Subgroups — #26369648

Order 12032.

682 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252256	C₂	1	1	yes	maximal, characteristic, complemented
2	#20252240	C₂	1	1	yes	maximal, characteristic
3	#20245253	C₂	1	4	yes	maximal, complemented
4	SD₃₂×Dic₄₇	C₂	1	4	yes	maximal, complemented
5	#20245172	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245108	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245104	C₂	1	1	yes	maximal, characteristic
8	#20244984	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
10	#15689601	—	2	1	no	—
11	#15689598	—	2	1	no	—
12	C₂×#13093964	C₂²	1	1	yes	characteristic
13	#15689235	—	2	2	no	—
14	#15689230	—	2	2	no	—
15	Q₁₆×Dic₄₇	C₂²	1	4	yes	complemented
16	D₈×Dic₄₇	C₂²	1	4	yes	—
17	#15689161	C₂²	1	1	yes	characteristic
18	#15689161	C₂²	1	1	yes	characteristic
19	#15689132	—	2	1	no	—
20	#15689128	—	2	1	no	—
21	#15689078	C₂²	1	1	yes	characteristic
22	#15688951	C₂²	1	2	yes	complemented
23	#15688949	C₂²	1	2	yes	complemented
24	#15688927	C₂²	1	4	yes	complemented
25	#15688924	C₂²	1	4	yes	—
26	#15688898	C₂²	1	2	yes	complemented
27	C₁₆×Dic₄₇	C₂²	1	2	yes	complemented
28	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×SD₃₂×C₄₇	C₄	1	8	yes	complemented
31	C₂×SD₃₂×C₄₇	C₂²	1	2	yes	—
32	C₂×SD₃₂×C₄₇	C₂²	1	2	yes	—
33	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×#10625329	D₄	1	1	yes	characteristic
35	Q₈×Dic₄₇	—	2	4	no	—
36	Q₈×Dic₄₇	—	4	2	no	—
37	D₄×Dic₄₇	—	2	4	no	—
38	D₄×Dic₄₇	—	4	2	no	—
39	#13093983	—	2	1	no	—
40	#13093983	—	2	1	no	—
41	#13093983	—	4	2	no	—
42	#13093983	D₄	1	1	yes	characteristic
43	C₂×C₂²⋊Dic₄₇	—	4	2	no	—
44	#13093964	—	2	2	no	—
45	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
46	#13093958	C₂³	1	2	yes	—
47	#13093958	C₂³	1	2	yes	—
48	#13093956	—	2	1	no	—
49	#13093956	—	2	1	no	—
50	#13093948	—	2	4	no	—
51	#13093945	—	2	4	no	—
52	#13093934	D₄	1	2	yes	—
53	C₈×Dic₄₇	C₂³	1	2	yes	—
54	C₂²×Q₈×C₄₇	—	2	1	no	—
55	C₂²×D₄×C₄₇	—	2	1	no	—
56	C₂×Q₁₆×C₄₇	—	2	1	no	—
57	C₂×Q₁₆×C₄₇	—	2	2	no	—
58	C₂×Q₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
59	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
60	C₂×D₈×C₄₇	—	2	1	no	—
61	C₂×D₈×C₄₇	—	2	2	no	—
62	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	—
63	C₂×D₈×C₄₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	SD₃₂×C₄₇	C₂×C₄	1	8	yes	complemented
66	SD₃₂×C₄₇	C₂×C₄	1	8	yes	complemented
67	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
68	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
70	C₂⁴×C₄₇	—	4	1	no	—
71	C₂×Q₈×C₄₇	—	2	4	no	—
72	C₂×Q₈×C₄₇	—	2	2	no	—
73	C₂×Q₈×C₄₇	—	4	1	no	—
74	C₂×Q₈×C₄₇	—	4	2	no	—
75	C₂×D₄×C₄₇	—	2	4	no	—
76	C₂×D₄×C₄₇	—	2	2	no	—
77	C₂×D₄×C₄₇	—	4	1	no	—
78	C₂×D₄×C₄₇	—	4	2	no	—
79	C₂²×C₄×C₄₇	—	4	1	no	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×Dic₄₇	—	4	4	no	—
82	Q₁₆×C₄₇	—	2	4	no	—
83	Q₁₆×C₄₇	—	2	2	no	—
84	Q₁₆×C₄₇	C₂²×C₄	1	4	yes	—
85	D₈×C₄₇	—	2	4	no	—
86	D₈×C₄₇	—	2	2	no	—
87	D₈×C₄₇	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₇	#41	1	2	yes	—
91	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
92	C₂²⋊Dic₄₇	—	4	1	no	—
93	C₂²⋊Dic₄₇	—	4	1	no	—
94	C₂²⋊Dic₄₇	—	4	4	no	—
95	C₂²⋊Dic₄₇	D₈	1	2	yes	—
96	C₄⋊Dic₄₇	—	2	2	no	—
97	C₄⋊Dic₄₇	—	2	2	no	—
98	C₄⋊Dic₄₇	—	4	4	no	—
99	C₄⋊Dic₄₇	D₈	1	2	yes	—
100	C₄×Dic₄₇	—	4	1	no	—
101	C₄×Dic₄₇	—	4	1	no	—
102	C₄×Dic₄₇	—	4	4	no	—
103	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
104	#10625329	—	2	2	no	—
105	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
106	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
107	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
108	C₄₇⋊C₁₆	—	2	2	no	—
109	C₂³×C₄₇	—	4	2	no	—
110	C₂³×C₄₇	—	4	4	no	—
111	C₂³×C₄₇	—	16	2	no	—
112	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
113	Q₈×C₄₇	—	2	4	no	—
114	Q₈×C₄₇	—	4	2	no	—
115	Q₈×C₄₇	—	4	4	no	—
116	D₄×C₄₇	—	2	4	no	—
117	D₄×C₄₇	—	4	2	no	—
118	D₄×C₄₇	—	4	4	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	4	no	—
121	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#135	1	2	yes	—
124	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	16	4	no	—
128	C₂×Dic₄₇	C₂×D₈	1	4	yes	—
129	C₈×C₄₇	C₄×D₄	1	2	yes	—
130	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
131	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
132	C₄₇⋊C₈	—	2	2	no	—
133	#33457	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	4	4	no	—
136	C₂²×C₄₇	—	8	4	no	—
137	C₂²×C₄₇	—	16	4	no	—
138	C₂²×C₄₇	C₄×D₈	1	2	yes	—
139	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
140	C₄×C₄₇	—	4	4	no	—
141	C₄×C₄₇	C₄×D₈	1	2	yes	—
142	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
143	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
144	Dic₄₇	—	2	4	no	—
145	Dic₄₇	—	8	2	no	—
146	Dic₄₇	—	8	2	no	—
147	C₂²×SD₃₂	D₄₇	1	1	yes	characteristic
148	#2950	—	47	1	no	—
149	#2948	—	47	1	no	—
150	#2175	—	47	4	no	—
151	C₄×SD₃₂	—	47	4	no	—
152	#2158	—	47	1	no	—
153	#2141	—	47	1	no	—
154	#2140	—	47	1	no	—
155	#2108	—	47	1	no	—
156	C₂×C₄₇	—	2	2	no	—
157	C₂×C₄₇	—	8	4	no	—
158	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
159	C₂×C₄₇	#3416	1	2	yes	—
160	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
161	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
162	#519	—	94	1	no	—
163	#518	—	94	1	no	—
164	C₂×SD₃₂	Dic₄₇	1	8	yes	complemented
165	C₂×SD₃₂	C₂×D₄₇	1	2	yes	—
166	C₂×SD₃₂	C₂×D₄₇	1	2	yes	—
167	C₂×M₃₂	—	47	1	no	—
168	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
169	#442	—	94	2	no	—
170	#441	—	94	2	no	—
171	C₄×Q₁₆	—	47	4	no	—
172	C₄×D₈	—	47	4	no	—
173	#427	—	47	1	no	—
174	#427	—	47	1	no	—
175	#419	—	94	1	no	—
176	#418	—	94	1	no	—
177	#405	—	47	1	no	—
178	#367	—	47	2	no	—
179	#366	—	47	2	no	—
180	#358	—	47	4	no	—
181	#357	—	47	4	no	—
182	#346	—	47	2	no	—
183	C₄×C₁₆	—	47	2	no	—
184	C₄₇	#33457	1	1	yes	characteristic, complemented, Sylow
185	C₂²×Q₈	—	2	1	no	—
186	C₂²×D₄	—	2	1	no	—
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	—	2	2	no	—
189	C₂×Q₁₆	C₂×Dic₄₇	1	4	yes	complemented
190	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	—	2	2	no	—
193	C₂×D₈	C₂×Dic₄₇	1	4	yes	—
194	C₂×D₈	C₂²×D₄₇	1	2	yes	—
195	C₂×M₁₆	—	47	1	no	—
196	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
197	C₄×Q₈	—	94	4	no	—
198	C₄×Q₈	—	188	2	no	—
199	C₄×D₄	—	94	4	no	—
200	C₄×D₄	—	188	2	no	—
201	#117	—	47	1	no	—
202	#117	—	94	1	no	—
203	#117	—	94	1	no	—
204	#117	—	188	2	no	—
205	C₂×C₂²⋊C₄	—	188	2	no	—
206	SD₃₂	C₂×Dic₄₇	1	8	yes	complemented
207	SD₃₂	C₂×Dic₄₇	1	8	yes	complemented
208	M₃₂	—	94	2	no	—
209	C₂×C₁₆	—	47	2	no	—
210	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	complemented
211	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
212	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
213	#107	—	47	2	no	—
214	#107	—	47	2	no	—
215	#106	—	94	1	no	—
216	#106	—	94	1	no	—
217	#103	—	94	4	no	—
218	#102	—	94	4	no	—
219	#97	—	47	2	no	—
220	C₄×C₈	—	47	2	no	—
221	C₂⁴	—	4	1	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	4	1	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	188	4	no	—
232	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
233	Q₁₆	—	2	4	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	C₂²×Dic₄₇	1	4	yes	—
236	D₈	—	2	4	no	—
237	D₈	—	2	2	no	—
238	D₈	C₂²×Dic₄₇	1	4	yes	—
239	M₁₆	—	94	2	no	—
240	C₂×C₈	—	47	2	no	—
241	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
242	C₂×C₈	#10625352	1	2	yes	—
243	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
244	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
245	C₄⋊C₄	—	47	2	no	—
246	C₄⋊C₄	—	94	2	no	—
247	C₄⋊C₄	—	94	2	no	—
248	C₄⋊C₄	—	188	4	no	—
249	C₂²⋊C₄	—	47	2	no	—
250	C₂²⋊C₄	—	188	4	no	—
251	C₂²⋊C₄	—	188	1	no	—
252	C₂²⋊C₄	—	188	1	no	—
253	C₄²	—	47	2	no	—
254	C₄²	—	188	4	no	—
255	C₄²	—	188	1	no	—
256	C₄²	—	188	1	no	—
257	C₁₆	—	94	2	no	—
258	C₁₆	C₂²×Dic₄₇	1	2	yes	—
259	C₁₆	C₂²×Dic₄₇	1	2	yes	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	16	2	no	—
263	C₂³	D₈×D₄₇	1	1	yes	characteristic
264	Q₈	—	2	4	no	—
265	Q₈	—	4	2	no	—
266	Q₈	—	4	4	no	—
267	D₄	—	2	4	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	47	4	no	—
273	C₂×C₄	—	188	2	no	—
274	C₂×C₄	—	188	2	no	—
275	C₂×C₄	—	752	4	no	—
276	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
277	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
278	C₂×C₄	#13094043	1	2	yes	—
279	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
280	C₈	—	94	2	no	—
281	C₈	D₄×Dic₄₇	1	2	yes	—
282	C₈	C₂³×Dic₄₇	1	1	yes	characteristic
283	C₈	C₂×#10625352	1	1	yes	characteristic
284	C₂²	—	2	2	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	—	16	4	no	—
288	C₂²	D₈×Dic₄₇	1	2	yes	—
289	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
290	C₄	—	4	4	no	—
291	C₄	—	94	4	no	—
292	C₄	—	376	2	no	—
293	C₄	—	376	2	no	—
294	C₄	D₈×Dic₄₇	1	2	yes	—
295	C₄	C₂×D₄×Dic₄₇	1	1	yes	characteristic
296	C₄	C₂×#13094043	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	8	4	no	—
299	C₂	C₂×D₈×Dic₄₇	1	1	yes	characteristic
300	C₂	#20256530	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂²×SD₃₂×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂²×D₈×C₄₇	C₂²
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂²×SD₃₂	D₄₇
hypocenter	C₄₇	#33457
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×SD₃₂

Complements

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

N	H ≅ G/N	type
1. #20252256	C₂	semidirect
3. #20245253	C₂	semidirect
4. SD₃₂×Dic₄₇	C₂	semidirect
5. #20245172	C₂	semidirect
6. #20245108	C₂	semidirect
8. #20244984	C₂	semidirect
15. Q₁₆×Dic₄₇	C₂²	semidirect
22. #15688951	C₂²	semidirect
23. #15688949	C₂²	semidirect
24. #15688927	C₂²	semidirect
26. #15688898	C₂²	semidirect
27. C₁₆×Dic₄₇	C₂²	semidirect
30. C₂×SD₃₂×C₄₇	C₄	semidirect
58. C₂×Q₁₆×C₄₇	C₂×C₄	semidirect
65. SD₃₂×C₄₇	C₂×C₄	semidirect
66. SD₃₂×C₄₇	C₂×C₄	semidirect
67. C₂×C₁₆×C₄₇	C₂×C₄	semidirect
164. C₂×SD₃₂	Dic₄₇	semidirect
184. C₄₇	#33457	semidirect
189. C₂×Q₁₆	C₂×Dic₄₇	semidirect
206. SD₃₂	C₂×Dic₄₇	semidirect
207. SD₃₂	C₂×Dic₄₇	semidirect
210. C₂×C₁₆	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20252256	2
#20252240	2
#20245253	2
SD₃₂×Dic₄₇	2
#20245172	2
#20245108	2
#20245104	2
#20244984	2
C₂²×SD₃₂×C₄₇	2
#33457	47