Subgroups — C₇×#8793 #13186009

Order 1792.

294 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1848	C₂	1	1	yes	maximal, characteristic
2	C₇×#1846	C₂	1	1	yes	maximal, characteristic
3	C₇×#1450	C₂	1	1	yes	maximal, characteristic
4	C₇×#1286	C₂	1	4	yes	maximal
5	C₇×#431	C₂²	1	2	yes	—
6	C₂×C₇×#108	—	2	1	no	—
7	C₂×C₇×#108	—	2	1	no	—
8	C₂×C₇×#107	—	2	1	no	—
9	C₂×C₇×#106	—	2	1	no	—
10	C₇×#423	C₂²	1	2	yes	—
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×#382	C₂²	1	2	yes	—
13	C₇×#322	C₄	1	2	yes	complemented
14	C₇×C₈²	C₄	1	2	yes	complemented
15	#8793	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂×C₇×M₁₆	—	2	2	no	—
17	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₄	—	2	2	no	—
20	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
21	C₇×#108	—	4	2	no	—
22	C₇×#108	—	4	2	no	—
23	C₇×#107	—	4	2	no	—
24	C₇×#106	—	4	2	no	—
25	C₇×C₄⋊C₈	—	2	2	no	—
26	C₇×C₄⋊C₈	—	2	4	no	—
27	C₇×C₂²⋊C₈	—	2	2	no	—
28	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
29	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
32	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
33	C₇×#95	—	2	4	no	—
34	#1848	C₂×C₇	1	1	yes	characteristic
35	#1846	C₂×C₇	1	1	yes	characteristic
36	#1450	C₂×C₇	1	1	yes	characteristic
37	#1286	C₂×C₇	1	4	yes	—
38	C₂²×C₄×C₇	—	4	2	no	—
39	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇	#41	1	2	yes	—
41	C₇×M₁₆	—	4	2	no	—
42	C₇×M₁₆	—	4	2	no	—
43	C₂×C₇×C₈	—	2	4	no	—
44	C₂×C₇×C₈	—	4	2	no	—
45	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₇×C₈	D₈	1	2	yes	—
50	C₂×C₇×C₈	SD₁₆	1	2	yes	—
51	C₂×C₇×C₈	SD₁₆	1	2	yes	—
52	C₂×C₇×C₈	Q₁₆	1	2	yes	—
53	C₇×C₄⋊C₄	—	4	2	no	—
54	C₇×C₄⋊C₄	—	4	2	no	—
55	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₇	#41	1	2	yes	—
58	#431	C₂²×C₇	1	2	yes	—
59	C₂×#108	—	2	1	no	—
60	C₂×#108	—	2	1	no	—
61	C₂×#107	—	2	1	no	—
62	C₂×#106	—	2	1	no	—
63	#423	C₂²×C₇	1	2	yes	—
64	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
65	#382	C₂²×C₇	1	2	yes	—
66	#322	C₄×C₇	1	2	yes	complemented
67	C₈²	C₄×C₇	1	2	yes	complemented
68	C₂³×C₇	#124	1	1	yes	characteristic
69	C₂×C₄×C₇	—	4	2	no	—
70	C₂×C₄×C₇	—	8	4	no	—
71	C₂×C₄×C₇	—	8	2	no	—
72	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₇	#117	1	2	yes	—
74	C₂×C₄×C₇	#124	1	1	yes	characteristic
75	C₂×C₄×C₇	#125	1	2	yes	—
76	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
80	C₇×C₈	—	4	4	no	—
81	C₇×C₈	—	8	2	no	—
82	C₇×C₈	#106	1	2	yes	—
83	C₇×C₈	#106	1	2	yes	—
84	C₇×C₈	#107	1	2	yes	—
85	C₇×C₈	#107	1	2	yes	—
86	C₂×M₁₆	—	2	2	no	—
87	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
88	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
89	C₂×C₄⋊C₄	—	2	2	no	—
90	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
91	#108	—	4	2	no	—
92	#108	—	4	2	no	—
93	#107	—	4	2	no	—
94	#106	—	4	2	no	—
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	2	4	no	—
97	C₂²⋊C₈	—	2	2	no	—
98	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
99	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
100	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
101	C₄×C₈	C₇×Q₈	1	1	yes	characteristic
102	C₄×C₈	C₇×Q₈	1	1	yes	characteristic
103	#95	—	2	4	no	—
104	C₂²×C₇	—	2	2	no	—
105	C₂²×C₇	#382	1	1	yes	characteristic
106	C₂²×C₇	#486	1	1	yes	characteristic
107	C₂²×C₇	#487	1	1	yes	characteristic
108	C₄×C₇	—	2	2	no	—
109	C₄×C₇	—	2	2	no	—
110	C₄×C₇	—	16	2	no	—
111	C₄×C₇	C₂×#106	1	1	yes	characteristic
112	C₄×C₇	C₂×#107	1	1	yes	characteristic
113	C₄×C₇	#486	1	1	yes	characteristic
114	C₄×C₇	#487	1	1	yes	characteristic
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
117	C₂²×C₄	C₇×#41	1	2	yes	—
118	M₁₆	—	4	2	no	—
119	M₁₆	—	4	2	no	—
120	C₂×C₈	—	2	4	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₈	C₇×D₈	1	2	yes	—
127	C₂×C₈	C₇×SD₁₆	1	2	yes	—
128	C₂×C₈	C₇×SD₁₆	1	2	yes	—
129	C₂×C₈	C₇×Q₁₆	1	2	yes	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
133	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
134	C₄²	C₇×#41	1	2	yes	—
135	C₂×C₇	—	2	1	no	—
136	C₂×C₇	—	2	1	no	—
137	C₂×C₇	#1833	1	2	yes	—
138	C₂×C₇	#1834	1	1	yes	characteristic
139	C₂³	C₇×#124	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	8	4	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄	C₇×#117	1	2	yes	—
145	C₂×C₄	C₇×#124	1	1	yes	characteristic
146	C₂×C₄	C₇×#125	1	2	yes	—
147	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
148	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
149	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
150	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
151	C₈	—	4	4	no	—
152	C₈	—	8	2	no	—
153	C₈	C₇×#106	1	2	yes	—
154	C₈	C₇×#106	1	2	yes	—
155	C₈	C₇×#107	1	2	yes	—
156	C₈	C₇×#107	1	2	yes	—
157	C₇	#8793	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²	—	2	2	no	—
159	C₂²	C₇×#382	1	1	yes	characteristic
160	C₂²	C₇×#486	1	1	yes	characteristic
161	C₂²	C₇×#487	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	16	2	no	—
165	C₄	C₂×C₇×#106	1	1	yes	characteristic
166	C₄	C₂×C₇×#107	1	1	yes	characteristic
167	C₄	C₇×#486	1	1	yes	characteristic
168	C₄	C₇×#487	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	C₇×#1833	1	2	yes	—
172	C₂	C₇×#1834	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	#382
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₇×#322	C₄	semidirect
14. C₇×C₈²	C₄	semidirect
15. #8793	C₇	direct (N × H)
66. #322	C₄×C₇	semidirect
67. C₈²	C₄×C₇	semidirect
157. C₇	#8793	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×#1848	2
C₇×#1846	2
C₇×#1450	2
C₇×#1286	2
#8793	7

Subgroups — C7×#8793 #13186009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#8793 #13186009