Subgroups — C₇×#19652 #13196868

Order 1792.

882 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3208	C₂	1	4	yes	maximal, complemented
2	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic
4	C₇×#1715	C₂	1	4	yes	maximal, complemented
5	C₇×#1706	C₂	1	4	yes	maximal, complemented
6	C₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₇×D₈	—	2	2	no	—
8	C₇×#533	C₂²	1	2	yes	—
9	C₂×C₇×#127	C₂²	1	2	yes	—
10	C₇×#500	C₂²	1	2	yes	complemented
11	C₇×#499	—	2	2	no	—
12	C₇×#498	C₂²	1	2	yes	complemented
13	C₇×#493	—	2	2	no	—
14	C₇×#493	C₂²	1	2	yes	—
15	C₇×#493	C₂²	1	2	yes	—
16	C₇×#492	C₂²	1	8	yes	complemented
17	C₇×#486	C₂²	1	8	yes	complemented
18	C₇×#460	—	2	8	no	—
19	C₇×#450	—	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	complemented
24	C₇×C₈²	C₂²	1	2	yes	complemented
25	#19652	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₇×D₄	—	2	2	no	—
27	C₂²×C₇×D₄	—	2	4	no	—
28	C₂×C₇×SD₁₆	—	2	16	no	—
29	C₂×C₇×D₈	—	2	4	no	—
30	C₂×C₇×D₈	—	2	4	no	—
31	C₂×C₇×D₈	—	4	8	no	—
32	C₂²×C₇×C₈	D₄	1	2	yes	—
33	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
34	C₇×#127	—	2	2	no	—
35	C₇×#127	—	2	4	no	—
36	C₇×#127	C₂³	1	4	yes	—
37	C₇×#125	—	2	2	no	—
38	C₇×C₂²⋊Q₈	—	2	4	no	—
39	C₄×C₇×Q₈	—	2	4	no	—
40	C₇×#117	—	2	2	no	—
41	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
42	C₇×#107	—	2	4	no	—
43	C₇×#106	—	2	4	no	—
44	C₇×C₄⋊C₈	D₄	1	4	yes	—
45	C₇×#102	—	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	#3208	C₂×C₇	1	4	yes	complemented
52	#3160	C₂×C₇	1	1	yes	characteristic, complemented
53	C₂×#493	C₂×C₇	1	1	yes	characteristic
54	#1715	C₂×C₇	1	4	yes	complemented
55	#1706	C₂×C₇	1	4	yes	complemented
56	#1450	C₂×C₇	1	1	yes	characteristic, complemented
57	C₂⁴×C₇	—	4	2	no	—
58	C₂×C₇×Q₈	—	2	4	no	—
59	C₂×C₇×D₄	—	2	8	no	—
60	C₂×C₇×D₄	—	2	4	no	—
61	C₂×C₇×D₄	—	4	4	no	—
62	C₂×C₇×D₄	—	4	2	no	—
63	C₂×C₇×D₄	—	4	8	no	—
64	C₂×C₇×D₄	—	8	8	no	—
65	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
67	C₇×SD₁₆	—	8	16	no	—
68	C₇×D₈	—	4	8	no	—
69	C₇×D₈	—	4	8	no	—
70	C₂×C₇×C₈	SD₁₆	1	8	yes	complemented
71	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
72	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
73	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
74	C₇×C₄⋊C₄	—	2	4	no	—
75	C₇×C₄⋊C₄	—	2	4	no	—
76	C₇×C₄⋊C₄	—	4	4	no	—
77	C₇×C₂²⋊C₄	—	4	2	no	—
78	C₄²×C₇	—	4	2	no	—
79	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₇	C₂×D₄	1	2	yes	—
81	C₄²×C₇	C₂⁴	1	1	yes	characteristic
82	C₂²×D₈	—	2	2	no	—
83	#533	C₂²×C₇	1	2	yes	—
84	C₂×#127	C₂²×C₇	1	2	yes	—
85	#500	C₂²×C₇	1	2	yes	complemented
86	#499	—	2	2	no	—
87	#498	C₂²×C₇	1	2	yes	complemented
88	#493	—	2	2	no	—
89	#493	C₂²×C₇	1	2	yes	—
90	#493	C₂²×C₇	1	2	yes	—
91	#492	C₂²×C₇	1	8	yes	complemented
92	#486	C₂²×C₇	1	8	yes	complemented
93	#460	—	2	8	no	—
94	#450	—	2	8	no	—
95	#431	C₂²×C₇	1	2	yes	—
96	#427	—	2	2	no	—
97	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
98	#322	C₂²×C₇	1	2	yes	complemented
99	C₈²	C₂²×C₇	1	2	yes	complemented
100	C₂³×C₇	—	4	4	no	—
101	C₂³×C₇	—	8	8	no	—
102	C₂³×C₇	—	8	4	no	—
103	C₂³×C₇	#127	1	1	yes	characteristic
104	C₇×Q₈	—	4	8	no	—
105	C₇×D₄	—	4	8	no	—
106	C₇×D₄	—	4	16	no	—
107	C₇×D₄	—	8	8	no	—
108	C₇×D₄	—	8	4	no	—
109	C₂×C₄×C₇	—	2	4	no	—
110	C₂×C₄×C₇	—	4	4	no	—
111	C₂×C₄×C₇	#127	1	2	yes	—
112	C₂×C₄×C₇	#127	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
114	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
116	C₇×C₈	—	2	8	no	—
117	C₇×C₈	C₂×SD₁₆	1	4	yes	—
118	C₇×C₈	C₂×SD₁₆	1	4	yes	—
119	C₂²×D₄	—	2	2	no	—
120	C₂²×D₄	—	2	4	no	—
121	C₂×SD₁₆	—	2	16	no	—
122	C₂×D₈	—	2	4	no	—
123	C₂×D₈	—	2	4	no	—
124	C₂×D₈	—	4	8	no	—
125	C₂²×C₈	C₇×D₄	1	2	yes	—
126	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
127	#127	—	2	4	no	—
128	#127	—	2	2	no	—
129	#127	C₂³×C₇	1	4	yes	—
130	#125	—	2	2	no	—
131	C₂²⋊Q₈	—	2	4	no	—
132	C₄×Q₈	—	2	4	no	—
133	#117	—	2	2	no	—
134	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
135	#107	—	2	4	no	—
136	#106	—	2	4	no	—
137	C₄⋊C₈	C₇×D₄	1	4	yes	—
138	#102	—	2	16	no	—
139	C₂²⋊C₈	C₇×D₄	1	4	yes	—
140	C₄×C₈	C₇×D₄	1	2	yes	—
141	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
142	C₄×C₈	C₂³×C₇	1	4	yes	—
143	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
144	C₂²×C₇	—	2	2	no	—
145	C₂²×C₇	—	8	4	no	—
146	C₂²×C₇	—	8	8	no	—
147	C₂²×C₇	—	16	8	no	—
148	C₂²×C₇	#492	1	2	yes	—
149	C₂²×C₇	C₂×#127	1	1	yes	characteristic
150	C₄×C₇	—	2	2	no	—
151	C₄×C₇	—	8	4	no	—
152	C₄×C₇	#492	1	2	yes	—
153	C₄×C₇	C₂²×SD₁₆	1	2	yes	—
154	C₄×C₇	#576	1	4	yes	—
155	C₂⁴	—	4	2	no	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×D₄	—	2	8	no	—
158	C₂×D₄	—	2	4	no	—
159	C₂×D₄	—	4	4	no	—
160	C₂×D₄	—	4	2	no	—
161	C₂×D₄	—	4	8	no	—
162	C₂×D₄	—	8	8	no	—
163	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
164	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
165	SD₁₆	—	8	16	no	—
166	D₈	—	4	8	no	—
167	D₈	—	4	8	no	—
168	C₂×C₈	C₇×SD₁₆	1	8	yes	complemented
169	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
170	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
171	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₂²⋊C₄	—	4	2	no	—
176	C₄²	—	4	2	no	—
177	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
178	C₄²	C₂×C₇×D₄	1	2	yes	—
179	C₄²	C₂⁴×C₇	1	1	yes	characteristic
180	C₂×C₇	—	2	2	no	—
181	C₂×C₇	—	16	4	no	—
182	C₂×C₇	C₂×#492	1	1	yes	characteristic
183	C₂×C₇	#3155	1	2	yes	—
184	C₂³	—	4	4	no	—
185	C₂³	—	8	8	no	—
186	C₂³	—	8	4	no	—
187	C₂³	C₇×#127	1	1	yes	characteristic
188	Q₈	—	4	8	no	—
189	D₄	—	4	8	no	—
190	D₄	—	4	16	no	—
191	D₄	—	8	8	no	—
192	D₄	—	8	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	C₇×#127	1	2	yes	—
196	C₂×C₄	C₇×#127	1	1	yes	characteristic
197	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
198	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
199	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
200	C₈	—	2	8	no	—
201	C₈	C₂×C₇×SD₁₆	1	4	yes	—
202	C₈	C₂×C₇×SD₁₆	1	4	yes	—
203	C₇	#19652	1	1	yes	characteristic, direct product factor, Sylow
204	C₂²	—	2	2	no	—
205	C₂²	—	8	8	no	—
206	C₂²	—	8	4	no	—
207	C₂²	—	16	8	no	—
208	C₂²	C₇×#492	1	2	yes	—
209	C₂²	C₂×C₇×#127	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	8	4	no	—
212	C₄	C₇×#492	1	2	yes	—
213	C₄	C₂²×C₇×SD₁₆	1	2	yes	—
214	C₄	C₇×#576	1	4	yes	—
215	C₂	—	2	2	no	—
216	C₂	—	16	4	no	—
217	C₂	C₂×C₇×#492	1	1	yes	characteristic
218	C₂	C₇×#3155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂×C₇×#493	C₂
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
1. C₇×#3208	C₂	semidirect
2. C₇×#3160	C₂	semidirect
4. C₇×#1715	C₂	semidirect
5. C₇×#1706	C₂	semidirect
6. C₇×#1450	C₂	semidirect
10. C₇×#500	C₂²	semidirect
12. C₇×#498	C₂²	semidirect
16. C₇×#492	C₂²	semidirect
17. C₇×#486	C₂²	semidirect
23. C₇×#322	C₂²	semidirect
24. C₇×C₈²	C₂²	semidirect
25. #19652	C₇	direct (N × H)
51. #3208	C₂×C₇	semidirect
52. #3160	C₂×C₇	semidirect
54. #1715	C₂×C₇	semidirect
55. #1706	C₂×C₇	semidirect
56. #1450	C₂×C₇	semidirect
70. C₂×C₇×C₈	SD₁₆	semidirect
85. #500	C₂²×C₇	semidirect
87. #498	C₂²×C₇	semidirect
91. #492	C₂²×C₇	semidirect
92. #486	C₂²×C₇	semidirect
98. #322	C₂²×C₇	semidirect
99. C₈²	C₂²×C₇	semidirect
168. C₂×C₈	C₇×SD₁₆	semidirect
203. C₇	#19652	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×#3208	2
C₇×#3160	2
C₂×C₇×#493	2
C₇×#1715	2
C₇×#1706	2
C₇×#1450	2
#19652	7

Subgroups — C7×#19652 #13196868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19652 #13196868