Subgroups — C₂²×C₇⋊#226 #13062112

Order 1344.

306 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇⋊#226	C₂	1	6	yes	maximal, direct product factor
3	C₂²×#1144	—	3	1	no	maximal, Hall
4	C₂²×Dic₂₁	—	4	1	no	maximal
5	C₂×C₇×SL(2,𝔽₃)	C₂²	1	3	yes	—
6	C₇⋊#226	C₂²	1	4	yes	direct product factor
7	C₂²×C₇×Q₈	D₃	1	1	yes	characteristic
8	C₂²×Dic₁₄	—	3	1	no	—
9	C₂×#1144	—	3	6	no	—
10	C₂×#1144	—	3	6	no	—
11	C₂²×C₇⋊C₈	—	3	1	no	—
12	C₂²×#226	—	7	1	no	maximal, Hall
13	C₂³×C₃×C₇	—	4	1	no	—
14	C₂×Dic₂₁	—	4	6	no	—
15	C₇×SL(2,𝔽₃)	C₂³	1	1	yes	characteristic
16	C₂×C₇×Q₈	—	3	3	no	—
17	C₂×C₇×Q₈	C₂×D₃	1	3	yes	—
18	C₂²×C₄×C₇	—	3	1	no	—
19	C₂²×Dic₇	—	6	1	no	—
20	C₂×Dic₁₄	—	3	6	no	—
21	C₂×Dic₁₄	—	6	3	no	—
22	#1144	—	3	12	no	—
23	#1144	—	3	4	no	—
24	C₂×C₇⋊C₈	—	3	6	no	—
25	C₂²×SL(2,𝔽₃)	D₇	1	1	yes	characteristic
26	C₂×#226	—	7	6	no	—
27	C₂²×C₃×C₇	—	4	4	no	—
28	C₂²×C₃×C₇	—	4	3	no	—
29	Dic₂₁	—	4	4	no	—
30	C₂²×Q₁₆	—	21	1	no	Sylow
31	C₂³×C₇	S₄	1	1	yes	characteristic
32	C₇×Q₈	—	3	3	no	—
33	C₇×Q₈	—	6	1	no	—
34	C₇×Q₈	C₂²×D₃	1	1	yes	characteristic
35	C₂×C₄×C₇	—	3	3	no	—
36	C₂×C₄×C₇	—	3	3	no	—
37	C₂×Dic₇	—	6	6	no	—
38	Dic₁₄	—	3	4	no	—
39	Dic₁₄	—	6	6	no	—
40	C₇⋊C₈	—	3	4	no	—
41	C₂²×Dic₃	—	28	1	no	—
42	C₂×SL(2,𝔽₃)	C₂×D₇	1	3	yes	—
43	#226	—	7	4	no	—
44	C₂×C₃×C₇	—	4	6	no	—
45	C₂×C₃×C₇	—	4	1	no	—
46	C₂²×Q₈	—	21	1	no	—
47	C₂²×Q₈	D₂₁	1	1	yes	characteristic
48	C₂×Q₁₆	—	21	6	no	—
49	C₂×Q₁₆	—	21	6	no	—
50	C₂²×C₈	—	21	1	no	—
51	C₂²×C₇	#226	1	4	yes	complemented
52	C₂²×C₇	C₂×S₄	1	3	yes	—
53	C₄×C₇	—	3	1	no	—
54	C₄×C₇	—	3	3	no	—
55	Dic₇	—	6	4	no	—
56	C₂³×C₃	—	4	1	no	—
57	C₂×Dic₃	—	28	6	no	—
58	SL(2,𝔽₃)	C₂²×D₇	1	1	yes	characteristic
59	C₃×C₇	—	4	1	no	Hall
60	C₂×Q₈	—	3	3	no	—
61	C₂×Q₈	—	21	6	no	—
62	C₂×Q₈	—	42	3	no	—
63	C₂×Q₈	C₂×D₂₁	1	3	yes	—
64	C₂²×C₄	—	3	1	no	—
65	C₂²×C₄	—	42	1	no	—
66	Q₁₆	—	21	12	no	—
67	Q₁₆	—	21	4	no	—
68	C₂×C₈	—	21	6	no	—
69	C₂×C₇	C₂×#226	1	6	yes	complemented
70	C₂×C₇	C₂²×S₄	1	1	yes	characteristic
71	C₂²×C₃	—	4	4	no	—
72	C₂²×C₃	—	4	3	no	—
73	Dic₃	—	28	4	no	—
74	C₂³	C₇⋊S₄	1	1	yes	characteristic
75	Q₈	—	3	3	no	—
76	Q₈	—	6	1	no	—
77	Q₈	—	21	4	no	—
78	Q₈	—	42	6	no	—
79	Q₈	C₂²×D₂₁	1	1	yes	characteristic
80	C₂×C₄	—	3	3	no	—
81	C₂×C₄	—	3	3	no	—
82	C₂×C₄	—	42	6	no	—
83	C₈	—	21	4	no	—
84	C₇	C₂²×#226	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₃	—	4	6	no	—
86	C₂×C₃	—	4	1	no	—
87	C₂²	C₇⋊#226	1	4	yes	direct product factor
88	C₂²	C₂×C₇⋊S₄	1	3	yes	—
89	C₄	—	3	1	no	—
90	C₄	—	3	3	no	—
91	C₄	—	42	4	no	—
92	C₃	—	4	1	no	Sylow
93	C₂	C₂×C₇⋊#226	1	6	yes	direct product factor
94	C₂	C₂²×C₇⋊S₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	C₇⋊S₄
commutator, cosocle, hypocenter	C₇×SL(2,𝔽₃)	C₂³
Frattini	C₂	C₂²×C₇⋊S₄
Fitting	C₂²×C₇×Q₈	D₃
socle	C₂³×C₇	S₄
prime radical	C₂²×C₇×SL(2,𝔽₃)	C₂
LCS	G ▷ C₇×SL(2,𝔽₃)
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂×C₇⋊#226	C₂	direct (N × H)
6. C₇⋊#226	C₂²	direct (N × H)
51. C₂²×C₇	#226	semidirect
69. C₂×C₇	C₂×#226	semidirect
84. C₇	C₂²×#226	semidirect
87. C₂²	C₇⋊#226	direct (N × H)
93. C₂	C₂×C₇⋊#226	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×SL(2,𝔽₃)	2
C₂×C₇⋊#226	2
C₂²×#1144	3
C₂²×Dic₂₁	4
C₂²×#226	7

Subgroups — C22×C7⋊#226 #13062112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇⋊#226 #13062112