Subgroups — C₇×#20762 #13197978

Order 1792.

910 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	C₂²×C₇×#97	C₂	1	1	yes	maximal, characteristic
2	C₇×#2285	C₂	1	2	yes	maximal
3	C₇×#1836	C₂	1	8	yes	maximal, complemented
4	C₇×#1735	C₂	1	4	yes	maximal
5	C₂³×C₇×C₈	C₂²	1	2	yes	—
6	C₂×C₇×#117	C₂²	1	4	yes	—
7	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₇×#97	C₄	1	8	yes	complemented
9	C₂×C₇×#97	C₂²	1	4	yes	—
10	C₇×#384	C₂²	1	4	yes	—
11	C₇×#382	C₂²	1	4	yes	—
12	C₇×#380	—	2	4	no	—
13	C₄×C₇×C₄⋊C₄	—	2	4	no	—
14	C₇×#340	C₂²	1	16	yes	—
15	#20762	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₄×C₇	D₄	1	2	yes	—
17	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₂²×C₇×C₈	—	2	8	no	—
19	C₂²×C₇×C₈	C₂×C₄	1	8	yes	—
20	C₂²×C₇×C₈	C₂³	1	4	yes	—
21	C₇×#117	—	2	16	no	—
22	C₂×C₇×C₄⋊C₄	—	2	8	no	—
23	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
24	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
25	C₂×C₄²×C₇	—	2	4	no	—
26	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₇	D₄	1	2	yes	—
28	C₂×C₄²×C₇	Q₈	1	4	yes	—
29	C₂×C₄²×C₇	C₂³	1	2	yes	—
30	C₇×#97	C₂×C₄	1	16	yes	—
31	C₇×#95	—	2	8	no	—
32	C₂²×#97	C₂×C₇	1	1	yes	characteristic
33	#2285	C₂×C₇	1	2	yes	—
34	#1836	C₂×C₇	1	8	yes	complemented
35	#1735	C₂×C₇	1	4	yes	—
36	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₇	—	2	8	no	—
38	C₂²×C₄×C₇	—	4	4	no	—
39	C₂²×C₄×C₇	C₄⋊C₄	1	8	yes	—
40	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
42	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
44	C₂²×C₄×C₇	#41	1	8	yes	—
45	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
46	C₂×C₇×C₈	—	2	16	no	—
47	C₂×C₇×C₈	—	2	8	no	—
48	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
49	C₇×C₄⋊C₄	—	2	16	no	—
50	C₇×C₄⋊C₄	—	4	16	no	—
51	C₇×C₂²⋊C₄	—	4	8	no	—
52	C₄²×C₇	—	4	8	no	—
53	C₄²×C₇	C₄⋊C₄	1	8	yes	—
54	C₄²×C₇	C₂²×C₄	1	4	yes	—
55	C₄²×C₇	C₂×Q₈	1	4	yes	—
56	C₂³×C₈	C₂²×C₇	1	2	yes	—
57	C₂×#117	C₂²×C₇	1	4	yes	—
58	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
59	C₂×#97	C₄×C₇	1	8	yes	complemented
60	C₂×#97	C₂²×C₇	1	4	yes	—
61	#384	C₂²×C₇	1	4	yes	—
62	#382	C₂²×C₇	1	4	yes	—
63	#380	—	2	4	no	—
64	C₄×C₄⋊C₄	—	2	4	no	—
65	#340	C₂²×C₇	1	16	yes	—
66	C₂³×C₇	—	2	4	no	—
67	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
68	C₂³×C₇	#124	1	4	yes	—
69	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇	—	2	8	no	—
71	C₂×C₄×C₇	—	4	8	no	—
72	C₂×C₄×C₇	—	4	8	no	—
73	C₂×C₄×C₇	—	4	16	no	—
74	C₂×C₄×C₇	C₂×C₄⋊C₄	1	8	yes	—
75	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
76	C₂×C₄×C₇	#117	1	8	yes	—
77	C₂×C₄×C₇	#124	1	4	yes	—
78	C₂×C₄×C₇	#125	1	8	yes	—
79	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₇	C₂×#41	1	4	yes	—
82	C₇×C₈	—	2	8	no	—
83	C₂³×C₄	C₇×D₄	1	2	yes	—
84	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
85	C₂²×C₈	—	2	8	no	—
86	C₂²×C₈	C₂×C₄×C₇	1	8	yes	—
87	C₂²×C₈	C₂³×C₇	1	4	yes	—
88	#117	—	2	16	no	—
89	C₂×C₄⋊C₄	—	2	8	no	—
90	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
91	C₂×C₂²⋊C₄	—	2	4	no	—
92	C₂×C₄²	—	2	4	no	—
93	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
94	C₂×C₄²	C₇×D₄	1	2	yes	—
95	C₂×C₄²	C₇×Q₈	1	4	yes	—
96	C₂×C₄²	C₂³×C₇	1	2	yes	—
97	#97	C₂×C₄×C₇	1	16	yes	—
98	#95	—	2	8	no	—
99	C₂²×C₇	—	2	8	no	—
100	C₂²×C₇	—	2	4	no	—
101	C₂²×C₇	#382	1	4	yes	—
102	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₇	C₂×#124	1	2	yes	—
104	C₂²×C₇	#575	1	4	yes	—
105	C₄×C₇	—	2	8	no	—
106	C₄×C₇	—	8	8	no	—
107	C₄×C₇	#382	1	4	yes	—
108	C₄×C₇	C₂×#117	1	2	yes	—
109	C₄×C₇	C₂×#125	1	2	yes	—
110	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
111	C₂²×C₄	—	2	8	no	—
112	C₂²×C₄	—	4	4	no	—
113	C₂²×C₄	C₇×C₄⋊C₄	1	8	yes	—
114	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
115	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
116	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
117	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
118	C₂²×C₄	C₇×#41	1	8	yes	—
119	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
120	C₂×C₈	—	2	16	no	—
121	C₂×C₈	—	2	8	no	—
122	C₂×C₈	C₂²×C₄×C₇	1	8	yes	—
123	C₄⋊C₄	—	2	16	no	—
124	C₄⋊C₄	—	4	16	no	—
125	C₂²⋊C₄	—	4	8	no	—
126	C₄²	—	4	8	no	—
127	C₄²	C₇×C₄⋊C₄	1	8	yes	—
128	C₄²	C₂²×C₄×C₇	1	4	yes	—
129	C₄²	C₂×C₇×Q₈	1	4	yes	—
130	C₂×C₇	—	2	4	no	—
131	C₂×C₇	C₂×#382	1	1	yes	characteristic
132	C₂×C₇	#2911	1	2	yes	—
133	C₂×C₇	#3135	1	4	yes	—
134	C₂³	—	2	4	no	—
135	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
136	C₂³	C₇×#124	1	4	yes	—
137	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	4	8	no	—
140	C₂×C₄	—	4	8	no	—
141	C₂×C₄	—	4	16	no	—
142	C₂×C₄	C₂×C₇×C₄⋊C₄	1	8	yes	—
143	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
144	C₂×C₄	C₇×#117	1	8	yes	—
145	C₂×C₄	C₇×#124	1	4	yes	—
146	C₂×C₄	C₇×#125	1	8	yes	—
147	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
148	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
149	C₂×C₄	C₂×C₇×#41	1	4	yes	—
150	C₈	—	2	8	no	—
151	C₇	#20762	1	1	yes	characteristic, direct product factor, Sylow
152	C₂²	—	2	4	no	—
153	C₂²	—	2	8	no	—
154	C₂²	C₇×#382	1	4	yes	—
155	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
156	C₂²	C₂×C₇×#124	1	2	yes	—
157	C₂²	C₇×#575	1	4	yes	—
158	C₄	—	2	8	no	—
159	C₄	—	8	8	no	—
160	C₄	C₇×#382	1	4	yes	—
161	C₄	C₂×C₇×#117	1	2	yes	—
162	C₄	C₂×C₇×#125	1	2	yes	—
163	C₂	—	2	4	no	—
164	C₂	C₂×C₇×#382	1	1	yes	characteristic
165	C₂	C₇×#2911	1	2	yes	—
166	C₂	C₇×#3135	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂²×D₄
prime radical	C₂⁴×C₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1836	C₂	semidirect
8. C₂×C₇×#97	C₄	semidirect
15. #20762	C₇	direct (N × H)
34. #1836	C₂×C₇	semidirect
59. C₂×#97	C₄×C₇	semidirect
151. C₇	#20762	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×#97	2
C₇×#2285	2
C₇×#1836	2
C₇×#1735	2
#20762	7

Subgroups — C7×#20762 #13197978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20762 #13197978