Subgroups — #30770540

Order 17152.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562438	C₂	1	4	yes	maximal
2	#22559827	C₂	1	1	yes	maximal, characteristic
3	#22558126	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1710	C₂	1	1	yes	maximal, characteristic
5	#16863801	—	2	2	no	—
6	#16863776	C₂²	1	2	yes	—
7	#16863199	—	2	8	no	—
8	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	#500×C₆₇	C₂²	1	2	yes	—
10	#487×C₆₇	—	2	2	no	—
11	C₈²×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₆₈	—	2	4	no	—
13	#14524043	—	2	2	no	—
14	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
15	#14523977	—	4	4	no	—
16	C₄⋊Q₈×C₆₇	—	2	2	no	—
17	#107×C₆₇	—	2	4	no	—
18	#103×C₆₇	—	4	4	no	—
19	C₄×C₈×C₆₇	D₄	1	2	yes	—
20	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₃₄	—	4	2	no	—
22	Dic₂₆₈	—	4	8	no	—
23	C₄⋊Dic₆₇	—	4	4	no	—
24	C₆₇⋊C₁₆	—	2	4	no	—
25	C₂×Q₈×C₆₇	—	4	2	no	—
26	C₂×C₈×C₆₇	D₈	1	4	yes	—
27	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
28	C₄⋊C₄×C₆₇	—	4	2	no	—
29	C₄⋊C₄×C₆₇	—	4	2	no	—
30	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₆₇	—	8	4	no	—
32	C₂×C₄×C₆₇	—	8	2	no	—
33	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
35	C₂×Dic₆₇	—	8	2	no	—
36	Dic₁₃₄	—	8	4	no	—
37	C₈×C₆₇	—	2	4	no	—
38	C₈×C₆₇	#135	1	4	yes	—
39	C₂²×C₆₇	#493	1	1	yes	characteristic
40	C₄×C₆₇	—	16	2	no	—
41	C₄×C₆₇	#495	1	2	yes	—
42	C₄×C₆₇	#510	1	4	yes	—
43	Dic₆₇	—	16	2	no	—
44	#12514	—	67	1	no	maximal, Sylow
45	C₂×C₆₇	#1709	1	1	yes	characteristic
46	C₂×C₆₇	#2251	1	2	yes	—
47	#2241	—	67	4	no	—
48	#1713	—	67	1	no	—
49	#1710	D₆₇	1	1	yes	characteristic
50	#1311	—	67	1	no	—
51	C₆₇	#12514	1	1	yes	characteristic, complemented, Sylow
52	#500	—	134	2	no	—
53	#500	C₂×D₆₇	1	2	yes	—
54	#494	—	67	2	no	—
55	#487	—	2	2	no	—
56	#358	—	134	8	no	—
57	C₄×C₁₆	—	67	2	no	—
58	C₈²	C₂×D₆₇	1	1	yes	characteristic
59	C₂×Q₁₆	—	134	4	no	—
60	C₄⋊Q₈	—	2	2	no	—
61	C₄⋊Q₈	—	134	2	no	—
62	C₂×C₁₆	—	67	4	no	—
63	#107	—	2	4	no	—
64	#107	—	268	4	no	—
65	#103	—	4	4	no	—
66	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
67	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
68	C₂×Q₈	—	4	2	no	—
69	C₂×Q₈	—	268	2	no	—
70	Q₁₆	—	268	8	no	—
71	C₂×C₈	#11761613	1	4	yes	—
72	C₂×C₈	D₄×D₆₇	1	2	yes	—
73	C₄⋊C₄	—	4	2	no	—
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	268	4	no	—
76	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
77	C₁₆	—	134	4	no	—
78	Q₈	—	8	4	no	—
79	Q₈	—	536	4	no	—
80	C₂×C₄	—	8	2	no	—
81	C₂×C₄	—	536	2	no	—
82	C₂×C₄	#14524039	1	1	yes	characteristic
83	C₂×C₄	C₂×#11761613	1	2	yes	—
84	C₈	—	2	4	no	—
85	C₈	#14524066	1	4	yes	—
86	C₂²	#16863772	1	1	yes	characteristic
87	C₄	—	16	2	no	—
88	C₄	—	1072	2	no	—
89	C₄	#16863781	1	2	yes	—
90	C₄	#16863836	1	4	yes	—
91	C₂	#22559816	1	1	yes	characteristic
92	C₂	#22562481	1	2	yes	—

	subgroup	G/H
center	C₂²	#16863772
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1710	C₂
socle, prime radical	C₂²×C₆₇	#493
hypercenter	#1710	D₆₇
hypocenter	C₆₇	#12514
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

Complements

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

N	H ≅ G/N	type
51. C₆₇	#12514	semidirect

Maximal subgroups

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

subgroup	index
#22562438	2
#22559827	2
#22558126	2
C₆₇×#1710	2
#12514	67