Subgroups — C₇×#12063 #13189279

Order 1792.

242 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×D₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₈×D₈	C₂	1	1	yes	maximal, characteristic
4	C₇×#1367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1312	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×M₃₂	—	2	1	no	—
9	C₂²×C₇×C₁₆	—	2	1	no	—
10	C₄×C₇×D₈	C₄	1	1	yes	characteristic
11	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
12	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
13	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₇×C₂²⋊C₁₆	—	2	1	no	—
16	C₇×C₂²⋊C₁₆	—	2	1	no	—
17	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
18	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
19	C₇×#335	C₄	1	1	yes	characteristic
20	C₇×D₄⋊C₈	C₄	1	1	yes	characteristic
21	C₇×D₄⋊C₈	C₄	1	1	yes	characteristic
22	C₇×C₈²	C₂²	1	1	yes	characteristic
23	#12063	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₇×D₈	C₈	1	1	yes	characteristic
25	C₂²×C₇×C₈	—	2	1	no	—
26	C₂²×C₇×C₈	—	2	1	no	—
27	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
28	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
29	C₇×M₃₂	—	4	2	no	—
30	C₂×C₇×C₁₆	—	2	1	no	—
31	C₂×C₇×C₁₆	—	2	1	no	—
32	C₂×C₇×C₁₆	—	2	1	no	—
33	C₂×C₇×C₁₆	—	4	2	no	—
34	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
35	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
36	C₇×#107	C₈	1	1	yes	characteristic
37	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₇×#102	C₈	1	1	yes	characteristic
40	C₇×#102	C₈	1	1	yes	characteristic
41	C₇×C₂²⋊C₈	—	2	1	no	—
42	C₇×C₂²⋊C₈	—	2	1	no	—
43	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
45	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
46	#2169	C₂×C₇	1	1	yes	characteristic, complemented
47	D₄×C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
48	C₈×D₈	C₂×C₇	1	1	yes	characteristic
49	#1367	C₂×C₇	1	1	yes	characteristic, complemented
50	D₄⋊C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
51	D₄⋊C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
52	#1312	C₂×C₇	1	1	yes	characteristic, complemented
53	C₂×C₇×D₄	C₂×C₈	1	1	yes	characteristic
54	C₂×C₇×D₄	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₇	—	2	1	no	—
56	C₂²×C₄×C₇	—	2	1	no	—
57	C₇×D₈	—	2	2	no	—
58	C₂×C₇×C₈	—	2	1	no	—
59	C₂×C₇×C₈	—	2	1	no	—
60	C₂×C₇×C₈	—	2	1	no	—
61	C₂×C₇×C₈	—	4	1	no	—
62	C₂×C₇×C₈	—	4	1	no	—
63	C₂×C₇×C₈	—	4	1	no	—
64	C₂×C₇×C₈	—	4	1	no	—
65	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
66	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
67	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₇×C₈	#41	1	1	yes	characteristic
69	C₇×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
70	C₇×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
71	C₇×C₂²⋊C₄	—	2	1	no	—
72	C₇×C₂²⋊C₄	—	2	1	no	—
73	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
74	C₇×C₁₆	—	2	1	no	—
75	C₇×C₁₆	—	2	1	no	—
76	C₇×C₁₆	—	4	1	no	—
77	C₇×C₁₆	—	4	1	no	—
78	C₇×C₁₆	—	4	1	no	—
79	C₂×M₃₂	—	2	1	no	—
80	C₂²×C₁₆	—	2	1	no	—
81	C₄×D₈	C₄×C₇	1	1	yes	characteristic
82	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
83	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
84	C₄⋊C₁₆	C₂²×C₇	1	1	yes	characteristic
85	C₄⋊C₁₆	C₂²×C₇	1	1	yes	characteristic
86	C₂²⋊C₁₆	—	2	1	no	—
87	C₂²⋊C₁₆	—	2	1	no	—
88	C₄×C₁₆	C₂²×C₇	1	1	yes	characteristic
89	C₄×C₁₆	C₂²×C₇	1	1	yes	characteristic
90	#335	C₄×C₇	1	1	yes	characteristic
91	D₄⋊C₈	C₄×C₇	1	1	yes	characteristic
92	D₄⋊C₈	C₄×C₇	1	1	yes	characteristic
93	C₈²	C₂²×C₇	1	1	yes	characteristic
94	C₂³×C₇	—	2	1	no	—
95	C₂³×C₇	—	2	1	no	—
96	C₇×D₄	—	2	1	no	—
97	C₇×D₄	—	2	1	no	—
98	C₇×D₄	—	2	1	no	—
99	C₇×D₄	M₃₂	1	2	yes	complemented
100	C₂×C₄×C₇	—	2	1	no	—
101	C₂×C₄×C₇	—	2	1	no	—
102	C₂×C₄×C₇	—	4	1	no	—
103	C₂×C₄×C₇	—	4	1	no	—
104	C₂×C₄×C₇	—	4	1	no	—
105	C₂×C₄×C₇	—	4	1	no	—
106	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₇	#131	1	1	yes	characteristic
109	C₇×C₈	—	2	2	no	—
110	C₇×C₈	—	2	1	no	—
111	C₇×C₈	—	2	1	no	—
112	C₇×C₈	—	2	1	no	—
113	C₇×C₈	—	4	1	no	—
114	C₇×C₈	—	4	1	no	—
115	C₇×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₇×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×D₈	C₇×C₈	1	1	yes	characteristic
118	C₂²×C₈	—	2	1	no	—
119	C₂²×C₈	—	2	1	no	—
120	C₄×D₄	C₂×C₄×C₇	1	1	yes	characteristic
121	C₄×D₄	C₂×C₄×C₇	1	1	yes	characteristic
122	M₃₂	—	4	2	no	—
123	C₂×C₁₆	—	2	1	no	—
124	C₂×C₁₆	—	2	1	no	—
125	C₂×C₁₆	—	2	1	no	—
126	C₂×C₁₆	—	4	2	no	—
127	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
128	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
129	#107	C₇×C₈	1	1	yes	characteristic
130	C₄⋊C₈	C₂×C₄×C₇	1	1	yes	characteristic
131	C₄⋊C₈	C₂×C₄×C₇	1	1	yes	characteristic
132	#102	C₇×C₈	1	1	yes	characteristic
133	#102	C₇×C₈	1	1	yes	characteristic
134	C₂²⋊C₈	—	2	1	no	—
135	C₂²⋊C₈	—	2	1	no	—
136	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
137	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
138	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
139	C₂²×C₇	—	2	2	no	—
140	C₂²×C₇	—	4	1	no	—
141	C₂²×C₇	—	4	1	no	—
142	C₂²×C₇	—	4	1	no	—
143	C₂²×C₇	—	4	1	no	—
144	C₂²×C₇	—	4	1	no	—
145	C₂²×C₇	C₈×D₄	1	1	yes	characteristic
146	C₄×C₇	—	2	1	no	—
147	C₄×C₇	—	4	1	no	—
148	C₄×C₇	—	4	1	no	—
149	C₄×C₇	#442	1	1	yes	characteristic
150	C₄×C₇	#444	1	1	yes	characteristic
151	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
152	C₄×C₇	#504	1	1	yes	characteristic
153	C₂×D₄	C₂×C₇×C₈	1	1	yes	characteristic
154	C₂×D₄	C₂×C₇×C₈	1	1	yes	characteristic
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	D₈	—	2	2	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	4	1	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	4	1	no	—
165	C₂×C₈	C₂×C₇×C₈	1	1	yes	characteristic
166	C₂×C₈	C₂×C₇×C₈	1	1	yes	characteristic
167	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
168	C₂×C₈	C₇×#41	1	1	yes	characteristic
169	C₄⋊C₄	C₂×C₇×C₈	1	1	yes	characteristic
170	C₄⋊C₄	C₂×C₇×C₈	1	1	yes	characteristic
171	C₂²⋊C₄	—	2	1	no	—
172	C₂²⋊C₄	—	2	1	no	—
173	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
174	C₁₆	—	2	1	no	—
175	C₁₆	—	2	1	no	—
176	C₁₆	—	4	1	no	—
177	C₁₆	—	4	1	no	—
178	C₁₆	—	4	1	no	—
179	C₂×C₇	—	4	2	no	—
180	C₂×C₇	—	8	1	no	—
181	C₂×C₇	#1580	1	1	yes	characteristic
182	C₂×C₇	#2168	1	1	yes	characteristic
183	C₂×C₇	#2170	1	1	yes	characteristic
184	C₂³	—	2	1	no	—
185	C₂³	—	2	1	no	—
186	D₄	—	2	1	no	—
187	D₄	—	2	1	no	—
188	D₄	—	2	1	no	—
189	D₄	C₇×M₃₂	1	2	yes	complemented
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
197	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
198	C₂×C₄	C₇×#131	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	4	1	no	—
204	C₈	—	4	1	no	—
205	C₈	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₈	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
207	C₇	#12063	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²	—	2	2	no	—
209	C₂²	—	4	1	no	—
210	C₂²	—	4	1	no	—
211	C₂²	—	4	1	no	—
212	C₂²	—	4	1	no	—
213	C₂²	—	4	1	no	—
214	C₂²	C₇×C₈×D₄	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	4	1	no	—
217	C₄	—	4	1	no	—
218	C₄	C₇×#442	1	1	yes	characteristic
219	C₄	C₇×#444	1	1	yes	characteristic
220	C₄	C₂×C₇×M₃₂	1	1	yes	characteristic
221	C₄	C₇×#504	1	1	yes	characteristic
222	C₂	—	4	2	no	—
223	C₂	—	8	1	no	—
224	C₂	C₇×#1580	1	1	yes	characteristic
225	C₂	C₇×#2168	1	1	yes	characteristic
226	C₂	C₇×#2170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂²×C₇	C₈×D₄
prime radical	C₂×C₇×D₈	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₄×C₇×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#2169	C₂	semidirect
2. C₇×D₄×C₁₆	C₂	semidirect
4. C₇×#1367	C₂	semidirect
5. C₇×D₄⋊C₁₆	C₂	semidirect
6. C₇×D₄⋊C₁₆	C₂	semidirect
7. C₇×#1312	C₂	semidirect
23. #12063	C₇	direct (N × H)
46. #2169	C₂×C₇	semidirect
47. D₄×C₁₆	C₂×C₇	semidirect
49. #1367	C₂×C₇	semidirect
50. D₄⋊C₁₆	C₂×C₇	semidirect
51. D₄⋊C₁₆	C₂×C₇	semidirect
52. #1312	C₂×C₇	semidirect
99. C₇×D₄	M₃₂	semidirect
189. D₄	C₇×M₃₂	semidirect
207. C₇	#12063	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×#2169	2
C₇×D₄×C₁₆	2
C₇×C₈×D₈	2
C₇×#1367	2
C₇×D₄⋊C₁₆	2
C₇×#1312	2
#12063	7

Subgroups — C7×#12063 #13189279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12063 #13189279