Subgroups — C₇×#2565 #11741874

Order 896.

452 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#398	C₂	1	4	yes	maximal
4	C₇×#391	C₂	1	1	yes	maximal, characteristic
5	C₇×#391	C₂	1	1	yes	maximal, characteristic
6	C₇×#387	C₂	1	4	yes	maximal
7	C₇×#384	C₂	1	1	yes	maximal, characteristic
8	C₇×#382	C₂	1	1	yes	maximal, characteristic
9	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₇×#125	—	2	4	no	—
12	C₄×C₇×Q₈	—	2	4	no	—
13	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
18	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
25	C₇×#95	C₂²	1	2	yes	—
26	C₇×#95	C₂²	1	4	yes	—
27	C₇×#95	C₂²	1	4	yes	—
28	C₇×#95	C₂²	1	4	yes	—
29	#2565	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₇×Q₈	—	2	2	no	—
31	C₂×C₇×Q₈	—	2	2	no	—
32	C₂×C₇×Q₈	Q₈	1	4	yes	—
33	C₂²×C₄×C₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₇	C₂³	1	2	yes	—
36	C₂²×C₄×C₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₇×C₄⋊C₄	—	2	4	no	—
43	C₇×C₄⋊C₄	—	2	2	no	—
44	C₇×C₄⋊C₄	—	2	4	no	—
45	C₇×C₄⋊C₄	—	2	4	no	—
46	C₇×C₄⋊C₄	—	2	2	no	—
47	C₇×C₄⋊C₄	—	2	2	no	—
48	C₇×C₄⋊C₄	—	2	4	no	—
49	C₇×C₄⋊C₄	—	2	8	no	—
50	C₄²×C₇	—	2	4	no	—
51	C₄²×C₇	—	2	2	no	—
52	C₄²×C₇	—	2	2	no	—
53	C₄²×C₇	D₄	1	4	yes	—
54	C₂×#125	C₂×C₇	1	1	yes	characteristic
55	C₂×C₄×Q₈	C₂×C₇	1	1	yes	characteristic
56	#398	C₂×C₇	1	4	yes	—
57	#391	C₂×C₇	1	1	yes	characteristic
58	#391	C₂×C₇	1	1	yes	characteristic
59	#387	C₂×C₇	1	4	yes	—
60	#384	C₂×C₇	1	1	yes	characteristic
61	#382	C₂×C₇	1	1	yes	characteristic
62	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
63	C₂³×C₇	C₂⁴	1	1	yes	characteristic
64	C₇×Q₈	—	2	8	no	—
65	C₂×C₄×C₇	—	2	2	no	—
66	C₂×C₄×C₇	—	2	2	no	—
67	C₂×C₄×C₇	—	2	2	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	4	no	—
76	C₂×C₄×C₇	—	2	4	no	—
77	C₂×C₄×C₇	—	2	2	no	—
78	C₂×C₄×C₇	—	2	4	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
84	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
85	C₂×C₄×C₇	#41	1	4	yes	—
86	C₂×C₄×C₇	#41	1	2	yes	—
87	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
88	#125	—	2	4	no	—
89	C₄×Q₈	—	2	4	no	—
90	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
91	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
92	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
93	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
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	4	yes	—
98	C₂×C₄²	C₂²×C₇	1	2	yes	—
99	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
100	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
101	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
102	#95	C₂²×C₇	1	2	yes	—
103	#95	C₂²×C₇	1	4	yes	—
104	#95	C₂²×C₇	1	4	yes	—
105	#95	C₂²×C₇	1	4	yes	—
106	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
107	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
108	C₂²×C₇	C₂×#41	1	1	yes	characteristic
109	C₂²×C₇	C₂×#41	1	2	yes	—
110	C₂²×C₇	2₋¹⁺⁴	1	2	yes	—
111	C₄×C₇	—	2	4	no	—
112	C₄×C₇	—	2	2	no	—
113	C₄×C₇	—	2	2	no	—
114	C₄×C₇	—	2	4	no	—
115	C₄×C₇	—	4	2	no	—
116	C₄×C₇	—	4	2	no	—
117	C₄×C₇	—	4	2	no	—
118	C₄×C₇	—	4	2	no	—
119	C₄×C₇	C₂²⋊Q₈	1	4	yes	—
120	C₂×Q₈	—	2	2	no	—
121	C₂×Q₈	—	2	2	no	—
122	C₂×Q₈	C₇×Q₈	1	4	yes	—
123	C₂²×C₄	C₂³×C₇	1	2	yes	—
124	C₂²×C₄	C₂³×C₇	1	2	yes	—
125	C₂²×C₄	C₂³×C₇	1	2	yes	—
126	C₂²×C₄	C₂³×C₇	1	2	yes	—
127	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
128	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
129	C₂²×C₄	C₂³×C₇	1	2	yes	—
130	C₂²×C₄	C₂³×C₇	1	2	yes	—
131	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	2	8	no	—
140	C₄²	—	2	4	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	C₇×D₄	1	4	yes	—
144	C₂×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
145	C₂×C₇	#532	1	1	yes	characteristic
146	C₂×C₇	#536	1	1	yes	characteristic
147	C₂×C₇	#552	1	2	yes	—
148	C₂×C₇	#557	1	2	yes	—
149	C₂³	C₂⁴×C₇	1	1	yes	characteristic
150	Q₈	—	2	8	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
167	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
168	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
169	C₂×C₄	C₂×C₇×Q₈	1	4	yes	—
170	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
171	C₂×C₄	C₇×#41	1	4	yes	—
172	C₂×C₄	C₇×#41	1	2	yes	—
173	C₇	#2565	1	1	yes	characteristic, direct product factor, Sylow
174	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
175	C₂²	C₂²×C₇×Q₈	1	1	yes	characteristic
176	C₂²	C₂×C₇×#41	1	1	yes	characteristic
177	C₂²	C₂×C₇×#41	1	2	yes	—
178	C₂²	C₇×2₋¹⁺⁴	1	2	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	4	no	—
183	C₄	—	4	2	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	2	no	—
186	C₄	—	4	2	no	—
187	C₄	C₇×C₂²⋊Q₈	1	4	yes	—
188	C₂	C₂×C₇×C₂²⋊Q₈	1	1	yes	characteristic
189	C₂	C₇×#532	1	1	yes	characteristic
190	C₂	C₇×#536	1	1	yes	characteristic
191	C₂	C₇×#552	1	2	yes	—
192	C₂	C₇×#557	1	2	yes	—

	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
29. #2565	C₇	direct (N × H)
173. C₇	#2565	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#125	2
C₂×C₄×C₇×Q₈	2
C₇×#398	2
C₇×#391	2
C₇×#387	2
C₇×#384	2
C₇×#382	2
C₄×C₇×C₄⋊C₄	2
#2565	7

Subgroups — C7×#2565 #11741874

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2565 #11741874