Subgroups — C₇×#5101 #13056136

Order 1344.

260 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#926	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#208	C₂	1	1	yes	maximal, characteristic
4	C₇×#815	C₂	1	2	yes	maximal, complemented
5	C₇×C₈×Dic₃	C₂	1	2	yes	maximal, complemented
6	C₇×#405	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₇×C₈	C₄	1	4	yes	complemented
9	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₇×#217	C₄	1	2	yes	—
12	C₇×C₄⋊Dic₃	C₄	1	2	yes	—
13	C₄×C₇×Dic₃	C₂²	1	2	yes	—
14	C₇×#208	C₄	1	4	yes	complemented
15	C₂×C₇×C₃⋊C₈	C₂²	1	2	yes	—
16	C₂×C₇×M₁₆	—	3	1	no	—
17	C₂²×C₇×C₈	D₃	1	1	yes	characteristic
18	C₇×#117	—	3	1	no	—
19	C₇×#97	—	3	2	no	—
20	C₄×C₇×C₈	—	3	2	no	—
21	#5101	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂×C₇×Dic₃	C₂×C₄	1	4	yes	—
28	C₃×C₇×C₈	C₂×C₄	1	2	yes	—
29	C₃×C₇×C₈	C₂×C₄	1	2	yes	—
30	C₇×C₃⋊C₈	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
32	C₇×M₁₆	—	3	4	no	—
33	C₂×C₇×C₈	—	3	2	no	—
34	C₂×C₇×C₈	Dic₃	1	4	yes	complemented
35	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
36	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
37	C₇×C₄⋊C₄	—	3	2	no	—
38	C₇×C₂²⋊C₄	—	3	2	no	—
39	C₄²×C₇	—	3	2	no	—
40	C₂²×C₃×C₈	C₂×C₇	1	1	yes	characteristic
41	#926	C₂×C₇	1	1	yes	characteristic
42	C₂×#208	C₂×C₇	1	1	yes	characteristic
43	#815	C₂×C₇	1	2	yes	complemented
44	C₈×Dic₃	C₂×C₇	1	2	yes	complemented
45	C₂²×C₃×C₇	—	2	2	no	—
46	C₂²×C₃×C₇	C₄²	1	2	yes	—
47	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₃×C₄×C₇	C₄²	1	2	yes	—
49	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
50	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₇×Dic₃	—	2	4	no	—
52	#405	—	3	1	no	Sylow
53	C₂³×C₇	C₄×D₃	1	1	yes	characteristic
54	C₂×C₄×C₇	—	3	4	no	—
55	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
56	C₂×C₄×C₇	C₄×D₃	1	2	yes	—
57	C₂×C₄×C₇	C₂×Dic₃	1	2	yes	—
58	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
59	C₇×C₈	—	3	4	no	—
60	C₇×C₈	C₂×Dic₃	1	2	yes	—
61	C₇×C₈	C₂×Dic₃	1	2	yes	—
62	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
63	C₂×C₃×C₈	C₄×C₇	1	4	yes	complemented
64	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
66	#217	C₄×C₇	1	2	yes	—
67	C₄⋊Dic₃	C₄×C₇	1	2	yes	—
68	C₄×Dic₃	C₂²×C₇	1	2	yes	—
69	#208	C₄×C₇	1	4	yes	complemented
70	C₂×C₃⋊C₈	C₂²×C₇	1	2	yes	—
71	C₂×C₃×C₇	—	2	2	no	—
72	C₂×C₃×C₇	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₃×C₇	#131	1	2	yes	—
74	C₂×M₁₆	—	3	1	no	—
75	C₂²×C₈	D₃×C₇	1	1	yes	characteristic
76	#117	—	3	1	no	—
77	#97	—	3	2	no	—
78	C₄×C₈	—	3	2	no	—
79	C₂²×C₇	—	2	2	no	—
80	C₂²×C₇	C₄×Dic₃	1	2	yes	—
81	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
82	C₄×C₇	—	6	4	no	—
83	C₄×C₇	C₄×Dic₃	1	2	yes	—
84	C₄×C₇	C₂×C₄×D₃	1	1	yes	characteristic
85	C₄×C₇	C₂²×Dic₃	1	1	yes	characteristic
86	C₂³×C₃	C₂×C₄×C₇	1	1	yes	characteristic
87	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
88	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
89	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
90	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
91	C₂×Dic₃	C₂×C₄×C₇	1	4	yes	—
92	C₃×C₈	C₂×C₄×C₇	1	2	yes	—
93	C₃×C₈	C₂×C₄×C₇	1	2	yes	—
94	C₃⋊C₈	C₂×C₄×C₇	1	4	yes	—
95	C₃×C₇	#405	1	1	yes	characteristic, complemented, Hall
96	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
97	M₁₆	—	3	4	no	—
98	C₂×C₈	—	3	2	no	—
99	C₂×C₈	C₇×Dic₃	1	4	yes	complemented
100	C₂×C₈	C₂×D₃×C₇	1	1	yes	characteristic
101	C₂×C₈	C₂×D₃×C₇	1	1	yes	characteristic
102	C₄⋊C₄	—	3	2	no	—
103	C₂²⋊C₄	—	3	2	no	—
104	C₄²	—	3	2	no	—
105	C₂×C₇	—	2	2	no	—
106	C₂×C₇	#901	1	2	yes	—
107	C₂×C₇	C₂×C₄×Dic₃	1	1	yes	characteristic
108	C₂²×C₃	—	2	2	no	—
109	C₂²×C₃	C₄²×C₇	1	2	yes	—
110	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
111	C₃×C₄	C₄²×C₇	1	2	yes	—
112	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
113	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
114	Dic₃	—	2	4	no	—
115	C₂³	C₄×D₃×C₇	1	1	yes	characteristic
116	C₂×C₄	—	3	4	no	—
117	C₂×C₄	C₄×D₃×C₇	1	1	yes	characteristic
118	C₂×C₄	C₄×D₃×C₇	1	2	yes	—
119	C₂×C₄	C₂×C₇×Dic₃	1	2	yes	—
120	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
121	C₈	—	3	4	no	—
122	C₈	C₂×C₇×Dic₃	1	2	yes	—
123	C₈	C₂×C₇×Dic₃	1	2	yes	—
124	C₇	#5101	1	1	yes	characteristic, direct product factor, Sylow
125	C₂×C₃	—	2	2	no	—
126	C₂×C₃	C₂×C₄²×C₇	1	1	yes	characteristic
127	C₂×C₃	C₇×#131	1	2	yes	—
128	C₂²	—	2	2	no	—
129	C₂²	C₄×C₇×Dic₃	1	2	yes	—
130	C₂²	C₂×C₄×D₃×C₇	1	1	yes	characteristic
131	C₄	—	6	4	no	—
132	C₄	C₄×C₇×Dic₃	1	2	yes	—
133	C₄	C₂×C₄×D₃×C₇	1	1	yes	characteristic
134	C₄	C₂²×C₇×Dic₃	1	1	yes	characteristic
135	C₃	C₇×#405	1	1	yes	characteristic, complemented, Sylow
136	C₂	—	2	2	no	—
137	C₂	C₇×#901	1	2	yes	—
138	C₂	C₂×C₄×C₇×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇×C₈	C₂×D₃
commutator	C₂×C₃	C₂×C₄²×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₂²×C₃×C₇×C₈	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂³×C₃×C₇	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₂²×C₇×C₈	D₃
hypocenter	C₃	C₇×#405
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₂²×C₇×C₈

Complements

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

N	H ≅ G/N	type
4. C₇×#815	C₂	semidirect
5. C₇×C₈×Dic₃	C₂	semidirect
8. C₂×C₃×C₇×C₈	C₄	semidirect
14. C₇×#208	C₄	semidirect
21. #5101	C₇	direct (N × H)
34. C₂×C₇×C₈	Dic₃	semidirect
43. #815	C₂×C₇	semidirect
44. C₈×Dic₃	C₂×C₇	semidirect
63. C₂×C₃×C₈	C₄×C₇	semidirect
69. #208	C₄×C₇	semidirect
95. C₃×C₇	#405	semidirect
99. C₂×C₈	C₇×Dic₃	semidirect
124. C₇	#5101	direct (N × H)
135. C₃	C₇×#405	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₇×C₈	2
C₇×#926	2
C₂×C₇×#208	2
C₇×#815	2
C₇×C₈×Dic₃	2
C₇×#405	3
#5101	7

Subgroups — C7×#5101 #13056136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5101 #13056136