Subgroups — C₇×#20355 #13197571

Order 1792.

2018 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#519	C₂	1	2	yes	maximal
2	C₂×C₇×#519	C₂	1	1	yes	maximal, characteristic
3	C₇×#1740	C₂	1	12	yes	maximal
4	C₂³×C₇×Q₈	C₂²	1	1	yes	characteristic
5	C₇×#519	—	2	8	no	—
6	C₇×#519	—	2	8	no	—
7	C₇×#519	C₄	1	16	yes	complemented
8	C₂×C₄×C₇×Q₈	—	2	8	no	—
9	C₂×C₄×C₇×Q₈	C₄	1	8	yes	complemented
10	C₂×C₇×#117	C₂²	1	12	yes	—
11	C₂×C₇×#117	C₂²	1	6	yes	—
12	C₇×#342	C₂²	1	16	yes	—
13	#20355	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂²×C₇×Q₈	C₂×C₄	1	4	yes	—
15	C₂²×C₇×Q₈	C₂×C₄	1	8	yes	—
16	C₂²×C₇×Q₈	D₄	1	8	yes	—
17	C₂²×C₇×Q₈	D₄	1	4	yes	—
18	C₂²×C₇×Q₈	Q₈	1	4	yes	—
19	C₂³×C₄×C₇	C₂³	1	3	yes	—
20	C₄×C₇×Q₈	—	2	64	no	—
21	C₄×C₇×Q₈	—	2	32	no	—
22	C₇×#117	—	2	24	no	—
23	C₇×#117	—	2	24	no	—
24	C₇×#117	C₂×C₄	1	48	yes	—
25	C₂×C₇×C₄⋊C₄	—	2	12	no	—
26	C₂×C₇×C₄⋊C₄	C₂×C₄	1	12	yes	—
27	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
28	C₂×C₇×C₂²⋊C₄	C₂³	1	8	yes	—
29	C₂×C₄²×C₇	—	2	12	no	—
30	C₂×C₄²×C₇	C₂×C₄	1	12	yes	—
31	C₂×#519	C₂×C₇	1	2	yes	—
32	C₂×#519	C₂×C₇	1	1	yes	characteristic
33	#1740	C₂×C₇	1	12	yes	—
34	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
35	C₂×C₇×Q₈	C₄²	1	16	yes	—
36	C₂×C₇×Q₈	C₂²⋊C₄	1	32	yes	—
37	C₂×C₇×Q₈	C₂²⋊C₄	1	16	yes	—
38	C₂×C₇×Q₈	C₄⋊C₄	1	16	yes	—
39	C₂×C₇×Q₈	C₄⋊C₄	1	32	yes	—
40	C₂²×C₄×C₇	—	2	8	no	—
41	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
42	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
43	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
44	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
45	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
46	C₂²×C₄×C₇	C₂×Q₈	1	6	yes	—
47	C₇×C₄⋊C₄	—	2	48	no	—
48	C₇×C₄⋊C₄	—	2	24	no	—
49	C₇×C₂²⋊C₄	—	2	8	no	—
50	C₇×C₂²⋊C₄	—	2	8	no	—
51	C₇×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
52	C₄²×C₇	—	2	48	no	—
53	C₄²×C₇	—	2	24	no	—
54	C₂³×Q₈	C₂²×C₇	1	1	yes	characteristic
55	#519	—	2	8	no	—
56	#519	—	2	8	no	—
57	#519	C₄×C₇	1	16	yes	complemented
58	C₂×C₄×Q₈	—	2	8	no	—
59	C₂×C₄×Q₈	C₄×C₇	1	8	yes	complemented
60	C₂×#117	C₂²×C₇	1	12	yes	—
61	C₂×#117	C₂²×C₇	1	6	yes	—
62	#342	C₂²×C₇	1	16	yes	—
63	C₂³×C₇	—	4	2	no	—
64	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
65	C₂³×C₇	C₂³×C₄	1	2	yes	—
66	C₂³×C₇	C₂²×D₄	1	2	yes	—
67	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
68	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
69	C₇×Q₈	#95	1	64	yes	—
70	C₂×C₄×C₇	—	2	16	no	—
71	C₂×C₄×C₇	—	2	8	no	—
72	C₂×C₄×C₇	—	4	16	no	—
73	C₂×C₄×C₇	—	4	8	no	—
74	C₂×C₄×C₇	C₂×C₄²	1	12	yes	—
75	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	24	yes	—
76	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	12	yes	—
77	C₂×C₄×C₇	C₂×C₄⋊C₄	1	12	yes	—
78	C₂×C₄×C₇	C₂×C₄⋊C₄	1	24	yes	—
79	C₂²×Q₈	C₂×C₄×C₇	1	4	yes	—
80	C₂²×Q₈	C₂×C₄×C₇	1	8	yes	—
81	C₂²×Q₈	C₇×D₄	1	8	yes	—
82	C₂²×Q₈	C₇×D₄	1	4	yes	—
83	C₂²×Q₈	C₇×Q₈	1	4	yes	—
84	C₂³×C₄	C₂³×C₇	1	3	yes	—
85	C₄×Q₈	—	2	64	no	—
86	C₄×Q₈	—	2	32	no	—
87	#117	—	2	24	no	—
88	#117	—	2	24	no	—
89	#117	C₂×C₄×C₇	1	48	yes	—
90	C₂×C₄⋊C₄	—	2	12	no	—
91	C₂×C₄⋊C₄	C₂×C₄×C₇	1	12	yes	—
92	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
93	C₂×C₂²⋊C₄	C₂³×C₇	1	8	yes	—
94	C₂×C₄²	—	2	12	no	—
95	C₂×C₄²	C₂×C₄×C₇	1	12	yes	—
96	C₂²×C₇	—	2	2	no	—
97	C₂²×C₇	—	2	4	no	—
98	C₂²×C₇	—	4	2	no	—
99	C₂²×C₇	—	4	1	no	—
100	C₂²×C₇	—	4	1	no	—
101	C₂²×C₇	C₂²×C₄²	1	1	yes	characteristic
102	C₂²×C₇	C₂²×C₂²⋊C₄	1	2	yes	—
103	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₇	C₂²×C₄⋊C₄	1	2	yes	—
106	C₄×C₇	—	4	16	no	—
107	C₄×C₇	—	4	8	no	—
108	C₄×C₇	C₂×#95	1	24	yes	—
109	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
110	C₂×Q₈	C₄²×C₇	1	16	yes	—
111	C₂×Q₈	C₇×C₂²⋊C₄	1	32	yes	—
112	C₂×Q₈	C₇×C₂²⋊C₄	1	16	yes	—
113	C₂×Q₈	C₇×C₄⋊C₄	1	16	yes	—
114	C₂×Q₈	C₇×C₄⋊C₄	1	32	yes	—
115	C₂²×C₄	—	2	8	no	—
116	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
117	C₂²×C₄	C₂²×C₄×C₇	1	6	yes	—
118	C₂²×C₄	C₂²×C₄×C₇	1	12	yes	—
119	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
120	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
121	C₂²×C₄	C₂×C₇×Q₈	1	6	yes	—
122	C₄⋊C₄	—	2	48	no	—
123	C₄⋊C₄	—	2	24	no	—
124	C₂²⋊C₄	—	2	8	no	—
125	C₂²⋊C₄	—	2	8	no	—
126	C₂²⋊C₄	C₂²×C₄×C₇	1	16	yes	—
127	C₄²	—	2	48	no	—
128	C₄²	—	2	24	no	—
129	C₂×C₇	—	2	2	no	—
130	C₂×C₇	—	2	1	no	—
131	C₂×C₇	—	2	2	no	—
132	C₂×C₇	—	2	1	no	—
133	C₂×C₇	C₂²×#95	1	1	yes	characteristic
134	C₂×C₇	#2884	1	2	yes	—
135	C₂³	—	4	2	no	—
136	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
137	C₂³	C₂³×C₄×C₇	1	2	yes	—
138	C₂³	C₂²×C₇×D₄	1	2	yes	—
139	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
140	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
141	Q₈	C₇×#95	1	64	yes	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	2	16	no	—
144	C₂×C₄	—	4	16	no	—
145	C₂×C₄	—	4	8	no	—
146	C₂×C₄	C₂×C₄²×C₇	1	12	yes	—
147	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	24	yes	—
148	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	12	yes	—
149	C₂×C₄	C₂×C₇×C₄⋊C₄	1	12	yes	—
150	C₂×C₄	C₂×C₇×C₄⋊C₄	1	24	yes	—
151	C₇	#20355	1	1	yes	characteristic, direct product factor, Sylow
152	C₂²	—	2	2	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	4	1	no	—
156	C₂²	—	4	1	no	—
157	C₂²	C₂²×C₄²×C₇	1	1	yes	characteristic
158	C₂²	C₂²×C₇×C₂²⋊C₄	1	2	yes	—
159	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
160	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
161	C₂²	C₂²×C₇×C₄⋊C₄	1	2	yes	—
162	C₄	—	4	16	no	—
163	C₄	—	4	8	no	—
164	C₄	C₂×C₇×#95	1	24	yes	—
165	C₂	—	2	2	no	—
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	2	2	no	—
169	C₂	C₂²×C₇×#95	1	1	yes	characteristic
170	C₂	C₇×#2884	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×C₂²⋊C₄
commutator	C₂²	C₂²×C₄²×C₇
Frattini, cosocle	C₂⁴	C₂⁴×C₇
prime radical	C₂⁴×C₇	C₂⁴
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₇×#519	C₄	semidirect
9. C₂×C₄×C₇×Q₈	C₄	semidirect
13. #20355	C₇	direct (N × H)
57. #519	C₄×C₇	semidirect
59. C₂×C₄×Q₈	C₄×C₇	semidirect
151. C₇	#20355	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#519	2
C₇×#1740	2
#20355	7

Subgroups — C7×#20355 #13197571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20355 #13197571