Subgroups — #26225128

Order 12032.

654 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253898	C₂	1	1	yes	maximal, characteristic
2	#20252328	C₂	1	1	yes	maximal, characteristic
3	#20244370	C₂	1	1	yes	maximal, characteristic
4	#20244368	C₂	1	1	yes	maximal, characteristic
5	#20244138	C₂	1	2	yes	maximal
6	#20244133	C₂	1	2	yes	maximal
7	#20243559	C₂	1	1	yes	maximal, characteristic
8	#20243273	C₂	1	2	yes	maximal
9	#20243268	C₂	1	2	yes	maximal
10	#20243076	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#2961	C₂	1	1	yes	maximal, characteristic
12	#15689680	C₂²	1	1	yes	characteristic
13	#15689526	C₄	1	2	yes	complemented
14	#15689522	C₄	1	2	yes	complemented
15	#15689522	C₄	1	2	yes	complemented
16	#15689519	C₄	1	2	yes	complemented
17	#15689251	—	2	4	no	—
18	C₂×#13093959	C₂²	1	2	yes	—
19	C₂×#13093959	C₂²	1	2	yes	—
20	#15689159	C₂²	1	2	yes	—
21	#15689159	C₂²	1	2	yes	—
22	#15689149	C₂²	1	2	yes	—
23	#15689147	C₂²	1	1	yes	characteristic
24	#15689078	C₂²	1	2	yes	—
25	C₂×#13093934	C₂²	1	1	yes	characteristic
26	C₂×#13093934	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
30	#15688969	C₂²	1	1	yes	characteristic
31	#15688884	C₂²	1	2	yes	—
32	#15688884	C₂²	1	2	yes	—
33	#15688882	C₂²	1	2	yes	—
34	#15688882	C₂²	1	2	yes	—
35	C₄₇×#533	C₂²	1	1	yes	characteristic
36	C₄₇×#446	—	2	4	no	—
37	C₄₇×#424	C₂²	1	2	yes	—
38	C₄₇×#423	C₂²	1	1	yes	characteristic
39	C₄₇×#405	C₂²	1	2	yes	—
40	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×#10625329	C₂³	1	2	yes	—
42	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
44	#13094024	C₂×C₄	1	2	yes	—
45	#13094016	C₂×C₄	1	2	yes	—
46	#13094003	—	2	2	no	—
47	C₄×Dic₉₄	—	2	2	no	—
48	#13093983	C₂³	1	2	yes	—
49	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
51	#13093959	—	2	4	no	—
52	#13093959	—	2	4	no	—
53	#13093957	C₂×C₄	1	4	yes	—
54	#13093957	C₂×C₄	1	4	yes	—
55	#13093955	C₂×C₄	1	4	yes	—
56	#13093955	C₂×C₄	1	4	yes	—
57	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
58	Dic₄₇⋊C₈	—	2	4	no	—
59	#13093934	—	2	2	no	—
60	#13093934	—	2	2	no	—
61	#13093934	Q₈	1	4	yes	complemented
62	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
64	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
65	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₄₇	—	2	2	no	—
67	C₈×Dic₄₇	—	2	2	no	—
68	C₈×Dic₄₇	Q₈	1	4	yes	complemented
69	C₄₇⋊#95	—	2	2	no	—
70	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
71	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
74	#125×C₄₇	C₂×C₄	1	2	yes	—
75	C₂²⋊Q₈×C₄₇	—	2	2	no	—
76	C₄×Q₈×C₄₇	—	2	2	no	—
77	#117×C₄₇	C₂³	1	2	yes	—
78	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₄₇	—	2	4	no	—
80	C₄⋊C₈×C₄₇	—	2	2	no	—
81	#97×C₄₇	—	2	2	no	—
82	#97×C₄₇	D₄	1	4	yes	—
83	C₄×C₈×C₄₇	—	2	2	no	—
84	C₂×Q₈×C₄₇	—	2	2	no	—
85	C₂²×C₄×C₄₇	#41	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₂²×Dic₄₇	—	2	1	no	—
88	C₂²×Dic₄₇	—	2	1	no	—
89	C₂×Dic₉₄	—	2	2	no	—
90	M₁₆×C₄₇	—	2	2	no	—
91	M₁₆×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	—	2	1	no	—
93	C₂×C₈×C₄₇	—	2	1	no	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
98	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
99	C₄⋊C₄×C₄₇	—	2	4	no	—
100	C₄⋊C₄×C₄₇	—	2	2	no	—
101	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
102	C₂²⋊C₄×C₄₇	—	2	2	no	—
103	C₄²×C₄₇	—	2	2	no	—
104	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₄₇	C₂×D₄	1	2	yes	—
107	C₂²⋊Dic₄₇	—	2	2	no	—
108	C₄⋊Dic₄₇	—	2	2	no	—
109	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
110	Dic₄₇⋊C₄	—	2	4	no	—
111	C₄×Dic₄₇	—	2	2	no	—
112	C₄×Dic₄₇	—	2	1	no	—
113	C₄×Dic₄₇	—	2	1	no	—
114	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
115	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
116	#10625329	—	2	2	no	—
117	#10625329	—	2	2	no	—
118	C₂×C₄₇⋊C₈	—	2	2	no	—
119	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
124	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
125	C₂×C₄₇⋊C₈	#41	1	2	yes	—
126	C₂×C₄₇⋊C₈	#41	1	2	yes	—
127	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
128	Q₈×C₄₇	—	8	2	no	—
129	C₂×C₄×C₄₇	—	2	4	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
139	C₂×Dic₄₇	—	2	4	no	—
140	C₂×Dic₄₇	—	2	2	no	—
141	C₂×Dic₄₇	—	2	2	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	4	2	no	—
144	Dic₉₄	—	8	2	no	—
145	C₈×C₄₇	—	2	2	no	—
146	C₈×C₄₇	—	4	2	no	—
147	C₈×C₄₇	C₄⋊Q₈	1	4	yes	—
148	C₄₇⋊C₈	—	4	2	no	—
149	C₄₇⋊C₈	C₄×Q₈	1	2	yes	—
150	C₄₇⋊C₈	C₄×Q₈	1	2	yes	—
151	C₄₇⋊C₈	C₄×Q₈	1	2	yes	—
152	C₄₇⋊C₈	C₄×Q₈	1	2	yes	—
153	#21733	—	47	1	no	maximal, Sylow
154	C₂²×C₄₇	—	2	2	no	—
155	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂²×C₄₇	#533	1	1	yes	characteristic
157	C₂²×C₄₇	#533	1	1	yes	characteristic
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	—	4	4	no	—
161	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
162	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
163	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
164	C₄×C₄₇	#532	1	1	yes	characteristic
165	Dic₄₇	—	4	2	no	—
166	Dic₄₇	—	4	2	no	—
167	Dic₄₇	—	4	4	no	—
168	#3160	—	47	1	no	—
169	#2961	—	47	1	no	—
170	#2961	D₄₇	1	1	yes	characteristic
171	#1995	—	47	1	no	—
172	#1995	—	47	1	no	—
173	#1946	—	47	2	no	—
174	#1945	—	47	2	no	—
175	#1848	—	47	1	no	—
176	#1773	—	47	2	no	—
177	#1772	—	47	2	no	—
178	#1726	—	47	1	no	—
179	C₂×C₄₇	—	2	1	no	—
180	C₂×C₄₇	—	2	1	no	—
181	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
182	C₂×C₄₇	#2957	1	2	yes	—
183	#533	—	47	1	no	—
184	#533	C₂×D₄₇	1	1	yes	characteristic
185	#500	—	47	2	no	—
186	#499	—	47	2	no	—
187	#499	—	47	2	no	—
188	#498	—	47	2	no	—
189	#446	—	2	4	no	—
190	#446	—	94	4	no	—
191	C₂×#108	—	47	2	no	—
192	C₂×#108	—	47	2	no	—
193	#427	—	47	2	no	—
194	#427	—	47	2	no	—
195	#424	—	47	2	no	—
196	#424	C₂×D₄₇	1	2	yes	—
197	#423	—	47	1	no	—
198	#423	C₂×D₄₇	1	1	yes	characteristic
199	#405	—	47	2	no	—
200	#405	C₂×D₄₇	1	2	yes	—
201	C₂×#97	—	47	1	no	—
202	C₂×#97	—	47	1	no	—
203	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
204	C₂×C₄×C₈	—	47	1	no	—
205	C₂×C₄×C₈	—	47	1	no	—
206	C₂×C₄×C₈	—	47	1	no	—
207	#376	—	47	1	no	—
208	#339	—	47	2	no	—
209	#339	—	47	2	no	—
210	#339	—	47	2	no	—
211	#339	—	47	2	no	—
212	C₄₇	#21733	1	1	yes	characteristic, complemented, Sylow
213	C₂×M₁₆	—	47	2	no	—
214	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
215	C₂²×C₈	—	47	1	no	—
216	C₂²×C₈	—	47	1	no	—
217	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
219	C₄⋊Q₈	—	47	2	no	—
220	C₄⋊Q₈	C₄×D₄₇	1	2	yes	—
221	#125	—	47	2	no	—
222	#125	C₄×D₄₇	1	2	yes	—
223	C₂²⋊Q₈	—	2	2	no	—
224	C₂²⋊Q₈	—	94	2	no	—
225	C₄×Q₈	—	2	2	no	—
226	C₄×Q₈	—	94	2	no	—
227	#117	—	47	2	no	—
228	#117	C₂²×D₄₇	1	2	yes	—
229	C₂×C₄²	—	47	1	no	—
230	C₂×C₄²	—	47	1	no	—
231	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
232	#108	—	94	4	no	—
233	#108	—	94	4	no	—
234	#107	—	47	4	no	—
235	#107	—	47	4	no	—
236	#106	—	47	4	no	—
237	#106	—	47	4	no	—
238	C₄⋊C₈	—	2	4	no	—
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	—	94	4	no	—
241	C₄⋊C₈	—	94	2	no	—
242	#97	—	2	2	no	—
243	#97	—	47	4	no	—
244	#97	—	94	2	no	—
245	#97	—	94	2	no	—
246	#97	C₂²⋊D₄₇	1	4	yes	—
247	C₄×C₈	—	2	2	no	—
248	C₄×C₈	—	47	4	no	—
249	C₄×C₈	—	47	1	no	—
250	C₄×C₈	—	47	1	no	—
251	C₄×C₈	—	47	1	no	—
252	C₄×C₈	—	47	1	no	—
253	C₄×C₈	—	94	2	no	—
254	C₄×C₈	—	94	2	no	—
255	#95	—	94	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	94	2	no	—
258	C₂²×C₄	—	94	1	no	—
259	C₂²×C₄	—	94	1	no	—
260	C₂²×C₄	#10625350	1	2	yes	—
261	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
262	M₁₆	—	2	2	no	—
263	M₁₆	—	2	2	no	—
264	M₁₆	—	94	2	no	—
265	M₁₆	—	94	2	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	47	1	no	—
270	C₂×C₈	—	47	1	no	—
271	C₂×C₈	—	47	1	no	—
272	C₂×C₈	—	47	1	no	—
273	C₂×C₈	—	47	2	no	—
274	C₂×C₈	—	47	2	no	—
275	C₂×C₈	—	47	2	no	—
276	C₂×C₈	—	47	2	no	—
277	C₂×C₈	—	94	2	no	—
278	C₂×C₈	Q₈×D₄₇	1	2	yes	—
279	C₂×C₈	Q₈×D₄₇	1	2	yes	—
280	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
281	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	47	4	no	—
285	C₄⋊C₄	—	94	4	no	—
286	C₄⋊C₄	—	94	2	no	—
287	C₄⋊C₄	C₂×C₄×D₄₇	1	4	yes	—
288	C₂²⋊C₄	—	2	2	no	—
289	C₂²⋊C₄	—	94	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	47	2	no	—
292	C₄²	—	47	2	no	—
293	C₄²	—	94	2	no	—
294	C₄²	—	94	1	no	—
295	C₄²	—	94	1	no	—
296	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
297	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
298	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
299	C₂³	C₂×#10625350	1	1	yes	characteristic
300	Q₈	—	8	2	no	—
301	Q₈	—	376	2	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	94	4	no	—
305	C₂×C₄	—	94	2	no	—
306	C₂×C₄	—	94	2	no	—
307	C₂×C₄	—	188	2	no	—
308	C₂×C₄	—	188	2	no	—
309	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
310	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
311	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
312	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
313	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
314	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
315	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
316	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
317	C₈	—	2	2	no	—
318	C₈	—	4	2	no	—
319	C₈	—	47	2	no	—
320	C₈	—	47	2	no	—
321	C₈	—	47	2	no	—
322	C₈	—	47	2	no	—
323	C₈	—	188	2	no	—
324	C₈	#13094022	1	4	yes	—
325	C₂²	—	2	2	no	—
326	C₂²	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
327	C₂²	#15689677	1	1	yes	characteristic
328	C₂²	#15689677	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	4	4	no	—
332	C₄	—	188	2	no	—
333	C₄	—	188	2	no	—
334	C₄	—	188	4	no	—
335	C₄	C₄×Q₈×D₄₇	1	1	yes	characteristic
336	C₄	C₄×Q₈×D₄₇	1	1	yes	characteristic
337	C₄	C₂×#13094022	1	1	yes	characteristic
338	C₄	#15689669	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	C₄×#13094022	1	1	yes	characteristic
342	C₂	#20252303	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	#21733
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. #15689526	C₄	semidirect
14. #15689522	C₄	semidirect
15. #15689522	C₄	semidirect
16. #15689519	C₄	semidirect
61. #13093934	Q₈	semidirect
68. C₈×Dic₄₇	Q₈	semidirect
212. C₄₇	#21733	semidirect

Maximal subgroups

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

subgroup	index
#20253898	2
#20252328	2
#20244370	2
#20244368	2
#20244138	2
#20244133	2
#20243559	2
#20243273	2
#20243268	2
#20243076	2
C₄₇×#2961	2
#21733	47