Subgroups — #26371781

Order 12032.

474 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253921	C₂	1	2	yes	maximal
2	#20245672	C₂	1	2	yes	maximal, complemented
3	#20245671	C₂	1	2	yes	maximal, complemented
4	#20245667	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245665	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245662	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245173	C₂	1	2	yes	maximal
8	#20245169	C₂	1	1	yes	maximal, characteristic
9	#366×D₄₇	C₂	1	1	yes	maximal, characteristic
10	#20244983	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#2252	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689750	—	2	2	no	—
13	C₂×#13093963	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₄₇	C₂²	1	1	yes	characteristic
15	#15689531	—	2	2	no	—
16	#15689529	—	2	2	no	—
17	#15689527	C₂²	1	2	yes	—
18	#15689525	C₂²	1	2	yes	—
19	#15689524	C₂²	1	2	yes	—
20	#15689164	—	2	2	no	—
21	#15689160	C₂²	1	2	yes	—
22	#107×D₄₇	C₂²	1	2	yes	—
23	#15689077	C₂²	1	1	yes	characteristic
24	#15688951	C₂²	1	2	yes	—
25	#15688950	C₂²	1	2	yes	—
26	#15688950	C₂²	1	2	yes	—
27	#15688949	C₂²	1	1	yes	characteristic
28	#15688949	C₂²	1	1	yes	characteristic
29	#15688948	C₂²	1	2	yes	—
30	#15688948	C₂²	1	2	yes	—
31	#15688899	C₂²	1	1	yes	characteristic
32	#15688898	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
34	C₄₇×#500	C₂²	1	2	yes	—
35	C₄₇×#366	C₂²	1	1	yes	characteristic
36	C₄₇×#366	C₂²	1	1	yes	characteristic
37	C₄₇×#366	C₂²	1	2	yes	—
38	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×#10625324	D₄	1	1	yes	characteristic
40	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
41	#13094023	—	2	2	no	—
42	#13094022	—	4	2	no	—
43	#13094016	—	2	2	no	—
44	#13094015	—	2	2	no	—
45	#13094014	—	4	2	no	—
46	#13094002	—	4	2	no	—
47	#13094000	—	4	2	no	—
48	#13093982	—	2	2	no	—
49	#13093981	D₄	1	1	yes	characteristic
50	C₄⋊C₄×D₄₇	—	2	2	no	—
51	#13093963	Q₈	1	4	yes	complemented
52	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
54	C₁₆×D₄₇	Q₈	1	4	yes	complemented
55	#13093958	C₂³	1	2	yes	—
56	#13093958	C₂³	1	2	yes	—
57	#13093957	—	2	2	no	—
58	#13093957	—	2	2	no	—
59	#13093956	—	2	2	no	—
60	#13093955	—	2	2	no	—
61	#13093955	—	2	2	no	—
62	#13093935	D₄	1	1	yes	characteristic
63	#13093934	D₄	1	1	yes	characteristic
64	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₄₇	—	2	2	no	—
66	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
68	#107×C₄₇	—	2	2	no	—
69	#107×C₄₇	C₂³	1	2	yes	—
70	#107×C₄₇	C₂³	1	2	yes	—
71	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×Q₈×C₄₇	—	4	2	no	—
73	C₂×C₄×D₄₇	—	4	2	no	—
74	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂×Dic₉₄	—	4	2	no	—
76	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₄₇	—	2	2	no	—
79	C₄⋊C₄×C₄₇	—	2	2	no	—
80	C₄⋊C₄×C₄₇	—	4	2	no	—
81	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
82	(C₂×D₄₇)⋊C₄	—	4	2	no	—
83	(C₂×D₄₇)⋊C₄	D₈	1	1	yes	characteristic
84	(C₂×D₄₇)⋊C₄	D₈	1	1	yes	characteristic
85	C₄⋊Dic₄₇	—	2	2	no	—
86	C₄⋊Dic₄₇	—	2	2	no	—
87	C₄⋊Dic₄₇	—	4	2	no	—
88	Dic₄₇⋊C₄	—	4	2	no	—
89	Dic₄₇⋊C₄	—	4	2	no	—
90	Dic₄₇⋊C₄	—	4	2	no	—
91	Dic₄₇⋊C₄	—	4	2	no	—
92	Dic₄₇⋊C₄	—	4	2	no	—
93	Dic₄₇⋊C₄	D₈	1	1	yes	characteristic
94	Dic₄₇⋊C₄	D₈	1	1	yes	characteristic
95	C₄×Dic₄₇	—	4	2	no	—
96	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
99	#10625324	Q₁₆	1	4	yes	—
100	C₈×D₄₇	C₂×Q₈	1	2	yes	—
101	C₈×D₄₇	C₂×Q₈	1	2	yes	—
102	C₁₆×C₄₇	C₂×Q₈	1	2	yes	—
103	C₁₆×C₄₇	C₂×Q₈	1	2	yes	—
104	C₄₇⋊C₁₆	C₂×Q₈	1	2	yes	—
105	C₄₇⋊C₁₆	C₂×Q₈	1	2	yes	—
106	C₂²×D₄₇	C₂×D₈	1	1	yes	characteristic
107	Q₈×C₄₇	—	16	2	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×Dic₄₇	—	4	2	no	—
114	C₂×Dic₄₇	—	4	2	no	—
115	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
116	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
117	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
118	C₄×D₄₇	—	16	2	no	—
119	C₄×D₄₇	C₄⋊Q₈	1	2	yes	—
120	C₄×D₄₇	C₂×Q₁₆	1	2	yes	—
121	Dic₉₄	—	16	2	no	—
122	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
123	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
124	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
125	C₄₇⋊C₈	C₄⋊Q₈	1	2	yes	—
126	C₄₇⋊C₈	C₂×Q₁₆	1	2	yes	—
127	#33729	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
129	C₂×D₄₇	—	2	1	no	—
130	C₂×D₄₇	—	2	1	no	—
131	C₂×D₄₇	#500	1	2	yes	—
132	C₄×C₄₇	—	2	1	no	—
133	C₄×C₄₇	—	2	1	no	—
134	C₄×C₄₇	—	8	2	no	—
135	C₄×C₄₇	—	8	2	no	—
136	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
137	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
138	Dic₄₇	—	2	1	no	—
139	Dic₄₇	—	2	1	no	—
140	Dic₄₇	—	8	2	no	—
141	Dic₄₇	—	8	2	no	—
142	Dic₄₇	#500	1	2	yes	—
143	#3164	—	47	2	no	—
144	#2255	—	47	2	no	—
145	#2255	—	47	2	no	—
146	#2253	—	47	1	no	—
147	#2253	—	47	1	no	—
148	#2252	—	47	1	no	—
149	#2252	D₄₇	1	1	yes	characteristic, complemented
150	#2159	—	47	2	no	—
151	#2158	—	47	1	no	—
152	C₂×#366	—	47	1	no	—
153	#2108	—	47	1	no	—
154	C₂×C₄₇	C₂×#500	1	1	yes	characteristic
155	C₂×C₄₇	#3415	1	1	yes	characteristic
156	C₂×C₄₇	#3417	1	1	yes	characteristic
157	D₄₇	—	2	2	no	—
158	#544	—	94	2	no	—
159	C₂×M₃₂	—	47	1	no	—
160	C₂²×C₁₆	—	47	1	no	—
161	#501	—	94	2	no	—
162	#501	—	94	2	no	—
163	#500	—	47	2	no	—
164	#500	C₂×D₄₇	1	2	yes	—
165	#499	—	47	2	no	—
166	#499	—	47	2	no	—
167	#428	—	94	2	no	—
168	#427	—	47	2	no	—
169	C₂×#107	—	47	2	no	—
170	#405	—	47	1	no	—
171	#367	—	47	2	no	—
172	#367	—	47	2	no	—
173	#367	—	47	2	no	—
174	#366	—	47	1	no	—
175	#366	—	47	1	no	—
176	#366	—	47	2	no	—
177	#366	—	47	2	no	—
178	#366	C₂×D₄₇	1	1	yes	characteristic
179	#366	C₂×D₄₇	1	1	yes	characteristic
180	#366	C₂×D₄₇	1	2	yes	—
181	#346	—	47	1	no	—
182	#346	—	47	1	no	—
183	C₄×C₁₆	—	47	1	no	—
184	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
185	C₄₇	#33729	1	1	yes	characteristic, complemented, Sylow
186	C₂×M₁₆	—	47	1	no	—
187	C₂²×C₈	—	47	1	no	—
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	94	2	no	—
190	C₄⋊Q₈	—	188	2	no	—
191	#125	—	94	2	no	—
192	#125	—	94	2	no	—
193	#125	—	188	2	no	—
194	C₂²⋊Q₈	—	188	2	no	—
195	C₂²⋊Q₈	—	188	2	no	—
196	#117	—	47	1	no	—
197	#117	—	94	2	no	—
198	C₂×C₄⋊C₄	—	94	2	no	—
199	M₃₂	—	47	4	no	—
200	C₂×C₁₆	—	47	1	no	—
201	C₂×C₁₆	—	47	1	no	—
202	C₂×C₁₆	—	47	4	no	—
203	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
204	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
205	#107	—	2	2	no	—
206	#107	—	47	2	no	—
207	#107	—	47	2	no	—
208	#107	—	94	2	no	—
209	#107	—	94	2	no	—
210	#107	C₂²×D₄₇	1	2	yes	—
211	#107	C₂²×D₄₇	1	2	yes	—
212	#106	—	94	2	no	—
213	#106	—	94	2	no	—
214	#106	—	94	2	no	—
215	#97	—	47	1	no	—
216	#97	—	47	1	no	—
217	C₄×C₈	—	47	1	no	—
218	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	188	2	no	—
221	C₂²×C₄	—	47	1	no	—
222	C₂²×C₄	—	188	2	no	—
223	M₁₆	—	47	4	no	—
224	C₂×C₈	—	47	1	no	—
225	C₂×C₈	—	47	1	no	—
226	C₂×C₈	—	47	2	no	—
227	C₂×C₈	—	47	2	no	—
228	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
229	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	47	1	no	—
234	C₄⋊C₄	—	47	1	no	—
235	C₄⋊C₄	—	94	2	no	—
236	C₄⋊C₄	—	94	2	no	—
237	C₄⋊C₄	—	188	2	no	—
238	C₄⋊C₄	—	188	2	no	—
239	C₄⋊C₄	—	188	2	no	—
240	C₄⋊C₄	—	188	2	no	—
241	C₄⋊C₄	—	188	2	no	—
242	C₄⋊C₄	—	188	2	no	—
243	C₂²⋊C₄	—	47	1	no	—
244	C₂²⋊C₄	—	47	1	no	—
245	C₂²⋊C₄	—	188	2	no	—
246	C₄²	—	47	1	no	—
247	C₄²	—	188	2	no	—
248	C₄²	D₄×D₄₇	1	1	yes	characteristic
249	C₁₆	—	47	2	no	—
250	C₁₆	—	47	2	no	—
251	C₁₆	Q₈×D₄₇	1	2	yes	—
252	C₁₆	Q₈×D₄₇	1	2	yes	—
253	C₂³	—	47	1	no	—
254	Q₈	—	16	2	no	—
255	Q₈	—	752	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	47	1	no	—
259	C₂×C₄	—	47	2	no	—
260	C₂×C₄	—	47	2	no	—
261	C₂×C₄	—	47	1	no	—
262	C₂×C₄	—	47	1	no	—
263	C₂×C₄	—	188	2	no	—
264	C₂×C₄	—	188	2	no	—
265	C₂×C₄	—	752	2	no	—
266	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
267	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
269	C₈	—	47	2	no	—
270	C₈	—	47	2	no	—
271	C₈	Q₁₆×D₄₇	1	2	yes	—
272	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
273	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
274	C₂²	—	47	2	no	—
275	C₂²	—	94	1	no	—
276	C₂²	—	94	1	no	—
277	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	47	2	no	—
283	C₄	—	94	1	no	—
284	C₄	—	94	1	no	—
285	C₄	—	376	2	no	—
286	C₄	—	376	2	no	—
287	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
288	C₄	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
289	C₂	—	94	2	no	—
290	C₂	#500×D₄₇	1	1	yes	characteristic
291	C₂	#20256525	1	1	yes	characteristic
292	C₂	#20256537	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2252	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂²×D₄₇	C₂×D₈
hypercenter	#2252	D₄₇
hypocenter	C₄₇	#33729
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
2. #20245672	C₂	semidirect
3. #20245671	C₂	semidirect
4. #20245667	C₂	semidirect
5. #20245665	C₂	semidirect
6. #20245662	C₂	semidirect
11. C₄₇×#2252	C₂	semidirect
51. #13093963	Q₈	semidirect
54. C₁₆×D₄₇	Q₈	semidirect
149. #2252	D₄₇	semidirect
185. C₄₇	#33729	semidirect

Maximal subgroups

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

subgroup	index
#20253921	2
#20245672	2
#20245671	2
#20245667	2
#20245665	2
#20245662	2
#20245173	2
#20245169	2
#366×D₄₇	2
#20244983	2
C₄₇×#2252	2
#33729	47