Subgroups — C₇×#36701 #13213917

Order 1792.

1304 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2845	C₂	1	1	yes	maximal, characteristic
2	C₇×#2839	C₂	1	12	yes	maximal
3	C₇×#2822	C₂	1	3	yes	maximal
4	C₇×#2402	C₂	1	12	yes	maximal
5	C₄×C₇×#125	C₂	1	3	yes	maximal
6	C₂×C₇×#125	C₂²	1	3	yes	—
7	C₇×#400	C₂²	1	12	yes	—
8	C₇×#400	C₂²	1	12	yes	—
9	C₇×#400	C₂²	1	12	yes	—
10	C₇×#400	C₂²	1	12	yes	—
11	C₇×#398	C₂²	1	12	yes	—
12	C₇×#398	C₂²	1	12	yes	—
13	C₇×#398	C₂²	1	12	yes	—
14	C₇×#398	C₂²	1	12	yes	—
15	C₇×#389	C₂²	1	6	yes	—
16	C₇×#387	C₂²	1	12	yes	—
17	C₇×#382	C₂²	1	3	yes	—
18	C₄×C₇×C₄⋊C₄	C₂²	1	12	yes	—
19	C₄×C₇×C₄⋊C₄	C₂²	1	12	yes	—
20	C₄×C₇×C₄⋊C₄	C₂²	1	6	yes	—
21	C₇×#376	C₂²	1	4	yes	—
22	C₄³×C₇	C₂²	1	1	yes	characteristic
23	#36701	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₇×#125	—	2	12	no	—
25	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
26	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
27	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
28	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
29	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
30	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
31	C₂×C₄²×C₇	C₂³	1	3	yes	—
32	C₂×C₄²×C₇	C₂³	1	12	yes	—
33	C₂×C₄²×C₇	C₂³	1	3	yes	—
34	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
35	C₇×#95	C₂³	1	12	yes	—
36	C₇×#95	C₂³	1	4	yes	—
37	C₇×#95	C₂³	1	12	yes	—
38	C₇×#95	C₂³	1	12	yes	—
39	C₇×#95	C₂³	1	12	yes	—
40	C₇×#95	C₂³	1	12	yes	—
41	C₇×#95	C₂³	1	12	yes	—
42	#2845	C₂×C₇	1	1	yes	characteristic
43	#2839	C₂×C₇	1	12	yes	—
44	#2822	C₂×C₇	1	3	yes	—
45	#2402	C₂×C₇	1	12	yes	—
46	C₄×#125	C₂×C₇	1	3	yes	—
47	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
48	C₂²×C₄×C₇	C₂⁴	1	12	yes	—
49	C₂²×C₄×C₇	C₂⁴	1	12	yes	—
50	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
51	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
52	C₇×C₄⋊C₄	—	2	24	no	—
53	C₇×C₄⋊C₄	—	2	12	no	—
54	C₇×C₄⋊C₄	—	4	6	no	—
55	C₇×C₄⋊C₄	—	4	12	no	—
56	C₇×C₄⋊C₄	—	4	12	no	—
57	C₇×C₄⋊C₄	—	4	12	no	—
58	C₄²×C₇	—	2	6	no	—
59	C₄²×C₇	—	2	24	no	—
60	C₄²×C₇	—	4	3	no	—
61	C₄²×C₇	#41	1	4	yes	—
62	C₂×#125	C₂²×C₇	1	3	yes	—
63	#400	C₂²×C₇	1	12	yes	—
64	#400	C₂²×C₇	1	12	yes	—
65	#400	C₂²×C₇	1	12	yes	—
66	#400	C₂²×C₇	1	12	yes	—
67	#398	C₂²×C₇	1	12	yes	—
68	#398	C₂²×C₇	1	12	yes	—
69	#398	C₂²×C₇	1	12	yes	—
70	#398	C₂²×C₇	1	12	yes	—
71	#389	C₂²×C₇	1	6	yes	—
72	#387	C₂²×C₇	1	12	yes	—
73	#382	C₂²×C₇	1	3	yes	—
74	C₄×C₄⋊C₄	C₂²×C₇	1	12	yes	—
75	C₄×C₄⋊C₄	C₂²×C₇	1	6	yes	—
76	C₄×C₄⋊C₄	C₂²×C₇	1	12	yes	—
77	#376	C₂²×C₇	1	4	yes	—
78	C₄³	C₂²×C₇	1	1	yes	characteristic
79	C₂³×C₇	C₂⁵	1	1	yes	characteristic
80	C₂×C₄×C₇	—	2	3	no	—
81	C₂×C₄×C₇	—	2	3	no	—
82	C₂×C₄×C₇	—	2	12	no	—
83	C₂×C₄×C₇	—	2	12	no	—
84	C₂×C₄×C₇	—	2	3	no	—
85	C₂×C₄×C₇	—	2	12	no	—
86	C₂×C₄×C₇	—	2	6	no	—
87	C₂×C₄×C₇	—	2	12	no	—
88	C₂×C₄×C₇	—	2	12	no	—
89	C₂×C₄×C₇	—	2	12	no	—
90	C₂×C₄×C₇	C₂×#41	1	6	yes	—
91	C₂×C₄×C₇	2₋¹⁺⁴	1	6	yes	—
92	#125	—	2	12	no	—
93	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
94	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
95	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
96	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
97	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
98	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
99	C₂×C₄²	C₂³×C₇	1	3	yes	—
100	C₂×C₄²	C₂³×C₇	1	12	yes	—
101	C₂×C₄²	C₂³×C₇	1	3	yes	—
102	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
103	#95	C₂³×C₇	1	12	yes	—
104	#95	C₂³×C₇	1	4	yes	—
105	#95	C₂³×C₇	1	12	yes	—
106	#95	C₂³×C₇	1	12	yes	—
107	#95	C₂³×C₇	1	12	yes	—
108	#95	C₂³×C₇	1	12	yes	—
109	#95	C₂³×C₇	1	12	yes	—
110	C₂²×C₇	C₂²×#41	1	1	yes	characteristic
111	C₂²×C₇	C₂×2₋¹⁺⁴	1	3	yes	—
112	C₂²×C₇	#585	1	3	yes	—
113	C₄×C₇	—	2	6	no	—
114	C₄×C₇	—	4	12	no	—
115	C₄×C₇	—	4	3	no	—
116	C₄×C₇	—	4	12	no	—
117	C₄×C₇	#564	1	4	yes	—
118	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
119	C₂²×C₄	C₂⁴×C₇	1	12	yes	—
120	C₂²×C₄	C₂⁴×C₇	1	12	yes	—
121	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
122	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
123	C₄⋊C₄	—	2	24	no	—
124	C₄⋊C₄	—	2	12	no	—
125	C₄⋊C₄	—	4	6	no	—
126	C₄⋊C₄	—	4	12	no	—
127	C₄⋊C₄	—	4	12	no	—
128	C₄⋊C₄	—	4	12	no	—
129	C₄²	—	2	6	no	—
130	C₄²	—	2	24	no	—
131	C₄²	—	4	3	no	—
132	C₄²	C₇×#41	1	4	yes	—
133	C₂×C₇	#3461	1	3	yes	—
134	C₂×C₇	C₂×#564	1	1	yes	characteristic
135	C₂×C₇	#3531	1	3	yes	—
136	C₂³	C₂⁵×C₇	1	1	yes	characteristic
137	C₂×C₄	—	2	3	no	—
138	C₂×C₄	—	2	3	no	—
139	C₂×C₄	—	2	12	no	—
140	C₂×C₄	—	2	12	no	—
141	C₂×C₄	—	2	3	no	—
142	C₂×C₄	—	2	12	no	—
143	C₂×C₄	—	2	6	no	—
144	C₂×C₄	—	2	12	no	—
145	C₂×C₄	—	2	12	no	—
146	C₂×C₄	—	2	12	no	—
147	C₂×C₄	C₂×C₇×#41	1	6	yes	—
148	C₂×C₄	C₇×2₋¹⁺⁴	1	6	yes	—
149	C₇	#36701	1	1	yes	characteristic, direct product factor, Sylow
150	C₂²	C₂²×C₇×#41	1	1	yes	characteristic
151	C₂²	C₂×C₇×2₋¹⁺⁴	1	3	yes	—
152	C₂²	C₇×#585	1	3	yes	—
153	C₄	—	2	6	no	—
154	C₄	—	4	12	no	—
155	C₄	—	4	3	no	—
156	C₄	—	4	12	no	—
157	C₄	C₇×#564	1	4	yes	—
158	C₂	C₇×#3461	1	3	yes	—
159	C₂	C₂×C₇×#564	1	1	yes	characteristic
160	C₂	C₇×#3531	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇
socle, prime radical	C₂³×C₇	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
23. #36701	C₇	direct (N × H)
149. C₇	#36701	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×#2845	2
C₇×#2839	2
C₇×#2822	2
C₇×#2402	2
C₄×C₇×#125	2
#36701	7

Subgroups — C7×#36701 #13213917

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#36701 #13213917