Subgroups — C₂²×C₇×#226 #13062091

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×C₇×Q₁₆	—	3	1	no	maximal, Hall
4	C₂²×C₇×Dic₃	—	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	—	3	1	no	—
8	C₂²×C₇×Q₈	D₃	1	1	yes	characteristic
9	C₂×C₇×Q₁₆	—	3	6	no	—
10	C₂×C₇×Q₁₆	—	3	6	no	—
11	C₂²×C₇×C₈	—	3	1	no	—
12	C₂²×#226	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂³×C₃×C₇	—	4	1	no	—
14	C₂×C₇×Dic₃	—	4	6	no	—
15	C₇×SL(2,𝔽₃)	C₂³	1	1	yes	characteristic
16	C₂×C₇×Q₈	—	3	3	no	—
17	C₂×C₇×Q₈	—	3	6	no	—
18	C₂×C₇×Q₈	—	6	3	no	—
19	C₂×C₇×Q₈	C₂×D₃	1	3	yes	—
20	C₂²×C₄×C₇	—	3	1	no	—
21	C₂²×C₄×C₇	—	6	1	no	—
22	C₇×Q₁₆	—	3	12	no	—
23	C₇×Q₁₆	—	3	4	no	—
24	C₂×C₇×C₈	—	3	6	no	—
25	C₂²×SL(2,𝔽₃)	C₂×C₇	1	1	yes	characteristic
26	C₂×#226	C₂×C₇	1	6	yes	direct product factor
27	C₂²×C₃×C₇	—	4	4	no	—
28	C₂²×C₃×C₇	—	4	3	no	—
29	C₇×Dic₃	—	4	4	no	—
30	C₂²×Q₁₆	—	3	1	no	Sylow
31	C₂³×C₇	S₄	1	1	yes	characteristic
32	C₇×Q₈	—	3	3	no	—
33	C₇×Q₈	—	3	4	no	—
34	C₇×Q₈	—	6	1	no	—
35	C₇×Q₈	—	6	6	no	—
36	C₇×Q₈	C₂²×D₃	1	1	yes	characteristic
37	C₂×C₄×C₇	—	3	3	no	—
38	C₂×C₄×C₇	—	3	3	no	—
39	C₂×C₄×C₇	—	6	6	no	—
40	C₇×C₈	—	3	4	no	—
41	C₂²×Dic₃	—	4	1	no	—
42	C₂×SL(2,𝔽₃)	C₂²×C₇	1	3	yes	—
43	#226	C₂²×C₇	1	4	yes	direct product factor
44	C₂×C₃×C₇	—	4	6	no	—
45	C₂×C₃×C₇	—	4	1	no	—
46	C₂²×Q₈	—	3	1	no	—
47	C₂²×Q₈	D₃×C₇	1	1	yes	characteristic
48	C₂×Q₁₆	—	3	6	no	—
49	C₂×Q₁₆	—	3	6	no	—
50	C₂²×C₈	—	3	1	no	—
51	C₂²×C₇	#226	1	4	yes	direct product factor
52	C₂²×C₇	C₂×S₄	1	3	yes	—
53	C₄×C₇	—	3	1	no	—
54	C₄×C₇	—	3	3	no	—
55	C₄×C₇	—	6	4	no	—
56	C₂³×C₃	—	4	1	no	—
57	C₂×Dic₃	—	4	6	no	—
58	SL(2,𝔽₃)	C₂³×C₇	1	1	yes	characteristic
59	C₃×C₇	—	4	1	no	Hall
60	C₂×Q₈	—	3	3	no	—
61	C₂×Q₈	—	3	6	no	—
62	C₂×Q₈	—	6	3	no	—
63	C₂×Q₈	C₂×D₃×C₇	1	3	yes	—
64	C₂²×C₄	—	3	1	no	—
65	C₂²×C₄	—	6	1	no	—
66	Q₁₆	—	3	12	no	—
67	Q₁₆	—	3	4	no	—
68	C₂×C₈	—	3	6	no	—
69	C₂×C₇	C₂×#226	1	6	yes	direct product factor
70	C₂×C₇	C₂²×S₄	1	1	yes	characteristic
71	C₂²×C₃	—	4	4	no	—
72	C₂²×C₃	—	4	3	no	—
73	Dic₃	—	4	4	no	—
74	C₂³	C₇×S₄	1	1	yes	characteristic
75	Q₈	—	3	3	no	—
76	Q₈	—	3	4	no	—
77	Q₈	—	6	1	no	—
78	Q₈	—	6	6	no	—
79	Q₈	C₂²×D₃×C₇	1	1	yes	characteristic
80	C₂×C₄	—	3	3	no	—
81	C₂×C₄	—	3	3	no	—
82	C₂×C₄	—	6	6	no	—
83	C₈	—	3	4	no	—
84	C₇	C₂²×#226	1	1	yes	characteristic, direct product factor, 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	—	6	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, socle, hypercenter	C₂³×C₇	S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂³×C₇
Frattini	C₂	C₂²×C₇×S₄
Fitting	C₂²×C₇×Q₈	D₃
prime radical	C₂²×C₇×SL(2,𝔽₃)	C₂
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂³×C₇

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)
12. C₂²×#226	C₇	direct (N × H)
26. C₂×#226	C₂×C₇	direct (N × H)
43. #226	C₂²×C₇	direct (N × H)
51. C₂²×C₇	#226	direct (N × H)
69. C₂×C₇	C₂×#226	direct (N × H)
84. C₇	C₂²×#226	direct (N × H)
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×C₇×Q₁₆	3
C₂²×C₇×Dic₃	4
C₂²×#226	7

Subgroups — C22×C7×#226 #13062091

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#226 #13062091