Subgroups — C₇×#2512 #11741821

Order 896.

496 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#126	C₂	1	2	yes	maximal, complemented
2	C₇×#397	C₂	1	2	yes	maximal, complemented
3	C₇×#396	C₂	1	2	yes	maximal, complemented
4	C₇×#390	C₂	1	1	yes	maximal, characteristic
5	C₇×#390	C₂	1	1	yes	maximal, characteristic
6	C₇×#390	C₂	1	1	yes	maximal, characteristic
7	C₇×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#388	C₂	1	2	yes	maximal, complemented
9	C₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
13	C₇×#126	—	2	8	no	—
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	1	yes	characteristic
18	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₇	C₂²	1	2	yes	—
31	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
32	C₇×#95	C₂²	1	1	yes	characteristic
33	C₇×#95	C₂²	1	2	yes	—
34	C₇×#95	C₂²	1	1	yes	characteristic
35	C₇×#95	C₂²	1	2	yes	—
36	#2512	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂⁴×C₇	C₂³	1	2	yes	—
38	C₂×C₇×D₄	—	2	2	no	—
39	C₂×C₇×D₄	—	2	2	no	—
40	C₂×C₇×D₄	—	2	2	no	—
41	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₇×C₄⋊C₄	—	2	4	no	—
52	C₇×C₄⋊C₄	—	2	4	no	—
53	C₇×C₄⋊C₄	—	2	2	no	—
54	C₇×C₄⋊C₄	—	2	2	no	—
55	C₇×C₄⋊C₄	—	2	2	no	—
56	C₇×C₄⋊C₄	—	2	2	no	—
57	C₇×C₄⋊C₄	—	4	2	no	—
58	C₇×C₄⋊C₄	D₄	1	4	yes	complemented
59	C₇×C₂²⋊C₄	—	2	4	no	—
60	C₇×C₂²⋊C₄	—	2	4	no	—
61	C₇×C₂²⋊C₄	—	2	4	no	—
62	C₇×C₂²⋊C₄	—	2	4	no	—
63	C₇×C₂²⋊C₄	—	4	2	no	—
64	C₇×C₂²⋊C₄	—	4	2	no	—
65	C₄²×C₇	—	2	4	no	—
66	C₄²×C₇	—	2	2	no	—
67	C₄²×C₇	—	2	2	no	—
68	C₄²×C₇	—	2	2	no	—
69	C₂×#126	C₂×C₇	1	2	yes	complemented
70	#397	C₂×C₇	1	2	yes	complemented
71	#396	C₂×C₇	1	2	yes	complemented
72	#390	C₂×C₇	1	1	yes	characteristic
73	#390	C₂×C₇	1	1	yes	characteristic
74	#390	C₂×C₇	1	1	yes	characteristic
75	#389	C₂×C₇	1	1	yes	characteristic, complemented
76	#388	C₂×C₇	1	2	yes	complemented
77	#384	C₂×C₇	1	1	yes	characteristic, complemented
78	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
79	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
80	C₂³×C₇	—	2	2	no	—
81	C₂³×C₇	—	2	2	no	—
82	C₂³×C₇	—	2	2	no	—
83	C₂³×C₇	—	2	2	no	—
84	C₂³×C₇	—	2	2	no	—
85	C₂³×C₇	—	2	2	no	—
86	C₂³×C₇	—	2	2	no	—
87	C₂³×C₇	C₂⁴	1	1	yes	characteristic
88	C₇×D₄	—	4	4	no	—
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	2	1	no	—
91	C₂×C₄×C₇	—	2	1	no	—
92	C₂×C₄×C₇	—	2	1	no	—
93	C₂×C₄×C₇	—	2	1	no	—
94	C₂×C₄×C₇	—	2	1	no	—
95	C₂×C₄×C₇	—	2	1	no	—
96	C₂×C₄×C₇	—	2	1	no	—
97	C₂×C₄×C₇	—	2	2	no	—
98	C₂×C₄×C₇	—	2	2	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	2	1	no	—
101	C₂×C₄×C₇	—	2	1	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	2	2	no	—
110	C₂×C₄×C₇	—	2	2	no	—
111	C₂×C₄×C₇	—	2	1	no	—
112	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₇	#41	1	2	yes	—
116	C₂×C₄×C₇	#41	1	2	yes	—
117	C₂×C₄×C₇	#41	1	2	yes	—
118	C₂×C₄×C₇	#41	1	2	yes	—
119	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
120	#126	—	2	8	no	—
121	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
122	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
125	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
127	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
128	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
129	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
130	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
131	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
133	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
135	C₂×C₄²	C₂²×C₇	1	2	yes	—
136	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
137	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
138	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
139	#95	C₂²×C₇	1	1	yes	characteristic
140	#95	C₂²×C₇	1	2	yes	—
141	#95	C₂²×C₇	1	1	yes	characteristic
142	#95	C₂²×C₇	1	2	yes	—
143	C₂²×C₇	—	4	2	no	—
144	C₂²×C₇	—	4	2	no	—
145	C₂²×C₇	—	4	2	no	—
146	C₂²×C₇	—	4	2	no	—
147	C₂²×C₇	—	4	2	no	—
148	C₂²×C₇	—	4	2	no	—
149	C₂²×C₇	—	4	2	no	—
150	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
151	C₂²×C₇	C₂×#41	1	1	yes	characteristic
152	C₂²×C₇	C₂×#41	1	1	yes	characteristic
153	C₂²×C₇	C₂×#41	1	1	yes	characteristic
154	C₂²×C₇	C₂×#41	1	1	yes	characteristic
155	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
157	C₄×C₇	—	2	2	no	—
158	C₄×C₇	—	2	2	no	—
159	C₄×C₇	—	2	2	no	—
160	C₄×C₇	—	2	2	no	—
161	C₄×C₇	—	4	2	no	—
162	C₄×C₇	—	4	1	no	—
163	C₄×C₇	—	4	2	no	—
164	C₄×C₇	—	4	2	no	—
165	C₄×C₇	—	4	1	no	—
166	C₄×C₇	#126	1	4	yes	complemented
167	C₂⁴	C₂³×C₇	1	2	yes	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
172	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₇	1	2	yes	—
174	C₂²×C₄	C₂³×C₇	1	2	yes	—
175	C₂²×C₄	C₂³×C₇	1	2	yes	—
176	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
180	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
181	C₄⋊C₄	—	2	4	no	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	C₇×D₄	1	4	yes	complemented
189	C₂²⋊C₄	—	2	4	no	—
190	C₂²⋊C₄	—	2	4	no	—
191	C₂²⋊C₄	—	2	4	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₂²⋊C₄	—	4	2	no	—
194	C₂²⋊C₄	—	4	2	no	—
195	C₄²	—	2	4	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₂×C₇	—	8	2	no	—
200	C₂×C₇	C₂×#126	1	1	yes	characteristic
201	C₂×C₇	#532	1	1	yes	characteristic
202	C₂×C₇	#542	1	1	yes	characteristic
203	C₂×C₇	#547	1	1	yes	characteristic
204	C₂×C₇	#550	1	1	yes	characteristic
205	C₂×C₇	#553	1	1	yes	characteristic
206	C₂×C₇	#556	1	1	yes	characteristic
207	C₂³	—	2	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₂⁴×C₇	1	1	yes	characteristic
215	D₄	—	4	4	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
240	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
241	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
242	C₂×C₄	C₇×#41	1	2	yes	—
243	C₂×C₄	C₇×#41	1	2	yes	—
244	C₂×C₄	C₇×#41	1	2	yes	—
245	C₂×C₄	C₇×#41	1	2	yes	—
246	C₇	#2512	1	1	yes	characteristic, direct product factor, Sylow
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
255	C₂²	C₂×C₇×#41	1	1	yes	characteristic
256	C₂²	C₂×C₇×#41	1	1	yes	characteristic
257	C₂²	C₂×C₇×#41	1	1	yes	characteristic
258	C₂²	C₂×C₇×#41	1	1	yes	characteristic
259	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
260	C₂²	C₇×2₋¹⁺⁴	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	1	no	—
270	C₄	C₇×#126	1	4	yes	complemented
271	C₂	—	8	2	no	—
272	C₂	C₂×C₇×#126	1	1	yes	characteristic
273	C₂	C₇×#532	1	1	yes	characteristic
274	C₂	C₇×#542	1	1	yes	characteristic
275	C₂	C₇×#547	1	1	yes	characteristic
276	C₂	C₇×#550	1	1	yes	characteristic
277	C₂	C₇×#553	1	1	yes	characteristic
278	C₂	C₇×#556	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂²
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. C₂×C₇×#126	C₂	semidirect
2. C₇×#397	C₂	semidirect
3. C₇×#396	C₂	semidirect
7. C₇×#389	C₂	semidirect
8. C₇×#388	C₂	semidirect
9. C₇×#384	C₂	semidirect
10. C₄×C₇×C₄⋊C₄	C₂	semidirect
11. C₄×C₇×C₄⋊C₄	C₂	semidirect
36. #2512	C₇	direct (N × H)
58. C₇×C₄⋊C₄	D₄	semidirect
69. C₂×#126	C₂×C₇	semidirect
70. #397	C₂×C₇	semidirect
71. #396	C₂×C₇	semidirect
75. #389	C₂×C₇	semidirect
76. #388	C₂×C₇	semidirect
77. #384	C₂×C₇	semidirect
78. C₄×C₄⋊C₄	C₂×C₇	semidirect
79. C₄×C₄⋊C₄	C₂×C₇	semidirect
166. C₄×C₇	#126	semidirect
188. C₄⋊C₄	C₇×D₄	semidirect
246. C₇	#2512	direct (N × H)
270. C₄	C₇×#126	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#126	2
C₇×#397	2
C₇×#396	2
C₇×#390	2
C₇×#389	2
C₇×#388	2
C₇×#384	2
C₄×C₇×C₄⋊C₄	2
#2512	7

Subgroups — C7×#2512 #11741821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2512 #11741821