Subgroups — #13274310

Order 1792.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#89301	C₂	1	2	yes	maximal
3	#11725399	C₂	1	4	yes	maximal, complemented
4	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
5	C₂×#6393	—	2	2	no	—
6	C₇×M₆₄	C₄	1	4	yes	—
7	C₂×C₇×C₃₂	C₂²	1	2	yes	—
8	#89301	—	2	2	no	—
9	#89301	C₂²	1	4	yes	—
10	#13636	—	7	1	no	maximal, Sylow
11	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
12	C₂×#1136	—	4	2	no	—
13	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
14	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
15	C₂×C₇×C₁₆	Q₈	1	2	yes	—
16	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
17	#6393	—	2	4	no	—
18	#6393	—	4	2	no	—
19	C₇×C₃₂	C₂×C₄	1	4	yes	—
20	C₂×M₆₄	D₇	1	1	yes	characteristic
21	C₂×#368	—	7	2	no	—
22	#1425	—	7	4	no	—
23	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
24	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
25	C₂×C₇×C₈	D₈	1	1	yes	characteristic
26	C₂×C₇×C₈	Q₁₆	1	2	yes	—
27	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
28	#1136	—	4	4	no	—
29	#1136	—	8	2	no	—
30	C₂×C₇⋊C₈	—	8	2	no	—
31	C₇×C₁₆	C₄⋊C₄	1	2	yes	—
32	C₇×C₁₆	C₂²×C₄	1	1	yes	characteristic
33	C₇×C₁₆	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
35	C₂×#108	—	14	2	no	—
36	M₆₄	Dic₇	1	4	yes	—
37	C₂×C₃₂	C₂×D₇	1	2	yes	—
38	#368	—	7	4	no	—
39	#368	—	14	2	no	—
40	C₂³×C₇	Q₃₂	1	1	yes	characteristic
41	C₂×C₄×C₇	#107	1	2	yes	—
42	C₂×C₄×C₇	D₁₆	1	2	yes	—
43	C₂×C₄×C₇	Q₃₂	1	1	yes	characteristic
44	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
45	C₇×C₈	#107	1	2	yes	—
46	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
48	C₇⋊C₈	—	8	4	no	—
49	C₂×M₁₆	—	28	2	no	—
50	C₂²×C₈	D₂₈	1	1	yes	characteristic
51	C₂×C₁₆	Dic₁₄	1	2	yes	—
52	C₂×C₁₆	D₂₈	1	1	yes	characteristic
53	C₂×C₁₆	C₂×Dic₇	1	2	yes	—
54	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
55	#108	—	14	4	no	—
56	#108	—	28	2	no	—
57	C₃₂	C₂×Dic₇	1	4	yes	—
58	C₂²×C₇	—	4	1	no	—
59	C₂²×C₇	#366	1	2	yes	—
60	C₂²×C₇	C₂×Q₃₂	1	1	yes	characteristic
61	C₄×C₇	#366	1	2	yes	—
62	C₄×C₇	C₂×#107	1	1	yes	characteristic
63	C₄×C₇	C₂×D₁₆	1	1	yes	characteristic
64	C₂²×C₄	D₅₆	1	1	yes	characteristic
65	M₁₆	—	28	4	no	—
66	M₁₆	—	56	2	no	—
67	C₂×C₈	—	56	2	no	—
68	C₂×C₈	D₅₆	1	1	yes	characteristic
69	C₂×C₈	Dic₂₈	1	2	yes	—
70	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
71	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
72	C₁₆	C₄⋊Dic₇	1	2	yes	—
73	C₁₆	C₂×Dic₁₄	1	1	yes	characteristic
74	C₁₆	C₂²×Dic₇	1	1	yes	characteristic
75	C₂×C₇	—	2	2	no	—
76	C₂×C₇	—	2	1	no	—
77	C₂×C₇	C₂×#366	1	1	yes	characteristic
78	C₂³	Dic₅₆	1	1	yes	characteristic
79	C₂×C₄	D₁₁₂	1	2	yes	—
80	C₂×C₄	Dic₅₆	1	1	yes	characteristic
81	C₂×C₄	#6392	1	2	yes	—
82	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
83	C₈	—	56	4	no	—
84	C₈	#6392	1	2	yes	—
85	C₈	C₂×Dic₂₈	1	1	yes	characteristic
86	C₈	C₂×C₄⋊Dic₇	1	1	yes	characteristic
87	C₇	#13636	1	1	yes	characteristic, complemented, Sylow
88	C₂²	—	4	1	no	—
89	C₂²	#89299	1	2	yes	—
90	C₂²	C₂×Dic₅₆	1	1	yes	characteristic
91	C₄	#89299	1	2	yes	—
92	C₄	C₂×D₁₁₂	1	1	yes	characteristic
93	C₄	C₂×#6392	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	2	2	no	—
96	C₂	C₂×#89299	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₁₁₂
commutator	C₇×C₁₆	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇
Fitting	C₂×C₇×M₆₄	C₂
socle	C₂×C₇	C₂×#366
prime radical	C₂³×C₇	Q₃₂
cosocle	C₂×C₇×C₁₆	C₂³
hypercenter	C₂×M₆₄	D₇
hypocenter	C₇	#13636
LCS	G ▷ C₇×C₁₆ ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
3. #11725399	C₂	semidirect
87. C₇	#13636	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×M₆₄	2
C₂×#89301	2
#11725399	2
#13636	7