Subgroups — C₂×C₄₇×#2206 #26073271

Order 12032.

870 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₆×C₄₇	C₂	1	2	yes	maximal, complemented
2	C₂×C₄₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#2206	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₇×#357	C₂	1	2	yes	maximal, complemented
7	C₂²×D₈×C₄₇	—	2	1	no	—
8	C₂²×D₈×C₄₇	C₂²	1	2	yes	—
9	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×#127×C₄₇	—	2	1	no	—
11	C₂×C₄×D₄×C₄₇	—	2	1	no	—
12	C₂×D₁₆×C₄₇	—	2	8	no	—
13	C₂×D₁₆×C₄₇	C₂²	1	8	yes	complemented
14	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
15	C₄₇×#493	—	2	2	no	—
16	C₄₇×#493	C₂²	1	4	yes	complemented
17	C₄×D₈×C₄₇	—	2	2	no	—
18	C₄×D₈×C₄₇	C₂²	1	4	yes	complemented
19	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#102×C₄₇	—	2	1	no	—
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×C₄⋊C₁₆	C₂²	1	4	yes	complemented
23	C₄₇×#357	C₂²	1	8	yes	complemented
24	C₂²×D₄×C₄₇	—	2	1	no	—
25	C₂²×D₄×C₄₇	—	2	2	no	—
26	C₂²×D₄×C₄₇	—	4	1	no	—
27	C₂²×D₄×C₄₇	—	4	1	no	—
28	C₂³×C₄×C₄₇	—	4	1	no	—
29	C₂×D₈×C₄₇	—	2	1	no	—
30	C₂×D₈×C₄₇	—	2	4	no	—
31	C₂×D₈×C₄₇	—	2	2	no	—
32	C₂×D₈×C₄₇	—	2	4	no	—
33	C₂×D₈×C₄₇	—	2	2	no	—
34	C₂×D₈×C₄₇	—	2	4	no	—
35	C₂×D₈×C₄₇	—	4	4	no	—
36	C₂×D₈×C₄₇	D₄	1	4	yes	complemented
37	C₂×D₈×C₄₇	C₂³	1	4	yes	—
38	C₂×D₈×C₄₇	C₂³	1	2	yes	—
39	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
41	#127×C₄₇	—	2	4	no	—
42	#127×C₄₇	—	4	2	no	—
43	C₄×D₄×C₄₇	—	2	4	no	—
44	C₄×D₄×C₄₇	—	4	2	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
46	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	D₁₆×C₄₇	—	2	16	no	—
49	C₂×C₁₆×C₄₇	—	2	4	no	—
50	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
51	#107×C₄₇	—	2	1	no	—
52	#107×C₄₇	C₂³	1	2	yes	—
53	#102×C₄₇	—	2	4	no	—
54	C₄×C₈×C₄₇	D₄	1	2	yes	—
55	C₄×C₈×C₄₇	C₂³	1	2	yes	—
56	C₂⁴×C₄₇	—	4	1	no	—
57	C₂⁴×C₄₇	—	4	2	no	—
58	C₂×D₄×C₄₇	—	2	4	no	—
59	C₂×D₄×C₄₇	—	2	2	no	—
60	C₂×D₄×C₄₇	—	2	4	no	—
61	C₂×D₄×C₄₇	—	4	4	no	—
62	C₂×D₄×C₄₇	—	4	1	no	—
63	C₂×D₄×C₄₇	—	4	4	no	—
64	C₂×D₄×C₄₇	—	4	4	no	—
65	C₂×D₄×C₄₇	—	4	2	no	—
66	C₂×D₄×C₄₇	—	4	2	no	—
67	C₂×D₄×C₄₇	—	4	4	no	—
68	C₂×D₄×C₄₇	—	8	4	no	—
69	C₂×D₄×C₄₇	—	8	4	no	—
70	C₂²×C₄×C₄₇	—	4	1	no	—
71	C₂²×C₄×C₄₇	—	4	4	no	—
72	C₂²×C₄×C₄₇	—	8	4	no	—
73	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	D₈×C₄₇	—	2	4	no	—
77	D₈×C₄₇	—	2	4	no	—
78	D₈×C₄₇	—	2	2	no	—
79	D₈×C₄₇	—	2	4	no	—
80	D₈×C₄₇	—	4	8	no	—
81	D₈×C₄₇	—	4	4	no	—
82	D₈×C₄₇	C₂×D₄	1	4	yes	complemented
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₇	#41	1	2	yes	—
88	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₄₇	—	2	2	no	—
90	C₄⋊C₄×C₄₇	—	4	1	no	—
91	C₂²⋊C₄×C₄₇	—	4	4	no	—
92	C₄²×C₄₇	D₈	1	2	yes	—
93	C₄²×C₄₇	C₂×D₄	1	2	yes	—
94	C₁₆×C₄₇	—	2	4	no	—
95	C₂³×C₄₇	—	4	2	no	—
96	C₂³×C₄₇	—	4	4	no	—
97	C₂³×C₄₇	—	4	4	no	—
98	C₂³×C₄₇	—	8	2	no	—
99	C₂³×C₄₇	—	8	2	no	—
100	C₂³×C₄₇	—	8	4	no	—
101	C₂³×C₄₇	—	8	4	no	—
102	C₂³×C₄₇	—	8	4	no	—
103	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
104	D₄×C₄₇	—	2	4	no	—
105	D₄×C₄₇	—	4	4	no	—
106	D₄×C₄₇	—	4	2	no	—
107	D₄×C₄₇	—	4	4	no	—
108	D₄×C₄₇	—	4	4	no	—
109	D₄×C₄₇	—	8	8	no	—
110	D₄×C₄₇	—	8	4	no	—
111	D₄×C₄₇	—	8	8	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	8	2	no	—
115	C₂×C₄×C₄₇	—	8	8	no	—
116	C₂×C₄×C₄₇	D₁₆	1	4	yes	complemented
117	C₂×C₄×C₄₇	#121	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₄₇	—	2	2	no	—
124	C₈×C₄₇	#121	1	2	yes	—
125	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
127	C₂×#2206	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₄₇	—	4	4	no	—
129	C₂²×C₄₇	—	8	2	no	—
130	C₂²×C₄₇	—	8	2	no	—
131	C₂²×C₄₇	—	8	8	no	—
132	C₂²×C₄₇	—	8	4	no	—
133	C₂²×C₄₇	—	8	4	no	—
134	C₂²×C₄₇	—	8	4	no	—
135	C₂²×C₄₇	—	16	8	no	—
136	C₂²×C₄₇	#459	1	2	yes	—
137	C₂²×C₄₇	C₂×D₁₆	1	2	yes	—
138	C₂²×C₄₇	#509	1	2	yes	—
139	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	8	2	no	—
142	C₄×C₄₇	#459	1	2	yes	—
143	C₄×C₄₇	C₂×D₁₆	1	4	yes	complemented
144	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
145	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂²×D₁₆	C₂×C₄₇	1	2	yes	complemented
147	C₂×#493	C₂×C₄₇	1	1	yes	characteristic, complemented
148	C₂×C₄×D₈	C₂×C₄₇	1	1	yes	characteristic, complemented
149	#2206	C₂×C₄₇	1	8	yes	direct product factor
150	C₂×C₄⋊C₁₆	C₂×C₄₇	1	1	yes	characteristic, complemented
151	C₂×#357	C₂×C₄₇	1	2	yes	complemented
152	C₂×C₄₇	—	8	4	no	—
153	C₂×C₄₇	—	16	4	no	—
154	C₂×C₄₇	#2206	1	4	yes	direct product factor
155	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
156	C₂×C₄₇	C₂²×D₁₆	1	1	yes	characteristic
157	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
158	C₂²×D₈	—	2	1	no	—
159	C₂²×D₈	C₂²×C₄₇	1	2	yes	—
160	C₂²×D₈	C₂²×C₄₇	1	1	yes	characteristic
161	C₂×#127	—	2	1	no	—
162	C₂×C₄×D₄	—	2	1	no	—
163	C₂×D₁₆	—	2	8	no	—
164	C₂×D₁₆	C₂²×C₄₇	1	8	yes	complemented
165	C₂²×C₁₆	C₂²×C₄₇	1	2	yes	—
166	#493	—	2	2	no	—
167	#493	C₂²×C₄₇	1	4	yes	complemented
168	C₄×D₈	—	2	2	no	—
169	C₄×D₈	C₂²×C₄₇	1	4	yes	complemented
170	C₂×#107	C₂²×C₄₇	1	1	yes	characteristic
171	C₂×#102	—	2	1	no	—
172	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
173	C₄⋊C₁₆	C₂²×C₄₇	1	4	yes	complemented
174	#357	C₂²×C₄₇	1	8	yes	complemented
175	C₄₇	C₂×#2206	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²×D₄	—	2	1	no	—
177	C₂²×D₄	—	2	2	no	—
178	C₂²×D₄	—	4	1	no	—
179	C₂²×D₄	—	4	1	no	—
180	C₂³×C₄	—	4	1	no	—
181	C₂×D₈	—	2	1	no	—
182	C₂×D₈	—	2	4	no	—
183	C₂×D₈	—	2	4	no	—
184	C₂×D₈	—	2	2	no	—
185	C₂×D₈	—	2	2	no	—
186	C₂×D₈	—	2	4	no	—
187	C₂×D₈	—	4	4	no	—
188	C₂×D₈	D₄×C₄₇	1	4	yes	complemented
189	C₂×D₈	C₂³×C₄₇	1	4	yes	—
190	C₂×D₈	C₂³×C₄₇	1	2	yes	—
191	C₂²×C₈	D₄×C₄₇	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
193	#127	—	2	4	no	—
194	#127	—	4	2	no	—
195	C₄×D₄	—	2	4	no	—
196	C₄×D₄	—	4	2	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₂²⋊C₄	—	4	1	no	—
199	C₂×C₄²	D₄×C₄₇	1	1	yes	characteristic
200	D₁₆	—	2	16	no	—
201	C₂×C₁₆	—	2	4	no	—
202	C₂×C₁₆	C₂³×C₄₇	1	4	yes	—
203	#107	—	2	1	no	—
204	#107	C₂³×C₄₇	1	2	yes	—
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	C₂⁴	—	4	2	no	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	2	4	no	—
213	C₂×D₄	—	4	4	no	—
214	C₂×D₄	—	4	1	no	—
215	C₂×D₄	—	4	4	no	—
216	C₂×D₄	—	4	4	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	4	no	—
220	C₂×D₄	—	8	4	no	—
221	C₂×D₄	—	8	4	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	—	8	4	no	—
225	C₂²×C₄	D₈×C₄₇	1	1	yes	characteristic
226	C₂²×C₄	D₈×C₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
228	D₈	—	2	4	no	—
229	D₈	—	2	4	no	—
230	D₈	—	2	2	no	—
231	D₈	—	2	4	no	—
232	D₈	—	4	8	no	—
233	D₈	—	4	4	no	—
234	D₈	C₂×D₄×C₄₇	1	4	yes	complemented
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
237	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
239	C₂×C₈	#41×C₄₇	1	2	yes	—
240	C₂×C₈	C₂⁴×C₄₇	1	1	yes	characteristic
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₂²⋊C₄	—	4	4	no	—
244	C₄²	D₈×C₄₇	1	2	yes	—
245	C₄²	C₂×D₄×C₄₇	1	2	yes	—
246	C₁₆	—	2	4	no	—
247	C₂³	—	4	4	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	4	no	—
250	C₂³	—	8	2	no	—
251	C₂³	—	8	2	no	—
252	C₂³	—	8	4	no	—
253	C₂³	—	8	4	no	—
254	C₂³	—	8	4	no	—
255	C₂³	C₂×D₈×C₄₇	1	1	yes	characteristic
256	D₄	—	2	4	no	—
257	D₄	—	4	4	no	—
258	D₄	—	4	4	no	—
259	D₄	—	4	2	no	—
260	D₄	—	4	4	no	—
261	D₄	—	8	8	no	—
262	D₄	—	8	4	no	—
263	D₄	—	8	8	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	—	8	8	no	—
268	C₂×C₄	D₁₆×C₄₇	1	4	yes	complemented
269	C₂×C₄	#121×C₄₇	1	2	yes	—
270	C₂×C₄	C₂×D₈×C₄₇	1	2	yes	—
271	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
272	C₂×C₄	C₂×D₈×C₄₇	1	2	yes	—
273	C₂×C₄	AGL(1,ℤ/8)×C₄₇	1	2	yes	—
274	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	#121×C₄₇	1	2	yes	—
277	C₈	C₂²×D₄×C₄₇	1	1	yes	characteristic
278	C₈	C₂×#41×C₄₇	1	1	yes	characteristic
279	C₂²	—	4	4	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	—	8	8	no	—
283	C₂²	—	8	4	no	—
284	C₂²	—	8	4	no	—
285	C₂²	—	8	4	no	—
286	C₂²	—	16	8	no	—
287	C₂²	C₄₇×#459	1	2	yes	—
288	C₂²	C₂×D₁₆×C₄₇	1	2	yes	—
289	C₂²	C₄₇×#509	1	2	yes	—
290	C₂²	C₂²×D₈×C₄₇	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	8	2	no	—
293	C₄	C₄₇×#459	1	2	yes	—
294	C₄	C₂×D₁₆×C₄₇	1	4	yes	complemented
295	C₄	C₂×#121×C₄₇	1	1	yes	characteristic
296	C₄	C₂×AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
297	C₂	—	8	4	no	—
298	C₂	—	16	4	no	—
299	C₂	C₄₇×#2206	1	4	yes	direct product factor
300	C₂	C₂×C₄₇×#459	1	1	yes	characteristic
301	C₂	C₂²×D₁₆×C₄₇	1	1	yes	characteristic
302	C₂	C₂×C₄₇×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₇
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. C₂²×D₁₆×C₄₇	C₂	semidirect
2. C₂×C₄₇×#493	C₂	semidirect
3. C₂×C₄×D₈×C₄₇	C₂	semidirect
4. C₄₇×#2206	C₂	direct (N × H)
5. C₂×C₄₇×C₄⋊C₁₆	C₂	semidirect
6. C₂×C₄₇×#357	C₂	semidirect
13. C₂×D₁₆×C₄₇	C₂²	semidirect
16. C₄₇×#493	C₂²	semidirect
18. C₄×D₈×C₄₇	C₂²	semidirect
22. C₄₇×C₄⋊C₁₆	C₂²	semidirect
23. C₄₇×#357	C₂²	semidirect
36. C₂×D₈×C₄₇	D₄	semidirect
82. D₈×C₄₇	C₂×D₄	semidirect
116. C₂×C₄×C₄₇	D₁₆	semidirect
127. C₂×#2206	C₄₇	direct (N × H)
143. C₄×C₄₇	C₂×D₁₆	semidirect
146. C₂²×D₁₆	C₂×C₄₇	semidirect
147. C₂×#493	C₂×C₄₇	semidirect
148. C₂×C₄×D₈	C₂×C₄₇	semidirect
149. #2206	C₂×C₄₇	direct (N × H)
150. C₂×C₄⋊C₁₆	C₂×C₄₇	semidirect
151. C₂×#357	C₂×C₄₇	semidirect
154. C₂×C₄₇	#2206	direct (N × H)
164. C₂×D₁₆	C₂²×C₄₇	semidirect
167. #493	C₂²×C₄₇	semidirect
169. C₄×D₈	C₂²×C₄₇	semidirect
173. C₄⋊C₁₆	C₂²×C₄₇	semidirect
174. #357	C₂²×C₄₇	semidirect
175. C₄₇	C₂×#2206	direct (N × H)
188. C₂×D₈	D₄×C₄₇	semidirect
234. D₈	C₂×D₄×C₄₇	semidirect
268. C₂×C₄	D₁₆×C₄₇	semidirect
294. C₄	C₂×D₁₆×C₄₇	semidirect
299. C₂	C₄₇×#2206	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₆×C₄₇	2
C₂×C₄₇×#493	2
C₂×C₄×D₈×C₄₇	2
C₄₇×#2206	2
C₂×C₄₇×C₄⋊C₁₆	2
C₂×C₄₇×#357	2
C₂×#2206	47

Subgroups — C2×C47×#2206 #26073271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#2206 #26073271