Subgroups — C₇×#4708 #13055744

Order 1344.

260 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
2	C₂×D₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#872	C₂	1	1	yes	maximal, characteristic
4	C₃×C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#815	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₇×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#405	—	3	1	no	maximal, Hall
9	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
13	C₇×(C₂×D₃)⋊C₄	C₄	1	2	yes	—
14	C₇×Dic₃⋊C₄	C₄	1	2	yes	—
15	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₇×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₇×ΓL(1,𝔽₂₅)	C₄	1	4	yes	complemented
19	D₃×C₇×C₈	C₄	1	4	yes	complemented
20	C₂×C₇×M₁₆	—	3	1	no	—
21	C₂²×C₇×C₈	—	3	1	no	—
22	C₇×#117	—	3	1	no	—
23	C₇×#97	—	3	1	no	—
24	C₇×#97	D₃	1	1	yes	characteristic, complemented
25	C₄×C₇×C₈	—	3	1	no	—
26	C₄×C₇×C₈	—	3	1	no	—
27	#4708	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₂²×D₃×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₂×C₇×Dic₃	C₂×C₄	1	2	yes	—
32	C₂×C₇×Dic₃	C₂×C₄	1	1	yes	characteristic
33	C₄×D₃×C₇	C₂×C₄	1	2	yes	—
34	C₄×D₃×C₇	C₂×C₄	1	2	yes	—
35	C₃×C₇×C₈	C₂×C₄	1	2	yes	—
36	C₃×C₇×C₈	C₂×C₄	1	2	yes	—
37	C₇×C₃⋊C₈	C₂×C₄	1	2	yes	—
38	C₇×C₃⋊C₈	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₇	—	3	1	no	—
40	C₇×M₁₆	—	3	4	no	—
41	C₂×C₇×C₈	—	3	1	no	—
42	C₂×C₇×C₈	—	3	1	no	—
43	C₂×C₇×C₈	—	3	4	no	—
44	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
45	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
46	C₇×C₄⋊C₄	—	3	2	no	—
47	C₇×C₂²⋊C₄	—	3	2	no	—
48	C₄²×C₇	—	3	1	no	—
49	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
50	C₂×ΓL(1,𝔽₂₅)	C₂×C₇	1	1	yes	characteristic
51	C₂×D₃×C₈	C₂×C₇	1	1	yes	characteristic
52	#872	C₂×C₇	1	1	yes	characteristic
53	C₃×#97	C₂×C₇	1	1	yes	characteristic, complemented
54	#815	C₂×C₇	1	1	yes	characteristic, complemented
55	C₈×Dic₃	C₂×C₇	1	1	yes	characteristic, complemented
56	C₄×C₃⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
57	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×D₃×C₇	—	2	2	no	—
59	C₂×D₃×C₇	C₄²	1	2	yes	—
60	C₃×C₄×C₇	—	2	2	no	—
61	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
62	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
63	C₇×Dic₃	—	2	2	no	—
64	C₇×Dic₃	C₄²	1	2	yes	—
65	#405	—	3	1	no	Sylow
66	C₂³×C₇	—	3	1	no	—
67	C₂×C₄×C₇	—	3	1	no	—
68	C₂×C₄×C₇	—	3	2	no	—
69	C₂×C₄×C₇	—	3	2	no	—
70	C₂×C₄×C₇	—	3	2	no	—
71	C₂×C₄×C₇	C₄×D₃	1	2	yes	—
72	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
73	C₇×C₈	—	3	2	no	—
74	C₇×C₈	—	3	2	no	—
75	C₇×C₈	C₄×D₃	1	2	yes	—
76	C₇×C₈	C₄×D₃	1	2	yes	—
77	C₂×C₄×D₃	C₂²×C₇	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
80	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
81	(C₂×D₃)⋊C₄	C₄×C₇	1	2	yes	—
82	Dic₃⋊C₄	C₄×C₇	1	2	yes	—
83	C₄×Dic₃	C₂²×C₇	1	1	yes	characteristic
84	C₂×C₃⋊C₈	C₂²×C₇	1	1	yes	characteristic
85	C₂×C₃⋊C₈	C₂²×C₇	1	1	yes	characteristic
86	ΓL(1,𝔽₂₅)	C₄×C₇	1	4	yes	complemented
87	D₃×C₈	C₄×C₇	1	4	yes	complemented
88	C₂×C₃×C₇	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₃×C₇	#131	1	2	yes	—
90	D₃×C₇	—	2	2	no	—
91	C₂×M₁₆	—	3	1	no	—
92	C₂²×C₈	—	3	1	no	—
93	#117	—	3	1	no	—
94	#97	—	3	1	no	—
95	#97	D₃×C₇	1	1	yes	characteristic, complemented
96	C₄×C₈	—	3	1	no	—
97	C₄×C₈	—	3	1	no	—
98	C₂²×C₇	—	3	2	no	—
99	C₂²×C₇	—	6	2	no	—
100	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
101	C₄×C₇	—	2	2	no	—
102	C₄×C₇	—	3	2	no	—
103	C₄×C₇	—	6	2	no	—
104	C₄×C₇	C₂×C₄×D₃	1	1	yes	characteristic
105	C₄×C₇	C₂×C₄×D₃	1	1	yes	characteristic
106	C₂²×D₃	C₂×C₄×C₇	1	1	yes	characteristic
107	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
108	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
109	C₂×Dic₃	C₂×C₄×C₇	1	2	yes	—
110	C₂×Dic₃	C₂×C₄×C₇	1	1	yes	characteristic
111	C₄×D₃	C₂×C₄×C₇	1	2	yes	—
112	C₄×D₃	C₂×C₄×C₇	1	2	yes	—
113	C₃×C₈	C₂×C₄×C₇	1	2	yes	—
114	C₃×C₈	C₂×C₄×C₇	1	2	yes	—
115	C₃⋊C₈	C₂×C₄×C₇	1	2	yes	—
116	C₃⋊C₈	C₂×C₄×C₇	1	2	yes	—
117	C₃×C₇	#405	1	1	yes	characteristic, complemented, Hall
118	C₂²×C₄	—	3	1	no	—
119	M₁₆	—	3	4	no	—
120	C₂×C₈	—	3	1	no	—
121	C₂×C₈	—	3	1	no	—
122	C₂×C₈	—	3	4	no	—
123	C₂×C₈	C₂×D₃×C₇	1	1	yes	characteristic
124	C₂×C₈	C₂×D₃×C₇	1	1	yes	characteristic
125	C₄⋊C₄	—	3	2	no	—
126	C₂²⋊C₄	—	3	2	no	—
127	C₄²	—	3	1	no	—
128	C₄²	C₂×D₃×C₇	1	1	yes	characteristic
129	C₂×C₇	—	6	2	no	—
130	C₂×C₇	C₄²×D₃	1	1	yes	characteristic
131	C₂×C₇	#907	1	2	yes	—
132	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
133	C₂×D₃	—	2	2	no	—
134	C₂×D₃	C₄²×C₇	1	2	yes	—
135	C₃×C₄	—	2	2	no	—
136	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
137	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
138	Dic₃	—	2	2	no	—
139	Dic₃	C₄²×C₇	1	2	yes	—
140	C₂³	—	3	1	no	—
141	C₂×C₄	—	3	1	no	—
142	C₂×C₄	—	3	2	no	—
143	C₂×C₄	—	3	2	no	—
144	C₂×C₄	—	3	2	no	—
145	C₂×C₄	C₄×D₃×C₇	1	2	yes	—
146	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
147	C₈	—	3	2	no	—
148	C₈	—	3	2	no	—
149	C₈	C₄×D₃×C₇	1	2	yes	—
150	C₈	C₄×D₃×C₇	1	2	yes	—
151	C₇	#4708	1	1	yes	characteristic, direct product factor, Sylow
152	C₂×C₃	C₂×C₄²×C₇	1	1	yes	characteristic
153	C₂×C₃	C₇×#131	1	2	yes	—
154	D₃	—	2	2	no	—
155	C₂²	—	3	2	no	—
156	C₂²	—	6	2	no	—
157	C₂²	C₂×C₄×D₃×C₇	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	3	2	no	—
160	C₄	—	6	2	no	—
161	C₄	C₂×C₄×D₃×C₇	1	1	yes	characteristic
162	C₄	C₂×C₄×D₃×C₇	1	1	yes	characteristic
163	C₃	C₇×#405	1	1	yes	characteristic, complemented, Sylow
164	C₂	—	6	2	no	—
165	C₂	C₄²×D₃×C₇	1	1	yes	characteristic
166	C₂	C₇×#907	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₃
commutator	C₂×C₃	C₂×C₄²×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#97	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂²×D₃×C₇	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₇×#97	D₃
hypocenter	C₃	C₇×#405
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₇×#97

Complements

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

N	H ≅ G/N	type
4. C₃×C₇×#97	C₂	semidirect
5. C₇×#815	C₂	semidirect
6. C₇×C₈×Dic₃	C₂	semidirect
7. C₄×C₇×C₃⋊C₈	C₂	semidirect
18. C₇×ΓL(1,𝔽₂₅)	C₄	semidirect
19. D₃×C₇×C₈	C₄	semidirect
24. C₇×#97	D₃	semidirect
27. #4708	C₇	direct (N × H)
53. C₃×#97	C₂×C₇	semidirect
54. #815	C₂×C₇	semidirect
55. C₈×Dic₃	C₂×C₇	semidirect
56. C₄×C₃⋊C₈	C₂×C₇	semidirect
86. ΓL(1,𝔽₂₅)	C₄×C₇	semidirect
87. D₃×C₈	C₄×C₇	semidirect
95. #97	D₃×C₇	semidirect
117. C₃×C₇	#405	semidirect
151. C₇	#4708	direct (N × H)
163. 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
C₂×C₇×ΓL(1,𝔽₂₅)	2
C₂×D₃×C₇×C₈	2
C₇×#872	2
C₃×C₇×#97	2
C₇×#815	2
C₇×C₈×Dic₃	2
C₄×C₇×C₃⋊C₈	2
C₇×#405	3
#4708	7

Subgroups — C7×#4708 #13055744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#4708 #13055744