Subgroups — C₇×#11405 #13188621

Order 1792.

282 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic
3	C₇×#1450	C₂	1	1	yes	maximal, characteristic
4	C₇×#1313	C₂	1	4	yes	maximal, complemented
5	C₂×C₇×M₃₂	C₄	1	2	yes	—
6	C₂²×C₇×C₁₆	C₄	1	2	yes	—
7	C₇×#431	C₄	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₇×#346	C₄	1	2	yes	—
10	C₇×#346	C₂²	1	2	yes	—
11	C₇×#346	C₂²	1	2	yes	—
12	C₄×C₇×C₁₆	C₄	1	2	yes	—
13	C₇×#322	C₂²	1	2	yes	—
14	C₇×C₈²	C₄	1	2	yes	—
15	#11405	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
19	C₇×M₃₂	—	2	4	no	—
20	C₂×C₇×C₁₆	C₈	1	8	yes	complemented
21	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	—
22	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	—
23	C₇×C₄⋊C₈	—	2	2	no	—
24	C₇×C₂²⋊C₈	—	2	2	no	—
25	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
28	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₄×M₃₂	C₂×C₇	1	1	yes	characteristic
31	C₂×#346	C₂×C₇	1	1	yes	characteristic
32	#1450	C₂×C₇	1	1	yes	characteristic
33	#1313	C₂×C₇	1	4	yes	complemented
34	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
35	C₂²×C₄×C₇	M₁₆	1	2	yes	—
36	C₂×C₇×C₈	—	2	4	no	—
37	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
38	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
39	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
40	C₂×C₇×C₈	M₁₆	1	4	yes	—
41	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₇	C₄²	1	1	yes	characteristic
44	C₄²×C₇	M₁₆	1	2	yes	—
45	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
46	C₇×C₁₆	—	2	4	no	—
47	C₇×C₁₆	C₂×C₈	1	8	yes	—
48	C₂×M₃₂	C₄×C₇	1	2	yes	—
49	C₂²×C₁₆	C₄×C₇	1	2	yes	—
50	#431	C₄×C₇	1	2	yes	—
51	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
52	#346	C₄×C₇	1	2	yes	—
53	#346	C₂²×C₇	1	2	yes	—
54	#346	C₂²×C₇	1	2	yes	—
55	C₄×C₁₆	C₄×C₇	1	2	yes	—
56	#322	C₂²×C₇	1	2	yes	—
57	C₈²	C₄×C₇	1	2	yes	—
58	C₂³×C₇	#97	1	1	yes	characteristic
59	C₂×C₄×C₇	—	4	2	no	—
60	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
61	C₂×C₄×C₇	#97	1	2	yes	—
62	C₂×C₄×C₇	#97	1	1	yes	characteristic
63	C₂×C₄×C₇	#97	1	2	yes	—
64	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
66	C₇×C₈	—	4	4	no	—
67	C₇×C₈	C₄×C₈	1	2	yes	—
68	C₇×C₈	#97	1	2	yes	—
69	C₇×C₈	C₂²×C₈	1	2	yes	—
70	C₇×C₈	C₂×M₁₆	1	2	yes	—
71	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
72	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
73	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
74	M₃₂	—	2	4	no	—
75	C₂×C₁₆	C₇×C₈	1	8	yes	complemented
76	C₂×C₁₆	C₂×C₄×C₇	1	4	yes	—
77	C₂×C₁₆	C₂×C₄×C₇	1	4	yes	—
78	C₄⋊C₈	—	2	2	no	—
79	C₂²⋊C₈	—	2	2	no	—
80	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
81	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
82	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
83	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
84	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
85	C₂²×C₇	—	2	2	no	—
86	C₂²×C₇	#322	1	2	yes	—
87	C₂²×C₇	C₂×#97	1	1	yes	characteristic
88	C₄×C₇	—	2	2	no	—
89	C₄×C₇	—	2	2	no	—
90	C₄×C₇	#322	1	2	yes	—
91	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
92	C₄×C₇	C₂×#97	1	1	yes	characteristic
93	C₂²×C₄	C₄²×C₇	1	1	yes	characteristic
94	C₂²×C₄	C₇×M₁₆	1	2	yes	—
95	C₂×C₈	—	2	4	no	—
96	C₂×C₈	C₄²×C₇	1	1	yes	characteristic
97	C₂×C₈	C₄²×C₇	1	1	yes	characteristic
98	C₂×C₈	C₂×C₇×C₈	1	4	yes	—
99	C₂×C₈	C₇×M₁₆	1	4	yes	—
100	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
101	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
102	C₄²	C₄²×C₇	1	1	yes	characteristic
103	C₄²	C₇×M₁₆	1	2	yes	—
104	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
105	C₁₆	—	2	4	no	—
106	C₁₆	C₂×C₇×C₈	1	8	yes	—
107	C₂×C₇	—	2	2	no	—
108	C₂×C₇	C₂×#322	1	1	yes	characteristic
109	C₂×C₇	#2110	1	2	yes	—
110	C₂³	C₇×#97	1	1	yes	characteristic
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	C₄×C₇×C₈	1	2	yes	—
113	C₂×C₄	C₇×#97	1	2	yes	—
114	C₂×C₄	C₇×#97	1	1	yes	characteristic
115	C₂×C₄	C₇×#97	1	2	yes	—
116	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
117	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
118	C₈	—	4	4	no	—
119	C₈	C₄×C₇×C₈	1	2	yes	—
120	C₈	C₇×#97	1	2	yes	—
121	C₈	C₂²×C₇×C₈	1	2	yes	—
122	C₈	C₂×C₇×M₁₆	1	2	yes	—
123	C₇	#11405	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²	—	2	2	no	—
125	C₂²	C₇×#322	1	2	yes	—
126	C₂²	C₂×C₇×#97	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	C₇×#322	1	2	yes	—
130	C₄	C₂×C₄×C₇×C₈	1	1	yes	characteristic
131	C₄	C₂×C₇×#97	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×C₇×#322	1	1	yes	characteristic
134	C₂	C₇×#2110	1	2	yes	—

	subgroup	G/H
center	C₂×C₇×C₈	C₄²
commutator	C₄	C₂×C₄×C₇×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×#97
prime radical	C₂³×C₇	#97
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1313	C₂	semidirect
15. #11405	C₇	direct (N × H)
20. C₂×C₇×C₁₆	C₈	semidirect
33. #1313	C₂×C₇	semidirect
75. C₂×C₁₆	C₇×C₈	semidirect
123. C₇	#11405	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×M₃₂	2
C₂×C₇×#346	2
C₇×#1450	2
C₇×#1313	2
#11405	7

Subgroups — C7×#11405 #13188621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11405 #13188621