Subgroups — C₇×#8808 #13186024

Order 1792.

418 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1842	C₂	1	1	yes	maximal, characteristic
2	C₇×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₇×#97	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#335	C₂	1	2	yes	maximal
5	C₂×C₇×#334	C₂	1	2	yes	maximal
6	C₂×C₇×C₄⋊C₈	—	2	2	no	—
7	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×#97	C₄	1	2	yes	—
11	C₂×C₇×#97	C₄	1	1	yes	characteristic
12	C₂×C₇×#97	C₄	1	1	yes	characteristic
13	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
14	C₄³×C₇	C₂²	1	1	yes	characteristic
15	C₇×#336	—	2	2	no	—
16	C₇×#335	—	2	4	no	—
17	C₇×#334	—	2	4	no	—
18	#8808	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂²×C₇×C₈	—	2	1	no	—
20	C₂²×C₇×C₈	—	2	1	no	—
21	C₂²×C₇×C₈	—	2	2	no	—
22	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₇	D₄	1	2	yes	—
27	C₂×C₄²×C₇	Q₈	1	2	yes	—
28	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
29	C₇×C₄⋊C₈	—	2	2	no	—
30	C₇×C₄⋊C₈	—	2	2	no	—
31	C₇×C₄⋊C₈	—	2	2	no	—
32	C₇×C₄⋊C₈	—	2	2	no	—
33	C₇×C₄⋊C₈	—	4	4	no	—
34	C₇×#97	—	2	4	no	—
35	C₇×#97	—	2	2	no	—
36	C₇×#97	C₈	1	4	yes	complemented
37	C₄×C₇×C₈	D₄	1	4	yes	—
38	C₄×C₇×C₈	Q₈	1	4	yes	—
39	#1842	C₂×C₇	1	1	yes	characteristic
40	#1831	C₂×C₇	1	1	yes	characteristic
41	C₄×#97	C₂×C₇	1	1	yes	characteristic
42	C₂×#335	C₂×C₇	1	2	yes	—
43	C₂×#334	C₂×C₇	1	2	yes	—
44	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₇×C₈	—	2	2	no	—
50	C₂×C₇×C₈	—	2	2	no	—
51	C₂×C₇×C₈	—	4	2	no	—
52	C₂×C₇×C₈	—	4	1	no	—
53	C₂×C₇×C₈	—	4	1	no	—
54	C₂×C₇×C₈	—	4	4	no	—
55	C₂×C₇×C₈	—	4	2	no	—
56	C₂×C₇×C₈	—	4	2	no	—
57	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
58	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
59	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
60	C₂×C₇×C₈	M₁₆	1	4	yes	—
61	C₄²×C₇	—	2	2	no	—
62	C₄²×C₇	—	2	2	no	—
63	C₄²×C₇	—	2	2	no	—
64	C₄²×C₇	—	4	2	no	—
65	C₄²×C₇	C₂×C₈	1	2	yes	—
66	C₄²×C₇	M₁₆	1	2	yes	—
67	C₄²×C₇	C₂×D₄	1	2	yes	—
68	C₄²×C₇	C₂×Q₈	1	2	yes	—
69	C₂×C₄⋊C₈	—	2	2	no	—
70	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
71	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
72	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
73	C₂×#97	C₄×C₇	1	2	yes	—
74	C₂×#97	C₄×C₇	1	1	yes	characteristic
75	C₂×#97	C₄×C₇	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
77	C₄³	C₂²×C₇	1	1	yes	characteristic
78	#336	—	2	2	no	—
79	#335	—	2	4	no	—
80	#334	—	2	4	no	—
81	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	C₄⋊C₈	1	4	yes	—
89	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₇	#127	1	1	yes	characteristic
91	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
92	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
95	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×C₄×C₇	#137	1	2	yes	—
98	C₇×C₈	—	2	4	no	—
99	C₇×C₈	—	8	2	no	—
100	C₇×C₈	—	8	1	no	—
101	C₇×C₈	—	8	1	no	—
102	C₇×C₈	C₄⋊C₈	1	8	yes	complemented
103	C₂²×C₈	—	2	1	no	—
104	C₂²×C₈	—	2	1	no	—
105	C₂²×C₈	—	2	2	no	—
106	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
107	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
108	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
109	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
110	C₂×C₄²	C₇×D₄	1	2	yes	—
111	C₂×C₄²	C₇×Q₈	1	2	yes	—
112	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	4	4	no	—
118	#97	—	2	4	no	—
119	#97	—	2	2	no	—
120	#97	C₇×C₈	1	4	yes	complemented
121	C₄×C₈	C₇×D₄	1	4	yes	—
122	C₄×C₈	C₇×Q₈	1	4	yes	—
123	C₂²×C₇	#384	1	1	yes	characteristic
124	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
125	C₂²×C₇	#423	1	1	yes	characteristic
126	C₂²×C₇	#428	1	2	yes	—
127	C₂²×C₇	#430	1	2	yes	—
128	C₄×C₇	—	2	4	no	—
129	C₄×C₇	—	2	2	no	—
130	C₄×C₇	—	4	2	no	—
131	C₄×C₇	C₂×C₄⋊C₈	1	2	yes	—
132	C₄×C₇	#423	1	2	yes	—
133	C₄×C₇	#496	1	1	yes	characteristic
134	C₄×C₇	#497	1	1	yes	characteristic
135	C₄×C₇	#501	1	2	yes	—
136	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
137	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
138	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	4	1	no	—
145	C₂×C₈	—	4	1	no	—
146	C₂×C₈	—	4	4	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
150	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
151	C₂×C₈	C₂×C₇×C₈	1	4	yes	—
152	C₂×C₈	C₇×M₁₆	1	4	yes	—
153	C₄²	—	2	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	4	2	no	—
157	C₄²	C₂×C₇×C₈	1	2	yes	—
158	C₄²	C₇×M₁₆	1	2	yes	—
159	C₄²	C₂×C₇×D₄	1	2	yes	—
160	C₄²	C₂×C₇×Q₈	1	2	yes	—
161	C₂×C₇	#1565	1	2	yes	—
162	C₂×C₇	#1569	1	2	yes	—
163	C₂×C₇	#1842	1	1	yes	characteristic
164	C₂×C₇	#1847	1	1	yes	characteristic
165	C₂×C₇	#1849	1	1	yes	characteristic
166	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	C₇×C₄⋊C₈	1	4	yes	—
174	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
175	C₂×C₄	C₇×#127	1	1	yes	characteristic
176	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
177	C₂×C₄	C₇×C₄⋊Q₈	1	1	yes	characteristic
178	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
179	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
180	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
181	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
182	C₂×C₄	C₇×#137	1	2	yes	—
183	C₈	—	2	4	no	—
184	C₈	—	8	2	no	—
185	C₈	—	8	1	no	—
186	C₈	—	8	1	no	—
187	C₈	C₇×C₄⋊C₈	1	8	yes	complemented
188	C₇	#8808	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²	C₇×#384	1	1	yes	characteristic
190	C₂²	C₂×C₇×C₄⋊C₈	1	1	yes	characteristic
191	C₂²	C₇×#423	1	1	yes	characteristic
192	C₂²	C₇×#428	1	2	yes	—
193	C₂²	C₇×#430	1	2	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	2	2	no	—
196	C₄	—	4	2	no	—
197	C₄	C₂×C₇×C₄⋊C₈	1	2	yes	—
198	C₄	C₇×#423	1	2	yes	—
199	C₄	C₇×#496	1	1	yes	characteristic
200	C₄	C₇×#497	1	1	yes	characteristic
201	C₄	C₇×#501	1	2	yes	—
202	C₂	C₇×#1565	1	2	yes	—
203	C₂	C₇×#1569	1	2	yes	—
204	C₂	C₇×#1842	1	1	yes	characteristic
205	C₂	C₇×#1847	1	1	yes	characteristic
206	C₂	C₇×#1849	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle, prime radical	C₂³×C₇	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
18. #8808	C₇	direct (N × H)
36. C₇×#97	C₈	semidirect
102. C₇×C₈	C₄⋊C₈	semidirect
120. #97	C₇×C₈	semidirect
187. C₈	C₇×C₄⋊C₈	semidirect
188. C₇	#8808	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×#1842	2
C₇×#1831	2
C₄×C₇×#97	2
C₂×C₇×#335	2
C₂×C₇×#334	2
#8808	7

Subgroups — C7×#8808 #13186024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#8808 #13186024