Subgroups — C₇×#20351 #13197567

Order 1792.

1978 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#568	C₂	1	2	yes	maximal
2	C₂²×C₇×#131	C₂	1	1	yes	maximal, characteristic
3	C₇×#1738	C₂	1	6	yes	maximal, complemented
4	C₂×C₇×#341	C₂	1	6	yes	maximal, complemented
5	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
6	C₇×#568	—	2	8	no	—
7	C₇×#568	C₄	1	16	yes	—
8	C₂×C₇×#131	—	2	8	no	—
9	C₂×C₇×#131	—	2	8	no	—
10	C₂×C₇×#131	C₄	1	8	yes	—
11	C₂²×C₇×M₁₆	C₂²	1	12	yes	—
12	C₂²×C₇×M₁₆	C₂²	1	3	yes	—
13	C₂³×C₇×C₈	C₂²	1	3	yes	—
14	C₇×#341	C₂²	1	4	yes	—
15	C₇×#341	C₂²	1	12	yes	—
16	#20351	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂×C₇×#41	C₂×C₄	1	4	yes	—
18	C₂×C₇×#41	C₂×C₄	1	8	yes	—
19	C₂×C₇×#41	D₄	1	4	yes	—
20	C₂×C₇×#41	D₄	1	8	yes	—
21	C₂²×C₇×Q₈	Q₈	1	1	yes	characteristic
22	C₂²×C₇×D₄	Q₈	1	3	yes	—
23	C₂³×C₄×C₇	C₂³	1	3	yes	—
24	C₇×#131	—	2	32	no	—
25	C₇×#131	—	2	64	no	—
26	C₂×C₇×M₁₆	—	2	12	no	—
27	C₂×C₇×M₁₆	—	2	12	no	—
28	C₂×C₇×M₁₆	—	2	24	no	—
29	C₂×C₇×M₁₆	C₂×C₄	1	48	yes	—
30	C₂×C₇×M₁₆	C₂×C₄	1	12	yes	—
31	C₂×C₇×M₁₆	C₂³	1	8	yes	—
32	C₂²×C₇×C₈	—	2	12	no	—
33	C₂²×C₇×C₈	—	2	12	no	—
34	C₂²×C₇×C₈	C₂×C₄	1	12	yes	—
35	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₇×C₈	C₂³	1	3	yes	—
37	C₂×#568	C₂×C₇	1	2	yes	—
38	C₂²×#131	C₂×C₇	1	1	yes	characteristic
39	#1738	C₂×C₇	1	6	yes	complemented
40	C₂×#341	C₂×C₇	1	6	yes	complemented
41	C₂⁴×C₇	C₂×Q₈	1	3	yes	—
42	C₇×#41	C₄²	1	16	yes	—
43	C₇×#41	C₂²⋊C₄	1	16	yes	—
44	C₇×#41	C₂²⋊C₄	1	32	yes	—
45	C₂×C₇×Q₈	C₄⋊C₄	1	4	yes	—
46	C₂×C₇×Q₈	C₄⋊C₄	1	8	yes	—
47	C₂×C₇×D₄	C₄⋊C₄	1	12	yes	—
48	C₂×C₇×D₄	C₄⋊C₄	1	24	yes	—
49	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
50	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
51	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
52	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
53	C₂²×C₄×C₇	C₂×Q₈	1	3	yes	—
54	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
55	C₇×M₁₆	—	2	48	no	—
56	C₇×M₁₆	—	2	24	no	—
57	C₇×M₁₆	—	2	8	no	—
58	C₇×M₁₆	C₂²×C₄	1	16	yes	—
59	C₂×C₇×C₈	—	2	24	no	—
60	C₂×C₇×C₈	—	2	48	no	—
61	C₂×C₇×C₈	—	2	2	no	—
62	C₂×C₇×C₈	—	2	6	no	—
63	C₂×C₇×C₈	—	2	8	no	—
64	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
65	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
66	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
67	#568	—	2	8	no	—
68	#568	C₄×C₇	1	16	yes	—
69	C₂×#131	—	2	8	no	—
70	C₂×#131	—	2	8	no	—
71	C₂×#131	C₄×C₇	1	8	yes	—
72	C₂²×M₁₆	C₂²×C₇	1	12	yes	—
73	C₂²×M₁₆	C₂²×C₇	1	3	yes	—
74	C₂³×C₈	C₂²×C₇	1	3	yes	—
75	#341	C₂²×C₇	1	4	yes	—
76	#341	C₂²×C₇	1	12	yes	—
77	C₂³×C₇	—	4	6	no	—
78	C₂³×C₇	C₂×C₄⋊C₄	1	6	yes	—
79	C₂³×C₇	C₂×C₄⋊C₄	1	12	yes	—
80	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
81	C₇×Q₈	#95	1	16	yes	—
82	C₇×D₄	#95	1	48	yes	—
83	C₂×C₄×C₇	C₂×C₄²	1	12	yes	—
84	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	12	yes	—
85	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	24	yes	—
86	C₂×C₄×C₇	C₂×C₄⋊C₄	1	12	yes	—
87	C₂×C₄×C₇	C₂×C₄⋊C₄	1	6	yes	—
88	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
90	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
91	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
92	C₇×C₈	—	2	2	no	—
93	C₇×C₈	—	2	16	no	—
94	C₇×C₈	—	2	6	no	—
95	C₂×#41	C₂×C₄×C₇	1	4	yes	—
96	C₂×#41	C₂×C₄×C₇	1	8	yes	—
97	C₂×#41	C₇×D₄	1	4	yes	—
98	C₂×#41	C₇×D₄	1	8	yes	—
99	C₂²×Q₈	C₇×Q₈	1	1	yes	characteristic
100	C₂²×D₄	C₇×Q₈	1	3	yes	—
101	C₂³×C₄	C₂³×C₇	1	3	yes	—
102	#131	—	2	32	no	—
103	#131	—	2	64	no	—
104	C₂×M₁₆	—	2	12	no	—
105	C₂×M₁₆	—	2	12	no	—
106	C₂×M₁₆	—	2	24	no	—
107	C₂×M₁₆	C₂×C₄×C₇	1	48	yes	—
108	C₂×M₁₆	C₂×C₄×C₇	1	12	yes	—
109	C₂×M₁₆	C₂³×C₇	1	8	yes	—
110	C₂²×C₈	—	2	12	no	—
111	C₂²×C₈	—	2	12	no	—
112	C₂²×C₈	C₂×C₄×C₇	1	12	yes	—
113	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
114	C₂²×C₈	C₂³×C₇	1	3	yes	—
115	C₂²×C₇	—	2	2	no	—
116	C₂²×C₇	—	2	12	no	—
117	C₂²×C₇	—	4	12	no	—
118	C₂²×C₇	C₂×#95	1	12	yes	—
119	C₂²×C₇	C₂²×C₄⋊C₄	1	2	yes	—
120	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₄×C₇	C₂×#95	1	12	yes	—
122	C₄×C₇	C₂²×C₄²	1	1	yes	characteristic
123	C₄×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
124	C₄×C₇	C₂²×C₂²⋊C₄	1	2	yes	—
125	C₂⁴	C₂×C₇×Q₈	1	3	yes	—
126	#41	C₄²×C₇	1	16	yes	—
127	#41	C₇×C₂²⋊C₄	1	16	yes	—
128	#41	C₇×C₂²⋊C₄	1	32	yes	—
129	C₂×Q₈	C₇×C₄⋊C₄	1	4	yes	—
130	C₂×Q₈	C₇×C₄⋊C₄	1	8	yes	—
131	C₂×D₄	C₇×C₄⋊C₄	1	12	yes	—
132	C₂×D₄	C₇×C₄⋊C₄	1	24	yes	—
133	C₂²×C₄	C₂²×C₄×C₇	1	6	yes	—
134	C₂²×C₄	C₂²×C₄×C₇	1	12	yes	—
135	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
136	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
137	C₂²×C₄	C₂×C₇×Q₈	1	3	yes	—
138	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
139	M₁₆	—	2	48	no	—
140	M₁₆	—	2	24	no	—
141	M₁₆	—	2	8	no	—
142	M₁₆	C₂²×C₄×C₇	1	16	yes	—
143	C₂×C₈	—	2	24	no	—
144	C₂×C₈	—	2	48	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	8	no	—
147	C₂×C₈	—	2	6	no	—
148	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
149	C₂×C₈	C₂²×C₄×C₇	1	6	yes	—
150	C₂×C₇	—	2	2	no	—
151	C₂×C₇	—	2	12	no	—
152	C₂×C₇	C₂²×#95	1	1	yes	characteristic
153	C₂×C₇	#2916	1	2	yes	—
154	C₂³	—	4	6	no	—
155	C₂³	C₂×C₇×C₄⋊C₄	1	6	yes	—
156	C₂³	C₂×C₇×C₄⋊C₄	1	12	yes	—
157	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
158	Q₈	C₇×#95	1	16	yes	—
159	D₄	C₇×#95	1	48	yes	—
160	C₂×C₄	C₂×C₄²×C₇	1	12	yes	—
161	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	12	yes	—
162	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	24	yes	—
163	C₂×C₄	C₂×C₇×C₄⋊C₄	1	12	yes	—
164	C₂×C₄	C₂×C₇×C₄⋊C₄	1	6	yes	—
165	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
166	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
167	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
168	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
169	C₈	—	2	2	no	—
170	C₈	—	2	16	no	—
171	C₈	—	2	6	no	—
172	C₇	#20351	1	1	yes	characteristic, direct product factor, Sylow
173	C₂²	—	2	2	no	—
174	C₂²	—	2	12	no	—
175	C₂²	—	4	12	no	—
176	C₂²	C₂×C₇×#95	1	12	yes	—
177	C₂²	C₂²×C₇×C₄⋊C₄	1	2	yes	—
178	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
179	C₄	C₂×C₇×#95	1	12	yes	—
180	C₄	C₂²×C₄²×C₇	1	1	yes	characteristic
181	C₄	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
182	C₄	C₂²×C₇×C₂²⋊C₄	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	2	12	no	—
185	C₂	C₂²×C₇×#95	1	1	yes	characteristic
186	C₂	C₇×#2916	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂²×C₇×#41	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1738	C₂	semidirect
4. C₂×C₇×#341	C₂	semidirect
16. #20351	C₇	direct (N × H)
39. #1738	C₂×C₇	semidirect
40. C₂×#341	C₂×C₇	semidirect
172. C₇	#20351	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×#568	2
C₂²×C₇×#131	2
C₇×#1738	2
C₂×C₇×#341	2
#20351	7

Subgroups — C7×#20351 #13197567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20351 #13197567