Subgroups — #26224427

Order 12032.

802 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	C₄⋊Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#125×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20244358	C₂	1	1	yes	maximal, characteristic
4	#20244209	C₂	1	1	yes	maximal, characteristic
5	#20244066	C₂	1	4	yes	maximal
6	#20243522	C₂	1	1	yes	maximal, characteristic
7	#20243220	C₂	1	4	yes	maximal
8	#97×Dic₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₇×#491	C₂	1	1	yes	maximal, characteristic
10	C₂×Q₈×Dic₄₇	C₂²	1	2	yes	—
11	C₂×#13093948	C₂²	1	4	yes	—
12	C₂×#13093935	C₂²	1	1	yes	characteristic
13	C₂×#13093934	C₂²	1	2	yes	—
14	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
15	#15689030	C₂²	1	2	yes	—
16	#15689022	—	2	2	no	—
17	#15689022	C₂²	1	2	yes	—
18	#15689009	—	2	2	no	—
19	#15688993	C₂²	1	1	yes	characteristic
20	#15688989	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×Dic₄₇	—	2	2	no	—
22	C₄⋊C₄×Dic₄₇	—	2	2	no	—
23	C₄⋊C₄×Dic₄₇	C₂²	1	2	yes	—
24	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
25	#15688886	C₂²	1	4	yes	—
26	#15688885	C₂²	1	4	yes	—
27	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#125×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#491	C₄	1	8	yes	complemented
30	C₂×#103×C₄₇	C₂²	1	4	yes	—
31	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
32	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
33	Q₈×Dic₄₇	—	2	8	no	—
34	C₂×C₄⋊Dic₄₇	—	2	2	no	—
35	C₂×C₄⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄⋊Dic₄₇	—	2	2	no	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
39	C₂×C₄×Dic₄₇	—	2	2	no	—
40	C₂×C₄×Dic₄₇	—	2	2	no	—
41	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
42	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
43	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
44	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
45	#13093948	—	2	8	no	—
46	#13093935	—	2	2	no	—
47	#13093934	—	2	4	no	—
48	C₈×Dic₄₇	—	2	4	no	—
49	C₄₇⋊#95	—	2	2	no	—
50	C₄₇⋊#95	—	2	2	no	—
51	C₄₇⋊#95	—	2	4	no	—
52	C₄₇⋊#95	—	2	2	no	—
53	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
54	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
55	C₄⋊Q₈×C₄₇	—	2	2	no	—
56	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
57	#125×C₄₇	—	2	2	no	—
58	#125×C₄₇	C₂×C₄	1	4	yes	—
59	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	#103×C₄₇	C₂×C₄	1	16	yes	—
64	#97×C₄₇	C₂×C₄	1	4	yes	—
65	C₂×Q₈×C₄₇	—	2	2	no	—
66	C₂×Q₈×C₄₇	—	2	4	no	—
67	C₂×Q₈×C₄₇	—	2	4	no	—
68	C₂×Q₈×C₄₇	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	—	2	2	no	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₄₇	—	2	2	no	—
74	C₂²×Dic₄₇	—	2	2	no	—
75	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
81	C₄⋊C₄×C₄₇	—	2	8	no	—
82	C₄⋊C₄×C₄₇	—	2	8	no	—
83	C₄⋊C₄×C₄₇	—	2	2	no	—
84	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
85	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
86	C₄²×C₄₇	#41	1	2	yes	—
87	C₄⋊Dic₄₇	—	2	4	no	—
88	C₄⋊Dic₄₇	—	2	4	no	—
89	C₄⋊Dic₄₇	—	4	4	no	—
90	C₄⋊Dic₄₇	—	4	4	no	—
91	C₄⋊Dic₄₇	—	4	4	no	—
92	C₄×Dic₄₇	—	2	2	no	—
93	C₄×Dic₄₇	—	2	4	no	—
94	C₄×Dic₄₇	—	4	2	no	—
95	C₄×Dic₄₇	—	4	4	no	—
96	C₄×Dic₄₇	—	4	4	no	—
97	C₄×Dic₄₇	#41	1	4	yes	—
98	C₂×C₄₇⋊C₈	—	2	2	no	—
99	C₂×C₄₇⋊C₈	—	2	2	no	—
100	C₂×C₄₇⋊C₈	#41	1	4	yes	—
101	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
102	Q₈×C₄₇	—	2	8	no	—
103	Q₈×C₄₇	—	4	4	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	—	4	2	no	—
111	C₂×C₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
113	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
117	C₂×Dic₄₇	—	2	2	no	—
118	C₂×Dic₄₇	—	2	2	no	—
119	C₂×Dic₄₇	—	2	2	no	—
120	C₂×Dic₄₇	—	2	2	no	—
121	C₂×Dic₄₇	—	4	2	no	—
122	C₂×Dic₄₇	—	4	2	no	—
123	C₂×Dic₄₇	—	4	2	no	—
124	C₂×Dic₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	4	4	no	—
126	C₂×Dic₄₇	#124	1	2	yes	—
127	C₂×Dic₄₇	#124	1	2	yes	—
128	C₂×Dic₄₇	#137	1	4	yes	—
129	C₈×C₄₇	—	2	4	no	—
130	C₄₇⋊C₈	—	2	4	no	—
131	#21657	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
134	C₂²×C₄₇	#532	1	1	yes	characteristic
135	C₂²×C₄₇	C₂×#137	1	2	yes	—
136	C₂²×C₄₇	#575	1	2	yes	—
137	C₄×C₄₇	—	2	4	no	—
138	C₄×C₄₇	—	4	4	no	—
139	C₄×C₄₇	—	4	4	no	—
140	C₄×C₄₇	C₄×#41	1	2	yes	—
141	C₄×C₄₇	#529	1	2	yes	—
142	Dic₄₇	—	2	2	no	—
143	Dic₄₇	—	2	2	no	—
144	Dic₄₇	—	2	4	no	—
145	Dic₄₇	—	8	2	no	—
146	Dic₄₇	—	8	2	no	—
147	C₂×#491	D₄₇	1	1	yes	characteristic
148	C₄×C₄⋊Q₈	—	47	1	no	—
149	C₄×#125	—	47	1	no	—
150	#1992	—	47	1	no	—
151	#1960	—	47	1	no	—
152	#1933	—	47	4	no	—
153	#1836	—	47	1	no	—
154	#1763	—	47	4	no	—
155	C₄×#97	—	47	1	no	—
156	C₂×C₄₇	C₄×#124	1	1	yes	characteristic
157	C₂×C₄₇	C₄×#137	1	2	yes	—
158	C₂×C₄₇	#3139	1	1	yes	characteristic
159	C₂×C₄₇	#3142	1	1	yes	characteristic
160	C₂×C₄₇	#3181	1	2	yes	—
161	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
162	C₂×#125	C₂×D₄₇	1	1	yes	characteristic
163	C₂×C₄×Q₈	—	47	2	no	—
164	#491	Dic₄₇	1	8	yes	complemented
165	C₂×#103	—	47	4	no	—
166	C₂×#103	C₂×D₄₇	1	4	yes	—
167	C₂×#97	—	47	2	no	—
168	C₂×#97	—	47	1	no	—
169	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
170	C₂×C₄×C₈	—	47	2	no	—
171	#391	—	47	2	no	—
172	#389	—	47	2	no	—
173	#389	—	94	2	no	—
174	#387	—	94	2	no	—
175	#384	—	47	1	no	—
176	#382	—	47	1	no	—
177	C₄×C₄⋊C₄	—	47	2	no	—
178	C₄×C₄⋊C₄	—	94	2	no	—
179	C₄×C₄⋊C₄	—	94	2	no	—
180	C₄³	—	47	1	no	—
181	#340	—	47	4	no	—
182	#340	—	47	4	no	—
183	C₄₇	#21657	1	1	yes	characteristic, complemented, Sylow
184	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
185	C₂²×C₈	—	47	2	no	—
186	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
187	C₄⋊Q₈	—	2	2	no	—
188	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
189	#125	—	2	2	no	—
190	#125	C₂×Dic₄₇	1	4	yes	—
191	C₄×Q₈	—	94	8	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	47	2	no	—
195	C₂×C₄⋊C₄	—	47	2	no	—
196	C₂×C₄⋊C₄	—	94	2	no	—
197	C₂×C₄⋊C₄	—	94	2	no	—
198	C₂×C₄⋊C₄	—	94	2	no	—
199	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
200	C₂×C₄²	—	47	1	no	—
201	C₂×C₄²	—	47	2	no	—
202	C₂×C₄²	—	47	1	no	—
203	C₂×C₄²	—	47	2	no	—
204	C₂×C₄²	—	94	2	no	—
205	C₂×C₄²	—	94	2	no	—
206	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
207	#103	—	94	8	no	—
208	#103	C₂×Dic₄₇	1	16	yes	—
209	#97	—	94	4	no	—
210	#97	—	94	2	no	—
211	#97	C₂×Dic₄₇	1	4	yes	—
212	C₄×C₈	—	94	4	no	—
213	#95	—	94	2	no	—
214	#95	—	94	2	no	—
215	#95	—	94	4	no	—
216	#95	—	94	2	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	2	4	no	—
219	C₂×Q₈	—	2	4	no	—
220	C₂×Q₈	C₂²×Dic₄₇	1	4	yes	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	47	2	no	—
224	C₂²×C₄	—	47	1	no	—
225	C₂²×C₄	—	47	1	no	—
226	C₂²×C₄	—	94	2	no	—
227	C₂²×C₄	—	94	2	no	—
228	C₂²×C₄	D₄×D₄₇	1	2	yes	—
229	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	47	4	no	—
233	C₂×C₈	—	94	2	no	—
234	C₂×C₈	—	94	2	no	—
235	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	8	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	94	4	no	—
240	C₄⋊C₄	—	94	4	no	—
241	C₄⋊C₄	—	188	4	no	—
242	C₄⋊C₄	—	188	4	no	—
243	C₄⋊C₄	—	188	4	no	—
244	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
245	C₄²	—	47	4	no	—
246	C₄²	—	94	4	no	—
247	C₄²	—	94	2	no	—
248	C₄²	—	188	2	no	—
249	C₄²	—	188	4	no	—
250	C₄²	—	188	4	no	—
251	C₄²	#10625352	1	2	yes	—
252	C₄²	C₂²×Dic₄₇	1	2	yes	—
253	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
254	Q₈	—	2	8	no	—
255	Q₈	—	4	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	47	2	no	—
265	C₂×C₄	—	47	2	no	—
266	C₂×C₄	—	47	4	no	—
267	C₂×C₄	—	94	2	no	—
268	C₂×C₄	—	94	2	no	—
269	C₂×C₄	—	94	2	no	—
270	C₂×C₄	—	94	2	no	—
271	C₂×C₄	—	188	2	no	—
272	C₂×C₄	—	188	2	no	—
273	C₂×C₄	—	188	2	no	—
274	C₂×C₄	—	188	4	no	—
275	C₂×C₄	—	188	2	no	—
276	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
277	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
278	C₂×C₄	#41×D₄₇	1	2	yes	—
279	C₂×C₄	#41×D₄₇	1	2	yes	—
280	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	94	4	no	—
283	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
284	C₂²	#124×D₄₇	1	1	yes	characteristic
285	C₂²	#15689672	1	1	yes	characteristic
286	C₂²	#137×D₄₇	1	2	yes	—
287	C₂²	#15689922	1	2	yes	—
288	C₄	—	2	4	no	—
289	C₄	—	4	4	no	—
290	C₄	—	4	4	no	—
291	C₄	—	94	2	no	—
292	C₄	—	94	2	no	—
293	C₄	—	94	4	no	—
294	C₄	—	376	2	no	—
295	C₄	—	376	2	no	—
296	C₄	#41×Dic₄₇	1	2	yes	—
297	C₄	#15689652	1	2	yes	—
298	C₂	#124×Dic₄₇	1	1	yes	characteristic
299	C₂	#137×Dic₄₇	1	2	yes	—
300	C₂	#20253767	1	1	yes	characteristic
301	C₂	#20253790	1	1	yes	characteristic
302	C₂	#20254096	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#491	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#491	D₄₇
hypocenter	C₄₇	#21657
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#491

Complements

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

N	H ≅ G/N	type
29. C₄₇×#491	C₄	semidirect
164. #491	Dic₄₇	semidirect
183. C₄₇	#21657	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×Dic₄₇	2
#125×Dic₄₇	2
#20244358	2
#20244209	2
#20244066	2
#20243522	2
#20243220	2
#97×Dic₄₇	2
C₂×C₄₇×#491	2
#21657	47