Subgroups — C₆₇×#7451 #30686702

Order 17152.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2252	C₂	1	1	yes	maximal, characteristic
2	C₆₇×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1371	C₂	1	1	yes	maximal, characteristic
4	#500×C₆₇	—	2	1	no	—
5	C₂×C₃₂×C₆₇	—	2	1	no	—
6	#366×C₆₇	—	2	1	no	—
7	#366×C₆₇	C₄	1	2	yes	—
8	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	#335×C₆₇	—	2	1	no	—
10	C₄⋊Q₈×C₆₇	—	4	1	no	—
11	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
13	#107×C₆₇	—	2	2	no	—
14	#107×C₆₇	—	4	1	no	—
15	C₄⋊C₈×C₆₇	—	4	1	no	—
16	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
17	C₃₂×C₆₇	—	4	1	no	—
18	C₂×Q₈×C₆₇	—	8	1	no	—
19	C₂×C₈×C₆₇	—	8	1	no	—
20	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
22	C₄⋊C₄×C₆₇	—	4	2	no	—
23	C₄⋊C₄×C₆₇	—	8	1	no	—
24	C₄²×C₆₇	D₈	1	1	yes	characteristic
25	C₁₆×C₆₇	—	2	1	no	—
26	C₁₆×C₆₇	SD₁₆	1	1	yes	characteristic
27	C₁₆×C₆₇	Q₁₆	1	1	yes	characteristic
28	Q₈×C₆₇	—	16	2	no	—
29	C₂×C₄×C₆₇	—	8	2	no	—
30	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
31	C₂×C₄×C₆₇	D₁₆	1	1	yes	characteristic
32	C₂×C₄×C₆₇	SD₃₂	1	1	yes	characteristic
33	C₈×C₆₇	—	2	1	no	—
34	C₈×C₆₇	—	16	1	no	—
35	C₈×C₆₇	#101	1	1	yes	characteristic
36	C₈×C₆₇	#103	1	1	yes	characteristic
37	C₂²×C₆₇	#357	1	1	yes	characteristic
38	C₄×C₆₇	—	2	1	no	—
39	C₄×C₆₇	—	16	2	no	—
40	C₄×C₆₇	#332	1	1	yes	characteristic
41	C₄×C₆₇	#362	1	1	yes	characteristic
42	C₄×C₆₇	SD₆₄	1	1	yes	characteristic
43	C₄×C₆₇	Q₆₄	1	1	yes	characteristic
44	#7451	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂×C₆₇	#1364	1	1	yes	characteristic
46	C₂×C₆₇	#1416	1	1	yes	characteristic
47	C₂×C₆₇	#1419	1	1	yes	characteristic
48	#2252	C₂×C₆₇	1	1	yes	characteristic
49	C₄⋊C₃₂	C₂×C₆₇	1	1	yes	characteristic
50	#1371	C₂×C₆₇	1	1	yes	characteristic
51	C₆₇	#7451	1	1	yes	characteristic, direct product factor, Sylow
52	#500	—	2	1	no	—
53	C₂×C₃₂	—	2	1	no	—
54	#366	—	2	1	no	—
55	#366	C₄×C₆₇	1	2	yes	—
56	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
57	#335	—	2	1	no	—
58	C₄⋊Q₈	—	4	1	no	—
59	C₂×C₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
60	C₂×C₁₆	D₄×C₆₇	1	1	yes	characteristic
61	#107	—	2	2	no	—
62	#107	—	4	1	no	—
63	C₄⋊C₈	—	4	1	no	—
64	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
65	C₃₂	—	4	1	no	—
66	C₂×Q₈	—	8	1	no	—
67	C₂×C₈	—	8	1	no	—
68	C₂×C₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
69	C₂×C₈	SD₁₆×C₆₇	1	1	yes	characteristic
70	C₄⋊C₄	—	4	2	no	—
71	C₄⋊C₄	—	8	1	no	—
72	C₄²	D₈×C₆₇	1	1	yes	characteristic
73	C₁₆	—	2	1	no	—
74	C₁₆	SD₁₆×C₆₇	1	1	yes	characteristic
75	C₁₆	Q₁₆×C₆₇	1	1	yes	characteristic
76	Q₈	—	16	2	no	—
77	C₂×C₄	—	8	2	no	—
78	C₂×C₄	#102×C₆₇	1	1	yes	characteristic
79	C₂×C₄	D₁₆×C₆₇	1	1	yes	characteristic
80	C₂×C₄	SD₃₂×C₆₇	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	—	16	1	no	—
83	C₈	#101×C₆₇	1	1	yes	characteristic
84	C₈	#103×C₆₇	1	1	yes	characteristic
85	C₂²	#357×C₆₇	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	16	2	no	—
88	C₄	#332×C₆₇	1	1	yes	characteristic
89	C₄	#362×C₆₇	1	1	yes	characteristic
90	C₄	SD₆₄×C₆₇	1	1	yes	characteristic
91	C₄	Q₆₄×C₆₇	1	1	yes	characteristic
92	C₂	C₆₇×#1364	1	1	yes	characteristic
93	C₂	C₆₇×#1416	1	1	yes	characteristic
94	C₂	C₆₇×#1419	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₆₇	#357
commutator	C₂×C₁₆	C₂×C₄×C₆₇
Frattini, cosocle	C₄×C₁₆	C₂²×C₆₇
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ C₄×C₈×C₆₇ ◁ C₄×C₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
44. #7451	C₆₇	direct (N × H)
51. C₆₇	#7451	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2252	2
C₆₇×C₄⋊C₃₂	2
C₆₇×#1371	2
#7451	67

Subgroups — C67×#7451 #30686702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#7451 #30686702