Subgroups — C₇×#11411 #13188627

Order 1792.

290 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2162	C₂	1	2	yes	maximal
2	C₂×C₇×C₈²	C₂	1	1	yes	maximal, characteristic
3	C₇×#1312	C₂	1	2	yes	maximal, complemented
4	C₇×#1311	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₇×C₄⋊C₁₆	C₄	1	4	yes	—
9	C₇×C₂²⋊C₁₆	C₄	1	4	yes	—
10	C₄×C₇×C₁₆	C₂²	1	4	yes	—
11	C₇×C₈²	C₄	1	1	yes	characteristic
12	C₇×C₈²	C₄	1	1	yes	characteristic
13	C₇×C₈²	C₂²	1	1	yes	characteristic
14	C₇×C₈²	C₂²	1	1	yes	characteristic
15	#11411	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₇×C₈	C₈	1	2	yes	—
17	C₂²×C₇×C₈	C₈	1	2	yes	—
18	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₇×C₁₆	C₂×C₄	1	8	yes	—
21	C₄×C₇×C₈	C₈	1	2	yes	—
22	C₄×C₇×C₈	C₈	1	2	yes	—
23	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
24	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
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	1	yes	characteristic
28	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
30	#2162	C₂×C₇	1	2	yes	—
31	C₂×C₈²	C₂×C₇	1	1	yes	characteristic
32	#1312	C₂×C₇	1	2	yes	complemented
33	#1311	C₂×C₇	1	2	yes	complemented
34	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
35	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
37	C₂×C₇×C₈	—	2	4	no	—
38	C₂×C₇×C₈	—	2	4	no	—
39	C₂×C₇×C₈	—	2	4	no	—
40	C₂×C₇×C₈	C₄²	1	2	yes	—
41	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
42	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
43	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
44	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
45	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
46	C₄²×C₇	C₄²	1	1	yes	characteristic
47	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
50	C₇×C₁₆	—	2	8	no	—
51	C₂×C₄×C₈	C₄×C₇	1	1	yes	characteristic
52	C₂×C₄×C₈	C₄×C₇	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
54	C₄⋊C₁₆	C₄×C₇	1	4	yes	—
55	C₂²⋊C₁₆	C₄×C₇	1	4	yes	—
56	C₄×C₁₆	C₂²×C₇	1	4	yes	—
57	C₈²	C₄×C₇	1	1	yes	characteristic
58	C₈²	C₄×C₇	1	1	yes	characteristic
59	C₈²	C₂²×C₇	1	1	yes	characteristic
60	C₈²	C₂²×C₇	1	1	yes	characteristic
61	C₂³×C₇	C₄×C₈	1	1	yes	characteristic
62	C₂×C₄×C₇	—	2	2	no	—
63	C₂×C₄×C₇	—	2	2	no	—
64	C₂×C₄×C₇	C₄×C₈	1	1	yes	characteristic
65	C₂×C₄×C₇	C₄×C₈	1	1	yes	characteristic
66	C₂×C₄×C₇	C₄×C₈	1	1	yes	characteristic
67	C₂×C₄×C₇	M₃₂	1	4	yes	—
68	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
71	C₇×C₈	—	2	2	no	—
72	C₇×C₈	—	2	2	no	—
73	C₇×C₈	—	2	2	no	—
74	C₇×C₈	—	2	2	no	—
75	C₇×C₈	M₃₂	1	4	yes	—
76	C₇×C₈	#131	1	4	yes	—
77	C₂²×C₈	C₇×C₈	1	2	yes	—
78	C₂²×C₈	C₇×C₈	1	2	yes	—
79	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
80	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
81	C₂×C₁₆	C₂×C₄×C₇	1	8	yes	—
82	C₄×C₈	C₇×C₈	1	2	yes	—
83	C₄×C₈	C₇×C₈	1	2	yes	—
84	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
85	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
86	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
87	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
88	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
89	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
90	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
91	C₂²×C₇	—	2	1	no	—
92	C₂²×C₇	—	2	1	no	—
93	C₂²×C₇	#346	1	2	yes	—
94	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
95	C₄×C₇	—	2	1	no	—
96	C₄×C₇	—	2	1	no	—
97	C₄×C₇	#346	1	2	yes	—
98	C₄×C₇	#405	1	2	yes	—
99	C₄×C₇	C₂×M₃₂	1	2	yes	—
100	C₄×C₇	C₂×M₃₂	1	2	yes	—
101	C₂²×C₄	C₄²×C₇	1	1	yes	characteristic
102	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
103	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	2	4	no	—
107	C₂×C₈	C₄²×C₇	1	2	yes	—
108	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
109	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
110	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
111	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
112	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
113	C₄²	C₄²×C₇	1	1	yes	characteristic
114	C₄²	C₂×C₇×C₈	1	1	yes	characteristic
115	C₄²	C₂×C₇×C₈	1	1	yes	characteristic
116	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
117	C₁₆	—	2	8	no	—
118	C₂×C₇	—	2	2	no	—
119	C₂×C₇	#1450	1	1	yes	characteristic
120	C₂×C₇	C₂×#346	1	1	yes	characteristic
121	C₂×C₇	C₄×M₃₂	1	1	yes	characteristic
122	C₂³	C₄×C₇×C₈	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	C₄×C₇×C₈	1	1	yes	characteristic
126	C₂×C₄	C₄×C₇×C₈	1	1	yes	characteristic
127	C₂×C₄	C₄×C₇×C₈	1	1	yes	characteristic
128	C₂×C₄	C₇×M₃₂	1	4	yes	—
129	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
130	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
131	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	2	2	no	—
134	C₈	—	2	2	no	—
135	C₈	—	2	2	no	—
136	C₈	C₇×M₃₂	1	4	yes	—
137	C₈	C₇×#131	1	4	yes	—
138	C₇	#11411	1	1	yes	characteristic, direct product factor, Sylow
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₇×#346	1	2	yes	—
142	C₂²	C₂×C₄×C₇×C₈	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	C₇×#346	1	2	yes	—
146	C₄	C₇×#405	1	2	yes	—
147	C₄	C₂×C₇×M₃₂	1	2	yes	—
148	C₄	C₂×C₇×M₃₂	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	C₇×#1450	1	1	yes	characteristic
151	C₂	C₂×C₇×#346	1	1	yes	characteristic
152	C₂	C₄×C₇×M₃₂	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₇×#1312	C₂	semidirect
4. C₇×#1311	C₂	semidirect
15. #11411	C₇	direct (N × H)
32. #1312	C₂×C₇	semidirect
33. #1311	C₂×C₇	semidirect
138. C₇	#11411	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#2162	2
C₂×C₇×C₈²	2
C₇×#1312	2
C₇×#1311	2
#11411	7

Subgroups — C7×#11411 #13188627

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11411 #13188627