Subgroups — C₇×#19653 #13196869

Order 1792.

730 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3229	C₂	1	4	yes	maximal
2	C₇×#3160	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic
4	C₇×#1716	C₂	1	4	yes	maximal, complemented
5	C₇×#1707	C₂	1	4	yes	maximal, complemented
6	C₇×#1450	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₇×Q₁₆	—	2	2	no	—
8	C₇×#533	C₂²	1	2	yes	—
9	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₇×#500	C₂²	1	2	yes	—
11	C₇×#499	—	2	2	no	—
12	C₇×#498	C₂²	1	8	yes	—
13	C₇×#498	C₂²	1	2	yes	—
14	C₇×#494	—	2	2	no	—
15	C₇×#494	C₂²	1	2	yes	—
16	C₇×#494	C₂²	1	2	yes	—
17	C₇×#487	C₂²	1	8	yes	—
18	C₇×#483	—	2	8	no	—
19	C₇×#475	—	2	8	no	—
20	C₇×#431	C₂²	1	2	yes	—
21	C₇×#427	—	2	2	no	—
22	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
23	C₇×#322	C₂²	1	2	yes	—
24	C₇×C₈²	C₂²	1	2	yes	—
25	#19653	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₇×Q₈	—	2	2	no	—
27	C₂×C₇×Q₁₆	—	2	4	no	—
28	C₂×C₇×Q₁₆	—	2	4	no	—
29	C₂×C₇×Q₁₆	—	4	8	no	—
30	C₂²×C₇×C₈	D₄	1	2	yes	—
31	C₇×C₄⋊Q₈	—	2	4	no	—
32	C₇×C₄⋊Q₈	—	2	2	no	—
33	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
34	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
35	C₇×#125	—	2	2	no	—
36	C₇×C₂²⋊Q₈	—	2	4	no	—
37	C₄×C₇×Q₈	—	2	4	no	—
38	C₇×#117	—	2	2	no	—
39	C₂×C₇×C₄⋊C₄	—	2	4	no	—
40	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
41	C₇×#107	—	2	4	no	—
42	C₇×#106	—	2	16	no	—
43	C₇×#106	—	2	4	no	—
44	C₇×C₄⋊C₈	D₄	1	4	yes	—
45	C₇×#103	—	2	16	no	—
46	C₇×C₂²⋊C₈	D₄	1	4	yes	—
47	C₄×C₇×C₈	D₄	1	2	yes	—
48	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
49	C₄×C₇×C₈	C₂³	1	4	yes	—
50	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
51	#3229	C₂×C₇	1	4	yes	—
52	#3160	C₂×C₇	1	1	yes	characteristic
53	C₂×#494	C₂×C₇	1	1	yes	characteristic
54	#1716	C₂×C₇	1	4	yes	complemented
55	#1707	C₂×C₇	1	4	yes	complemented
56	#1450	C₂×C₇	1	1	yes	characteristic
57	C₂×C₇×Q₈	—	2	4	no	—
58	C₂×C₇×Q₈	—	2	4	no	—
59	C₂×C₇×Q₈	—	4	2	no	—
60	C₂×C₇×Q₈	—	4	8	no	—
61	C₂²×C₄×C₇	—	4	2	no	—
62	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
64	C₇×Q₁₆	—	4	8	no	—
65	C₇×Q₁₆	—	4	8	no	—
66	C₂×C₇×C₈	SD₁₆	1	8	yes	—
67	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
68	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
69	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
70	C₇×C₄⋊C₄	—	2	8	no	—
71	C₇×C₄⋊C₄	—	2	4	no	—
72	C₇×C₄⋊C₄	—	2	4	no	—
73	C₇×C₄⋊C₄	—	4	4	no	—
74	C₇×C₄⋊C₄	—	4	4	no	—
75	C₇×C₂²⋊C₄	—	4	2	no	—
76	C₄²×C₇	—	4	2	no	—
77	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₇	C₂×D₄	1	2	yes	—
79	C₄²×C₇	C₂⁴	1	1	yes	characteristic
80	C₂²×Q₁₆	—	2	2	no	—
81	#533	C₂²×C₇	1	2	yes	—
82	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
83	#500	C₂²×C₇	1	2	yes	—
84	#499	—	2	2	no	—
85	#498	C₂²×C₇	1	2	yes	—
86	#498	C₂²×C₇	1	8	yes	—
87	#494	—	2	2	no	—
88	#494	C₂²×C₇	1	2	yes	—
89	#494	C₂²×C₇	1	2	yes	—
90	#487	C₂²×C₇	1	8	yes	—
91	#483	—	2	8	no	—
92	#475	—	2	8	no	—
93	#431	C₂²×C₇	1	2	yes	—
94	#427	—	2	2	no	—
95	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
96	#322	C₂²×C₇	1	2	yes	—
97	C₈²	C₂²×C₇	1	2	yes	—
98	C₂³×C₇	#127	1	1	yes	characteristic
99	C₇×Q₈	—	4	8	no	—
100	C₇×Q₈	—	4	8	no	—
101	C₇×Q₈	—	8	4	no	—
102	C₂×C₄×C₇	—	2	4	no	—
103	C₂×C₄×C₇	—	4	4	no	—
104	C₂×C₄×C₇	—	4	4	no	—
105	C₂×C₄×C₇	—	8	4	no	—
106	C₂×C₄×C₇	#127	1	2	yes	—
107	C₂×C₄×C₇	#127	1	1	yes	characteristic
108	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
109	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
111	C₇×C₈	—	2	8	no	—
112	C₇×C₈	C₂×SD₁₆	1	4	yes	—
113	C₇×C₈	C₂×SD₁₆	1	4	yes	—
114	C₂²×Q₈	—	2	2	no	—
115	C₂×Q₁₆	—	2	4	no	—
116	C₂×Q₁₆	—	2	4	no	—
117	C₂×Q₁₆	—	4	8	no	—
118	C₂²×C₈	C₇×D₄	1	2	yes	—
119	C₄⋊Q₈	—	2	4	no	—
120	C₄⋊Q₈	—	2	2	no	—
121	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
122	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
123	#125	—	2	2	no	—
124	C₂²⋊Q₈	—	2	4	no	—
125	C₄×Q₈	—	2	4	no	—
126	#117	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	4	no	—
128	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
129	#107	—	2	4	no	—
130	#106	—	2	4	no	—
131	#106	—	2	16	no	—
132	C₄⋊C₈	C₇×D₄	1	4	yes	—
133	#103	—	2	16	no	—
134	C₂²⋊C₈	C₇×D₄	1	4	yes	—
135	C₄×C₈	C₇×D₄	1	2	yes	—
136	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
137	C₄×C₈	C₂³×C₇	1	4	yes	—
138	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
139	C₂²×C₇	—	2	2	no	—
140	C₂²×C₇	#492	1	2	yes	—
141	C₂²×C₇	C₂×#127	1	1	yes	characteristic
142	C₄×C₇	—	2	2	no	—
143	C₄×C₇	—	8	4	no	—
144	C₄×C₇	—	8	4	no	—
145	C₄×C₇	#492	1	2	yes	—
146	C₄×C₇	C₂²×SD₁₆	1	2	yes	—
147	C₄×C₇	#578	1	4	yes	—
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	—	2	4	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	4	8	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
154	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
155	Q₁₆	—	4	8	no	—
156	Q₁₆	—	4	8	no	—
157	C₂×C₈	C₇×SD₁₆	1	8	yes	—
158	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
159	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
160	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
161	C₄⋊C₄	—	2	8	no	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₂²⋊C₄	—	4	2	no	—
167	C₄²	—	4	2	no	—
168	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
169	C₄²	C₂×C₇×D₄	1	2	yes	—
170	C₄²	C₂⁴×C₇	1	1	yes	characteristic
171	C₂×C₇	—	2	2	no	—
172	C₂×C₇	C₂×#492	1	1	yes	characteristic
173	C₂×C₇	#3156	1	2	yes	—
174	C₂³	C₇×#127	1	1	yes	characteristic
175	Q₈	—	4	8	no	—
176	Q₈	—	4	8	no	—
177	Q₈	—	8	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	8	4	no	—
182	C₂×C₄	C₇×#127	1	2	yes	—
183	C₂×C₄	C₇×#127	1	1	yes	characteristic
184	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
185	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
186	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
187	C₈	—	2	8	no	—
188	C₈	C₂×C₇×SD₁₆	1	4	yes	—
189	C₈	C₂×C₇×SD₁₆	1	4	yes	—
190	C₇	#19653	1	1	yes	characteristic, direct product factor, Sylow
191	C₂²	—	2	2	no	—
192	C₂²	C₇×#492	1	2	yes	—
193	C₂²	C₂×C₇×#127	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	8	4	no	—
196	C₄	—	8	4	no	—
197	C₄	C₇×#492	1	2	yes	—
198	C₄	C₂²×C₇×SD₁₆	1	2	yes	—
199	C₄	C₇×#578	1	4	yes	—
200	C₂	—	2	2	no	—
201	C₂	C₂×C₇×#492	1	1	yes	characteristic
202	C₂	C₇×#3156	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂³×C₇	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1716	C₂	semidirect
5. C₇×#1707	C₂	semidirect
25. #19653	C₇	direct (N × H)
54. #1716	C₂×C₇	semidirect
55. #1707	C₂×C₇	semidirect
190. C₇	#19653	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×#3229	2
C₇×#3160	2
C₂×C₇×#494	2
C₇×#1716	2
C₇×#1707	2
C₇×#1450	2
#19653	7

Subgroups — C7×#19653 #13196869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19653 #13196869