Subgroups — C₇×#19668 #13196884

Order 1792.

644 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3231	C₂	1	4	yes	maximal
2	C₇×#3170	C₂	1	2	yes	maximal
3	C₇×#1718	C₂	1	4	yes	maximal, complemented
4	C₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1710	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1707	C₂	1	2	yes	maximal, complemented
7	C₇×#1450	C₂	1	1	yes	maximal, characteristic
8	C₇×#533	C₂²	1	4	yes	—
9	C₇×#500	C₂²	1	2	yes	—
10	C₇×#500	C₂²	1	4	yes	—
11	C₇×#498	C₂²	1	4	yes	—
12	C₇×#494	C₂²	1	2	yes	—
13	C₇×#487	C₂²	1	4	yes	—
14	C₇×#487	C₂²	1	4	yes	—
15	C₇×#487	C₂²	1	4	yes	—
16	C₇×#485	—	2	8	no	—
17	C₇×#479	—	2	8	no	—
18	C₇×#467	—	2	4	no	—
19	C₄×C₇×Q₁₆	—	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	#19668	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₇×Q₁₆	—	2	4	no	—
27	C₂²×C₇×C₈	D₄	1	2	yes	—
28	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
29	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
30	C₇×#125	—	2	4	no	—
31	C₇×C₂²⋊Q₈	—	2	4	no	—
32	C₇×C₂²⋊Q₈	—	2	4	no	—
33	C₄×C₇×Q₈	—	2	4	no	—
34	C₄×C₇×Q₈	—	2	4	no	—
35	C₇×#117	—	2	4	no	—
36	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
37	C₇×#107	—	2	4	no	—
38	C₇×#107	—	2	8	no	—
39	C₇×#106	—	2	8	no	—
40	C₇×C₄⋊C₈	D₄	1	4	yes	—
41	C₇×#103	—	2	8	no	—
42	C₇×#103	—	2	8	no	—
43	C₇×#103	—	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	#3231	C₂×C₇	1	4	yes	—
51	#3170	C₂×C₇	1	2	yes	—
52	#1718	C₂×C₇	1	4	yes	complemented
53	#1713	C₂×C₇	1	1	yes	characteristic, complemented
54	#1710	C₂×C₇	1	1	yes	characteristic, complemented
55	#1707	C₂×C₇	1	2	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	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
61	C₇×Q₁₆	—	4	8	no	—
62	C₂×C₇×C₈	—	2	4	no	—
63	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
64	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
65	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
66	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
67	C₇×C₄⋊C₄	—	2	4	no	—
68	C₇×C₄⋊C₄	—	2	4	no	—
69	C₇×C₄⋊C₄	—	2	4	no	—
70	C₇×C₄⋊C₄	—	2	4	no	—
71	C₇×C₄⋊C₄	—	4	4	no	—
72	C₇×C₄⋊C₄	—	4	4	no	—
73	C₇×C₂²⋊C₄	—	4	4	no	—
74	C₄²×C₇	—	4	4	no	—
75	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₇	C₂×D₄	1	2	yes	—
77	C₄²×C₇	C₂⁴	1	1	yes	characteristic
78	#533	C₂²×C₇	1	4	yes	—
79	#500	C₂²×C₇	1	2	yes	—
80	#500	C₂²×C₇	1	4	yes	—
81	#498	C₂²×C₇	1	4	yes	—
82	#494	C₂²×C₇	1	2	yes	—
83	#487	C₂²×C₇	1	4	yes	—
84	#487	C₂²×C₇	1	4	yes	—
85	#487	C₂²×C₇	1	4	yes	—
86	#485	—	2	8	no	—
87	#479	—	2	8	no	—
88	#467	—	2	4	no	—
89	C₄×Q₁₆	—	2	4	no	—
90	#431	C₂²×C₇	1	2	yes	—
91	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
92	#322	C₂²×C₇	1	2	yes	—
93	C₈²	C₂²×C₇	1	1	yes	characteristic
94	C₈²	C₂²×C₇	1	1	yes	characteristic
95	C₂³×C₇	#127	1	1	yes	characteristic
96	C₇×Q₈	—	4	8	no	—
97	C₇×Q₈	—	4	8	no	—
98	C₂×C₄×C₇	—	2	2	no	—
99	C₂×C₄×C₇	—	2	4	no	—
100	C₂×C₄×C₇	—	4	4	no	—
101	C₂×C₄×C₇	—	4	4	no	—
102	C₂×C₄×C₇	#127	1	2	yes	—
103	C₂×C₄×C₇	#127	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
106	C₇×C₈	—	2	2	no	—
107	C₇×C₈	—	2	4	no	—
108	C₇×C₈	—	2	4	no	—
109	C₇×C₈	#135	1	4	yes	—
110	C₂×Q₁₆	—	2	4	no	—
111	C₂²×C₈	C₇×D₄	1	2	yes	—
112	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
113	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
114	#125	—	2	4	no	—
115	C₂²⋊Q₈	—	2	4	no	—
116	C₂²⋊Q₈	—	2	4	no	—
117	C₄×Q₈	—	2	4	no	—
118	C₄×Q₈	—	2	4	no	—
119	#117	—	2	4	no	—
120	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
121	#107	—	2	4	no	—
122	#107	—	2	8	no	—
123	#106	—	2	8	no	—
124	C₄⋊C₈	C₇×D₄	1	4	yes	—
125	#103	—	2	8	no	—
126	#103	—	2	8	no	—
127	#103	—	2	8	no	—
128	C₂²⋊C₈	C₇×D₄	1	4	yes	—
129	C₄×C₈	C₇×D₄	1	2	yes	—
130	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
131	C₄×C₈	C₂³×C₇	1	2	yes	—
132	C₄×C₈	C₂³×C₇	1	2	yes	—
133	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
134	C₂²×C₇	—	2	1	no	—
135	C₂²×C₇	—	2	2	no	—
136	C₂²×C₇	C₂×#127	1	1	yes	characteristic
137	C₄×C₇	—	2	1	no	—
138	C₄×C₇	—	2	2	no	—
139	C₄×C₇	—	8	4	no	—
140	C₄×C₇	—	8	4	no	—
141	C₄×C₇	C₂×#135	1	2	yes	—
142	C₄×C₇	#578	1	4	yes	—
143	C₂×Q₈	—	2	4	no	—
144	C₂×Q₈	—	2	4	no	—
145	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
146	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
147	Q₁₆	—	4	8	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
150	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
151	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
152	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₂²⋊C₄	—	4	4	no	—
160	C₄²	—	4	4	no	—
161	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
162	C₄²	C₂×C₇×D₄	1	2	yes	—
163	C₄²	C₂⁴×C₇	1	1	yes	characteristic
164	C₂×C₇	—	4	1	no	—
165	C₂×C₇	#3147	1	1	yes	characteristic
166	C₂×C₇	#3156	1	2	yes	—
167	C₂³	C₇×#127	1	1	yes	characteristic
168	Q₈	—	4	8	no	—
169	Q₈	—	4	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	C₇×#127	1	2	yes	—
175	C₂×C₄	C₇×#127	1	1	yes	characteristic
176	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
177	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	2	4	no	—
180	C₈	—	2	4	no	—
181	C₈	C₇×#135	1	4	yes	—
182	C₇	#19668	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²	—	2	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	C₂×C₇×#127	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	2	1	no	—
188	C₄	—	8	4	no	—
189	C₄	—	8	4	no	—
190	C₄	C₂×C₇×#135	1	2	yes	—
191	C₄	C₇×#578	1	4	yes	—
192	C₂	—	4	1	no	—
193	C₂	C₇×#3147	1	1	yes	characteristic
194	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
3. C₇×#1718	C₂	semidirect
4. C₇×#1713	C₂	semidirect
5. C₇×#1710	C₂	semidirect
6. C₇×#1707	C₂	semidirect
25. #19668	C₇	direct (N × H)
52. #1718	C₂×C₇	semidirect
53. #1713	C₂×C₇	semidirect
54. #1710	C₂×C₇	semidirect
55. #1707	C₂×C₇	semidirect
182. C₇	#19668	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×#3231	2
C₇×#3170	2
C₇×#1718	2
C₇×#1713	2
C₇×#1710	2
C₇×#1707	2
C₇×#1450	2
#19668	7

Subgroups — C7×#19668 #13196884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19668 #13196884