Subgroups — C₂×C₇×#2140 #13210593

Order 1792.

1406 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2140	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#357	C₂	1	4	yes	maximal, complemented
6	C₂³×C₇×D₄	—	2	1	no	—
7	C₂²×C₇×D₈	—	2	1	no	—
8	C₂²×C₇×D₈	—	2	2	no	—
9	C₂²×C₇×D₈	—	2	4	no	—
10	C₂²×C₇×D₈	C₄	1	8	yes	complemented
11	C₂²×C₇×D₈	C₂²	1	2	yes	—
12	C₂²×C₇×D₈	C₂²	1	4	yes	—
13	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₇×#117	—	2	1	no	—
15	C₂×C₇×M₃₂	—	2	4	no	—
16	C₂×C₇×M₃₂	C₂²	1	4	yes	complemented
17	C₂²×C₇×C₁₆	C₂²	1	2	yes	complemented
18	C₇×#427	—	2	2	no	—
19	C₇×#427	C₂²	1	4	yes	complemented
20	C₂×C₇×#107	C₂²	1	2	yes	complemented
21	C₂×C₇×#106	—	2	1	no	—
22	C₇×#357	C₂²	1	16	yes	complemented
23	C₂×#2140	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂⁵×C₇	—	4	1	no	—
25	C₂²×C₇×D₄	—	2	8	no	—
26	C₂²×C₇×D₄	—	2	4	no	—
27	C₂²×C₇×D₄	—	2	2	no	—
28	C₂²×C₇×D₄	—	4	2	no	—
29	C₂²×C₇×D₄	—	4	1	no	—
30	C₂²×C₇×D₄	—	4	4	no	—
31	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
32	C₂×C₇×D₈	—	2	16	no	—
33	C₂×C₇×D₈	—	2	8	no	—
34	C₂×C₇×D₈	—	2	4	no	—
35	C₂×C₇×D₈	—	2	2	no	—
36	C₂×C₇×D₈	—	2	4	no	—
37	C₂×C₇×D₈	—	2	8	no	—
38	C₂×C₇×D₈	C₂×C₄	1	8	yes	—
39	C₂×C₇×D₈	C₂×C₄	1	16	yes	complemented
40	C₂×C₇×D₈	C₂³	1	4	yes	—
41	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
42	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
43	C₂²×C₇×C₈	D₄	1	2	yes	—
44	C₂²×C₇×C₈	D₄	1	4	yes	—
45	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₇×C₈	C₂³	1	2	yes	—
47	C₇×#117	—	2	4	no	—
48	C₇×#117	—	4	2	no	—
49	C₂×C₇×C₄⋊C₄	—	2	2	no	—
50	C₂×C₇×C₂²⋊C₄	—	4	1	no	—
51	C₂×C₄²×C₇	—	4	1	no	—
52	C₇×M₃₂	—	2	8	no	—
53	C₂×C₇×C₁₆	—	2	4	no	—
54	C₂×C₇×C₁₆	C₂³	1	4	yes	complemented
55	C₇×#107	—	2	2	no	—
56	C₇×#107	C₂³	1	4	yes	complemented
57	C₇×#106	—	2	4	no	—
58	C₂³×D₈	C₂×C₇	1	1	yes	characteristic
59	C₂²×M₃₂	C₂×C₇	1	1	yes	characteristic, complemented
60	C₂×#427	C₂×C₇	1	1	yes	characteristic, complemented
61	#2140	C₂×C₇	1	8	yes	direct product factor
62	C₂×#357	C₂×C₇	1	4	yes	complemented
63	C₂⁴×C₇	—	4	8	no	—
64	C₂⁴×C₇	—	4	4	no	—
65	C₂⁴×C₇	—	4	2	no	—
66	C₂⁴×C₇	—	8	8	no	—
67	C₂⁴×C₇	D₈	1	1	yes	characteristic
68	C₂×C₇×D₄	—	2	8	no	—
69	C₂×C₇×D₄	—	2	16	no	—
70	C₂×C₇×D₄	—	2	4	no	—
71	C₂×C₇×D₄	—	4	16	no	—
72	C₂×C₇×D₄	—	4	8	no	—
73	C₂×C₇×D₄	—	4	4	no	—
74	C₂×C₇×D₄	—	4	2	no	—
75	C₂×C₇×D₄	—	4	8	no	—
76	C₂×C₇×D₄	—	4	4	no	—
77	C₂²×C₄×C₇	—	4	2	no	—
78	C₂²×C₄×C₇	C₂²⋊C₄	1	4	yes	—
79	C₂²×C₄×C₇	D₈	1	2	yes	—
80	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₇	SD₁₆	1	4	yes	—
82	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
84	C₇×D₈	—	2	8	no	—
85	C₇×D₈	—	2	16	no	—
86	C₇×D₈	—	2	4	no	—
87	C₇×D₈	C₂²×C₄	1	8	yes	—
88	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
89	C₂×C₇×C₈	C₂²⋊C₄	1	8	yes	—
90	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
92	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
93	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
95	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
96	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
97	C₇×C₄⋊C₄	—	2	4	no	—
98	C₇×C₄⋊C₄	—	4	2	no	—
99	C₇×C₂²⋊C₄	—	4	4	no	—
100	C₄²×C₇	—	4	4	no	—
101	C₇×C₁₆	—	2	4	no	—
102	C₂³×D₄	—	2	1	no	—
103	C₂²×D₈	—	2	1	no	—
104	C₂²×D₈	—	2	2	no	—
105	C₂²×D₈	—	2	4	no	—
106	C₂²×D₈	C₄×C₇	1	8	yes	complemented
107	C₂²×D₈	C₂²×C₇	1	2	yes	—
108	C₂²×D₈	C₂²×C₇	1	4	yes	—
109	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
110	C₂×#117	—	2	1	no	—
111	C₂×M₃₂	—	2	4	no	—
112	C₂×M₃₂	C₂²×C₇	1	4	yes	complemented
113	C₂²×C₁₆	C₂²×C₇	1	2	yes	complemented
114	#427	—	2	2	no	—
115	#427	C₂²×C₇	1	4	yes	complemented
116	C₂×#107	C₂²×C₇	1	2	yes	complemented
117	C₂×#106	—	2	1	no	—
118	#357	C₂²×C₇	1	16	yes	complemented
119	C₂³×C₇	—	2	4	no	—
120	C₂³×C₇	—	4	4	no	—
121	C₂³×C₇	—	4	16	no	—
122	C₂³×C₇	—	4	8	no	—
123	C₂³×C₇	—	8	16	no	—
124	C₂³×C₇	—	8	32	no	—
125	C₂³×C₇	—	8	8	no	—
126	C₂³×C₇	#102	1	4	yes	—
127	C₂³×C₇	C₂×D₈	1	2	yes	—
128	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
129	C₇×D₄	—	2	8	no	—
130	C₇×D₄	—	4	4	no	—
131	C₇×D₄	—	4	8	no	—
132	C₇×D₄	—	4	16	no	—
133	C₂×C₄×C₇	—	4	4	no	—
134	C₂×C₄×C₇	—	8	4	no	—
135	C₂×C₄×C₇	#102	1	8	yes	—
136	C₂×C₄×C₇	#102	1	4	yes	—
137	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
138	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
139	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
142	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
143	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
144	C₇×C₈	C₂×C₂²⋊C₄	1	2	yes	—
145	C₇×C₈	C₂×C₂²⋊C₄	1	4	yes	—
146	C₇×C₈	C₂³×C₄	1	1	yes	characteristic
147	C₇×C₈	C₂²×D₄	1	1	yes	characteristic
148	C₂⁵	—	4	1	no	—
149	C₂²×D₄	—	2	8	no	—
150	C₂²×D₄	—	2	2	no	—
151	C₂²×D₄	—	2	4	no	—
152	C₂²×D₄	—	4	1	no	—
153	C₂²×D₄	—	4	2	no	—
154	C₂²×D₄	—	4	4	no	—
155	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
156	C₂×D₈	—	2	16	no	—
157	C₂×D₈	—	2	8	no	—
158	C₂×D₈	—	2	4	no	—
159	C₂×D₈	—	2	2	no	—
160	C₂×D₈	—	2	4	no	—
161	C₂×D₈	—	2	8	no	—
162	C₂×D₈	C₂×C₄×C₇	1	8	yes	—
163	C₂×D₈	C₂×C₄×C₇	1	16	yes	complemented
164	C₂×D₈	C₂³×C₇	1	4	yes	—
165	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
166	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
167	C₂²×C₈	C₇×D₄	1	2	yes	—
168	C₂²×C₈	C₇×D₄	1	4	yes	—
169	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₇	1	2	yes	—
171	#117	—	2	4	no	—
172	#117	—	4	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₂²⋊C₄	—	4	1	no	—
175	C₂×C₄²	—	4	1	no	—
176	M₃₂	—	2	8	no	—
177	C₂×C₁₆	—	2	4	no	—
178	C₂×C₁₆	C₂³×C₇	1	4	yes	complemented
179	#107	—	2	2	no	—
180	#107	C₂³×C₇	1	4	yes	complemented
181	#106	—	2	4	no	—
182	C₂²×C₇	—	2	4	no	—
183	C₂²×C₇	—	2	8	no	—
184	C₂²×C₇	—	4	8	no	—
185	C₂²×C₇	—	8	32	no	—
186	C₂²×C₇	—	8	8	no	—
187	C₂²×C₇	—	8	16	no	—
188	C₂²×C₇	C₂×#102	1	4	yes	—
189	C₂²×C₇	C₂×#102	1	2	yes	—
190	C₂²×C₇	#509	1	4	yes	—
191	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
192	C₄×C₇	—	8	4	no	—
193	C₄×C₇	C₂×#102	1	4	yes	—
194	C₄×C₇	C₂×#102	1	2	yes	—
195	C₄×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
196	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
197	C₂⁴	—	4	8	no	—
198	C₂⁴	—	4	4	no	—
199	C₂⁴	—	4	2	no	—
200	C₂⁴	—	8	8	no	—
201	C₂⁴	C₇×D₈	1	1	yes	characteristic
202	C₂×D₄	—	2	8	no	—
203	C₂×D₄	—	2	16	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	4	16	no	—
206	C₂×D₄	—	4	8	no	—
207	C₂×D₄	—	4	4	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	8	no	—
210	C₂×D₄	—	4	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	C₇×C₂²⋊C₄	1	4	yes	—
213	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
214	C₂²×C₄	C₇×D₈	1	2	yes	—
215	C₂²×C₄	C₇×SD₁₆	1	4	yes	—
216	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
218	D₈	—	2	8	no	—
219	D₈	—	2	16	no	—
220	D₈	—	2	4	no	—
221	D₈	C₂²×C₄×C₇	1	8	yes	—
222	C₂×C₈	C₇×C₂²⋊C₄	1	4	yes	—
223	C₂×C₈	C₇×C₂²⋊C₄	1	8	yes	—
224	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
225	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
226	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
227	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
228	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
229	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
230	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₄²	—	4	4	no	—
235	C₁₆	—	2	4	no	—
236	C₂×C₇	—	2	4	no	—
237	C₂×C₇	—	8	8	no	—
238	C₂×C₇	#2140	1	4	yes	direct product factor
239	C₂×C₇	C₂²×#102	1	1	yes	characteristic
240	C₂×C₇	C₂×#509	1	2	yes	—
241	C₂³	—	2	4	no	—
242	C₂³	—	4	16	no	—
243	C₂³	—	4	8	no	—
244	C₂³	—	4	4	no	—
245	C₂³	—	8	16	no	—
246	C₂³	—	8	32	no	—
247	C₂³	—	8	8	no	—
248	C₂³	C₇×#102	1	4	yes	—
249	C₂³	C₂×C₇×D₈	1	2	yes	—
250	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
251	D₄	—	2	8	no	—
252	D₄	—	4	8	no	—
253	D₄	—	4	16	no	—
254	D₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	C₇×#102	1	8	yes	—
258	C₂×C₄	C₇×#102	1	4	yes	—
259	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
260	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
261	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
262	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
263	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
264	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
265	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
266	C₈	C₂×C₇×C₂²⋊C₄	1	2	yes	—
267	C₈	C₂×C₇×C₂²⋊C₄	1	4	yes	—
268	C₈	C₂³×C₄×C₇	1	1	yes	characteristic
269	C₈	C₂²×C₇×D₄	1	1	yes	characteristic
270	C₇	C₂×#2140	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²	—	2	4	no	—
272	C₂²	—	2	8	no	—
273	C₂²	—	4	8	no	—
274	C₂²	—	8	32	no	—
275	C₂²	—	8	8	no	—
276	C₂²	—	8	16	no	—
277	C₂²	C₂×C₇×#102	1	4	yes	—
278	C₂²	C₂×C₇×#102	1	2	yes	—
279	C₂²	C₇×#509	1	4	yes	—
280	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
281	C₄	—	8	4	no	—
282	C₄	C₂×C₇×#102	1	4	yes	—
283	C₄	C₂×C₇×#102	1	2	yes	—
284	C₄	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
285	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
286	C₂	—	2	4	no	—
287	C₂	—	8	8	no	—
288	C₂	C₇×#2140	1	4	yes	direct product factor
289	C₂	C₂²×C₇×#102	1	1	yes	characteristic
290	C₂	C₂×C₇×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂×D₈
commutator	C₈	C₂³×C₄×C₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₇
prime radical	C₂³×C₇×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂³×C₄×C₇ ◁ C₂³×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×M₃₂	C₂	semidirect
3. C₂×C₇×#427	C₂	semidirect
4. C₇×#2140	C₂	direct (N × H)
5. C₂×C₇×#357	C₂	semidirect
10. C₂²×C₇×D₈	C₄	semidirect
16. C₂×C₇×M₃₂	C₂²	semidirect
17. C₂²×C₇×C₁₆	C₂²	semidirect
19. C₇×#427	C₂²	semidirect
20. C₂×C₇×#107	C₂²	semidirect
22. C₇×#357	C₂²	semidirect
23. C₂×#2140	C₇	direct (N × H)
39. C₂×C₇×D₈	C₂×C₄	semidirect
54. C₂×C₇×C₁₆	C₂³	semidirect
56. C₇×#107	C₂³	semidirect
59. C₂²×M₃₂	C₂×C₇	semidirect
60. C₂×#427	C₂×C₇	semidirect
61. #2140	C₂×C₇	direct (N × H)
62. C₂×#357	C₂×C₇	semidirect
106. C₂²×D₈	C₄×C₇	semidirect
112. C₂×M₃₂	C₂²×C₇	semidirect
113. C₂²×C₁₆	C₂²×C₇	semidirect
115. #427	C₂²×C₇	semidirect
116. C₂×#107	C₂²×C₇	semidirect
118. #357	C₂²×C₇	semidirect
163. C₂×D₈	C₂×C₄×C₇	semidirect
178. C₂×C₁₆	C₂³×C₇	semidirect
180. #107	C₂³×C₇	semidirect
238. C₂×C₇	#2140	direct (N × H)
270. C₇	C₂×#2140	direct (N × H)
288. C₂	C₇×#2140	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇×D₈	2
C₂²×C₇×M₃₂	2
C₂×C₇×#427	2
C₇×#2140	2
C₂×C₇×#357	2
C₂×#2140	7

Subgroups — C2×C7×#2140 #13210593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2140 #13210593