Subgroups — C₂×#66633 #10621416

Order 672.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₄₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#66633	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#1142	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₇	—	2	1	no	—
7	C₃×C₇×D₄	—	2	1	no	—
8	C₃×C₇×D₄	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₁	—	2	1	no	—
11	Dic₄₂	—	2	1	no	—
12	Dic₄₂	C₂²	1	2	yes	complemented
13	C₃⋊(C₇⋊C₈)	C₂²	1	2	yes	complemented
14	C₂×C₇×D₄	D₃	1	1	yes	characteristic
15	C₂×Dic₁₄	—	3	1	no	—
16	#1142	—	3	4	no	—
17	C₂×C₇⋊C₈	—	3	1	no	—
18	C₂×#214	—	7	1	no	maximal, Hall
19	C₂²×C₃×C₇	—	2	2	no	—
20	C₂²×C₃×C₇	—	4	2	no	—
21	C₂²×C₃×C₇	D₄	1	1	yes	characteristic
22	C₃×C₄×C₇	D₄	1	1	yes	characteristic
23	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
24	Dic₂₁	—	2	2	no	—
25	C₂³×C₇	—	2	1	no	—
26	C₇×D₄	—	2	1	no	—
27	C₇×D₄	C₂×D₃	1	2	yes	—
28	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
29	C₂×Dic₇	—	6	1	no	—
30	Dic₁₄	—	3	2	no	—
31	Dic₁₄	—	6	1	no	—
32	C₇⋊C₈	—	3	2	no	—
33	C₂×C₃×D₄	D₇	1	1	yes	characteristic
34	C₂×Dic₆	—	7	1	no	—
35	#214	—	7	4	no	—
36	C₂×C₃⋊C₈	—	7	1	no	—
37	C₂×C₃×C₇	—	4	2	no	—
38	C₂×C₃×C₇	SD₁₆	1	2	yes	complemented
39	C₂×C₃×C₇	C₂×D₄	1	1	yes	characteristic
40	C₂×SD₁₆	—	21	1	no	Sylow
41	C₂²×C₇	—	2	2	no	—
42	C₂²×C₇	—	4	2	no	—
43	C₂²×C₇	#67	1	1	yes	characteristic
44	C₄×C₇	#67	1	1	yes	characteristic
45	C₄×C₇	C₂²×D₃	1	1	yes	characteristic
46	Dic₇	—	6	2	no	—
47	C₂³×C₃	—	2	1	no	—
48	C₃×D₄	—	2	1	no	—
49	C₃×D₄	C₂×D₇	1	2	yes	—
50	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
51	C₂×Dic₃	—	14	1	no	—
52	Dic₆	—	7	2	no	—
53	Dic₆	—	14	1	no	—
54	C₃⋊C₈	—	7	2	no	—
55	C₃×C₇	C₂×SD₁₆	1	1	yes	characteristic, complemented, Hall
56	C₂×Q₈	—	21	1	no	—
57	C₂×D₄	D₂₁	1	1	yes	characteristic
58	SD₁₆	—	21	4	no	—
59	C₂×C₈	—	21	1	no	—
60	C₂×C₇	—	4	2	no	—
61	C₂×C₇	#214	1	2	yes	complemented
62	C₂×C₇	C₂×#67	1	1	yes	characteristic
63	C₂²×C₃	—	2	2	no	—
64	C₂²×C₃	—	4	2	no	—
65	C₂²×C₃	C₂²⋊D₇	1	1	yes	characteristic
66	C₃×C₄	C₂²⋊D₇	1	1	yes	characteristic
67	C₃×C₄	C₂²×D₇	1	1	yes	characteristic
68	Dic₃	—	14	2	no	—
69	C₂³	—	2	1	no	—
70	Q₈	—	21	2	no	—
71	Q₈	—	42	1	no	—
72	D₄	—	2	1	no	—
73	D₄	C₂×D₂₁	1	2	yes	—
74	C₂×C₄	—	42	1	no	—
75	C₂×C₄	C₂×D₂₁	1	1	yes	characteristic
76	C₈	—	21	2	no	—
77	C₇	C₂×#214	1	1	yes	characteristic, complemented, Sylow
78	C₂×C₃	—	4	2	no	—
79	C₂×C₃	#1142	1	2	yes	complemented
80	C₂×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
81	C₂²	—	2	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	#4268	1	1	yes	characteristic
84	C₄	—	42	2	no	—
85	C₄	#4268	1	1	yes	characteristic
86	C₄	C₂²×D₂₁	1	1	yes	characteristic
87	C₃	C₂×#1142	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	4	2	no	—
89	C₂	#66633	1	2	yes	direct product factor
90	C₂	C₂×#4268	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#4268
commutator, cosocle	C₃×C₄×C₇	C₂³
Frattini	C₄	C₂²×D₂₁
Fitting, prime radical	C₂×C₃×C₇×D₄	C₂
socle	C₂²×C₃×C₇	D₄
hypercenter	C₂×D₄	D₂₁
hypocenter	C₃×C₇	C₂×SD₁₆
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₄₂	C₂	semidirect
3. #66633	C₂	direct (N × H)
4. C₂×C₃⋊(C₇⋊C₈)	C₂	semidirect
12. Dic₄₂	C₂²	semidirect
13. C₃⋊(C₇⋊C₈)	C₂²	semidirect
38. C₂×C₃×C₇	SD₁₆	semidirect
55. C₃×C₇	C₂×SD₁₆	semidirect
61. C₂×C₇	#214	semidirect
77. C₇	C₂×#214	semidirect
79. C₂×C₃	#1142	semidirect
87. C₃	C₂×#1142	semidirect
89. C₂	#66633	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×C₃×C₇×D₄	2
C₂×Dic₄₂	2
#66633	2
C₂×C₃⋊(C₇⋊C₈)	2
C₂×#1142	3
C₂×#214	7

Subgroups — C2×#66633 #10621416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#66633 #10621416