Subgroups — #26225151

Order 12032.

670 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253910	C₂	1	1	yes	maximal, characteristic
2	#20252328	C₂	1	1	yes	maximal, characteristic
3	#20244366	C₂	1	1	yes	maximal, characteristic
4	#20244364	C₂	1	1	yes	maximal, characteristic
5	#20244138	C₂	1	2	yes	maximal
6	#20244133	C₂	1	2	yes	maximal
7	#20243562	C₂	1	1	yes	maximal, characteristic
8	#20243273	C₂	1	2	yes	maximal
9	#20243268	C₂	1	2	yes	maximal
10	#97×Dic₄₇	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#2961	C₂	1	1	yes	maximal, characteristic
12	#15689680	C₂²	1	1	yes	characteristic
13	#15689533	C₄	1	4	yes	complemented
14	#15689529	C₄	1	4	yes	complemented
15	#15689251	—	2	4	no	—
16	C₂×#13093959	C₂²	1	2	yes	—
17	C₂×#13093959	C₂²	1	2	yes	—
18	#15689159	C₂²	1	2	yes	—
19	#15689159	C₂²	1	2	yes	—
20	#15689149	C₂²	1	2	yes	—
21	#15689147	C₂²	1	1	yes	characteristic
22	#15689078	C₂²	1	2	yes	—
23	C₂×#13093935	C₂²	1	1	yes	characteristic
24	C₂×#13093934	C₂²	1	1	yes	characteristic
25	C₂×#13093934	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
28	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
29	#15688884	C₂²	1	2	yes	—
30	#15688884	C₂²	1	2	yes	—
31	#15688882	C₂²	1	2	yes	—
32	#15688882	C₂²	1	2	yes	—
33	C₄₇×#533	C₂²	1	1	yes	characteristic
34	C₄₇×#446	—	2	4	no	—
35	C₄₇×#424	C₂²	1	2	yes	—
36	C₄₇×#423	C₂²	1	1	yes	characteristic
37	C₄₇×#405	C₂²	1	2	yes	—
38	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×#10625329	C₂³	1	2	yes	—
40	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	#13094024	C₂×C₄	1	2	yes	—
43	#13094016	C₂×C₄	1	2	yes	—
44	#13094003	—	2	2	no	—
45	C₄×Dic₉₄	—	2	2	no	—
46	#13093983	C₂³	1	2	yes	—
47	C₂×C₄×Dic₄₇	—	2	2	no	—
48	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
50	#13093959	—	2	4	no	—
51	#13093959	—	2	4	no	—
52	#13093957	C₂×C₄	1	4	yes	—
53	#13093957	C₂×C₄	1	4	yes	—
54	#13093955	C₂×C₄	1	4	yes	—
55	#13093955	C₂×C₄	1	4	yes	—
56	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
57	Dic₄₇⋊C₈	—	2	4	no	—
58	#13093935	C₂×C₄	1	2	yes	—
59	#13093935	C₂×C₄	1	2	yes	—
60	#13093934	—	2	2	no	—
61	#13093934	—	2	2	no	—
62	#13093934	Q₈	1	4	yes	complemented
63	C₈×Dic₄₇	—	2	2	no	—
64	C₈×Dic₄₇	—	2	2	no	—
65	C₈×Dic₄₇	Q₈	1	4	yes	complemented
66	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
67	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
70	#125×C₄₇	C₂×C₄	1	2	yes	—
71	C₂²⋊Q₈×C₄₇	—	2	2	no	—
72	C₄×Q₈×C₄₇	—	2	2	no	—
73	#117×C₄₇	C₂³	1	2	yes	—
74	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₄₇	—	2	4	no	—
76	C₄⋊C₈×C₄₇	—	2	2	no	—
77	#97×C₄₇	—	2	2	no	—
78	#97×C₄₇	D₄	1	4	yes	—
79	C₄×C₈×C₄₇	—	2	2	no	—
80	C₂×Q₈×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	#41	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
83	C₂²×Dic₄₇	—	2	1	no	—
84	C₂²×Dic₄₇	—	2	1	no	—
85	C₂×Dic₉₄	—	2	2	no	—
86	M₁₆×C₄₇	—	2	2	no	—
87	M₁₆×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	—	2	1	no	—
89	C₂×C₈×C₄₇	—	2	1	no	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
95	C₄⋊C₄×C₄₇	—	2	4	no	—
96	C₄⋊C₄×C₄₇	—	2	2	no	—
97	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
98	C₂²⋊C₄×C₄₇	—	2	2	no	—
99	C₄²×C₄₇	—	2	2	no	—
100	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₄₇	C₂×D₄	1	2	yes	—
103	C₂²⋊Dic₄₇	—	2	2	no	—
104	C₄⋊Dic₄₇	—	2	2	no	—
105	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
106	Dic₄₇⋊C₄	—	2	4	no	—
107	C₄×Dic₄₇	—	2	2	no	—
108	C₄×Dic₄₇	—	2	1	no	—
109	C₄×Dic₄₇	—	2	1	no	—
110	C₄×Dic₄₇	—	4	4	no	—
111	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
112	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
113	#10625329	—	2	2	no	—
114	#10625329	—	2	2	no	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
117	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
118	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
119	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
120	C₂×C₄₇⋊C₈	#41	1	2	yes	—
121	C₂×C₄₇⋊C₈	#41	1	2	yes	—
122	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
123	Q₈×C₄₇	—	8	2	no	—
124	C₂×C₄×C₄₇	—	2	4	no	—
125	C₂×C₄×C₄₇	—	4	2	no	—
126	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
134	C₂×Dic₄₇	—	2	4	no	—
135	C₂×Dic₄₇	—	2	2	no	—
136	C₂×Dic₄₇	—	2	4	no	—
137	C₂×Dic₄₇	—	2	2	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	Dic₉₄	—	8	2	no	—
140	C₈×C₄₇	—	2	2	no	—
141	C₈×C₄₇	—	4	2	no	—
142	C₈×C₄₇	C₄⋊Q₈	1	4	yes	—
143	C₄₇⋊C₈	—	4	2	no	—
144	C₄₇⋊C₈	C₄×Q₈	1	4	yes	—
145	C₄₇⋊C₈	C₄×Q₈	1	4	yes	—
146	#21736	—	47	1	no	maximal, Sylow
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂²×C₄₇	#533	1	1	yes	characteristic
150	C₂²×C₄₇	#533	1	1	yes	characteristic
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	4	4	no	—
154	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
155	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
156	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
157	C₄×C₄₇	#532	1	1	yes	characteristic
158	Dic₄₇	—	2	4	no	—
159	Dic₄₇	—	4	2	no	—
160	Dic₄₇	—	4	4	no	—
161	#3162	—	47	1	no	—
162	#2961	—	47	1	no	—
163	#2961	D₄₇	1	1	yes	characteristic
164	#1994	—	47	1	no	—
165	#1994	—	47	1	no	—
166	#1946	—	47	2	no	—
167	#1945	—	47	2	no	—
168	#1849	—	47	1	no	—
169	#1773	—	47	2	no	—
170	#1772	—	47	2	no	—
171	C₄×#97	—	47	1	no	—
172	C₂×C₄₇	—	2	1	no	—
173	C₂×C₄₇	—	2	1	no	—
174	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
175	C₂×C₄₇	#2956	1	2	yes	—
176	#533	—	47	1	no	—
177	#533	C₂×D₄₇	1	1	yes	characteristic
178	#501	—	47	4	no	—
179	#501	—	47	4	no	—
180	#446	—	2	4	no	—
181	#446	—	94	4	no	—
182	C₂×#108	—	47	2	no	—
183	C₂×#108	—	47	2	no	—
184	#427	—	47	2	no	—
185	#427	—	47	2	no	—
186	#424	—	47	2	no	—
187	#424	C₂×D₄₇	1	2	yes	—
188	#423	—	47	1	no	—
189	#423	C₂×D₄₇	1	1	yes	characteristic
190	#405	—	47	2	no	—
191	#405	C₂×D₄₇	1	2	yes	—
192	C₂×#97	—	47	1	no	—
193	C₂×#97	—	47	1	no	—
194	C₂×#97	—	47	1	no	—
195	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
196	C₂×C₄×C₈	—	47	1	no	—
197	C₂×C₄×C₈	—	47	1	no	—
198	C₄³	—	47	1	no	—
199	#339	—	47	2	no	—
200	#339	—	47	2	no	—
201	#339	—	47	2	no	—
202	#339	—	47	2	no	—
203	C₄₇	#21736	1	1	yes	characteristic, complemented, Sylow
204	C₂×M₁₆	—	47	2	no	—
205	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
206	C₂²×C₈	—	47	1	no	—
207	C₂²×C₈	—	47	1	no	—
208	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
210	C₄⋊Q₈	—	47	2	no	—
211	C₄⋊Q₈	C₄×D₄₇	1	2	yes	—
212	#125	—	47	2	no	—
213	#125	C₄×D₄₇	1	2	yes	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	94	2	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×Q₈	—	94	2	no	—
218	#117	—	47	2	no	—
219	#117	C₂²×D₄₇	1	2	yes	—
220	C₂×C₄²	—	47	1	no	—
221	C₂×C₄²	—	47	1	no	—
222	C₂×C₄²	—	94	2	no	—
223	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
224	#108	—	94	4	no	—
225	#108	—	94	4	no	—
226	#107	—	47	4	no	—
227	#107	—	47	4	no	—
228	#106	—	47	4	no	—
229	#106	—	47	4	no	—
230	C₄⋊C₈	—	2	4	no	—
231	C₄⋊C₈	—	2	2	no	—
232	C₄⋊C₈	—	94	4	no	—
233	C₄⋊C₈	—	94	2	no	—
234	#97	—	2	2	no	—
235	#97	—	47	4	no	—
236	#97	—	47	2	no	—
237	#97	—	47	2	no	—
238	#97	—	94	2	no	—
239	#97	—	94	2	no	—
240	#97	C₂²⋊D₄₇	1	4	yes	—
241	C₄×C₈	—	2	2	no	—
242	C₄×C₈	—	47	4	no	—
243	C₄×C₈	—	94	2	no	—
244	C₄×C₈	—	94	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	94	2	no	—
247	C₂²×C₄	—	94	1	no	—
248	C₂²×C₄	—	94	1	no	—
249	C₂²×C₄	#10625350	1	2	yes	—
250	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
251	M₁₆	—	2	2	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	94	2	no	—
254	M₁₆	—	94	2	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	47	2	no	—
259	C₂×C₈	—	47	2	no	—
260	C₂×C₈	—	47	2	no	—
261	C₂×C₈	—	47	2	no	—
262	C₂×C₈	—	47	2	no	—
263	C₂×C₈	—	47	2	no	—
264	C₂×C₈	—	94	2	no	—
265	C₂×C₈	Q₈×D₄₇	1	2	yes	—
266	C₂×C₈	Q₈×D₄₇	1	2	yes	—
267	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
268	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	47	4	no	—
272	C₄⋊C₄	—	94	4	no	—
273	C₄⋊C₄	—	94	2	no	—
274	C₄⋊C₄	C₂×C₄×D₄₇	1	4	yes	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	94	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	47	2	no	—
279	C₄²	—	47	2	no	—
280	C₄²	—	94	2	no	—
281	C₄²	—	94	1	no	—
282	C₄²	—	94	1	no	—
283	C₄²	—	188	4	no	—
284	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
286	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
287	C₂³	C₂×#10625350	1	1	yes	characteristic
288	Q₈	—	8	2	no	—
289	Q₈	—	376	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	94	4	no	—
293	C₂×C₄	—	94	2	no	—
294	C₂×C₄	—	94	4	no	—
295	C₂×C₄	—	94	2	no	—
296	C₂×C₄	—	188	2	no	—
297	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
298	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
299	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
301	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
302	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
303	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
304	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	4	2	no	—
307	C₈	—	47	4	no	—
308	C₈	—	47	4	no	—
309	C₈	—	188	2	no	—
310	C₈	#13094022	1	4	yes	—
311	C₂²	—	2	2	no	—
312	C₂²	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
313	C₂²	#15689677	1	1	yes	characteristic
314	C₂²	#15689677	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	4	4	no	—
318	C₄	—	94	4	no	—
319	C₄	—	188	2	no	—
320	C₄	—	188	4	no	—
321	C₄	C₄×Q₈×D₄₇	1	1	yes	characteristic
322	C₄	C₄×Q₈×D₄₇	1	1	yes	characteristic
323	C₄	C₂×#13094022	1	1	yes	characteristic
324	C₄	#15689669	1	1	yes	characteristic
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	C₄×#13094022	1	1	yes	characteristic
328	C₂	#20252295	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2961	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₂×#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2961	D₄₇
hypocenter	C₄₇	#21736
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #2961

Complements

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

N	H ≅ G/N	type
13. #15689533	C₄	semidirect
14. #15689529	C₄	semidirect
62. #13093934	Q₈	semidirect
65. C₈×Dic₄₇	Q₈	semidirect
203. C₄₇	#21736	semidirect

Maximal subgroups

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

subgroup	index
#20253910	2
#20252328	2
#20244366	2
#20244364	2
#20244138	2
#20244133	2
#20243562	2
#20243273	2
#20243268	2
#97×Dic₄₇	2
C₄₇×#2961	2
#21736	47