Subgroups — C₇×#2500 #11741809

Order 896.

568 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#121	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#394	C₂	1	4	yes	maximal, complemented
4	C₇×#390	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#390	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#388	C₂	1	4	yes	maximal, complemented
7	C₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₇×D₄	C₂²	1	2	yes	—
12	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₇×#121	—	2	4	no	—
14	C₇×#117	—	2	4	no	—
15	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
19	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₂×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
24	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₇	C₂²	1	2	yes	—
27	C₇×#95	C₂²	1	2	yes	—
28	C₇×#95	C₂²	1	4	yes	—
29	#2500	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂⁴×C₇	C₂³	1	2	yes	—
31	C₂⁴×C₇	C₂³	1	1	yes	characteristic
32	C₂×C₇×D₄	—	2	4	no	—
33	C₂×C₇×D₄	—	2	2	no	—
34	C₂×C₇×D₄	—	2	2	no	—
35	C₂×C₇×D₄	—	2	2	no	—
36	C₂×C₇×D₄	—	4	4	no	—
37	C₂²×C₄×C₇	—	2	2	no	—
38	C₂²×C₄×C₇	—	2	2	no	—
39	C₂²×C₄×C₇	D₄	1	4	yes	—
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	1	yes	characteristic
45	C₂²×C₄×C₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₇×C₄⋊C₄	—	2	2	no	—
48	C₇×C₄⋊C₄	—	2	2	no	—
49	C₇×C₄⋊C₄	—	2	2	no	—
50	C₇×C₄⋊C₄	—	4	4	no	—
51	C₇×C₂²⋊C₄	—	2	4	no	—
52	C₇×C₂²⋊C₄	—	2	4	no	—
53	C₇×C₂²⋊C₄	—	2	4	no	—
54	C₇×C₂²⋊C₄	—	2	8	no	—
55	C₇×C₂²⋊C₄	—	4	4	no	—
56	C₄²×C₇	—	2	4	no	—
57	C₄²×C₇	—	2	2	no	—
58	C₄²×C₇	—	2	2	no	—
59	C₂×#121	C₂×C₇	1	1	yes	characteristic
60	C₂×#117	C₂×C₇	1	1	yes	characteristic, complemented
61	#394	C₂×C₇	1	4	yes	complemented
62	#390	C₂×C₇	1	1	yes	characteristic, complemented
63	#390	C₂×C₇	1	1	yes	characteristic, complemented
64	#388	C₂×C₇	1	4	yes	complemented
65	#384	C₂×C₇	1	1	yes	characteristic, complemented
66	#382	C₂×C₇	1	1	yes	characteristic, complemented
67	C₄×C₂²⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
68	C₂³×C₇	—	2	2	no	—
69	C₂³×C₇	—	2	1	no	—
70	C₂³×C₇	—	2	1	no	—
71	C₂³×C₇	—	2	2	no	—
72	C₂³×C₇	—	2	2	no	—
73	C₂³×C₇	—	2	4	no	—
74	C₂³×C₇	—	2	2	no	—
75	C₂³×C₇	—	2	2	no	—
76	C₂³×C₇	—	2	4	no	—
77	C₂³×C₇	C₂×D₄	1	2	yes	—
78	C₂³×C₇	C₂⁴	1	1	yes	characteristic
79	C₇×D₄	—	4	8	no	—
80	C₇×D₄	—	4	4	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	—	2	1	no	—
83	C₂×C₄×C₇	—	2	1	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	2	8	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	2	4	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	—	2	4	no	—
94	C₂×C₄×C₇	—	2	4	no	—
95	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₇	#41	1	4	yes	—
98	C₂×C₄×C₇	#41	1	2	yes	—
99	C₂×C₄×C₇	#41	1	2	yes	—
100	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
101	C₂²×D₄	C₂²×C₇	1	2	yes	—
102	C₂³×C₄	C₂²×C₇	1	1	yes	characteristic
103	#121	—	2	4	no	—
104	#117	—	2	4	no	—
105	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
106	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
107	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
108	C₂×C₄⋊C₄	C₂²×C₇	1	4	yes	—
109	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
110	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
111	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
112	C₂×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
113	C₂×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
114	C₂×C₄²	C₂²×C₇	1	2	yes	—
115	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
116	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
117	#95	C₂²×C₇	1	2	yes	—
118	#95	C₂²×C₇	1	4	yes	—
119	C₂²×C₇	—	2	2	no	—
120	C₂²×C₇	—	2	2	no	—
121	C₂²×C₇	—	2	2	no	—
122	C₂²×C₇	—	4	2	no	—
123	C₂²×C₇	—	4	2	no	—
124	C₂²×C₇	—	4	2	no	—
125	C₂²×C₇	—	4	2	no	—
126	C₂²×C₇	—	4	2	no	—
127	C₂²×C₇	—	4	4	no	—
128	C₂²×C₇	—	4	4	no	—
129	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
130	C₂²×C₇	C₂×#41	1	1	yes	characteristic
131	C₂²×C₇	C₂×#41	1	1	yes	characteristic
132	C₂²×C₇	C₂×#41	1	2	yes	—
133	C₂²×C₇	H(5,𝔽₂)	1	2	yes	—
134	C₄×C₇	—	2	2	no	—
135	C₄×C₇	—	2	4	no	—
136	C₄×C₇	—	4	2	no	—
137	C₄×C₇	—	4	2	no	—
138	C₄×C₇	—	4	2	no	—
139	C₄×C₇	—	4	2	no	—
140	C₄×C₇	#123	1	4	yes	—
141	C₂⁴	C₂³×C₇	1	2	yes	—
142	C₂⁴	C₂³×C₇	1	1	yes	characteristic
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	—	2	2	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	—	4	4	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	C₇×D₄	1	4	yes	—
151	C₂²×C₄	C₂³×C₇	1	2	yes	—
152	C₂²×C₄	C₂³×C₇	1	2	yes	—
153	C₂²×C₄	C₂³×C₇	1	2	yes	—
154	C₂²×C₄	C₂³×C₇	1	2	yes	—
155	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
156	C₂²×C₄	C₂³×C₇	1	2	yes	—
157	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₂²⋊C₄	—	2	4	no	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₂²⋊C₄	—	2	4	no	—
165	C₂²⋊C₄	—	2	8	no	—
166	C₂²⋊C₄	—	4	4	no	—
167	C₄²	—	2	4	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	2	no	—
170	C₂×C₇	—	4	2	no	—
171	C₂×C₇	—	8	2	no	—
172	C₂×C₇	C₂×#123	1	1	yes	characteristic
173	C₂×C₇	#532	1	1	yes	characteristic
174	C₂×C₇	#535	1	1	yes	characteristic
175	C₂×C₇	#553	1	2	yes	—
176	C₂×C₇	#555	1	2	yes	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	1	no	—
179	C₂³	—	2	1	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	4	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	4	no	—
186	C₂³	C₂×C₇×D₄	1	2	yes	—
187	C₂³	C₂⁴×C₇	1	1	yes	characteristic
188	D₄	—	4	8	no	—
189	D₄	—	4	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
205	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
206	C₂×C₄	C₇×#41	1	4	yes	—
207	C₂×C₄	C₇×#41	1	2	yes	—
208	C₂×C₄	C₇×#41	1	2	yes	—
209	C₇	#2500	1	1	yes	characteristic, direct product factor, Sylow
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	4	no	—
219	C₂²	—	4	4	no	—
220	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
221	C₂²	C₂×C₇×#41	1	1	yes	characteristic
222	C₂²	C₂×C₇×#41	1	1	yes	characteristic
223	C₂²	C₂×C₇×#41	1	2	yes	—
224	C₂²	C₇×H(5,𝔽₂)	1	2	yes	—
225	C₄	—	2	2	no	—
226	C₄	—	2	4	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	2	no	—
231	C₄	C₇×#123	1	4	yes	—
232	C₂	—	4	2	no	—
233	C₂	—	8	2	no	—
234	C₂	C₂×C₇×#123	1	1	yes	characteristic
235	C₂	C₇×#532	1	1	yes	characteristic
236	C₂	C₇×#535	1	1	yes	characteristic
237	C₂	C₇×#553	1	2	yes	—
238	C₂	C₇×#555	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂×C₇×#121	C₂
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
2. C₂×C₇×#117	C₂	semidirect
3. C₇×#394	C₂	semidirect
4. C₇×#390	C₂	semidirect
5. C₇×#390	C₂	semidirect
6. C₇×#388	C₂	semidirect
7. C₇×#384	C₂	semidirect
8. C₇×#382	C₂	semidirect
9. C₄×C₇×C₂²⋊C₄	C₂	semidirect
29. #2500	C₇	direct (N × H)
60. C₂×#117	C₂×C₇	semidirect
61. #394	C₂×C₇	semidirect
62. #390	C₂×C₇	semidirect
63. #390	C₂×C₇	semidirect
64. #388	C₂×C₇	semidirect
65. #384	C₂×C₇	semidirect
66. #382	C₂×C₇	semidirect
67. C₄×C₂²⋊C₄	C₂×C₇	semidirect
209. C₇	#2500	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×#121	2
C₂×C₇×#117	2
C₇×#394	2
C₇×#390	2
C₇×#388	2
C₇×#384	2
C₇×#382	2
C₄×C₇×C₂²⋊C₄	2
#2500	7

Subgroups — C7×#2500 #11741809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2500 #11741809