Subgroups — C₇×#1849 #11730318

Order 896.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#108	C₂	1	4	yes	maximal
2	C₇×#423	C₂	1	2	yes	maximal
3	C₂×C₇×#97	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×M₁₆	C₂²	1	4	yes	—
5	C₂²×C₇×C₈	C₂²	1	2	yes	—
6	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₇×#108	—	2	8	no	—
8	C₇×C₄⋊C₈	—	2	4	no	—
9	C₇×#97	C₄	1	4	yes	—
10	#1849	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₄×C₇	Q₈	1	2	yes	—
12	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
13	C₇×M₁₆	—	2	8	no	—
14	C₂×C₇×C₈	—	2	4	no	—
15	C₂×C₇×C₈	C₂×C₄	1	4	yes	—
16	C₂×C₇×C₈	D₄	1	4	yes	—
17	C₂×C₇×C₈	Q₈	1	4	yes	—
18	C₄²×C₇	C₂×C₄	1	2	yes	—
19	C₄²×C₇	D₄	1	2	yes	—
20	C₂×#108	C₂×C₇	1	4	yes	—
21	#423	C₂×C₇	1	2	yes	—
22	C₂×#97	C₂×C₇	1	1	yes	characteristic
23	C₂³×C₇	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₄×C₇	—	4	2	no	—
25	C₂×C₄×C₇	C₄⋊C₄	1	4	yes	—
26	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
28	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
30	C₇×C₈	—	4	4	no	—
31	C₇×C₈	C₄⋊C₄	1	8	yes	—
32	C₂×M₁₆	C₂²×C₇	1	4	yes	—
33	C₂²×C₈	C₂²×C₇	1	2	yes	—
34	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
35	#108	—	2	8	no	—
36	C₄⋊C₈	—	2	4	no	—
37	#97	C₄×C₇	1	4	yes	—
38	C₂²×C₇	—	2	2	no	—
39	C₂²×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₇	C₄⋊Q₈	1	2	yes	—
41	C₄×C₇	—	2	4	no	—
42	C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
43	C₄×C₇	#127	1	1	yes	characteristic
44	C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₄	C₇×Q₈	1	2	yes	—
46	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
47	M₁₆	—	2	8	no	—
48	C₂×C₈	—	2	4	no	—
49	C₂×C₈	C₂×C₄×C₇	1	4	yes	—
50	C₂×C₈	C₇×D₄	1	4	yes	—
51	C₂×C₈	C₇×Q₈	1	4	yes	—
52	C₄²	C₂×C₄×C₇	1	2	yes	—
53	C₄²	C₇×D₄	1	2	yes	—
54	C₂×C₇	—	2	2	no	—
55	C₂×C₇	#384	1	1	yes	characteristic
56	C₂×C₇	#430	1	2	yes	—
57	C₂³	C₂×C₇×Q₈	1	1	yes	characteristic
58	C₂×C₄	—	4	2	no	—
59	C₂×C₄	C₇×C₄⋊C₄	1	4	yes	—
60	C₂×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
61	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
62	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
63	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
64	C₈	—	4	4	no	—
65	C₈	C₇×C₄⋊C₄	1	8	yes	—
66	C₇	#1849	1	1	yes	characteristic, direct product factor, Sylow
67	C₂²	—	2	2	no	—
68	C₂²	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
69	C₂²	C₇×C₄⋊Q₈	1	2	yes	—
70	C₄	—	2	4	no	—
71	C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
72	C₄	C₇×#127	1	1	yes	characteristic
73	C₄	C₇×C₄⋊Q₈	1	1	yes	characteristic
74	C₂	—	2	2	no	—
75	C₂	C₇×#384	1	1	yes	characteristic
76	C₂	C₇×#430	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂³×C₇
socle	C₂²×C₇	C₂×C₄⋊C₄
prime radical	C₂³×C₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
10. #1849	C₇	direct (N × H)
66. C₇	#1849	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×C₇×#108	2
C₇×#423	2
C₂×C₇×#97	2
#1849	7

Subgroups — C7×#1849 #11730318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#1849 #11730318