Subgroups — C₇×#13170 #13190386

Order 1792.

278 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#368	C₂	1	1	yes	maximal, characteristic
2	C₇×#2158	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic
4	C₇×#1945	C₂	1	2	yes	maximal
5	C₇×#1382	C₂	1	2	yes	maximal
6	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₇×#108	C₂²	1	2	yes	—
9	C₇×#427	C₂²	1	2	yes	—
10	C₇×#424	—	2	2	no	—
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×#368	—	2	2	no	—
13	C₇×#367	C₄	1	2	yes	complemented
14	C₇×#366	C₄	1	2	yes	complemented
15	C₇×#346	—	2	2	no	—
16	C₇×#339	—	2	2	no	—
17	#13170	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂×C₇×M₁₆	—	2	2	no	—
19	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
21	C₇×#117	—	2	2	no	—
22	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
23	C₂×C₇×C₁₆	—	2	1	no	—
24	C₂×C₇×C₁₆	—	2	1	no	—
25	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₇×C₁₆	D₄	1	2	yes	—
28	C₂×C₇×C₁₆	Q₈	1	2	yes	—
29	C₂×C₇×C₁₆	Q₈	1	2	yes	—
30	C₇×#108	—	2	2	no	—
31	C₇×#108	—	2	2	no	—
32	C₇×#107	C₂×C₄	1	4	yes	—
33	C₇×#106	—	2	2	no	—
34	C₇×C₄⋊C₈	—	4	4	no	—
35	C₄×C₇×C₈	—	2	1	no	—
36	C₄×C₇×C₈	—	2	1	no	—
37	C₂×#368	C₂×C₇	1	1	yes	characteristic
38	#2158	C₂×C₇	1	1	yes	characteristic
39	C₂×#346	C₂×C₇	1	1	yes	characteristic
40	#1945	C₂×C₇	1	2	yes	—
41	#1382	C₂×C₇	1	2	yes	—
42	C₂²×C₄×C₇	—	2	1	no	—
43	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
44	C₇×M₁₆	—	4	2	no	—
45	C₇×M₁₆	—	4	2	no	—
46	C₂×C₇×C₈	—	2	1	no	—
47	C₂×C₇×C₈	—	2	1	no	—
48	C₂×C₇×C₈	—	4	2	no	—
49	C₂×C₇×C₈	D₈	1	2	yes	—
50	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
52	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₇×C₈	#41	1	1	yes	characteristic
55	C₂×C₇×C₈	#41	1	1	yes	characteristic
56	C₇×C₄⋊C₄	—	2	4	no	—
57	C₇×C₂²⋊C₄	—	4	2	no	—
58	C₄²×C₇	—	2	1	no	—
59	C₄²×C₇	—	4	2	no	—
60	C₄²×C₇	Q₁₆	1	2	yes	—
61	C₇×C₁₆	—	2	2	no	—
62	C₇×C₁₆	C₄⋊C₄	1	2	yes	—
63	C₇×C₁₆	C₄⋊C₄	1	2	yes	—
64	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
65	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
66	C₂×#108	C₂²×C₇	1	2	yes	—
67	#427	C₂²×C₇	1	2	yes	—
68	#424	—	2	2	no	—
69	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
70	#368	—	2	2	no	—
71	#367	C₄×C₇	1	2	yes	complemented
72	#366	C₄×C₇	1	2	yes	complemented
73	#346	—	2	2	no	—
74	#339	—	2	2	no	—
75	C₂³×C₇	#135	1	1	yes	characteristic
76	C₂×C₄×C₇	—	2	2	no	—
77	C₂×C₄×C₇	—	4	2	no	—
78	C₂×C₄×C₇	—	4	4	no	—
79	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₇	#121	1	1	yes	characteristic
81	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
84	C₂×C₄×C₇	#135	1	1	yes	characteristic
85	C₇×C₈	—	2	2	no	—
86	C₇×C₈	—	8	2	no	—
87	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
89	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
90	C₇×C₈	#125	1	1	yes	characteristic
91	C₂×M₁₆	—	2	2	no	—
92	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
93	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
94	#117	—	2	2	no	—
95	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
96	C₂×C₁₆	—	2	1	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
99	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
100	C₂×C₁₆	C₇×D₄	1	2	yes	—
101	C₂×C₁₆	C₇×Q₈	1	2	yes	—
102	C₂×C₁₆	C₇×Q₈	1	2	yes	—
103	#108	—	2	2	no	—
104	#108	—	2	2	no	—
105	#107	C₂×C₄×C₇	1	4	yes	—
106	#106	—	2	2	no	—
107	C₄⋊C₈	—	4	4	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	—	2	1	no	—
110	C₂²×C₇	—	2	2	no	—
111	C₂²×C₇	C₄×D₈	1	1	yes	characteristic
112	C₂²×C₇	#467	1	1	yes	characteristic
113	C₂²×C₇	#499	1	1	yes	characteristic
114	C₄×C₇	—	4	2	no	—
115	C₄×C₇	—	8	4	no	—
116	C₄×C₇	#389	1	1	yes	characteristic
117	C₄×C₇	C₄×Q₁₆	1	1	yes	characteristic
118	C₄×C₇	#466	1	1	yes	characteristic
119	C₄×C₇	#500	1	1	yes	characteristic
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
122	M₁₆	—	4	2	no	—
123	M₁₆	—	4	2	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	C₇×D₈	1	2	yes	—
128	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
129	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
130	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
131	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
132	C₂×C₈	C₇×#41	1	1	yes	characteristic
133	C₂×C₈	C₇×#41	1	1	yes	characteristic
134	C₄⋊C₄	—	2	4	no	—
135	C₂²⋊C₄	—	4	2	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	4	2	no	—
138	C₄²	C₇×Q₁₆	1	2	yes	—
139	C₁₆	—	2	2	no	—
140	C₁₆	C₇×C₄⋊C₄	1	2	yes	—
141	C₁₆	C₇×C₄⋊C₄	1	2	yes	—
142	C₂×C₇	—	2	1	no	—
143	C₂×C₇	—	2	1	no	—
144	C₂×C₇	#1942	1	1	yes	characteristic
145	C₂×C₇	#2179	1	2	yes	—
146	C₂³	C₇×#135	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
151	C₂×C₄	C₇×#121	1	1	yes	characteristic
152	C₂×C₄	C₇×C₄⋊Q₈	1	1	yes	characteristic
153	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
154	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
155	C₂×C₄	C₇×#135	1	1	yes	characteristic
156	C₈	—	2	2	no	—
157	C₈	—	8	2	no	—
158	C₈	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
159	C₈	C₄×C₇×Q₈	1	1	yes	characteristic
160	C₈	C₇×C₂²⋊Q₈	1	1	yes	characteristic
161	C₈	C₇×#125	1	1	yes	characteristic
162	C₇	#13170	1	1	yes	characteristic, direct product factor, Sylow
163	C₂²	—	2	2	no	—
164	C₂²	C₄×C₇×D₈	1	1	yes	characteristic
165	C₂²	C₇×#467	1	1	yes	characteristic
166	C₂²	C₇×#499	1	1	yes	characteristic
167	C₄	—	4	2	no	—
168	C₄	—	8	4	no	—
169	C₄	C₇×#389	1	1	yes	characteristic
170	C₄	C₄×C₇×Q₁₆	1	1	yes	characteristic
171	C₄	C₇×#466	1	1	yes	characteristic
172	C₄	C₇×#500	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	C₇×#1942	1	1	yes	characteristic
176	C₂	C₇×#2179	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂²×C₇	C₄×D₈
prime radical	C₂³×C₇	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₇×#367	C₄	semidirect
14. C₇×#366	C₄	semidirect
17. #13170	C₇	direct (N × H)
71. #367	C₄×C₇	semidirect
72. #366	C₄×C₇	semidirect
162. C₇	#13170	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×#368	2
C₇×#2158	2
C₂×C₇×#346	2
C₇×#1945	2
C₇×#1382	2
#13170	7

Subgroups — C7×#13170 #13190386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13170 #13190386