Subgroups — #13268504

Order 1792.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11726819	C₂	1	4	yes	maximal
2	#11725819	C₂	1	1	yes	maximal, characteristic
3	#11725818	C₂	1	1	yes	maximal, characteristic
4	C₇×#1311	C₂	1	1	yes	maximal, characteristic
5	#89471	C₂²	1	2	yes	—
6	#89454	—	2	2	no	—
7	#89454	C₂²	1	2	yes	—
8	#89453	—	2	2	no	—
9	C₄×C₇×C₁₆	C₂²	1	2	yes	—
10	C₇×C₈²	C₂²	1	1	yes	characteristic
11	#89298	—	2	8	no	—
12	#12514	—	7	1	no	maximal, Sylow
13	C₂×Dic₂₈	—	2	4	no	—
14	#6432	—	2	2	no	—
15	#6432	—	2	2	no	—
16	C₂×C₇×C₁₆	D₄	1	4	yes	—
17	C₄×C₇×C₈	D₄	1	2	yes	—
18	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
19	#6392	—	2	4	no	—
20	#6392	—	4	4	no	—
21	#6390	—	4	4	no	—
22	#2241	—	7	4	no	—
23	#1713	—	7	1	no	—
24	#1710	—	7	1	no	—
25	#1311	D₇	1	1	yes	characteristic
26	C₂×Dic₁₄	—	4	2	no	—
27	C₂×Dic₁₄	—	4	2	no	—
28	C₂×C₇×C₈	D₈	1	4	yes	—
29	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
30	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
31	C₄⋊Dic₇	—	4	2	no	—
32	C₄⋊Dic₇	—	4	2	no	—
33	C₄⋊Dic₇	—	4	4	no	—
34	Dic₂₈	—	4	8	no	—
35	C₇×C₁₆	—	2	4	no	—
36	#500	—	7	2	no	—
37	#500	—	14	2	no	—
38	#494	—	7	2	no	—
39	#487	—	14	2	no	—
40	#358	—	14	8	no	—
41	C₄×C₁₆	C₂×D₇	1	2	yes	—
42	C₈²	C₂×D₇	1	1	yes	characteristic
43	C₂×C₄×C₇	#127	1	1	yes	characteristic
44	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
45	C₂×Dic₇	—	8	2	no	—
46	C₂×Dic₇	—	8	2	no	—
47	Dic₁₄	—	8	4	no	—
48	Dic₁₄	—	8	4	no	—
49	C₇×C₈	—	2	4	no	—
50	C₇×C₈	#135	1	4	yes	—
51	C₂×Q₁₆	—	14	4	no	—
52	C₄⋊Q₈	—	14	2	no	—
53	C₄⋊Q₈	—	14	2	no	—
54	C₂×C₁₆	D₂₈	1	4	yes	—
55	#107	—	14	4	no	—
56	#107	—	28	4	no	—
57	#103	—	28	4	no	—
58	C₄×C₈	D₂₈	1	2	yes	—
59	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
60	C₂²×C₇	#493	1	1	yes	characteristic
61	C₄×C₇	#495	1	2	yes	—
62	C₄×C₇	#510	1	4	yes	—
63	Dic₇	—	16	2	no	—
64	Dic₇	—	16	2	no	—
65	C₂×Q₈	—	28	2	no	—
66	C₂×Q₈	—	28	2	no	—
67	Q₁₆	—	28	8	no	—
68	C₂×C₈	D₅₆	1	4	yes	—
69	C₂×C₈	C₂×D₂₈	1	2	yes	—
70	C₄⋊C₄	—	28	2	no	—
71	C₄⋊C₄	—	28	2	no	—
72	C₄⋊C₄	—	28	4	no	—
73	C₄²	C₂×D₂₈	1	1	yes	characteristic
74	C₁₆	—	2	4	no	—
75	C₂×C₇	#1709	1	1	yes	characteristic
76	C₂×C₇	#2251	1	2	yes	—
77	Q₈	—	56	4	no	—
78	Q₈	—	56	4	no	—
79	C₂×C₄	—	56	2	no	—
80	C₂×C₄	—	56	2	no	—
81	C₂×C₄	#6437	1	1	yes	characteristic
82	C₂×C₄	C₂×D₅₆	1	2	yes	—
83	C₈	—	2	4	no	—
84	C₈	#6467	1	4	yes	—
85	C₇	#12514	1	1	yes	characteristic, complemented, Sylow
86	C₂²	#89467	1	1	yes	characteristic
87	C₄	—	112	2	no	—
88	C₄	—	112	2	no	—
89	C₄	#89468	1	2	yes	—
90	C₄	#89681	1	4	yes	—
91	C₂	#11725814	1	1	yes	characteristic
92	C₂	#11726838	1	2	yes	—

	subgroup	G/H
center	C₂²	#89467
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1311	C₂
socle, prime radical	C₂²×C₇	#493
hypercenter	#1311	D₇
hypocenter	C₇	#12514
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

Complements

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

N	H ≅ G/N	type
85. C₇	#12514	semidirect

Maximal subgroups

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

subgroup	index
#11726819	2
#11725819	2
#11725818	2
C₇×#1311	2
#12514	7