Subgroups — #26224739

Order 12032.

1062 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689502	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689227	C₂	1	4	yes	maximal
3	C₄⋊Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#20245955	C₂	1	1	yes	maximal, characteristic
5	#20244256	C₂	1	4	yes	maximal
6	#20244190	C₂	1	1	yes	maximal, characteristic
7	#20243544	C₂	1	1	yes	maximal, characteristic
8	C₄×#13093934	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1956	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10625337	C₂²	1	4	yes	—
11	C₂×#13094024	C₂²	1	1	yes	characteristic
12	C₂×Q₈×Dic₄₇	C₂²	1	2	yes	—
13	C₂×#13093989	C₂²	1	2	yes	—
14	#15689502	C₄	1	8	yes	complemented
15	#15689227	—	2	16	no	—
16	C₂×#13093955	C₂²	1	4	yes	—
17	C₂×#13093949	C₂²	1	4	yes	—
18	C₂×#13093934	C₂²	1	2	yes	—
19	C₂×#13093934	C₂²	1	2	yes	—
20	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
21	#15689030	C₂²	1	2	yes	—
22	#15689028	C₂²	1	2	yes	—
23	#15689022	—	2	2	no	—
24	#15689019	—	2	2	no	—
25	#15688993	C₂²	1	1	yes	characteristic
26	C₄⋊C₄×Dic₄₇	—	2	2	no	—
27	C₄⋊C₄×Dic₄₇	—	2	2	no	—
28	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×#103×C₄₇	C₂²	1	4	yes	—
31	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#384	C₂²	1	1	yes	characteristic
33	C₂²×Dic₉₄	C₂³	1	2	yes	—
34	C₂×#10625337	—	2	8	no	—
35	C₂×#10625337	—	2	8	no	—
36	C₂×#10625337	C₂×C₄	1	16	yes	—
37	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
38	#13094024	—	2	2	no	—
39	#13094024	C₂×C₄	1	4	yes	—
40	Q₈×Dic₄₇	—	2	8	no	—
41	#13093989	—	2	8	no	—
42	C₂×C₄⋊Dic₄₇	—	2	2	no	—
43	C₂×C₄⋊Dic₄₇	—	2	2	no	—
44	C₂×C₄⋊Dic₄₇	—	2	2	no	—
45	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
46	C₂×Dic₄₇⋊C₄	—	2	2	no	—
47	C₂×C₄×Dic₄₇	—	2	2	no	—
48	C₂×C₄×Dic₄₇	—	2	2	no	—
49	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
50	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
51	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
53	#13093955	—	2	8	no	—
54	#13093949	—	2	8	no	—
55	#13093934	—	2	4	no	—
56	#13093934	D₄	1	8	yes	—
57	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
58	C₄₇⋊#95	—	2	4	no	—
59	C₄₇⋊#95	—	2	4	no	—
60	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₄₇	—	2	2	no	—
63	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
64	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
65	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
66	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
67	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
68	#103×C₄₇	—	2	8	no	—
69	C₄×C₈×C₄₇	—	2	2	no	—
70	C₂×Q₈×C₄₇	—	2	2	no	—
71	C₂×Q₈×C₄₇	—	2	4	no	—
72	C₂×Q₈×C₄₇	—	2	4	no	—
73	C₂×Q₈×C₄₇	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₇	—	2	2	no	—
75	C₂²×C₄×C₄₇	—	2	2	no	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₄₇	—	2	2	no	—
79	C₂²×Dic₄₇	—	2	2	no	—
80	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
81	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₉₄	—	2	4	no	—
84	C₂×Dic₉₄	—	2	4	no	—
85	C₂×Dic₉₄	—	2	2	no	—
86	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
90	C₄⋊C₄×C₄₇	—	2	8	no	—
91	C₄⋊C₄×C₄₇	—	2	4	no	—
92	C₄⋊C₄×C₄₇	—	4	4	no	—
93	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₄₇	#41	1	2	yes	—
95	#10625337	—	2	32	no	—
96	C₄⋊Dic₄₇	—	2	8	no	—
97	C₄⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	—	4	4	no	—
99	C₄⋊Dic₄₇	—	4	4	no	—
100	Dic₄₇⋊C₄	—	4	4	no	—
101	C₄×Dic₄₇	—	2	4	no	—
102	C₄×Dic₄₇	—	2	4	no	—
103	C₄×Dic₄₇	—	2	2	no	—
104	C₄×Dic₄₇	—	4	4	no	—
105	C₄×Dic₄₇	—	4	4	no	—
106	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
107	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
108	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
109	C₂×C₄₇⋊C₈	#41	1	4	yes	—
110	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₄₇	—	2	8	no	—
112	Q₈×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
119	C₂×C₄×C₄₇	#137	1	4	yes	—
120	C₂×C₄×C₄₇	#137	1	4	yes	—
121	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₄₇	—	2	2	no	—
126	C₂×Dic₄₇	—	2	2	no	—
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	2	2	no	—
129	C₂×Dic₄₇	—	2	2	no	—
130	C₂×Dic₄₇	—	2	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	—	4	2	no	—
134	C₂×Dic₄₇	—	4	4	no	—
135	C₂×Dic₄₇	#127	1	2	yes	—
136	C₂×Dic₄₇	#127	1	2	yes	—
137	Dic₉₄	—	2	8	no	—
138	Dic₉₄	—	4	4	no	—
139	C₈×C₄₇	—	2	4	no	—
140	C₄₇⋊C₈	C₄×D₄	1	8	yes	—
141	#21691	—	47	1	no	maximal, Sylow
142	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
144	C₂²×C₄₇	#532	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×#137	1	2	yes	—
146	C₂²×C₄₇	C₂×#137	1	2	yes	—
147	C₄×C₄₇	—	4	4	no	—
148	C₄×C₄₇	—	8	2	no	—
149	C₄×C₄₇	#441	1	8	yes	complemented
150	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
151	C₄×C₄₇	#532	1	2	yes	—
152	Dic₄₇	—	2	2	no	—
153	Dic₄₇	—	2	2	no	—
154	Dic₄₇	—	4	4	no	—
155	Dic₄₇	—	4	2	no	—
156	Dic₄₇	—	8	2	no	—
157	C₂×#494	—	47	1	no	—
158	C₂×#441	—	47	4	no	—
159	C₄×C₄⋊Q₈	—	47	1	no	—
160	C₄×C₄⋊Q₈	—	47	1	no	—
161	#1971	—	47	4	no	—
162	#1956	—	47	1	no	—
163	#1956	D₄₇	1	1	yes	characteristic
164	#1844	—	47	1	no	—
165	C₄×#97	—	47	1	no	—
166	C₂×C₄₇	C₄×#127	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#441	1	2	yes	—
168	C₂×C₄₇	#3152	1	1	yes	characteristic
169	C₂×C₄₇	#3152	1	1	yes	characteristic
170	C₂×C₄₇	#3185	1	2	yes	—
171	C₂²×Q₁₆	—	47	4	no	—
172	C₂×C₄⋊Q₈	—	47	1	no	—
173	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
174	C₂×C₄×Q₈	—	47	2	no	—
175	C₂×C₄×Q₈	—	47	2	no	—
176	#494	—	47	8	no	—
177	#441	—	94	16	no	—
178	C₂×#106	—	47	4	no	—
179	C₂×#103	—	47	4	no	—
180	C₂×#103	C₂×D₄₇	1	4	yes	—
181	C₂×#97	—	47	2	no	—
182	C₂×#97	—	47	2	no	—
183	C₂×C₄×C₈	—	47	1	no	—
184	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
185	#391	—	47	2	no	—
186	#391	—	47	2	no	—
187	#389	—	94	2	no	—
188	#389	—	94	2	no	—
189	#384	—	47	1	no	—
190	#384	C₂×D₄₇	1	1	yes	characteristic
191	C₄×C₄⋊C₄	—	94	2	no	—
192	C₄×C₄⋊C₄	—	94	2	no	—
193	C₄³	—	47	1	no	—
194	C₄₇	#21691	1	1	yes	characteristic, complemented, Sylow
195	C₂²×Q₈	—	47	2	no	—
196	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
197	C₂×Q₁₆	—	47	16	no	—
198	C₂×Q₁₆	—	94	8	no	—
199	C₂×Q₁₆	—	94	8	no	—
200	C₂²×C₈	—	47	2	no	—
201	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
202	C₄⋊Q₈	—	2	2	no	—
203	C₄⋊Q₈	—	47	4	no	—
204	C₄⋊Q₈	—	94	2	no	—
205	C₄⋊Q₈	C₄×D₄₇	1	4	yes	—
206	C₄×Q₈	—	94	8	no	—
207	C₄×Q₈	—	94	8	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	47	2	no	—
211	C₂×C₄⋊C₄	—	94	2	no	—
212	C₂×C₄⋊C₄	—	94	2	no	—
213	C₂×C₄⋊C₄	—	94	2	no	—
214	C₂×C₄⋊C₄	—	94	2	no	—
215	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
216	C₂×C₄²	—	47	1	no	—
217	C₂×C₄²	—	47	2	no	—
218	C₂×C₄²	—	47	2	no	—
219	C₂×C₄²	—	47	1	no	—
220	C₂×C₄²	—	94	2	no	—
221	C₂×C₄²	—	94	2	no	—
222	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
223	#106	—	94	8	no	—
224	#103	—	2	8	no	—
225	#103	—	94	8	no	—
226	#97	—	47	8	no	—
227	#97	—	94	4	no	—
228	C₄×C₈	—	2	2	no	—
229	C₄×C₈	—	47	4	no	—
230	#95	—	94	4	no	—
231	#95	—	94	4	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	47	4	no	—
236	C₂×Q₈	—	94	2	no	—
237	C₂×Q₈	—	94	4	no	—
238	C₂×Q₈	—	94	4	no	—
239	C₂×Q₈	C₂×C₄×D₄₇	1	4	yes	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	47	1	no	—
243	C₂²×C₄	—	47	1	no	—
244	C₂²×C₄	—	47	2	no	—
245	C₂²×C₄	—	94	2	no	—
246	C₂²×C₄	—	94	2	no	—
247	C₂²×C₄	D₄×D₄₇	1	2	yes	—
248	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
249	Q₁₆	—	94	32	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	47	4	no	—
253	C₂×C₈	—	47	4	no	—
254	C₂×C₈	—	47	4	no	—
255	C₂×C₈	D₄×D₄₇	1	4	yes	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	4	4	no	—
259	C₄⋊C₄	—	94	8	no	—
260	C₄⋊C₄	—	94	4	no	—
261	C₄⋊C₄	—	188	4	no	—
262	C₄⋊C₄	—	188	4	no	—
263	C₄⋊C₄	—	188	4	no	—
264	C₄²	—	47	4	no	—
265	C₄²	—	94	4	no	—
266	C₄²	—	94	4	no	—
267	C₄²	—	94	2	no	—
268	C₄²	—	188	4	no	—
269	C₄²	—	188	4	no	—
270	C₄²	C₂×C₄×D₄₇	1	2	yes	—
271	C₄²	#10625352	1	2	yes	—
272	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
273	Q₈	—	2	8	no	—
274	Q₈	—	4	4	no	—
275	Q₈	—	94	8	no	—
276	Q₈	—	188	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	47	2	no	—
283	C₂×C₄	—	47	2	no	—
284	C₂×C₄	—	94	2	no	—
285	C₂×C₄	—	94	2	no	—
286	C₂×C₄	—	94	4	no	—
287	C₂×C₄	—	94	2	no	—
288	C₂×C₄	—	94	2	no	—
289	C₂×C₄	—	94	2	no	—
290	C₂×C₄	—	188	2	no	—
291	C₂×C₄	—	188	4	no	—
292	C₂×C₄	—	188	2	no	—
293	C₂×C₄	—	188	2	no	—
294	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	4	yes	—
295	C₂×C₄	#13094053	1	4	yes	—
296	C₂×C₄	#13094056	1	4	yes	—
297	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
299	C₂×C₄	#41×D₄₇	1	2	yes	—
300	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
301	C₈	—	2	4	no	—
302	C₈	—	47	8	no	—
303	C₂²	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
304	C₂²	#127×D₄₇	1	1	yes	characteristic
305	C₂²	#15689672	1	1	yes	characteristic
306	C₂²	C₂×#13094053	1	2	yes	—
307	C₂²	C₂×#13094056	1	2	yes	—
308	C₄	—	4	4	no	—
309	C₄	—	8	2	no	—
310	C₄	—	94	2	no	—
311	C₄	—	94	2	no	—
312	C₄	—	376	2	no	—
313	C₄	—	188	2	no	—
314	C₄	—	188	4	no	—
315	C₄	#15689227	1	8	yes	complemented
316	C₄	C₄×D₄×D₄₇	1	2	yes	—
317	C₄	#15689668	1	2	yes	—
318	C₂	#20245952	1	1	yes	characteristic
319	C₂	C₂×#15689227	1	2	yes	—
320	C₂	#20253842	1	1	yes	characteristic
321	C₂	#20253846	1	1	yes	characteristic
322	C₂	#20254132	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1956	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1956	D₄₇
hypocenter	C₄₇	#21691
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

Complements

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

N	H ≅ G/N	type
14. #15689502	C₄	semidirect
149. C₄×C₄₇	#441	semidirect
194. C₄₇	#21691	semidirect
315. C₄	#15689227	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689502	2
C₂×#15689227	2
C₄⋊Q₈×Dic₄₇	2
#20245955	2
#20244256	2
#20244190	2
#20243544	2
C₄×#13093934	2
C₄₇×#1956	2
#21691	47