⋊C₈	1	1	yes	characteristic
84	C₄×C₈	C₇⋊C₈	1	1	yes	characteristic
85	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
87	C₃₂	—	14	4	no	—
88	C₂²×C₇	—	2	1	no	—
89	C₂²×C₇	—	2	1	no	—
90	C₂²×C₇	C₂×C₃₂	1	2	yes	—
91	C₂²×C₇	C₂²×C₁₆	1	1	yes	characteristic
92	C₄×C₇	—	2	1	no	—
93	C₄×C₇	—	2	1	no	—
94	C₄×C₇	C₂×C₃₂	1	2	yes	—
95	C₄×C₇	C₂×C₃₂	1	2	yes	—
96	C₄×C₇	M₆₄	1	2	yes	—
97	C₄×C₇	#431	1	1	yes	characteristic
98	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
99	C₂²×C₄	C₇⋊C₁₆	1	1	yes	characteristic
100	C₂²×C₄	C₇⋊C₁₆	1	1	yes	characteristic
101	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	C₂×C₇⋊C₈	1	1	yes	characteristic
106	C₂×C₈	C₂×C₇⋊C₈	1	1	yes	characteristic
107	C₂×C₈	C₂×C₇⋊C₈	1	1	yes	characteristic
108	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
109	C₄²	C₇⋊C₁₆	1	1	yes	characteristic
110	C₄²	C₇⋊C₁₆	1	1	yes	characteristic
111	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
112	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
113	C₁₆	—	2	1	no	—
114	C₁₆	—	2	1	no	—
115	C₁₆	#1136	1	2	yes	—
116	C₁₆	#1157	1	2	yes	—
117	C₂×C₇	—	2	2	no	—
118	C₂×C₇	#2162	1	1	yes	characteristic
119	C₂×C₇	C₂²×C₃₂	1	1	yes	characteristic
120	C₂×C₇	C₂×M₆₄	1	1	yes	characteristic
121	C₂³	C₂×C₇⋊C₁₆	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	C₇⋊C₃₂	1	4	yes	complemented
126	C₂×C₄	C₂×C₇⋊C₁₆	1	1	yes	characteristic
127	C₂×C₄	C₂×C₇⋊C₁₆	1	1	yes	characteristic
128	C₂×C₄	C₂×C₇⋊C₁₆	1	1	yes	characteristic
129	C₂×C₄	C₂×C₇⋊C₁₆	1	1	yes	characteristic
130	C₂×C₄	C₂×C₇⋊C₁₆	1	1	yes	characteristic
131	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	#6386	1	2	yes	—
135	C₈	C₂×#1136	1	1	yes	characteristic
136	C₈	#6489	1	1	yes	characteristic
137	C₇	#13639	1	1	yes	characteristic, complemented, Sylow
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	C₂×C₇⋊C₃₂	1	2	yes	—
141	C₂²	C₂²×C₇⋊C₁₆	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	C₂×C₇⋊C₃₂	1	2	yes	—
145	C₄	C₂×C₇⋊C₃₂	1	2	yes	—
146	C₄	#89294	1	2	yes	—
147	C₄	#89697	1	1	yes	characteristic
148	C₄	C₂×#6386	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	#11727498	1	1	yes	characteristic
151	C₂	C₂²×C₇⋊C₃₂	1	1	yes	characteristic
152	C₂	C₂×#89294	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₆	C₂×D₇
commutator	C₂×C₇	C₂²×C₃₂
Frattini	C₂×C₁₆	C₂²×D₇
Fitting	C₂×C₄×C₇×C₁₆	C₂
socle	C₂²×C₇	C₂²×C₁₆
prime radical	C₂³×C₇	C₂×C₁₆
cosocle	C₂×C₇×C₁₆	C₂³
hypercenter	C₂×C₄×C₁₆	D₇
hypocenter	C₇	#13639
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₇⋊C₃₂)	C₂	semidirect
4. C₄×C₇⋊C₃₂	C₂	semidirect
62. C₂×C₄×C₇	C₃₂	semidirect
125. C₂×C₄	C₇⋊C₃₂	semidirect
137. C₇	#13639	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×C₁₆	2
C₂²⋊(C₇⋊C₃₂)	2
C₄⋊(C₇⋊C₃₂)	2
C₄×C₇⋊C₃₂	2
#13639	7