Subgroups — C₂²×#15689356 #27083367

Order 12032.

3694 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13093995	C₂	1	2	yes	maximal, complemented
3	C₂×#15689356	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13093958	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13093945	C₂	1	2	yes	maximal, complemented
6	C₂³×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₂²⋊D₄₇	—	2	2	no	—
8	C₂³×C₄×D₄₇	C₂²	1	1	yes	characteristic
9	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₈×D₄₇	—	2	8	no	—
11	C₂²×C₈×D₄₇	C₂²	1	12	yes	complemented
12	C₂×#13093995	—	2	12	no	—
13	C₂×#13093995	C₂²	1	24	yes	complemented
14	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
15	C₂²×C₂²⋊Dic₄₇	—	2	2	no	—
16	#15689356	C₂²	1	64	yes	direct product factor
17	C₂×#13093958	C₂²	1	12	yes	complemented
18	C₂×#13093945	C₂²	1	24	yes	complemented
19	C₂³×D₄×C₄₇	C₂²	1	2	yes	—
20	C₂²×D₈×C₄₇	—	2	8	no	—
21	C₂²×D₈×C₄₇	C₂²	1	12	yes	complemented
22	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂⁴×D₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₂²⋊D₄₇	—	2	24	no	—
25	C₂²×C₂²⋊D₄₇	—	4	16	no	—
26	C₂³×Dic₄₇	—	2	2	no	—
27	C₂³×Dic₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×D₄₇	—	2	4	no	—
29	C₂²×C₄×D₄₇	—	2	4	no	—
30	C₂²×C₄×D₄₇	D₄	1	6	yes	—
31	C₂²×C₄×D₄₇	C₂³	1	6	yes	—
32	C₂²×C₄₇⋊C₈	—	2	4	no	—
33	C₂²×C₄₇⋊C₈	C₂³	1	6	yes	—
34	C₂×C₈×D₄₇	—	2	48	no	—
35	C₂×C₈×D₄₇	C₂³	1	16	yes	complemented
36	#13093995	—	2	48	no	—
37	#13093995	C₂³	1	32	yes	complemented
38	C₂×C₄⋊Dic₄₇	—	2	6	no	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
40	C₂×C₂²⋊Dic₄₇	—	2	24	no	—
41	#13093958	C₂³	1	16	yes	complemented
42	#13093945	C₂³	1	32	yes	complemented
43	C₂⁵×C₄₇	—	2	2	no	—
44	C₂²×D₄×C₄₇	—	2	8	no	—
45	C₂²×D₄×C₄₇	—	2	6	no	—
46	C₂²×D₄×C₄₇	—	2	8	no	—
47	C₂²×D₄×C₄₇	C₂³	1	12	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×D₈×C₄₇	—	2	48	no	—
50	C₂×D₈×C₄₇	C₂³	1	16	yes	complemented
51	C₂²×C₈×C₄₇	D₄	1	8	yes	complemented
52	C₂²×C₈×C₄₇	C₂³	1	6	yes	—
53	C₂⁴×C₄₇	—	2	12	no	—
54	C₂⁴×C₄₇	—	4	8	no	—
55	C₂⁴×C₄₇	—	4	8	no	—
56	C₂⁴×C₄₇	—	4	8	no	—
57	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂³×D₄₇	—	2	4	no	—
59	C₂³×D₄₇	—	2	4	no	—
60	C₂³×D₄₇	D₈	1	8	yes	complemented
61	C₂³×D₄₇	C₂×D₄	1	6	yes	—
62	C₂×D₄×C₄₇	—	2	24	no	—
63	C₂×D₄×C₄₇	—	2	12	no	—
64	C₂×D₄×C₄₇	—	2	24	no	—
65	C₂×D₄×C₄₇	—	4	24	no	—
66	C₂×D₄×C₄₇	C₂⁴	1	8	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
69	C₂²×C₄×C₄₇	#41	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
71	C₂×C₂²⋊D₄₇	—	2	32	no	—
72	C₂×C₂²⋊D₄₇	—	4	96	no	—
73	C₂²×Dic₄₇	—	2	4	no	—
74	C₂²×Dic₄₇	—	2	12	no	—
75	C₂²×Dic₄₇	—	4	8	no	—
76	C₂²×Dic₄₇	C₂×D₄	1	6	yes	—
77	C₂×C₄×D₄₇	—	2	24	no	—
78	C₂×C₄×D₄₇	—	2	12	no	—
79	C₂×C₄×D₄₇	—	2	12	no	—
80	C₂×C₄×D₄₇	C₂×D₄	1	12	yes	—
81	C₂×C₄×D₄₇	C₂⁴	1	4	yes	—
82	D₈×C₄₇	—	2	32	no	—
83	C₂×C₈×C₄₇	C₂×D₄	1	24	yes	complemented
84	C₂×C₈×C₄₇	C₂⁴	1	4	yes	—
85	C₂²⋊Dic₄₇	—	2	32	no	—
86	C₄⋊Dic₄₇	—	2	12	no	—
87	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
88	C₂×C₄₇⋊C₈	—	2	12	no	—
89	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
90	C₈×D₄₇	—	2	32	no	—
91	C₂³×C₄₇	—	2	8	no	—
92	C₂³×C₄₇	—	4	24	no	—
93	C₂³×C₄₇	—	4	24	no	—
94	C₂³×C₄₇	—	4	24	no	—
95	C₂³×C₄₇	—	8	32	no	—
96	C₂³×C₄₇	#121	1	4	yes	—
97	C₂³×C₄₇	C₂×D₈	1	4	yes	—
98	C₂³×C₄₇	#135	1	4	yes	—
99	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
100	C₂²×D₄₇	—	2	12	no	—
101	C₂²×D₄₇	—	2	12	no	—
102	C₂²×D₄₇	—	2	32	no	—
103	C₂²×D₄₇	C₂×D₈	1	24	yes	complemented
104	C₂²×D₄₇	C₂²×D₄	1	4	yes	—
105	D₄×C₄₇	—	2	8	no	—
106	D₄×C₄₇	—	2	8	no	—
107	D₄×C₄₇	—	4	24	no	—
108	C₂×C₄×C₄₇	#121	1	12	yes	—
109	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
110	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
111	C₂×C₄×C₄₇	C₂×#41	1	6	yes	—
112	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
113	C₂²⋊D₄₇	—	4	64	no	—
114	C₂×Dic₄₇	—	2	12	no	—
115	C₂×Dic₄₇	—	2	8	no	—
116	C₂×Dic₄₇	—	4	24	no	—
117	C₂×Dic₄₇	C₂²×D₄	1	4	yes	—
118	C₄×D₄₇	—	2	24	no	—
119	C₄×D₄₇	—	2	4	no	—
120	C₄×D₄₇	—	2	4	no	—
121	C₈×C₄₇	C₂²×D₄	1	8	yes	complemented
122	C₄₇⋊C₈	—	2	4	no	—
123	C₂²×#466	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	4	8	no	—
125	C₂²×C₄₇	—	4	8	no	—
126	C₂²×C₄₇	—	4	8	no	—
127	C₂²×C₄₇	—	8	48	no	—
128	C₂²×C₄₇	#466	1	16	yes	complemented
129	C₂²×C₄₇	C₂×#121	1	6	yes	—
130	C₂²×C₄₇	C₂²×D₈	1	6	yes	—
131	C₂²×C₄₇	C₂×#135	1	6	yes	—
132	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
133	C₂×D₄₇	—	2	4	no	—
134	C₂×D₄₇	—	2	48	no	—
135	C₂×D₄₇	—	2	4	no	—
136	C₂×D₄₇	C₂²×D₈	1	8	yes	complemented
137	C₄×C₄₇	C₂×#121	1	6	yes	—
138	C₄×C₄₇	C₂³×D₄	1	1	yes	characteristic
139	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
140	Dic₄₇	—	2	4	no	—
141	Dic₄₇	—	4	8	no	—
142	C₂³×D₈	D₄₇	1	1	yes	characteristic, complemented
143	C₂⁴×C₈	—	47	1	no	—
144	C₂²×#121	—	47	2	no	—
145	C₂×#466	—	47	24	no	—
146	C₂²×#107	—	47	1	no	—
147	C₂²×#102	—	47	2	no	—
148	C₂×C₄₇	—	8	8	no	—
149	C₂×C₄₇	C₂×#466	1	12	yes	complemented
150	C₂×C₄₇	C₂²×#121	1	1	yes	characteristic
151	C₂×C₄₇	C₂³×D₈	1	1	yes	characteristic
152	C₂×C₄₇	C₂²×#135	1	1	yes	characteristic
153	D₄₇	—	2	8	no	—
154	C₂³×D₄	—	94	2	no	—
155	C₂³×D₄	C₂×D₄₇	1	2	yes	—
156	C₂⁴×C₄	—	47	1	no	—
157	C₂²×D₈	—	2	8	no	—
158	C₂²×D₈	C₂×D₄₇	1	12	yes	complemented
159	C₂³×C₈	—	47	1	no	—
160	C₂³×C₈	—	47	12	no	—
161	C₂³×C₈	—	94	8	no	—
162	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
163	C₂×#121	—	47	24	no	—
164	C₂×#121	—	94	12	no	—
165	C₂²×C₄⋊C₄	—	47	2	no	—
166	C₂²×C₂²⋊C₄	—	94	2	no	—
167	#466	—	47	64	no	—
168	C₂×#107	—	47	12	no	—
169	C₂×#102	—	47	24	no	—
170	C₄₇	C₂²×#466	1	1	yes	characteristic, complemented, Sylow
171	C₂⁵	—	2	2	no	—
172	C₂⁵	—	47	1	no	—
173	C₂²×D₄	—	2	6	no	—
174	C₂²×D₄	—	2	8	no	—
175	C₂²×D₄	—	2	8	no	—
176	C₂²×D₄	—	94	24	no	—
177	C₂²×D₄	—	188	16	no	—
178	C₂²×D₄	C₂²×D₄₇	1	12	yes	—
179	C₂³×C₄	—	47	1	no	—
180	C₂³×C₄	—	47	6	no	—
181	C₂³×C₄	—	47	6	no	—
182	C₂³×C₄	—	94	4	no	—
183	C₂³×C₄	—	94	4	no	—
184	C₂³×C₄	—	94	2	no	—
185	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
186	C₂×D₈	—	2	48	no	—
187	C₂×D₈	C₂²×D₄₇	1	16	yes	complemented
188	C₂²×C₈	—	47	6	no	—
189	C₂²×C₈	—	47	16	no	—
190	C₂²×C₈	—	94	4	no	—
191	C₂²×C₈	—	94	48	no	—
192	C₂²×C₈	C₂²⋊D₄₇	1	8	yes	complemented
193	C₂²×C₈	C₂²×D₄₇	1	6	yes	—
194	#121	—	47	32	no	—
195	#121	—	94	48	no	—
196	C₂×C₄⋊C₄	—	47	12	no	—
197	C₂×C₄⋊C₄	—	94	6	no	—
198	C₂×C₂²⋊C₄	—	94	24	no	—
199	#107	—	47	16	no	—
200	#102	—	47	32	no	—
201	C₂⁴	—	2	12	no	—
202	C₂⁴	—	4	8	no	—
203	C₂⁴	—	4	8	no	—
204	C₂⁴	—	4	8	no	—
205	C₂⁴	—	47	8	no	—
206	C₂⁴	—	47	6	no	—
207	C₂⁴	—	94	4	no	—
208	C₂⁴	—	94	4	no	—
209	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
210	C₂×D₄	—	2	12	no	—
211	C₂×D₄	—	2	24	no	—
212	C₂×D₄	—	2	24	no	—
213	C₂×D₄	—	4	24	no	—
214	C₂×D₄	—	94	32	no	—
215	C₂×D₄	—	188	96	no	—
216	C₂×D₄	C₂³×D₄₇	1	8	yes	—
217	C₂²×C₄	—	47	6	no	—
218	C₂²×C₄	—	47	12	no	—
219	C₂²×C₄	—	47	4	no	—
220	C₂²×C₄	—	94	4	no	—
221	C₂²×C₄	—	94	24	no	—
222	C₂²×C₄	—	94	12	no	—
223	C₂²×C₄	—	94	12	no	—
224	C₂²×C₄	—	94	12	no	—
225	C₂²×C₄	—	188	8	no	—
226	C₂²×C₄	D₄×D₄₇	1	3	yes	—
227	C₂²×C₄	#10625352	1	4	yes	—
228	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
229	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
230	D₈	—	2	32	no	—
231	C₂×C₈	—	47	4	no	—
232	C₂×C₈	—	94	12	no	—
233	C₂×C₈	—	94	32	no	—
234	C₂×C₈	C₂×C₂²⋊D₄₇	1	24	yes	complemented
235	C₂×C₈	C₂³×D₄₇	1	4	yes	—
236	C₄⋊C₄	—	47	8	no	—
237	C₄⋊C₄	—	94	12	no	—
238	C₂²⋊C₄	—	94	32	no	—
239	C₂³	—	2	8	no	—
240	C₂³	—	4	24	no	—
241	C₂³	—	4	24	no	—
242	C₂³	—	4	24	no	—
243	C₂³	—	8	32	no	—
244	C₂³	—	47	24	no	—
245	C₂³	—	47	4	no	—
246	C₂³	—	94	32	no	—
247	C₂³	—	94	12	no	—
248	C₂³	—	94	12	no	—
249	C₂³	#13093995	1	4	yes	—
250	C₂³	D₈×D₄₇	1	4	yes	—
251	C₂³	#13094043	1	4	yes	—
252	C₂³	C₂×D₄×D₄₇	1	3	yes	—
253	D₄	—	2	8	no	—
254	D₄	—	2	8	no	—
255	D₄	—	4	24	no	—
256	D₄	—	188	64	no	—
257	C₂×C₄	—	47	4	no	—
258	C₂×C₄	—	94	12	no	—
259	C₂×C₄	—	94	24	no	—
260	C₂×C₄	—	94	4	no	—
261	C₂×C₄	—	94	4	no	—
262	C₂×C₄	—	94	8	no	—
263	C₂×C₄	—	188	24	no	—
264	C₂×C₄	#13093995	1	12	yes	—
265	C₂×C₄	C₂×D₄×D₄₇	1	3	yes	—
266	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
267	C₂×C₄	C₂×#10625352	1	6	yes	—
268	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
269	C₈	—	94	4	no	—
270	C₈	C₂²×C₂²⋊D₄₇	1	8	yes	complemented
271	C₂²	—	4	8	no	—
272	C₂²	—	4	8	no	—
273	C₂²	—	4	8	no	—
274	C₂²	—	8	48	no	—
275	C₂²	—	47	8	no	—
276	C₂²	—	94	48	no	—
277	C₂²	—	94	4	no	—
278	C₂²	—	94	4	no	—
279	C₂²	#15689356	1	16	yes	direct product factor
280	C₂²	C₂×#13093995	1	6	yes	—
281	C₂²	C₂×D₈×D₄₇	1	6	yes	—
282	C₂²	C₂×#13094043	1	6	yes	—
283	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
284	C₄	—	94	4	no	—
285	C₄	—	188	8	no	—
286	C₄	C₂×#13093995	1	6	yes	—
287	C₄	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
288	C₄	C₂²×#10625352	1	1	yes	characteristic
289	C₂	—	8	8	no	—
290	C₂	—	94	8	no	—
291	C₂	C₂×#15689356	1	12	yes	direct product factor
292	C₂	C₂²×#13093995	1	1	yes	characteristic
293	C₂	C₂²×D₈×D₄₇	1	1	yes	characteristic
294	C₂	C₂²×#13094043	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂³×D₈×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂³×D₈	D₄₇
hypocenter	C₄₇	C₂²×#466
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×D₈

Complements

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

N	H ≅ G/N	type
1. C₂³×C₈×D₄₇	C₂	semidirect
2. C₂²×#13093995	C₂	semidirect
3. C₂×#15689356	C₂	direct (N × H)
4. C₂²×#13093958	C₂	semidirect
5. C₂²×#13093945	C₂	semidirect
6. C₂³×D₈×C₄₇	C₂	semidirect
11. C₂²×C₈×D₄₇	C₂²	semidirect
13. C₂×#13093995	C₂²	semidirect
16. #15689356	C₂²	direct (N × H)
17. C₂×#13093958	C₂²	semidirect
18. C₂×#13093945	C₂²	semidirect
21. C₂²×D₈×C₄₇	C₂²	semidirect
35. C₂×C₈×D₄₇	C₂³	semidirect
37. #13093995	C₂³	semidirect
41. #13093958	C₂³	semidirect
42. #13093945	C₂³	semidirect
50. C₂×D₈×C₄₇	C₂³	semidirect
51. C₂²×C₈×C₄₇	D₄	semidirect
60. C₂³×D₄₇	D₈	semidirect
83. C₂×C₈×C₄₇	C₂×D₄	semidirect
103. C₂²×D₄₇	C₂×D₈	semidirect
121. C₈×C₄₇	C₂²×D₄	semidirect
128. C₂²×C₄₇	#466	semidirect
136. C₂×D₄₇	C₂²×D₈	semidirect
142. C₂³×D₈	D₄₇	semidirect
149. C₂×C₄₇	C₂×#466	semidirect
158. C₂²×D₈	C₂×D₄₇	semidirect
170. C₄₇	C₂²×#466	semidirect
187. C₂×D₈	C₂²×D₄₇	semidirect
192. C₂²×C₈	C₂²⋊D₄₇	semidirect
234. C₂×C₈	C₂×C₂²⋊D₄₇	semidirect
270. C₈	C₂²×C₂²⋊D₄₇	semidirect
279. C₂²	#15689356	direct (N × H)
291. C₂	C₂×#15689356	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×C₈×D₄₇	2
C₂²×#13093995	2
C₂×#15689356	2
C₂²×#13093958	2
C₂²×#13093945	2
C₂³×D₈×C₄₇	2
C₂²×#466	47

Subgroups — C22×#15689356 #27083367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689356 #27083367