Subgroups — C₇×#2495 #11741804

Order 896.

412 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#400	C₂	1	2	yes	maximal, complemented
2	C₇×#397	C₂	1	2	yes	maximal
3	C₇×#389	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#389	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#388	C₂	1	2	yes	maximal
6	C₇×#383	C₂	1	2	yes	maximal, complemented
7	C₇×#382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#380	C₂	1	1	yes	maximal, characteristic
9	C₇×#380	C₂	1	1	yes	maximal, characteristic
10	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	2	yes	—
18	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₇	C₂²	1	2	yes	—
26	C₇×#95	C₂²	1	1	yes	characteristic
27	C₇×#95	C₂²	1	1	yes	characteristic
28	C₇×#95	C₂²	1	2	yes	—
29	C₇×#95	C₂²	1	2	yes	—
30	C₇×#95	C₂²	1	2	yes	—
31	C₇×#95	C₂²	1	2	yes	—
32	C₇×#95	C₂²	1	2	yes	—
33	C₇×#95	C₂²	1	2	yes	—
34	C₇×#95	C₂²	1	2	yes	—
35	#2495	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂⁴×C₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₇	—	2	2	no	—
38	C₂²×C₄×C₇	—	2	2	no	—
39	C₂²×C₄×C₇	—	2	2	no	—
40	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
46	C₂²×C₄×C₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₇×C₄⋊C₄	—	2	2	no	—
51	C₇×C₄⋊C₄	—	2	2	no	—
52	C₇×C₄⋊C₄	—	2	2	no	—
53	C₇×C₄⋊C₄	—	2	2	no	—
54	C₇×C₄⋊C₄	—	4	2	no	—
55	C₇×C₄⋊C₄	—	4	2	no	—
56	C₇×C₂²⋊C₄	—	2	2	no	—
57	C₇×C₂²⋊C₄	—	2	4	no	—
58	C₇×C₂²⋊C₄	—	2	2	no	—
59	C₇×C₂²⋊C₄	—	4	2	no	—
60	C₄²×C₇	—	2	2	no	—
61	C₄²×C₇	—	2	4	no	—
62	C₄²×C₇	—	2	2	no	—
63	#400	C₂×C₇	1	2	yes	complemented
64	#397	C₂×C₇	1	2	yes	—
65	#389	C₂×C₇	1	1	yes	characteristic, complemented
66	#389	C₂×C₇	1	1	yes	characteristic, complemented
67	#388	C₂×C₇	1	2	yes	—
68	#383	C₂×C₇	1	2	yes	complemented
69	#382	C₂×C₇	1	1	yes	characteristic, complemented
70	#380	C₂×C₇	1	1	yes	characteristic
71	#380	C₂×C₇	1	1	yes	characteristic
72	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
73	C₄×C₂²⋊C₄	C₂×C₇	1	1	yes	characteristic
74	C₂³×C₇	—	2	1	no	—
75	C₂³×C₇	—	2	1	no	—
76	C₂³×C₇	—	2	1	no	—
77	C₂³×C₇	—	2	1	no	—
78	C₂³×C₇	—	2	1	no	—
79	C₂³×C₇	—	2	1	no	—
80	C₂³×C₇	#41	1	2	yes	—
81	C₂³×C₇	C₂⁴	1	1	yes	characteristic
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	1	no	—
86	C₂×C₄×C₇	—	2	1	no	—
87	C₂×C₄×C₇	—	2	1	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	2	1	no	—
91	C₂×C₄×C₇	—	2	1	no	—
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	—	2	1	no	—
94	C₂×C₄×C₇	—	2	2	no	—
95	C₂×C₄×C₇	—	2	2	no	—
96	C₂×C₄×C₇	—	2	2	no	—
97	C₂×C₄×C₇	—	2	2	no	—
98	C₂×C₄×C₇	—	2	1	no	—
99	C₂×C₄×C₇	—	2	1	no	—
100	C₂×C₄×C₇	—	2	1	no	—
101	C₂×C₄×C₇	—	2	2	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	2	1	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	2	1	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	4	4	no	—
109	C₂×C₄×C₇	#41	1	2	yes	—
110	C₂×C₄×C₇	#41	1	2	yes	—
111	C₂×C₄×C₇	#41	1	2	yes	—
112	C₂×C₄×C₇	#41	1	2	yes	—
113	C₂³×C₄	C₂²×C₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
116	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
118	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
119	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
120	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
121	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
122	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
123	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
124	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
125	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
126	C₂×C₄²	C₂²×C₇	1	2	yes	—
127	#95	C₂²×C₇	1	1	yes	characteristic
128	#95	C₂²×C₇	1	1	yes	characteristic
129	#95	C₂²×C₇	1	2	yes	—
130	#95	C₂²×C₇	1	2	yes	—
131	#95	C₂²×C₇	1	2	yes	—
132	#95	C₂²×C₇	1	2	yes	—
133	#95	C₂²×C₇	1	2	yes	—
134	#95	C₂²×C₇	1	2	yes	—
135	#95	C₂²×C₇	1	2	yes	—
136	C₂²×C₇	—	2	2	no	—
137	C₂²×C₇	—	2	2	no	—
138	C₂²×C₇	—	2	2	no	—
139	C₂²×C₇	—	4	1	no	—
140	C₂²×C₇	—	4	1	no	—
141	C₂²×C₇	—	4	1	no	—
142	C₂²×C₇	—	4	1	no	—
143	C₂²×C₇	C₂×#41	1	1	yes	characteristic
144	C₂²×C₇	C₂×#41	1	1	yes	characteristic
145	C₂²×C₇	C₂×#41	1	1	yes	characteristic
146	C₂²×C₇	C₂×#41	1	1	yes	characteristic
147	C₂²×C₇	C₂×#41	1	1	yes	characteristic
148	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
150	C₄×C₇	—	2	2	no	—
151	C₄×C₇	—	4	1	no	—
152	C₄×C₇	—	4	2	no	—
153	C₄×C₇	—	4	2	no	—
154	C₄×C₇	—	4	1	no	—
155	C₄×C₇	—	4	1	no	—
156	C₄×C₇	—	4	2	no	—
157	C₄×C₇	—	4	1	no	—
158	C₄×C₇	—	4	2	no	—
159	C₄×C₇	#126	1	4	yes	—
160	C₂⁴	C₂³×C₇	1	1	yes	characteristic
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₇	1	2	yes	—
167	C₂²×C₄	C₂³×C₇	1	2	yes	—
168	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₇	1	2	yes	—
171	C₂²×C₄	C₂³×C₇	1	2	yes	—
172	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	4	no	—
186	C₄²	—	2	2	no	—
187	C₂×C₇	—	4	2	no	—
188	C₂×C₇	C₂×#126	1	1	yes	characteristic
189	C₂×C₇	#529	1	1	yes	characteristic
190	C₂×C₇	#542	1	1	yes	characteristic
191	C₂×C₇	#552	1	1	yes	characteristic
192	C₂×C₇	#552	1	1	yes	characteristic
193	C₂×C₇	#553	1	1	yes	characteristic
194	C₂×C₇	#555	1	1	yes	characteristic
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	1	no	—
198	C₂³	—	2	1	no	—
199	C₂³	—	2	1	no	—
200	C₂³	—	2	1	no	—
201	C₂³	C₇×#41	1	2	yes	—
202	C₂³	C₂⁴×C₇	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	C₇×#41	1	2	yes	—
231	C₂×C₄	C₇×#41	1	2	yes	—
232	C₂×C₄	C₇×#41	1	2	yes	—
233	C₂×C₄	C₇×#41	1	2	yes	—
234	C₇	#2495	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	C₂×C₇×#41	1	1	yes	characteristic
243	C₂²	C₂×C₇×#41	1	1	yes	characteristic
244	C₂²	C₂×C₇×#41	1	1	yes	characteristic
245	C₂²	C₂×C₇×#41	1	1	yes	characteristic
246	C₂²	C₂×C₇×#41	1	1	yes	characteristic
247	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
248	C₂²	C₇×2₋¹⁺⁴	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	2	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	2	no	—
258	C₄	C₇×#126	1	4	yes	—
259	C₂	—	4	2	no	—
260	C₂	C₂×C₇×#126	1	1	yes	characteristic
261	C₂	C₇×#529	1	1	yes	characteristic
262	C₂	C₇×#542	1	1	yes	characteristic
263	C₂	C₇×#552	1	1	yes	characteristic
264	C₂	C₇×#552	1	1	yes	characteristic
265	C₂	C₇×#553	1	1	yes	characteristic
266	C₂	C₇×#555	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂⁴×C₇	C₂³
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
1. C₇×#400	C₂	semidirect
3. C₇×#389	C₂	semidirect
4. C₇×#389	C₂	semidirect
6. C₇×#383	C₂	semidirect
7. C₇×#382	C₂	semidirect
10. C₄×C₇×C₄⋊C₄	C₂	semidirect
35. #2495	C₇	direct (N × H)
63. #400	C₂×C₇	semidirect
65. #389	C₂×C₇	semidirect
66. #389	C₂×C₇	semidirect
68. #383	C₂×C₇	semidirect
69. #382	C₂×C₇	semidirect
72. C₄×C₄⋊C₄	C₂×C₇	semidirect
234. C₇	#2495	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×#400	2
C₇×#397	2
C₇×#389	2
C₇×#388	2
C₇×#383	2
C₇×#382	2
C₇×#380	2
C₄×C₇×C₄⋊C₄	2
C₄×C₇×C₂²⋊C₄	2
#2495	7

Subgroups — C7×#2495 #11741804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2495 #11741804