Subgroups — C₇×#13173 #13190389

Order 1792.

262 conjugacy classes of subgroups.

The subgroup lattice has 176 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	C₂×C₇×#368	C₂	1	1	yes	maximal, characteristic
3	C₇×#2108	C₂	1	1	yes	maximal, characteristic
4	C₇×#1953	C₂	1	2	yes	maximal
5	C₇×#1388	C₂	1	2	yes	maximal
6	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
8	C₇×#430	—	2	2	no	—
9	C₂×C₇×#108	C₂²	1	2	yes	—
10	C₂×C₇×#108	C₂²	1	2	yes	—
11	C₇×#405	C₂²	1	1	yes	characteristic
12	C₇×#368	—	2	1	no	—
13	C₇×#368	—	2	1	no	—
14	C₇×#368	C₄	1	4	yes	complemented
15	C₇×#346	—	2	1	no	—
16	C₄×C₇×C₁₆	—	2	1	no	—
17	C₇×#344	—	2	2	no	—
18	#13173	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂×C₇×M₁₆	—	2	2	no	—
20	C₂×C₇×M₁₆	—	2	2	no	—
21	C₂×C₇×M₁₆	D₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
23	C₇×#117	D₄	1	1	yes	characteristic
24	C₇×M₃₂	—	2	1	no	—
25	C₇×M₃₂	Q₈	1	2	yes	—
26	C₂×C₇×C₁₆	—	2	1	no	—
27	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
29	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
30	C₂×C₇×C₁₆	Q₈	1	2	yes	—
31	C₇×#108	—	2	2	no	—
32	C₇×#108	—	2	2	no	—
33	C₇×#108	—	2	2	no	—
34	C₇×#108	—	4	4	no	—
35	C₇×#108	C₂×C₄	1	4	yes	—
36	C₇×#97	—	2	1	no	—
37	C₄×C₇×C₈	—	2	1	no	—
38	C₂×#368	C₂×C₇	1	1	yes	characteristic
39	C₂×#368	C₂×C₇	1	1	yes	characteristic
40	#2108	C₂×C₇	1	1	yes	characteristic
41	#1953	C₂×C₇	1	2	yes	—
42	#1388	C₂×C₇	1	2	yes	—
43	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
44	C₇×M₁₆	—	2	1	no	—
45	C₇×M₁₆	—	2	4	no	—
46	C₇×M₁₆	—	4	2	no	—
47	C₇×M₁₆	—	4	2	no	—
48	C₇×M₁₆	—	4	2	no	—
49	C₇×M₁₆	D₈	1	2	yes	—
50	C₂×C₇×C₈	—	2	1	no	—
51	C₂×C₇×C₈	—	4	2	no	—
52	C₂×C₇×C₈	—	4	2	no	—
53	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
55	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₇×C₈	#41	1	1	yes	characteristic
58	C₂×C₇×C₈	#41	1	1	yes	characteristic
59	C₇×C₄⋊C₄	—	2	1	no	—
60	C₇×C₂²⋊C₄	Q₁₆	1	2	yes	—
61	C₄²×C₇	—	2	1	no	—
62	C₇×C₁₆	—	2	2	no	—
63	C₇×C₁₆	C₄⋊C₄	1	2	yes	—
64	C₇×C₁₆	C₄⋊C₄	1	2	yes	—
65	C₂×M₃₂	C₂²×C₇	1	1	yes	characteristic
66	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
67	#430	—	2	2	no	—
68	C₂×#108	C₂²×C₇	1	2	yes	—
69	C₂×#108	C₂²×C₇	1	2	yes	—
70	#405	C₂²×C₇	1	1	yes	characteristic
71	#368	—	2	1	no	—
72	#368	—	2	1	no	—
73	#368	C₄×C₇	1	4	yes	complemented
74	#346	—	2	1	no	—
75	C₄×C₁₆	—	2	1	no	—
76	#344	—	2	2	no	—
77	C₂³×C₇	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₇	—	2	2	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	#135	1	1	yes	characteristic
84	C₂×C₄×C₇	#135	1	1	yes	characteristic
85	C₇×C₈	—	2	2	no	—
86	C₇×C₈	—	4	4	no	—
87	C₇×C₈	—	8	2	no	—
88	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
90	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
91	C₇×C₈	#125	1	1	yes	characteristic
92	C₂×M₁₆	—	2	2	no	—
93	C₂×M₁₆	—	2	2	no	—
94	C₂×M₁₆	C₇×D₄	1	1	yes	characteristic
95	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
96	#117	C₇×D₄	1	1	yes	characteristic
97	M₃₂	—	2	1	no	—
98	M₃₂	C₇×Q₈	1	2	yes	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
101	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
102	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
103	C₂×C₁₆	C₇×Q₈	1	2	yes	—
104	#108	—	2	2	no	—
105	#108	—	2	2	no	—
106	#108	—	2	2	no	—
107	#108	—	4	4	no	—
108	#108	C₂×C₄×C₇	1	4	yes	—
109	#97	—	2	1	no	—
110	C₄×C₈	—	2	1	no	—
111	C₂²×C₇	—	4	1	no	—
112	C₂²×C₇	C₄×Q₁₆	1	1	yes	characteristic
113	C₂²×C₇	#467	1	1	yes	characteristic
114	C₂²×C₇	#500	1	1	yes	characteristic
115	C₄×C₇	—	4	2	no	—
116	C₄×C₇	#389	1	1	yes	characteristic
117	C₄×C₇	C₄×D₈	1	1	yes	characteristic
118	C₄×C₇	#466	1	1	yes	characteristic
119	C₄×C₇	#499	1	1	yes	characteristic
120	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
121	M₁₆	—	2	1	no	—
122	M₁₆	—	2	4	no	—
123	M₁₆	—	4	2	no	—
124	M₁₆	—	4	2	no	—
125	M₁₆	—	4	2	no	—
126	M₁₆	C₇×D₈	1	2	yes	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
131	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
132	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
133	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
134	C₂×C₈	C₇×#41	1	1	yes	characteristic
135	C₂×C₈	C₇×#41	1	1	yes	characteristic
136	C₄⋊C₄	—	2	1	no	—
137	C₂²⋊C₄	C₇×Q₁₆	1	2	yes	—
138	C₄²	—	2	1	no	—
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	—	2	1	no	—
145	C₂×C₇	#1942	1	1	yes	characteristic
146	C₂³	C₂×C₇×Q₁₆	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
149	C₂×C₄	C₇×#121	1	1	yes	characteristic
150	C₂×C₄	C₇×C₄⋊Q₈	1	1	yes	characteristic
151	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
152	C₂×C₄	C₇×#135	1	1	yes	characteristic
153	C₂×C₄	C₇×#135	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	4	4	no	—
156	C₈	—	8	2	no	—
157	C₈	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
158	C₈	C₄×C₇×Q₈	1	1	yes	characteristic
159	C₈	C₇×C₂²⋊Q₈	1	1	yes	characteristic
160	C₈	C₇×#125	1	1	yes	characteristic
161	C₇	#13173	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²	—	4	1	no	—
163	C₂²	C₄×C₇×Q₁₆	1	1	yes	characteristic
164	C₂²	C₇×#467	1	1	yes	characteristic
165	C₂²	C₇×#500	1	1	yes	characteristic
166	C₄	—	4	2	no	—
167	C₄	C₇×#389	1	1	yes	characteristic
168	C₄	C₄×C₇×D₈	1	1	yes	characteristic
169	C₄	C₇×#466	1	1	yes	characteristic
170	C₄	C₇×#499	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	C₇×#1942	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	#466
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#1942
prime radical	C₂³×C₇	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#405 ◁ G

Complements

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

N	H ≅ G/N	type
14. C₇×#368	C₄	semidirect
18. #13173	C₇	direct (N × H)
73. #368	C₄×C₇	semidirect
161. C₇	#13173	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×#2108	2
C₇×#1953	2
C₇×#1388	2
#13173	7

Subgroups — C7×#13173 #13190389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13173 #13190389