Subgroups — C₇×#13172 #13190388

Order 1792.

274 conjugacy classes of subgroups.

The subgroup lattice has 182 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×#367	C₂	1	1	yes	maximal, characteristic
3	C₇×#2108	C₂	1	1	yes	maximal, characteristic
4	C₇×#1952	C₂	1	2	yes	maximal
5	C₇×#1387	C₂	1	2	yes	maximal
6	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₇×#108	C₂²	1	2	yes	—
9	C₇×#428	—	2	2	no	—
10	C₂×C₇×#107	C₂²	1	2	yes	—
11	C₇×#405	C₂²	1	1	yes	characteristic
12	C₇×#368	C₄	1	4	yes	complemented
13	C₇×#367	—	2	1	no	—
14	C₇×#367	—	2	1	no	—
15	C₇×#346	—	2	1	no	—
16	C₄×C₇×C₁₆	—	2	1	no	—
17	C₇×#343	—	2	2	no	—
18	#13172	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂×C₇×M₁₆	—	2	2	no	—
20	C₂×C₇×M₁₆	D₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
22	C₇×#117	D₄	1	1	yes	characteristic
23	C₂×C₇×C₄⋊C₄	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
33	C₇×#107	—	2	2	no	—
34	C₇×#107	—	2	2	no	—
35	C₇×#107	—	4	2	no	—
36	C₇×#106	—	4	2	no	—
37	C₇×#97	—	2	1	no	—
38	C₄×C₇×C₈	—	2	1	no	—
39	C₂×#368	C₂×C₇	1	1	yes	characteristic
40	C₂×#367	C₂×C₇	1	1	yes	characteristic
41	#2108	C₂×C₇	1	1	yes	characteristic
42	#1952	C₂×C₇	1	2	yes	—
43	#1387	C₂×C₇	1	2	yes	—
44	C₂²×C₄×C₇	—	4	2	no	—
45	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
46	C₇×M₁₆	—	2	1	no	—
47	C₇×M₁₆	—	2	4	no	—
48	C₇×M₁₆	—	4	2	no	—
49	C₇×M₁₆	Q₁₆	1	2	yes	—
50	C₂×C₇×C₈	—	2	1	no	—
51	C₂×C₇×C₈	—	4	2	no	—
52	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₇×C₈	#41	1	1	yes	characteristic
57	C₂×C₇×C₈	#41	1	1	yes	characteristic
58	C₇×C₄⋊C₄	—	4	2	no	—
59	C₇×C₄⋊C₄	—	4	2	no	—
60	C₇×C₄⋊C₄	D₈	1	2	yes	—
61	C₇×C₂²⋊C₄	—	2	1	no	—
62	C₄²×C₇	—	2	1	no	—
63	C₇×C₁₆	—	2	2	no	—
64	C₇×C₁₆	C₄⋊C₄	1	2	yes	—
65	C₇×C₁₆	C₄⋊C₄	1	2	yes	—
66	C₂×M₃₂	C₂²×C₇	1	1	yes	characteristic
67	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
68	C₂×#108	C₂²×C₇	1	2	yes	—
69	#428	—	2	2	no	—
70	C₂×#107	C₂²×C₇	1	2	yes	—
71	#405	C₂²×C₇	1	1	yes	characteristic
72	#368	C₄×C₇	1	4	yes	complemented
73	#367	—	2	1	no	—
74	#367	—	2	1	no	—
75	#346	—	2	1	no	—
76	C₄×C₁₆	—	2	1	no	—
77	#343	—	2	2	no	—
78	C₂³×C₇	#135	1	1	yes	characteristic
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	8	2	no	—
81	C₂×C₄×C₇	—	16	2	no	—
82	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₇	#121	1	1	yes	characteristic
84	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇	#135	1	1	yes	characteristic
88	C₇×C₈	—	2	2	no	—
89	C₇×C₈	—	4	4	no	—
90	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
92	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
93	C₇×C₈	#125	1	1	yes	characteristic
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	C₇×D₄	1	1	yes	characteristic
96	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
97	#117	C₇×D₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄	—	2	2	no	—
99	M₃₂	—	2	1	no	—
100	M₃₂	C₇×Q₈	1	2	yes	—
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
103	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
104	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
105	C₂×C₁₆	C₇×Q₈	1	2	yes	—
106	#108	—	2	2	no	—
107	#108	C₂×C₄×C₇	1	4	yes	—
108	#107	—	2	2	no	—
109	#107	—	2	2	no	—
110	#107	—	4	2	no	—
111	#106	—	4	2	no	—
112	#97	—	2	1	no	—
113	C₄×C₈	—	2	1	no	—
114	C₂²×C₇	—	2	2	no	—
115	C₂²×C₇	C₄×Q₁₆	1	1	yes	characteristic
116	C₂²×C₇	#466	1	1	yes	characteristic
117	C₂²×C₇	#499	1	1	yes	characteristic
118	C₄×C₇	—	4	2	no	—
119	C₄×C₇	—	16	2	no	—
120	C₄×C₇	#389	1	1	yes	characteristic
121	C₄×C₇	C₄×D₈	1	1	yes	characteristic
122	C₄×C₇	#467	1	1	yes	characteristic
123	C₄×C₇	#500	1	1	yes	characteristic
124	C₂²×C₄	—	4	2	no	—
125	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
126	M₁₆	—	2	1	no	—
127	M₁₆	—	2	4	no	—
128	M₁₆	—	4	2	no	—
129	M₁₆	C₇×Q₁₆	1	2	yes	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
133	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
134	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
135	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
136	C₂×C₈	C₇×#41	1	1	yes	characteristic
137	C₂×C₈	C₇×#41	1	1	yes	characteristic
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	C₇×D₈	1	2	yes	—
141	C₂²⋊C₄	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₁₆	—	2	2	no	—
144	C₁₆	C₇×C₄⋊C₄	1	2	yes	—
145	C₁₆	C₇×C₄⋊C₄	1	2	yes	—
146	C₂×C₇	—	2	1	no	—
147	C₂×C₇	—	2	1	no	—
148	C₂×C₇	#1942	1	1	yes	characteristic
149	C₂×C₇	#2223	1	2	yes	—
150	C₂³	C₇×#135	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	16	2	no	—
154	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
155	C₂×C₄	C₇×#121	1	1	yes	characteristic
156	C₂×C₄	C₇×C₄⋊Q₈	1	1	yes	characteristic
157	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
158	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
159	C₂×C₄	C₇×#135	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	4	4	no	—
162	C₈	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
163	C₈	C₄×C₇×Q₈	1	1	yes	characteristic
164	C₈	C₇×C₂²⋊Q₈	1	1	yes	characteristic
165	C₈	C₇×#125	1	1	yes	characteristic
166	C₇	#13172	1	1	yes	characteristic, direct product factor, Sylow
167	C₂²	—	2	2	no	—
168	C₂²	C₄×C₇×Q₁₆	1	1	yes	characteristic
169	C₂²	C₇×#466	1	1	yes	characteristic
170	C₂²	C₇×#499	1	1	yes	characteristic
171	C₄	—	4	2	no	—
172	C₄	—	16	2	no	—
173	C₄	C₇×#389	1	1	yes	characteristic
174	C₄	C₄×C₇×D₈	1	1	yes	characteristic
175	C₄	C₇×#467	1	1	yes	characteristic
176	C₄	C₇×#500	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	C₇×#1942	1	1	yes	characteristic
180	C₂	C₇×#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	#466
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂³×C₇	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#405 ◁ G

Complements

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

N	H ≅ G/N	type
12. C₇×#368	C₄	semidirect
18. #13172	C₇	direct (N × H)
72. #368	C₄×C₇	semidirect
166. C₇	#13172	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
C₂×C₇×#367	2
C₇×#2108	2
C₇×#1952	2
C₇×#1387	2
#13172	7

Subgroups — C7×#13172 #13190388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13172 #13190388