Subgroups — C₇×#2586 #11741895

Order 896.

412 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#398	C₂	1	1	yes	maximal, characteristic
2	C₇×#395	C₂	1	1	yes	maximal, characteristic
3	C₇×#395	C₂	1	2	yes	maximal
4	C₇×#391	C₂	1	2	yes	maximal
5	C₇×#391	C₂	1	1	yes	maximal, characteristic
6	C₇×#389	C₂	1	2	yes	maximal
7	C₇×#387	C₂	1	2	yes	maximal
8	C₇×#387	C₂	1	2	yes	maximal
9	C₄×C₇×C₄⋊C₄	C₂	1	2	yes	maximal
10	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₄²×C₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₇	C₂²	1	2	yes	—
22	C₇×#95	C₂²	1	2	yes	—
23	C₇×#95	C₂²	1	2	yes	—
24	C₇×#95	C₂²	1	1	yes	characteristic
25	C₇×#95	C₂²	1	2	yes	—
26	C₇×#95	C₂²	1	1	yes	characteristic
27	C₇×#95	C₂²	1	2	yes	—
28	C₇×#95	C₂²	1	2	yes	—
29	C₇×#95	C₂²	1	2	yes	—
30	C₇×#95	C₂²	1	2	yes	—
31	#2586	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂×C₇×Q₈	—	2	4	no	—
33	C₂×C₇×Q₈	—	2	2	no	—
34	C₂×C₇×Q₈	—	4	1	no	—
35	C₂×C₇×Q₈	—	4	1	no	—
36	C₂×C₇×Q₈	—	4	1	no	—
37	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₇×C₄⋊C₄	—	2	4	no	—
47	C₇×C₄⋊C₄	—	2	4	no	—
48	C₇×C₄⋊C₄	—	4	2	no	—
49	C₇×C₄⋊C₄	—	4	2	no	—
50	C₇×C₄⋊C₄	—	4	1	no	—
51	C₇×C₄⋊C₄	—	4	1	no	—
52	C₇×C₄⋊C₄	Q₈	1	8	yes	complemented
53	C₄²×C₇	—	2	4	no	—
54	C₄²×C₇	—	4	1	no	—
55	C₄²×C₇	—	4	2	no	—
56	#398	C₂×C₇	1	1	yes	characteristic
57	#395	C₂×C₇	1	1	yes	characteristic
58	#395	C₂×C₇	1	2	yes	—
59	#391	C₂×C₇	1	2	yes	—
60	#391	C₂×C₇	1	1	yes	characteristic
61	#389	C₂×C₇	1	2	yes	—
62	#387	C₂×C₇	1	2	yes	—
63	#387	C₂×C₇	1	2	yes	—
64	C₄×C₄⋊C₄	C₂×C₇	1	2	yes	—
65	C₂³×C₇	C₂⁴	1	1	yes	characteristic
66	C₇×Q₈	—	4	4	no	—
67	C₇×Q₈	—	4	4	no	—
68	C₂×C₄×C₇	—	2	2	no	—
69	C₂×C₄×C₇	—	2	2	no	—
70	C₂×C₄×C₇	—	2	2	no	—
71	C₂×C₄×C₇	—	2	2	no	—
72	C₂×C₄×C₇	—	2	2	no	—
73	C₂×C₄×C₇	—	2	2	no	—
74	C₂×C₄×C₇	—	2	2	no	—
75	C₂×C₄×C₇	—	2	1	no	—
76	C₂×C₄×C₇	—	2	1	no	—
77	C₂×C₄×C₇	—	2	2	no	—
78	C₂×C₄×C₇	—	2	2	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	—	2	1	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	1	no	—
88	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
89	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
90	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
91	C₂×C₄×C₇	#41	1	4	yes	—
92	C₂×C₄×C₇	#41	1	2	yes	—
93	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
94	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
95	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
96	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
97	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
98	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
99	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
100	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
101	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
102	C₂×C₄²	C₂²×C₇	1	2	yes	—
103	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
104	C₂×C₄²	C₂²×C₇	1	2	yes	—
105	#95	C₂²×C₇	1	2	yes	—
106	#95	C₂²×C₇	1	2	yes	—
107	#95	C₂²×C₇	1	1	yes	characteristic
108	#95	C₂²×C₇	1	2	yes	—
109	#95	C₂²×C₇	1	1	yes	characteristic
110	#95	C₂²×C₇	1	2	yes	—
111	#95	C₂²×C₇	1	2	yes	—
112	#95	C₂²×C₇	1	2	yes	—
113	#95	C₂²×C₇	1	2	yes	—
114	C₂²×C₇	C₂²×Q₈	1	2	yes	—
115	C₂²×C₇	C₂×#41	1	1	yes	characteristic
116	C₂²×C₇	C₂×#41	1	2	yes	—
117	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
119	C₄×C₇	—	2	4	no	—
120	C₄×C₇	—	2	2	no	—
121	C₄×C₇	—	2	4	no	—
122	C₄×C₇	—	2	4	no	—
123	C₄×C₇	—	2	4	no	—
124	C₄×C₇	—	4	2	no	—
125	C₄×C₇	—	4	2	no	—
126	C₄×C₇	—	4	1	no	—
127	C₄×C₇	—	4	1	no	—
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	4	1	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂×Q₈	—	4	1	no	—
133	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
134	C₂²×C₄	C₂³×C₇	1	2	yes	—
135	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₇	1	2	yes	—
137	C₂²×C₄	C₂³×C₇	1	2	yes	—
138	C₂²×C₄	C₂³×C₇	1	2	yes	—
139	C₂²×C₄	C₂³×C₇	1	2	yes	—
140	C₂²×C₄	C₂³×C₇	1	2	yes	—
141	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	1	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄⋊C₄	C₇×Q₈	1	8	yes	complemented
149	C₄²	—	2	4	no	—
150	C₄²	—	4	1	no	—
151	C₄²	—	4	2	no	—
152	C₂×C₇	#533	1	2	yes	—
153	C₂×C₇	#538	1	1	yes	characteristic
154	C₂×C₇	#551	1	1	yes	characteristic
155	C₂×C₇	#554	1	2	yes	—
156	C₂×C₇	Q₈²	1	1	yes	characteristic
157	C₂³	C₂⁴×C₇	1	1	yes	characteristic
158	Q₈	—	4	4	no	—
159	Q₈	—	4	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	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	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	C₂×C₇×Q₈	1	4	yes	—
181	C₂×C₄	C₂×C₇×Q₈	1	4	yes	—
182	C₂×C₄	C₂×C₇×Q₈	1	4	yes	—
183	C₂×C₄	C₇×#41	1	4	yes	—
184	C₂×C₄	C₇×#41	1	2	yes	—
185	C₇	#2586	1	1	yes	characteristic, direct product factor, Sylow
186	C₂²	C₂²×C₇×Q₈	1	2	yes	—
187	C₂²	C₂×C₇×#41	1	1	yes	characteristic
188	C₂²	C₂×C₇×#41	1	2	yes	—
189	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
190	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	—	2	4	no	—
194	C₄	—	2	4	no	—
195	C₄	—	2	4	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	4	1	no	—
199	C₄	—	4	1	no	—
200	C₂	C₇×#533	1	2	yes	—
201	C₂	C₇×#538	1	1	yes	characteristic
202	C₂	C₇×#551	1	1	yes	characteristic
203	C₂	C₇×#554	1	2	yes	—
204	C₂	C₇×Q₈²	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
31. #2586	C₇	direct (N × H)
52. C₇×C₄⋊C₄	Q₈	semidirect
148. C₄⋊C₄	C₇×Q₈	semidirect
185. C₇	#2586	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×#398	2
C₇×#395	2
C₇×#391	2
C₇×#389	2
C₇×#387	2
C₄×C₇×C₄⋊C₄	2
#2586	7

Subgroups — C7×#2586 #11741895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2586 #11741895