Subgroups — C₇×#19667 #13196883

Order 1792.

780 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3207	C₂	1	4	yes	maximal, complemented
2	C₇×#3169	C₂	1	2	yes	maximal, complemented
3	C₇×#1717	C₂	1	4	yes	maximal, complemented
4	C₇×#1712	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1710	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1706	C₂	1	2	yes	maximal, complemented
7	C₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#532	C₂²	1	4	yes	—
9	C₇×#500	C₂²	1	2	yes	—
10	C₇×#493	C₂²	1	4	yes	—
11	C₇×#493	C₂²	1	2	yes	—
12	C₇×#492	C₂²	1	4	yes	—
13	C₇×#487	C₂²	1	4	yes	—
14	C₇×#486	C₂²	1	4	yes	—
15	C₇×#486	C₂²	1	4	yes	—
16	C₇×#466	—	2	4	no	—
17	C₇×#463	—	2	8	no	—
18	C₇×#449	—	2	8	no	—
19	C₄×C₇×D₈	—	2	4	no	—
20	C₇×#431	C₂²	1	2	yes	—
21	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
22	C₇×#322	C₂²	1	2	yes	—
23	C₇×C₈²	C₂²	1	1	yes	characteristic
24	C₇×C₈²	C₂²	1	1	yes	characteristic
25	#19667	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₇×#41	—	2	4	no	—
27	C₂×C₇×SD₁₆	—	2	8	no	—
28	C₂×C₇×D₈	—	2	4	no	—
29	C₂×C₇×D₈	—	2	8	no	—
30	C₂²×C₇×C₈	D₄	1	2	yes	—
31	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
32	C₇×#127	C₂³	1	4	yes	—
33	C₇×#124	—	2	4	no	—
34	C₇×#121	—	2	4	no	—
35	C₇×#121	—	2	4	no	—
36	C₄×C₇×D₄	—	2	4	no	—
37	C₄×C₇×D₄	—	2	4	no	—
38	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
39	C₇×#107	—	2	4	no	—
40	C₇×C₄⋊C₈	D₄	1	4	yes	—
41	C₇×#103	—	2	8	no	—
42	C₇×#102	—	2	8	no	—
43	C₇×#102	—	2	8	no	—
44	C₇×C₂²⋊C₈	D₄	1	4	yes	—
45	C₄×C₇×C₈	D₄	1	2	yes	—
46	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
47	C₄×C₇×C₈	C₂³	1	2	yes	—
48	C₄×C₇×C₈	C₂³	1	2	yes	—
49	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
50	#3207	C₂×C₇	1	4	yes	complemented
51	#3169	C₂×C₇	1	2	yes	complemented
52	#1717	C₂×C₇	1	4	yes	complemented
53	#1712	C₂×C₇	1	1	yes	characteristic, complemented
54	#1710	C₂×C₇	1	1	yes	characteristic, complemented
55	#1706	C₂×C₇	1	2	yes	complemented
56	#1450	C₂×C₇	1	1	yes	characteristic, complemented
57	C₇×#41	—	8	8	no	—
58	C₂×C₇×Q₈	—	2	4	no	—
59	C₂×C₇×D₄	—	2	4	no	—
60	C₂×C₇×D₄	—	2	4	no	—
61	C₂×C₇×D₄	—	2	4	no	—
62	C₂×C₇×D₄	—	4	4	no	—
63	C₂²×C₄×C₇	—	4	4	no	—
64	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
66	C₇×SD₁₆	—	8	8	no	—
67	C₇×D₈	—	4	8	no	—
68	C₇×D₈	—	8	8	no	—
69	C₂×C₇×C₈	—	2	4	no	—
70	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
71	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
72	C₂×C₇×C₈	C₂×D₄	1	4	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×C₂²⋊C₄	—	4	4	no	—
77	C₇×C₂²⋊C₄	—	4	4	no	—
78	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₇	C₂×D₄	1	2	yes	—
80	C₄²×C₇	C₂⁴	1	1	yes	characteristic
81	#532	C₂²×C₇	1	4	yes	—
82	#500	C₂²×C₇	1	2	yes	—
83	#493	C₂²×C₇	1	2	yes	—
84	#493	C₂²×C₇	1	4	yes	—
85	#492	C₂²×C₇	1	4	yes	—
86	#487	C₂²×C₇	1	4	yes	—
87	#486	C₂²×C₇	1	4	yes	—
88	#486	C₂²×C₇	1	4	yes	—
89	#466	—	2	4	no	—
90	#463	—	2	8	no	—
91	#449	—	2	8	no	—
92	C₄×D₈	—	2	4	no	—
93	#431	C₂²×C₇	1	2	yes	—
94	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
95	#322	C₂²×C₇	1	2	yes	—
96	C₈²	C₂²×C₇	1	1	yes	characteristic
97	C₈²	C₂²×C₇	1	1	yes	characteristic
98	C₂³×C₇	—	4	4	no	—
99	C₂³×C₇	#127	1	1	yes	characteristic
100	C₇×Q₈	—	4	8	no	—
101	C₇×D₄	—	4	8	no	—
102	C₇×D₄	—	4	8	no	—
103	C₇×D₄	—	4	8	no	—
104	C₇×D₄	—	8	8	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	4	no	—
107	C₂×C₄×C₇	—	4	4	no	—
108	C₂×C₄×C₇	—	8	8	no	—
109	C₂×C₄×C₇	#127	1	1	yes	characteristic
110	C₂×C₄×C₇	#127	1	2	yes	—
111	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
113	C₇×C₈	—	2	2	no	—
114	C₇×C₈	—	2	4	no	—
115	C₇×C₈	—	2	4	no	—
116	C₇×C₈	#135	1	4	yes	—
117	C₂×#41	—	2	4	no	—
118	C₂×SD₁₆	—	2	8	no	—
119	C₂×D₈	—	2	4	no	—
120	C₂×D₈	—	2	8	no	—
121	C₂²×C₈	C₇×D₄	1	2	yes	—
122	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
123	#127	C₂³×C₇	1	4	yes	—
124	#124	—	2	4	no	—
125	#121	—	2	4	no	—
126	#121	—	2	4	no	—
127	C₄×D₄	—	2	4	no	—
128	C₄×D₄	—	2	4	no	—
129	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
130	#107	—	2	4	no	—
131	C₄⋊C₈	C₇×D₄	1	4	yes	—
132	#103	—	2	8	no	—
133	#102	—	2	8	no	—
134	#102	—	2	8	no	—
135	C₂²⋊C₈	C₇×D₄	1	4	yes	—
136	C₄×C₈	C₇×D₄	1	2	yes	—
137	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
138	C₄×C₈	C₂³×C₇	1	2	yes	—
139	C₄×C₈	C₂³×C₇	1	2	yes	—
140	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
141	C₂²×C₇	—	2	1	no	—
142	C₂²×C₇	—	2	2	no	—
143	C₂²×C₇	—	8	4	no	—
144	C₂²×C₇	—	8	4	no	—
145	C₂²×C₇	—	8	4	no	—
146	C₂²×C₇	C₂×#127	1	1	yes	characteristic
147	C₄×C₇	—	2	1	no	—
148	C₄×C₇	—	2	2	no	—
149	C₄×C₇	—	8	4	no	—
150	C₄×C₇	C₂×#135	1	2	yes	—
151	C₄×C₇	#576	1	4	yes	—
152	#41	—	8	8	no	—
153	C₂×Q₈	—	2	4	no	—
154	C₂×D₄	—	2	4	no	—
155	C₂×D₄	—	2	4	no	—
156	C₂×D₄	—	2	4	no	—
157	C₂×D₄	—	4	4	no	—
158	C₂²×C₄	—	4	4	no	—
159	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
160	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
161	SD₁₆	—	8	8	no	—
162	D₈	—	4	8	no	—
163	D₈	—	8	8	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
166	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
167	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
168	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₂²⋊C₄	—	4	4	no	—
172	C₂²⋊C₄	—	4	4	no	—
173	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
174	C₄²	C₂×C₇×D₄	1	2	yes	—
175	C₄²	C₂⁴×C₇	1	1	yes	characteristic
176	C₂×C₇	—	4	1	no	—
177	C₂×C₇	—	16	4	no	—
178	C₂×C₇	#3147	1	1	yes	characteristic
179	C₂×C₇	#3155	1	2	yes	—
180	C₂³	—	4	4	no	—
181	C₂³	C₇×#127	1	1	yes	characteristic
182	Q₈	—	4	8	no	—
183	D₄	—	4	8	no	—
184	D₄	—	4	8	no	—
185	D₄	—	4	8	no	—
186	D₄	—	8	8	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	8	8	no	—
191	C₂×C₄	C₇×#127	1	1	yes	characteristic
192	C₂×C₄	C₇×#127	1	2	yes	—
193	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
194	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
195	C₈	—	2	2	no	—
196	C₈	—	2	4	no	—
197	C₈	—	2	4	no	—
198	C₈	C₇×#135	1	4	yes	—
199	C₇	#19667	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²	—	2	2	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	8	4	no	—
203	C₂²	—	8	4	no	—
204	C₂²	—	8	4	no	—
205	C₂²	C₂×C₇×#127	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	2	1	no	—
208	C₄	—	8	4	no	—
209	C₄	C₂×C₇×#135	1	2	yes	—
210	C₄	C₇×#576	1	4	yes	—
211	C₂	—	4	1	no	—
212	C₂	—	16	4	no	—
213	C₂	C₇×#3147	1	1	yes	characteristic
214	C₂	C₇×#3155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
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×#3207	C₂	semidirect
2. C₇×#3169	C₂	semidirect
3. C₇×#1717	C₂	semidirect
4. C₇×#1712	C₂	semidirect
5. C₇×#1710	C₂	semidirect
6. C₇×#1706	C₂	semidirect
7. C₇×#1450	C₂	semidirect
25. #19667	C₇	direct (N × H)
50. #3207	C₂×C₇	semidirect
51. #3169	C₂×C₇	semidirect
52. #1717	C₂×C₇	semidirect
53. #1712	C₂×C₇	semidirect
54. #1710	C₂×C₇	semidirect
55. #1706	C₂×C₇	semidirect
56. #1450	C₂×C₇	semidirect
199. C₇	#19667	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×#3207	2
C₇×#3169	2
C₇×#1717	2
C₇×#1712	2
C₇×#1710	2
C₇×#1706	2
C₇×#1450	2
#19667	7

Subgroups — C7×#19667 #13196883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19667 #13196883