Subgroups — C₂×C₄₇×#2228 #26073320

Order 12032.

606 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	C₂×C₄₇×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#2228	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄₇×#367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₇×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₇×#357	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#357	C₂	1	1	yes	maximal, characteristic
9	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×#125×C₄₇	—	2	1	no	—
11	C₂×C₄×D₄×C₄₇	—	2	1	no	—
12	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#499	—	2	2	no	—
15	C₄₇×#499	C₂²	1	4	yes	complemented
16	C₄×D₈×C₄₇	—	2	2	no	—
17	C₄×D₈×C₄₇	C₂²	1	4	yes	—
18	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#106×C₄₇	—	2	1	no	—
22	C₂×#102×C₄₇	—	2	1	no	—
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#367	C₂²	1	4	yes	complemented
25	C₄₇×#366	C₂²	1	4	yes	complemented
26	C₄₇×C₄⋊C₁₆	C₂²	1	4	yes	complemented
27	C₄₇×#357	C₂²	1	4	yes	—
28	C₄₇×#357	C₂²	1	4	yes	—
29	C₂²×D₄×C₄₇	—	2	1	no	—
30	C₂³×C₄×C₄₇	—	4	1	no	—
31	C₂×D₈×C₄₇	—	2	1	no	—
32	C₂×D₈×C₄₇	—	2	2	no	—
33	C₂×D₈×C₄₇	Q₈	1	4	yes	—
34	C₂×D₈×C₄₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
37	#125×C₄₇	—	2	4	no	—
38	#125×C₄₇	—	4	2	no	—
39	C₄×D₄×C₄₇	—	2	4	no	—
40	C₄×D₄×C₄₇	—	4	2	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
45	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
46	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₁₆×C₄₇	—	2	2	no	—
49	C₂×C₁₆×C₄₇	—	2	2	no	—
50	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
51	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
52	#107×C₄₇	—	2	1	no	—
53	#107×C₄₇	—	2	1	no	—
54	#107×C₄₇	—	2	1	no	—
55	#107×C₄₇	C₂³	1	2	yes	—
56	#107×C₄₇	C₂³	1	2	yes	—
57	#107×C₄₇	C₂³	1	2	yes	—
58	#106×C₄₇	—	2	4	no	—
59	#102×C₄₇	—	2	4	no	—
60	C₄×C₈×C₄₇	D₄	1	2	yes	—
61	C₄×C₈×C₄₇	C₂³	1	2	yes	—
62	C₂⁴×C₄₇	—	4	1	no	—
63	C₂×D₄×C₄₇	—	2	4	no	—
64	C₂×D₄×C₄₇	—	2	2	no	—
65	C₂×D₄×C₄₇	—	4	1	no	—
66	C₂×D₄×C₄₇	—	4	2	no	—
67	C₂²×C₄×C₄₇	—	4	1	no	—
68	C₂²×C₄×C₄₇	—	4	1	no	—
69	C₂²×C₄×C₄₇	—	4	1	no	—
70	C₂²×C₄×C₄₇	—	4	4	no	—
71	C₂²×C₄×C₄₇	—	8	4	no	—
72	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	D₈×C₄₇	—	2	4	no	—
76	D₈×C₄₇	—	2	2	no	—
77	D₈×C₄₇	C₂×Q₈	1	4	yes	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₄₇	#41	1	2	yes	—
83	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₄₇	—	2	2	no	—
85	C₄⋊C₄×C₄₇	—	2	2	no	—
86	C₄⋊C₄×C₄₇	—	2	2	no	—
87	C₄⋊C₄×C₄₇	—	4	1	no	—
88	C₄⋊C₄×C₄₇	—	4	1	no	—
89	C₄⋊C₄×C₄₇	—	4	1	no	—
90	C₄⋊C₄×C₄₇	—	4	4	no	—
91	C₄⋊C₄×C₄₇	—	4	4	no	—
92	C₂²⋊C₄×C₄₇	—	4	4	no	—
93	C₄²×C₄₇	D₈	1	2	yes	—
94	C₄²×C₄₇	C₂×D₄	1	2	yes	—
95	C₁₆×C₄₇	—	2	2	no	—
96	C₁₆×C₄₇	—	2	2	no	—
97	C₂³×C₄₇	—	4	2	no	—
98	C₂³×C₄₇	—	4	4	no	—
99	C₂³×C₄₇	—	8	2	no	—
100	C₂³×C₄₇	—	8	2	no	—
101	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
102	D₄×C₄₇	—	2	4	no	—
103	D₄×C₄₇	—	4	2	no	—
104	D₄×C₄₇	—	4	4	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	4	2	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	—	8	8	no	—
111	C₂×C₄×C₄₇	—	8	2	no	—
112	C₂×C₄×C₄₇	—	8	2	no	—
113	C₂×C₄×C₄₇	—	8	2	no	—
114	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#137	1	2	yes	—
119	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₄₇	—	2	2	no	—
121	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
122	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
123	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
124	C₂×#2228	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₄₇	—	4	4	no	—
126	C₂²×C₄₇	—	8	2	no	—
127	C₂²×C₄₇	—	8	2	no	—
128	C₂²×C₄₇	—	8	8	no	—
129	C₂²×C₄₇	#474	1	2	yes	—
130	C₂²×C₄₇	#508	1	2	yes	—
131	C₂²×C₄₇	#509	1	2	yes	—
132	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	2	2	no	—
135	C₄×C₄₇	—	8	2	no	—
136	C₄×C₄₇	—	8	2	no	—
137	C₄×C₄₇	—	8	2	no	—
138	C₄×C₄₇	#474	1	2	yes	—
139	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
140	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
141	C₂×#499	C₂×C₄₇	1	1	yes	characteristic, complemented
142	C₂×C₄×D₈	C₂×C₄₇	1	1	yes	characteristic
143	#2228	C₂×C₄₇	1	8	yes	direct product factor
144	C₂×#367	C₂×C₄₇	1	1	yes	characteristic, complemented
145	C₂×#366	C₂×C₄₇	1	1	yes	characteristic, complemented
146	C₂×C₄⋊C₁₆	C₂×C₄₇	1	1	yes	characteristic, complemented
147	C₂×#357	C₂×C₄₇	1	1	yes	characteristic
148	C₂×#357	C₂×C₄₇	1	1	yes	characteristic
149	C₂×C₄₇	—	8	4	no	—
150	C₂×C₄₇	#2228	1	4	yes	direct product factor
151	C₂×C₄₇	C₂×#474	1	1	yes	characteristic
152	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
153	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
154	C₂²×D₈	C₂²×C₄₇	1	1	yes	characteristic
155	C₂×#125	—	2	1	no	—
156	C₂×C₄×D₄	—	2	1	no	—
157	C₂²×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
158	C₂²×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
159	#499	—	2	2	no	—
160	#499	C₂²×C₄₇	1	4	yes	complemented
161	C₄×D₈	—	2	2	no	—
162	C₄×D₈	C₂²×C₄₇	1	4	yes	—
163	C₂×#107	C₂²×C₄₇	1	1	yes	characteristic
164	C₂×#107	C₂²×C₄₇	1	1	yes	characteristic
165	C₂×#107	C₂²×C₄₇	1	1	yes	characteristic
166	C₂×#106	—	2	1	no	—
167	C₂×#102	—	2	1	no	—
168	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
169	#367	C₂²×C₄₇	1	4	yes	complemented
170	#366	C₂²×C₄₇	1	4	yes	complemented
171	C₄⋊C₁₆	C₂²×C₄₇	1	4	yes	complemented
172	#357	C₂²×C₄₇	1	4	yes	—
173	#357	C₂²×C₄₇	1	4	yes	—
174	C₄₇	C₂×#2228	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²×D₄	—	2	1	no	—
176	C₂³×C₄	—	4	1	no	—
177	C₂×D₈	—	2	1	no	—
178	C₂×D₈	—	2	2	no	—
179	C₂×D₈	Q₈×C₄₇	1	4	yes	—
180	C₂×D₈	C₂³×C₄₇	1	2	yes	—
181	C₂²×C₈	D₄×C₄₇	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
183	#125	—	2	4	no	—
184	#125	—	4	2	no	—
185	C₄×D₄	—	2	4	no	—
186	C₄×D₄	—	4	2	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	4	1	no	—
191	C₂×C₄⋊C₄	—	4	1	no	—
192	C₂×C₂²⋊C₄	—	4	1	no	—
193	C₂×C₄²	D₄×C₄₇	1	1	yes	characteristic
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	—	2	2	no	—
196	C₂×C₁₆	C₂³×C₄₇	1	2	yes	—
197	C₂×C₁₆	C₂³×C₄₇	1	2	yes	—
198	#107	—	2	1	no	—
199	#107	—	2	1	no	—
200	#107	—	2	1	no	—
201	#107	C₂³×C₄₇	1	2	yes	—
202	#107	C₂³×C₄₇	1	2	yes	—
203	#107	C₂³×C₄₇	1	2	yes	—
204	#106	—	2	4	no	—
205	#102	—	2	4	no	—
206	C₄×C₈	D₄×C₄₇	1	2	yes	—
207	C₄×C₈	C₂³×C₄₇	1	2	yes	—
208	C₂⁴	—	4	1	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	4	no	—
217	C₂²×C₄	—	8	4	no	—
218	C₂²×C₄	D₈×C₄₇	1	1	yes	characteristic
219	C₂²×C₄	D₈×C₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
221	D₈	—	2	4	no	—
222	D₈	—	2	2	no	—
223	D₈	C₂×Q₈×C₄₇	1	4	yes	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
226	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
227	C₂×C₈	C₂×Q₈×C₄₇	1	2	yes	—
228	C₂×C₈	#41×C₄₇	1	2	yes	—
229	C₂×C₈	C₂⁴×C₄₇	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₂²⋊C₄	—	4	4	no	—
239	C₄²	D₈×C₄₇	1	2	yes	—
240	C₄²	C₂×D₄×C₄₇	1	2	yes	—
241	C₁₆	—	2	2	no	—
242	C₁₆	—	2	2	no	—
243	C₂³	—	4	4	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	8	2	no	—
246	C₂³	—	8	2	no	—
247	C₂³	C₂×D₈×C₄₇	1	1	yes	characteristic
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	8	8	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	8	2	no	—
260	C₂×C₄	C₂²⋊Q₈×C₄₇	1	2	yes	—
261	C₂×C₄	C₂×D₈×C₄₇	1	2	yes	—
262	C₂×C₄	C₂×D₈×C₄₇	1	2	yes	—
263	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
264	C₂×C₄	#137×C₄₇	1	2	yes	—
265	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	C₂²⋊Q₈×C₄₇	1	2	yes	—
268	C₈	C₂²×Q₈×C₄₇	1	1	yes	characteristic
269	C₈	C₂×#41×C₄₇	1	1	yes	characteristic
270	C₂²	—	4	4	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	8	no	—
274	C₂²	C₄₇×#474	1	2	yes	—
275	C₂²	C₄₇×#508	1	2	yes	—
276	C₂²	C₄₇×#509	1	2	yes	—
277	C₂²	C₂²×D₈×C₄₇	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	2	no	—
283	C₄	C₄₇×#474	1	2	yes	—
284	C₄	C₂×C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
285	C₄	C₂×#137×C₄₇	1	1	yes	characteristic
286	C₂	—	8	4	no	—
287	C₂	C₄₇×#2228	1	4	yes	direct product factor
288	C₂	C₂×C₄₇×#474	1	1	yes	characteristic
289	C₂	C₂×C₄₇×#508	1	1	yes	characteristic
290	C₂	C₂×C₄₇×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₇
prime radical	C₂²×D₈×C₄₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ C₂×C₄×C₈×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₇×#499	C₂	semidirect
3. C₄₇×#2228	C₂	direct (N × H)
4. C₂×C₄₇×#367	C₂	semidirect
5. C₂×C₄₇×#366	C₂	semidirect
6. C₂×C₄₇×C₄⋊C₁₆	C₂	semidirect
15. C₄₇×#499	C₂²	semidirect
24. C₄₇×#367	C₂²	semidirect
25. C₄₇×#366	C₂²	semidirect
26. C₄₇×C₄⋊C₁₆	C₂²	semidirect
124. C₂×#2228	C₄₇	direct (N × H)
141. C₂×#499	C₂×C₄₇	semidirect
143. #2228	C₂×C₄₇	direct (N × H)
144. C₂×#367	C₂×C₄₇	semidirect
145. C₂×#366	C₂×C₄₇	semidirect
146. C₂×C₄⋊C₁₆	C₂×C₄₇	semidirect
150. C₂×C₄₇	#2228	direct (N × H)
160. #499	C₂²×C₄₇	semidirect
169. #367	C₂²×C₄₇	semidirect
170. #366	C₂²×C₄₇	semidirect
171. C₄⋊C₁₆	C₂²×C₄₇	semidirect
174. C₄₇	C₂×#2228	direct (N × H)
287. C₂	C₄₇×#2228	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×C₄₇×#499	2
C₂×C₄×D₈×C₄₇	2
C₄₇×#2228	2
C₂×C₄₇×#367	2
C₂×C₄₇×#366	2
C₂×C₄₇×C₄⋊C₁₆	2
C₂×C₄₇×#357	2
C₂×#2228	47

Subgroups — C2×C47×#2228 #26073320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#2228 #26073320