Subgroups — C₂×C₇×#1589 #13195099

Order 1792.

822 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#436	C₂	1	2	yes	maximal, complemented
3	C₇×#1589	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₇×D₈	C₄	1	1	yes	characteristic
8	C₂²×C₇×M₁₆	—	2	2	no	—
9	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
10	C₄×C₇×D₈	C₄	1	4	yes	—
11	C₄×C₇×D₈	C₂²	1	4	yes	—
12	C₇×#436	—	2	4	no	—
13	C₇×#436	C₂²	1	8	yes	complemented
14	C₂×C₇×#107	C₄	1	1	yes	characteristic
15	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₇×#102	C₄	1	2	yes	—
17	C₂×C₇×C₂²⋊C₈	—	2	2	no	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
21	C₇×#335	C₂²	1	4	yes	complemented
22	C₇×D₄⋊C₈	C₂²	1	8	yes	complemented
23	C₇×C₈²	C₂²	1	4	yes	complemented
24	C₂×#1589	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₇×D₄	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₇	—	2	2	no	—
27	C₂×C₇×D₈	—	2	4	no	—
28	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
29	C₂×C₇×M₁₆	—	2	8	no	—
30	C₂×C₇×M₁₆	—	4	8	no	—
31	C₂²×C₇×C₈	—	2	1	no	—
32	C₂²×C₇×C₈	—	2	2	no	—
33	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
37	C₄×C₇×D₄	—	2	2	no	—
38	C₄×C₇×D₄	—	2	2	no	—
39	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
40	C₄×C₇×D₄	C₂³	1	4	yes	—
41	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
43	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
44	C₇×#107	C₂×C₄	1	4	yes	—
45	C₇×C₄⋊C₈	—	2	2	no	—
46	C₇×C₄⋊C₈	C₂³	1	4	yes	—
47	C₇×#102	C₂×C₄	1	8	yes	—
48	C₇×C₂²⋊C₈	—	2	8	no	—
49	C₄×C₇×C₈	—	2	1	no	—
50	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
51	C₄×C₇×C₈	D₄	1	2	yes	—
52	C₄×C₇×C₈	C₂³	1	2	yes	—
53	C₄×C₇×C₈	C₂³	1	2	yes	—
54	C₄×C₇×C₈	C₂³	1	2	yes	—
55	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic
56	C₂×#436	C₂×C₇	1	2	yes	complemented
57	#1589	C₂×C₇	1	8	yes	direct product factor
58	C₂×#335	C₂×C₇	1	1	yes	characteristic, complemented
59	C₂×D₄⋊C₈	C₂×C₇	1	2	yes	complemented
60	C₂×C₈²	C₂×C₇	1	1	yes	characteristic, complemented
61	C₂⁴×C₇	—	2	2	no	—
62	C₂×C₇×D₄	—	2	2	no	—
63	C₂×C₇×D₄	—	2	4	no	—
64	C₂×C₇×D₄	—	2	4	no	—
65	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₇	—	2	2	no	—
67	C₂²×C₄×C₇	—	2	4	no	—
68	C₂²×C₄×C₇	—	2	4	no	—
69	C₂²×C₄×C₇	—	4	4	no	—
70	C₂²×C₄×C₇	—	4	4	no	—
71	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇	#41	1	1	yes	characteristic
74	C₇×D₈	—	2	8	no	—
75	C₇×M₁₆	—	4	16	no	—
76	C₂×C₇×C₈	—	2	4	no	—
77	C₂×C₇×C₈	—	2	2	no	—
78	C₂×C₇×C₈	—	2	2	no	—
79	C₂×C₇×C₈	—	2	2	no	—
80	C₂×C₇×C₈	—	2	4	no	—
81	C₂×C₇×C₈	—	4	4	no	—
82	C₂×C₇×C₈	M₁₆	1	4	yes	complemented
83	C₂×C₇×C₈	D₈	1	4	yes	complemented
84	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
87	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
88	C₇×C₄⋊C₄	—	2	2	no	—
89	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
90	C₇×C₂²⋊C₄	—	2	4	no	—
91	C₇×C₂²⋊C₄	—	2	4	no	—
92	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₇	#41	1	1	yes	characteristic
95	C₄²×C₇	C₂⁴	1	1	yes	characteristic
96	C₂²×D₈	C₄×C₇	1	1	yes	characteristic
97	C₂²×M₁₆	—	2	2	no	—
98	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
99	C₄×D₈	C₄×C₇	1	4	yes	—
100	C₄×D₈	C₂²×C₇	1	4	yes	—
101	#436	—	2	4	no	—
102	#436	C₂²×C₇	1	8	yes	complemented
103	C₂×#107	C₄×C₇	1	1	yes	characteristic
104	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
105	C₂×#102	C₄×C₇	1	2	yes	—
106	C₂×C₂²⋊C₈	—	2	2	no	—
107	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
110	#335	C₂²×C₇	1	4	yes	complemented
111	D₄⋊C₈	C₂²×C₇	1	8	yes	complemented
112	C₈²	C₂²×C₇	1	4	yes	complemented
113	C₂³×C₇	—	2	4	no	—
114	C₂³×C₇	—	4	4	no	—
115	C₂³×C₇	—	4	4	no	—
116	C₂³×C₇	—	4	4	no	—
117	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
118	C₇×D₄	—	2	4	no	—
119	C₇×D₄	—	2	4	no	—
120	C₇×D₄	—	4	4	no	—
121	C₂×C₄×C₇	—	2	2	no	—
122	C₂×C₄×C₇	—	2	4	no	—
123	C₂×C₄×C₇	—	4	4	no	—
124	C₂×C₄×C₇	—	4	4	no	—
125	C₂×C₄×C₇	—	4	4	no	—
126	C₂×C₄×C₇	—	4	8	no	—
127	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
131	C₂×C₄×C₇	#131	1	2	yes	—
132	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₇	#135	1	2	yes	—
134	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
137	C₇×C₈	—	2	4	no	—
138	C₇×C₈	—	2	2	no	—
139	C₇×C₈	—	2	2	no	—
140	C₇×C₈	—	4	4	no	—
141	C₇×C₈	C₂×M₁₆	1	4	yes	complemented
142	C₇×C₈	C₂×D₈	1	4	yes	complemented
143	C₂²×D₄	C₂×C₄×C₇	1	2	yes	—
144	C₂³×C₄	—	2	2	no	—
145	C₂×D₈	—	2	4	no	—
146	C₂×D₈	C₂×C₄×C₇	1	4	yes	—
147	C₂×M₁₆	—	2	8	no	—
148	C₂×M₁₆	—	4	8	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
152	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
153	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
154	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
155	C₄×D₄	—	2	2	no	—
156	C₄×D₄	—	2	2	no	—
157	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
158	C₄×D₄	C₂³×C₇	1	4	yes	—
159	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
160	C₂×C₂²⋊C₄	—	2	2	no	—
161	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
162	#107	C₂×C₄×C₇	1	4	yes	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	C₂³×C₇	1	4	yes	—
165	#102	C₂×C₄×C₇	1	8	yes	—
166	C₂²⋊C₈	—	2	8	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
169	C₄×C₈	C₇×D₄	1	2	yes	—
170	C₄×C₈	C₂³×C₇	1	2	yes	—
171	C₄×C₈	C₂³×C₇	1	2	yes	—
172	C₄×C₈	C₂³×C₇	1	2	yes	—
173	C₂²×C₇	—	4	4	no	—
174	C₂²×C₇	—	4	4	no	—
175	C₂²×C₇	—	4	4	no	—
176	C₂²×C₇	—	8	8	no	—
177	C₂²×C₇	#436	1	2	yes	—
178	C₂²×C₇	C₄×D₈	1	2	yes	—
179	C₂²×C₇	#443	1	2	yes	—
180	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	1	no	—
183	C₄×C₇	—	4	4	no	—
184	C₄×C₇	#436	1	2	yes	—
185	C₄×C₇	C₄×D₈	1	2	yes	—
186	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
187	C₄×C₇	C₂×#131	1	1	yes	characteristic
188	C₄×C₇	C₂²×D₈	1	1	yes	characteristic
189	C₄×C₇	C₂×#135	1	1	yes	characteristic
190	C₂⁴	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
202	C₂²×C₄	C₇×#41	1	1	yes	characteristic
203	D₈	—	2	8	no	—
204	M₁₆	—	4	16	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	4	4	no	—
211	C₂×C₈	C₇×M₁₆	1	4	yes	complemented
212	C₂×C₈	C₇×D₈	1	4	yes	complemented
213	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
214	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
215	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
216	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
219	C₂²⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	2	4	no	—
221	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
222	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
223	C₄²	C₇×#41	1	1	yes	characteristic
224	C₄²	C₂⁴×C₇	1	1	yes	characteristic
225	C₂×C₇	—	8	4	no	—
226	C₂×C₇	#1589	1	4	yes	direct product factor
227	C₂×C₇	C₂×#436	1	1	yes	characteristic
228	C₂×C₇	C₂×C₄×D₈	1	1	yes	characteristic
229	C₂×C₇	C₂×#443	1	1	yes	characteristic
230	C₂³	—	2	4	no	—
231	C₂³	—	4	4	no	—
232	C₂³	—	4	4	no	—
233	C₂³	—	4	4	no	—
234	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
235	D₄	—	2	4	no	—
236	D₄	—	2	4	no	—
237	D₄	—	4	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	8	no	—
244	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
245	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
246	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
247	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
248	C₂×C₄	C₇×#131	1	2	yes	—
249	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
250	C₂×C₄	C₇×#135	1	2	yes	—
251	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
253	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
254	C₈	—	2	4	no	—
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₈	—	4	4	no	—
258	C₈	C₂×C₇×M₁₆	1	4	yes	complemented
259	C₈	C₂×C₇×D₈	1	4	yes	complemented
260	C₇	C₂×#1589	1	1	yes	characteristic, direct product factor, Sylow
261	C₂²	—	4	4	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	4	4	no	—
264	C₂²	—	8	8	no	—
265	C₂²	C₇×#436	1	2	yes	—
266	C₂²	C₄×C₇×D₈	1	2	yes	—
267	C₂²	C₇×#443	1	2	yes	—
268	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	4	no	—
272	C₄	C₇×#436	1	2	yes	—
273	C₄	C₄×C₇×D₈	1	2	yes	—
274	C₄	C₂²×C₇×M₁₆	1	1	yes	characteristic
275	C₄	C₂×C₇×#131	1	1	yes	characteristic
276	C₄	C₂²×C₇×D₈	1	1	yes	characteristic
277	C₄	C₂×C₇×#135	1	1	yes	characteristic
278	C₂	—	8	4	no	—
279	C₂	C₇×#1589	1	4	yes	direct product factor
280	C₂	C₂×C₇×#436	1	1	yes	characteristic
281	C₂	C₂×C₄×C₇×D₈	1	1	yes	characteristic
282	C₂	C₂×C₇×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂³×C₇	C₄×D₄
prime radical	C₂²×C₇×D₈	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#436	C₂	semidirect
3. C₇×#1589	C₂	direct (N × H)
4. C₂×C₇×#335	C₂	semidirect
5. C₂×C₇×D₄⋊C₈	C₂	semidirect
6. C₂×C₇×C₈²	C₂	semidirect
13. C₇×#436	C₂²	semidirect
21. C₇×#335	C₂²	semidirect
22. C₇×D₄⋊C₈	C₂²	semidirect
23. C₇×C₈²	C₂²	semidirect
24. C₂×#1589	C₇	direct (N × H)
56. C₂×#436	C₂×C₇	semidirect
57. #1589	C₂×C₇	direct (N × H)
58. C₂×#335	C₂×C₇	semidirect
59. C₂×D₄⋊C₈	C₂×C₇	semidirect
60. C₂×C₈²	C₂×C₇	semidirect
82. C₂×C₇×C₈	M₁₆	semidirect
83. C₂×C₇×C₈	D₈	semidirect
102. #436	C₂²×C₇	semidirect
110. #335	C₂²×C₇	semidirect
111. D₄⋊C₈	C₂²×C₇	semidirect
112. C₈²	C₂²×C₇	semidirect
141. C₇×C₈	C₂×M₁₆	semidirect
142. C₇×C₈	C₂×D₈	semidirect
211. C₂×C₈	C₇×M₁₆	semidirect
212. C₂×C₈	C₇×D₈	semidirect
226. C₂×C₇	#1589	direct (N × H)
258. C₈	C₂×C₇×M₁₆	semidirect
259. C₈	C₂×C₇×D₈	semidirect
260. C₇	C₂×#1589	direct (N × H)
279. C₂	C₇×#1589	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×C₄×C₇×D₈	2
C₂×C₇×#436	2
C₇×#1589	2
C₂×C₇×#335	2
C₂×C₇×D₄⋊C₈	2
C₂×C₇×C₈²	2
C₂×#1589	7

Subgroups — C2×C7×#1589 #13195099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1589 #13195099