Subgroups — C₂²×C₇×#469 #13237655

Order 1792.

3566 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	C₂³×C₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇×#121	C₂	1	2	yes	maximal, complemented
4	C₂×C₇×#469	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×#102	C₂	1	2	yes	maximal, complemented
7	C₂³×C₇×D₄	—	2	2	no	—
8	C₂³×C₇×D₄	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂²×C₇×D₈	—	2	8	no	—
11	C₂²×C₇×D₈	C₂²	1	12	yes	complemented
12	C₂²×C₇×M₁₆	—	2	8	no	—
13	C₂²×C₇×M₁₆	C₂²	1	12	yes	complemented
14	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₇×#121	—	2	12	no	—
17	C₂×C₇×#121	C₂²	1	24	yes	complemented
18	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂²×C₇×C₂²⋊C₄	—	2	2	no	—
20	C₇×#469	C₂²	1	64	yes	direct product factor
21	C₂×C₇×#106	C₂²	1	12	yes	complemented
22	C₂×C₇×#102	C₂²	1	24	yes	complemented
23	C₂²×#469	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂⁵×C₇	—	2	2	no	—
25	C₂⁵×C₇	D₄	1	1	yes	characteristic
26	C₂²×C₇×D₄	—	2	24	no	—
27	C₂²×C₇×D₄	—	2	8	no	—
28	C₂²×C₇×D₄	—	2	6	no	—
29	C₂²×C₇×D₄	—	2	8	no	—
30	C₂²×C₇×D₄	—	4	16	no	—
31	C₂²×C₇×D₄	C₂³	1	12	yes	—
32	C₂³×C₄×C₇	—	2	4	no	—
33	C₂³×C₄×C₇	—	2	4	no	—
34	C₂³×C₄×C₇	—	2	2	no	—
35	C₂³×C₄×C₇	D₄	1	6	yes	—
36	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₇	C₂³	1	6	yes	—
38	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
39	C₂×C₇×D₈	—	2	48	no	—
40	C₂×C₇×D₈	C₂³	1	16	yes	complemented
41	C₂×C₇×M₁₆	—	2	48	no	—
42	C₂×C₇×M₁₆	C₂³	1	16	yes	complemented
43	C₂²×C₇×C₈	—	2	4	no	—
44	C₂²×C₇×C₈	D₄	1	8	yes	complemented
45	C₂²×C₇×C₈	C₂³	1	6	yes	—
46	C₂²×C₇×C₈	C₂³	1	6	yes	—
47	C₇×#121	—	2	48	no	—
48	C₇×#121	C₂³	1	32	yes	complemented
49	C₂×C₇×C₄⋊C₄	—	2	6	no	—
50	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
51	C₂×C₇×C₂²⋊C₄	—	2	24	no	—
52	C₇×#106	C₂³	1	16	yes	complemented
53	C₇×#102	C₂³	1	32	yes	complemented
54	C₂³×D₈	C₂×C₇	1	1	yes	characteristic, complemented
55	C₂³×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
56	C₂²×#121	C₂×C₇	1	2	yes	complemented
57	C₂×#469	C₂×C₇	1	24	yes	direct product factor
58	C₂²×#106	C₂×C₇	1	1	yes	characteristic, complemented
59	C₂²×#102	C₂×C₇	1	2	yes	complemented
60	C₂⁴×C₇	—	2	4	no	—
61	C₂⁴×C₇	—	2	8	no	—
62	C₂⁴×C₇	—	2	12	no	—
63	C₂⁴×C₇	—	4	8	no	—
64	C₂⁴×C₇	—	4	8	no	—
65	C₂⁴×C₇	—	4	8	no	—
66	C₂⁴×C₇	C₂×D₄	1	6	yes	—
67	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₇×D₄	—	2	32	no	—
69	C₂×C₇×D₄	—	2	24	no	—
70	C₂×C₇×D₄	—	2	12	no	—
71	C₂×C₇×D₄	—	2	24	no	—
72	C₂×C₇×D₄	—	4	96	no	—
73	C₂×C₇×D₄	—	4	24	no	—
74	C₂×C₇×D₄	C₂⁴	1	8	yes	—
75	C₂²×C₄×C₇	—	2	4	no	—
76	C₂²×C₄×C₇	—	2	24	no	—
77	C₂²×C₄×C₇	—	2	12	no	—
78	C₂²×C₄×C₇	—	2	12	no	—
79	C₂²×C₄×C₇	—	2	12	no	—
80	C₂²×C₄×C₇	—	4	8	no	—
81	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
82	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
83	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
84	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₇	#41	1	4	yes	—
86	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
87	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
88	C₇×D₈	—	2	32	no	—
89	C₇×M₁₆	—	2	32	no	—
90	C₂×C₇×C₈	—	2	12	no	—
91	C₂×C₇×C₈	C₂×D₄	1	24	yes	complemented
92	C₂×C₇×C₈	C₂⁴	1	4	yes	—
93	C₂×C₇×C₈	C₂⁴	1	4	yes	—
94	C₇×C₄⋊C₄	—	2	12	no	—
95	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
96	C₇×C₂²⋊C₄	—	2	32	no	—
97	C₂³×D₄	—	2	2	no	—
98	C₂³×D₄	C₂²×C₇	1	2	yes	—
99	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
100	C₂²×D₈	—	2	8	no	—
101	C₂²×D₈	C₂²×C₇	1	12	yes	complemented
102	C₂²×M₁₆	—	2	8	no	—
103	C₂²×M₁₆	C₂²×C₇	1	12	yes	complemented
104	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
105	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
106	C₂×#121	—	2	12	no	—
107	C₂×#121	C₂²×C₇	1	24	yes	complemented
108	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
109	C₂²×C₂²⋊C₄	—	2	2	no	—
110	#469	C₂²×C₇	1	64	yes	direct product factor
111	C₂×#106	C₂²×C₇	1	12	yes	complemented
112	C₂×#102	C₂²×C₇	1	24	yes	complemented
113	C₂³×C₇	—	2	12	no	—
114	C₂³×C₇	—	2	24	no	—
115	C₂³×C₇	—	2	8	no	—
116	C₂³×C₇	—	4	16	no	—
117	C₂³×C₇	—	4	24	no	—
118	C₂³×C₇	—	4	24	no	—
119	C₂³×C₇	—	4	24	no	—
120	C₂³×C₇	—	8	32	no	—
121	C₂³×C₇	#121	1	4	yes	—
122	C₂³×C₇	AGL(1,ℤ/8)	1	8	yes	—
123	C₂³×C₇	C₂²×D₄	1	4	yes	—
124	C₂³×C₇	C₂²×D₄	1	3	yes	—
125	C₇×D₄	—	2	8	no	—
126	C₇×D₄	—	2	8	no	—
127	C₇×D₄	—	4	64	no	—
128	C₇×D₄	—	4	24	no	—
129	C₂×C₄×C₇	—	2	12	no	—
130	C₂×C₄×C₇	—	2	24	no	—
131	C₂×C₄×C₇	—	2	4	no	—
132	C₂×C₄×C₇	—	2	4	no	—
133	C₂×C₄×C₇	—	2	8	no	—
134	C₂×C₄×C₇	—	4	24	no	—
135	C₂×C₄×C₇	#121	1	12	yes	—
136	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
137	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
138	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
139	C₂×C₄×C₇	C₂×#41	1	6	yes	—
140	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
141	C₇×C₈	—	2	4	no	—
142	C₇×C₈	C₂²×D₄	1	8	yes	complemented
143	C₂⁵	—	2	2	no	—
144	C₂⁵	C₇×D₄	1	1	yes	characteristic
145	C₂²×D₄	—	2	24	no	—
146	C₂²×D₄	—	2	8	no	—
147	C₂²×D₄	—	2	6	no	—
148	C₂²×D₄	—	2	8	no	—
149	C₂²×D₄	—	4	16	no	—
150	C₂²×D₄	C₂³×C₇	1	12	yes	—
151	C₂³×C₄	—	2	4	no	—
152	C₂³×C₄	—	2	4	no	—
153	C₂³×C₄	—	2	2	no	—
154	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
155	C₂³×C₄	C₇×D₄	1	6	yes	—
156	C₂³×C₄	C₂³×C₇	1	6	yes	—
157	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
158	C₂×D₈	—	2	48	no	—
159	C₂×D₈	C₂³×C₇	1	16	yes	complemented
160	C₂×M₁₆	—	2	48	no	—
161	C₂×M₁₆	C₂³×C₇	1	16	yes	complemented
162	C₂²×C₈	—	2	4	no	—
163	C₂²×C₈	C₇×D₄	1	8	yes	complemented
164	C₂²×C₈	C₂³×C₇	1	6	yes	—
165	C₂²×C₈	C₂³×C₇	1	6	yes	—
166	#121	—	2	48	no	—
167	#121	C₂³×C₇	1	32	yes	complemented
168	C₂×C₄⋊C₄	—	2	6	no	—
169	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
170	C₂×C₂²⋊C₄	—	2	24	no	—
171	#106	C₂³×C₇	1	16	yes	complemented
172	#102	C₂³×C₇	1	32	yes	complemented
173	C₂²×C₇	—	2	4	no	—
174	C₂²×C₇	—	2	8	no	—
175	C₂²×C₇	—	4	24	no	—
176	C₂²×C₇	—	4	8	no	—
177	C₂²×C₇	—	4	8	no	—
178	C₂²×C₇	—	4	8	no	—
179	C₂²×C₇	—	8	48	no	—
180	C₂²×C₇	#469	1	16	yes	direct product factor
181	C₂²×C₇	C₂×#121	1	6	yes	—
182	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	12	yes	—
183	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
184	C₄×C₇	—	2	4	no	—
185	C₄×C₇	—	4	8	no	—
186	C₄×C₇	C₂×#121	1	6	yes	—
187	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
188	C₄×C₇	C₂²×#41	1	1	yes	characteristic
189	C₂⁴	—	2	4	no	—
190	C₂⁴	—	2	8	no	—
191	C₂⁴	—	2	12	no	—
192	C₂⁴	—	4	8	no	—
193	C₂⁴	—	4	8	no	—
194	C₂⁴	—	4	8	no	—
195	C₂⁴	C₂×C₇×D₄	1	6	yes	—
196	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
197	C₂×D₄	—	2	32	no	—
198	C₂×D₄	—	2	24	no	—
199	C₂×D₄	—	2	12	no	—
200	C₂×D₄	—	2	24	no	—
201	C₂×D₄	—	4	96	no	—
202	C₂×D₄	—	4	24	no	—
203	C₂×D₄	C₂⁴×C₇	1	8	yes	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	24	no	—
206	C₂²×C₄	—	2	12	no	—
207	C₂²×C₄	—	2	12	no	—
208	C₂²×C₄	—	2	12	no	—
209	C₂²×C₄	—	4	8	no	—
210	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
211	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
212	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
213	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
214	C₂²×C₄	C₇×#41	1	4	yes	—
215	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
216	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
217	D₈	—	2	32	no	—
218	M₁₆	—	2	32	no	—
219	C₂×C₈	—	2	12	no	—
220	C₂×C₈	C₂×C₇×D₄	1	24	yes	complemented
221	C₂×C₈	C₂⁴×C₇	1	4	yes	—
222	C₂×C₈	C₂⁴×C₇	1	4	yes	—
223	C₄⋊C₄	—	2	12	no	—
224	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
225	C₂²⋊C₄	—	2	32	no	—
226	C₂×C₇	—	4	4	no	—
227	C₂×C₇	—	8	8	no	—
228	C₂×C₇	C₂×#469	1	12	yes	direct product factor
229	C₂×C₇	C₂²×#121	1	1	yes	characteristic
230	C₂×C₇	C₂²×AGL(1,ℤ/8)	1	2	yes	—
231	C₂³	—	2	12	no	—
232	C₂³	—	2	24	no	—
233	C₂³	—	2	8	no	—
234	C₂³	—	4	16	no	—
235	C₂³	—	4	24	no	—
236	C₂³	—	4	24	no	—
237	C₂³	—	4	24	no	—
238	C₂³	—	8	32	no	—
239	C₂³	C₇×#121	1	4	yes	—
240	C₂³	C₇×AGL(1,ℤ/8)	1	8	yes	—
241	C₂³	C₂²×C₇×D₄	1	4	yes	—
242	C₂³	C₂²×C₇×D₄	1	3	yes	—
243	D₄	—	2	8	no	—
244	D₄	—	2	8	no	—
245	D₄	—	4	64	no	—
246	D₄	—	4	24	no	—
247	C₂×C₄	—	2	12	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	24	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	4	24	no	—
253	C₂×C₄	C₇×#121	1	12	yes	—
254	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
255	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
256	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
257	C₂×C₄	C₂×C₇×#41	1	6	yes	—
258	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
259	C₈	—	2	4	no	—
260	C₈	C₂²×C₇×D₄	1	8	yes	complemented
261	C₇	C₂²×#469	1	1	yes	characteristic, direct product factor, Sylow
262	C₂²	—	2	8	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	24	no	—
265	C₂²	—	4	8	no	—
266	C₂²	—	4	8	no	—
267	C₂²	—	4	8	no	—
268	C₂²	—	8	48	no	—
269	C₂²	C₇×#469	1	16	yes	direct product factor
270	C₂²	C₂×C₇×#121	1	6	yes	—
271	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	12	yes	—
272	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	4	8	no	—
275	C₄	C₂×C₇×#121	1	6	yes	—
276	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
277	C₄	C₂²×C₇×#41	1	1	yes	characteristic
278	C₂	—	4	4	no	—
279	C₂	—	8	8	no	—
280	C₂	C₂×C₇×#469	1	12	yes	direct product factor
281	C₂	C₂²×C₇×#121	1	1	yes	characteristic
282	C₂	C₂²×C₇×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂⁴×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₇×D₈	C₂	semidirect
2. C₂³×C₇×M₁₆	C₂	semidirect
3. C₂²×C₇×#121	C₂	semidirect
4. C₂×C₇×#469	C₂	direct (N × H)
5. C₂²×C₇×#106	C₂	semidirect
6. C₂²×C₇×#102	C₂	semidirect
11. C₂²×C₇×D₈	C₂²	semidirect
13. C₂²×C₇×M₁₆	C₂²	semidirect
17. C₂×C₇×#121	C₂²	semidirect
20. C₇×#469	C₂²	direct (N × H)
21. C₂×C₇×#106	C₂²	semidirect
22. C₂×C₇×#102	C₂²	semidirect
23. C₂²×#469	C₇	direct (N × H)
40. C₂×C₇×D₈	C₂³	semidirect
42. C₂×C₇×M₁₆	C₂³	semidirect
44. C₂²×C₇×C₈	D₄	semidirect
48. C₇×#121	C₂³	semidirect
52. C₇×#106	C₂³	semidirect
53. C₇×#102	C₂³	semidirect
54. C₂³×D₈	C₂×C₇	semidirect
55. C₂³×M₁₆	C₂×C₇	semidirect
56. C₂²×#121	C₂×C₇	semidirect
57. C₂×#469	C₂×C₇	direct (N × H)
58. C₂²×#106	C₂×C₇	semidirect
59. C₂²×#102	C₂×C₇	semidirect
91. C₂×C₇×C₈	C₂×D₄	semidirect
101. C₂²×D₈	C₂²×C₇	semidirect
103. C₂²×M₁₆	C₂²×C₇	semidirect
107. C₂×#121	C₂²×C₇	semidirect
110. #469	C₂²×C₇	direct (N × H)
111. C₂×#106	C₂²×C₇	semidirect
112. C₂×#102	C₂²×C₇	semidirect
142. C₇×C₈	C₂²×D₄	semidirect
159. C₂×D₈	C₂³×C₇	semidirect
161. C₂×M₁₆	C₂³×C₇	semidirect
163. C₂²×C₈	C₇×D₄	semidirect
167. #121	C₂³×C₇	semidirect
171. #106	C₂³×C₇	semidirect
172. #102	C₂³×C₇	semidirect
180. C₂²×C₇	#469	direct (N × H)
220. C₂×C₈	C₂×C₇×D₄	semidirect
228. C₂×C₇	C₂×#469	direct (N × H)
260. C₈	C₂²×C₇×D₄	semidirect
261. C₇	C₂²×#469	direct (N × H)
269. C₂²	C₇×#469	direct (N × H)
280. C₂	C₂×C₇×#469	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
C₂³×C₇×M₁₆	2
C₂²×C₇×#121	2
C₂×C₇×#469	2
C₂²×C₇×#106	2
C₂²×C₇×#102	2
C₂²×#469	7

Subgroups — C22×C7×#469 #13237655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#469 #13237655