Subgroups — #26316113

Order 12032.

754 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251751	C₂	1	1	yes	maximal, characteristic, complemented
2	#20244539	C₂	1	2	yes	maximal, complemented
3	#20244534	C₂	1	2	yes	maximal
4	#20244200	C₂	1	2	yes	maximal
5	#20243518	C₂	1	2	yes	maximal, complemented
6	#20243393	C₂	1	2	yes	maximal, complemented
7	#20243388	C₂	1	2	yes	maximal
8	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#124×C₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#13094016	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×#13093948	C₂²	1	4	yes	—
13	C₂×#13093945	C₂²	1	4	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	#15689021	—	2	2	no	—
16	#15689021	—	2	2	no	—
17	#15688993	C₂²	1	2	yes	—
18	#15688989	C₂²	1	2	yes	—
19	#15688885	C₂²	1	4	yes	—
20	#15688885	C₂²	1	4	yes	—
21	#15688878	C₂²	1	2	yes	—
22	C₂×#124×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#391	C₂²	1	1	yes	characteristic
26	C₄₇×#390	C₂²	1	1	yes	characteristic
27	C₄₇×#388	—	2	2	no	—
28	C₄₇×#382	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₄₇	—	2	2	no	—
30	C₄³×C₄₇	C₂²	1	1	yes	characteristic
31	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
32	#13094016	—	4	4	no	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	—	2	2	no	—
35	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
36	C₂×Dic₄₇⋊C₄	—	2	4	no	—
37	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
38	#13093948	—	4	4	no	—
39	#13093945	—	4	4	no	—
40	C₄×C₄₇⋊C₈	—	2	4	no	—
41	C₄₇⋊#95	—	2	4	no	—
42	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	—	2	1	no	—
45	#124×C₄₇	—	2	2	no	—
46	#124×C₄₇	Q₈	1	4	yes	—
47	C₄×Q₈×C₄₇	—	2	4	no	—
48	C₄×D₄×C₄₇	—	2	4	no	—
49	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
50	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
54	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
55	C₂×C₄²×C₄₇	—	2	1	no	—
56	C₂×C₄²×C₄₇	D₄	1	2	yes	—
57	C₂×C₄²×C₄₇	D₄	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	#95×C₄₇	—	2	2	no	—
62	#95×C₄₇	—	2	2	no	—
63	C₂⁴×C₄₇	—	2	1	no	—
64	C₂×Q₈×C₄₇	—	2	1	no	—
65	C₂×Q₈×C₄₇	—	2	2	no	—
66	C₂×Q₈×C₄₇	—	2	2	no	—
67	C₂×Q₈×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂×D₄×C₄₇	—	2	2	no	—
69	C₂×D₄×C₄₇	—	2	1	no	—
70	C₂×D₄×C₄₇	—	2	2	no	—
71	C₂×D₄×C₄₇	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₄₇	—	2	1	no	—
73	C₂²×C₄×C₄₇	—	2	1	no	—
74	C₂²×C₄×C₄₇	—	2	1	no	—
75	C₂²×C₄×C₄₇	—	4	2	no	—
76	C₂²×C₄×C₄₇	—	4	4	no	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₄₇	—	2	4	no	—
83	C₄⋊C₄×C₄₇	—	2	2	no	—
84	C₄⋊C₄×C₄₇	—	2	2	no	—
85	C₄⋊C₄×C₄₇	—	4	2	no	—
86	C₂²⋊C₄×C₄₇	—	2	8	no	—
87	C₂²⋊C₄×C₄₇	—	4	4	no	—
88	C₂²⋊C₄×C₄₇	—	4	2	no	—
89	C₄²×C₄₇	—	2	4	no	—
90	C₄²×C₄₇	—	2	2	no	—
91	C₄²×C₄₇	—	4	2	no	—
92	C₄²×C₄₇	—	4	2	no	—
93	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
94	C₄²×C₄₇	#41	1	2	yes	—
95	C₄²×C₄₇	#41	1	4	yes	—
96	C₄⋊Dic₄₇	—	2	4	no	—
97	C₄⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	—	4	4	no	—
99	C₄⋊Dic₄₇	—	4	4	no	—
100	Dic₄₇⋊C₄	—	4	8	no	—
101	C₂×C₄₇⋊C₈	—	2	4	no	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	C₂×C₄₇⋊C₈	—	2	4	no	—
104	C₂³×C₄₇	—	2	2	no	—
105	C₂³×C₄₇	—	4	1	no	—
106	C₂³×C₄₇	—	4	1	no	—
107	C₂³×C₄₇	—	4	2	no	—
108	C₂³×C₄₇	—	4	2	no	—
109	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
110	Q₈×C₄₇	—	2	4	no	—
111	Q₈×C₄₇	—	4	2	no	—
112	D₄×C₄₇	—	2	4	no	—
113	D₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	4	1	no	—
122	C₂×C₄×C₄₇	—	4	1	no	—
123	C₂×C₄×C₄₇	—	4	1	no	—
124	C₂×C₄×C₄₇	—	4	1	no	—
125	C₂×C₄×C₄₇	—	4	2	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	8	4	no	—
128	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
129	C₂×C₄×C₄₇	#124	1	2	yes	—
130	C₂×C₄×C₄₇	#124	1	2	yes	—
131	C₂×C₄×C₄₇	#135	1	4	yes	—
132	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
134	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
136	C₂×Dic₄₇	—	4	4	no	—
137	C₂×Dic₄₇	—	4	4	no	—
138	C₂×Dic₄₇	—	4	4	no	—
139	C₄₇⋊C₈	—	4	4	no	—
140	#29140	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	4	2	no	—
142	C₂²×C₄₇	—	4	4	no	—
143	C₂²×C₄₇	—	8	1	no	—
144	C₂²×C₄₇	—	8	1	no	—
145	C₂²×C₄₇	—	8	2	no	—
146	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
148	C₂²×C₄₇	#535	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×#135	1	2	yes	—
150	C₂²×C₄₇	#575	1	2	yes	—
151	C₄×C₄₇	—	2	4	no	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	4	2	no	—
154	C₄×C₄₇	—	4	2	no	—
155	C₄×C₄₇	—	8	1	no	—
156	C₄×C₄₇	—	8	1	no	—
157	C₄×C₄₇	#488	1	4	yes	—
158	C₄×C₄₇	#554	1	2	yes	—
159	C₄×C₄₇	#555	1	2	yes	—
160	Dic₄₇	—	8	4	no	—
161	#2859	—	47	1	no	—
162	C₄×#124	D₄₇	1	1	yes	characteristic
163	#2026	—	47	2	no	—
164	#2025	—	47	2	no	—
165	#1958	—	47	2	no	—
166	#1835	—	47	2	no	—
167	#1803	—	47	2	no	—
168	#1802	—	47	2	no	—
169	C₄×C₄⋊C₈	—	47	1	no	—
170	C₂×C₄₇	—	8	2	no	—
171	C₂×C₄₇	#2563	1	1	yes	characteristic
172	C₂×C₄₇	#3076	1	2	yes	—
173	C₂×C₄₇	C₂×#488	1	1	yes	characteristic
174	C₂×C₄₇	#3142	1	1	yes	characteristic
175	C₂×C₄₇	#3212	1	2	yes	—
176	C₂×#125	—	47	2	no	—
177	C₂×#124	C₂×D₄₇	1	1	yes	characteristic
178	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
179	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
180	C₂×C₄⋊C₈	—	47	2	no	—
181	C₂×#103	—	47	4	no	—
182	C₂×#102	—	47	4	no	—
183	C₂×C₄×C₈	—	47	2	no	—
184	#391	C₂×D₄₇	1	1	yes	characteristic
185	#390	C₂×D₄₇	1	1	yes	characteristic
186	#389	—	94	2	no	—
187	#389	—	94	2	no	—
188	#388	—	2	2	no	—
189	#384	—	47	2	no	—
190	#382	—	47	2	no	—
191	#382	C₂×D₄₇	1	1	yes	characteristic
192	C₄×C₂²⋊C₄	—	2	2	no	—
193	C₄³	C₂×D₄₇	1	1	yes	characteristic
194	#340	—	47	4	no	—
195	#340	—	47	4	no	—
196	#336	—	47	2	no	—
197	C₄₇	#29140	1	1	yes	characteristic, complemented, Sylow
198	C₂²×Q₈	C₂²×D₄₇	1	1	yes	characteristic
199	C₂²×D₄	C₂²×D₄₇	1	1	yes	characteristic
200	C₂³×C₄	—	2	1	no	—
201	C₂²×C₈	—	47	4	no	—
202	#125	—	188	4	no	—
203	#124	—	2	2	no	—
204	#124	Dic₉₄	1	4	yes	—
205	C₄×Q₈	—	2	4	no	—
206	C₄×D₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	47	4	no	—
209	C₂×C₄⋊C₄	—	94	4	no	—
210	C₂×C₄⋊C₄	—	94	2	no	—
211	C₂×C₄⋊C₄	—	94	2	no	—
212	C₂×C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
213	C₂×C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
214	C₂×C₂²⋊C₄	—	2	2	no	—
215	C₂×C₂²⋊C₄	—	2	1	no	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₄²	—	2	1	no	—
218	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
219	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
220	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
223	C₄⋊C₈	—	94	4	no	—
224	#103	—	188	4	no	—
225	#102	—	188	4	no	—
226	C₄×C₈	—	94	4	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	94	4	no	—
230	C₂⁴	—	2	1	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	C₂×Dic₉₄	1	2	yes	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	C₂×Dic₉₄	1	2	yes	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	—	94	4	no	—
245	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
246	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
247	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
249	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
250	C₂×C₈	—	94	4	no	—
251	C₂×C₈	—	94	4	no	—
252	C₂×C₈	—	94	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	94	4	no	—
257	C₄⋊C₄	—	94	4	no	—
258	C₄⋊C₄	—	188	8	no	—
259	C₄⋊C₄	—	188	4	no	—
260	C₄⋊C₄	—	188	4	no	—
261	C₂²⋊C₄	—	2	8	no	—
262	C₂²⋊C₄	—	4	4	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₄²	—	2	4	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	4	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	C₂×Dic₉₄	1	2	yes	—
269	C₄²	#10625350	1	2	yes	—
270	C₄²	#10625352	1	4	yes	—
271	C₂³	—	2	2	no	—
272	C₂³	—	4	1	no	—
273	C₂³	—	4	2	no	—
274	C₂³	—	4	2	no	—
275	C₂³	—	4	1	no	—
276	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
277	Q₈	—	2	4	no	—
278	Q₈	—	4	2	no	—
279	D₄	—	2	4	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	8	4	no	—
295	C₂×C₄	—	188	4	no	—
296	C₂×C₄	—	188	4	no	—
297	C₂×C₄	—	188	4	no	—
298	C₂×C₄	#13094003	1	4	yes	—
299	C₂×C₄	#13094011	1	2	yes	—
300	C₂×C₄	#13094011	1	2	yes	—
301	C₂×C₄	#13094044	1	4	yes	—
302	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
303	C₂×C₄	C₂×#10625352	1	2	yes	—
304	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
305	C₂×C₄	#13094070	1	2	yes	—
306	C₈	—	188	4	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	4	no	—
309	C₂²	—	8	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	C₂×#13094003	1	1	yes	characteristic
313	C₂²	C₂×#13094011	1	1	yes	characteristic
314	C₂²	#15689687	1	1	yes	characteristic
315	C₂²	C₂×#13094044	1	2	yes	—
316	C₂²	#15689921	1	2	yes	—
317	C₄	—	2	2	no	—
318	C₄	—	2	4	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	8	1	no	—
322	C₄	—	8	1	no	—
323	C₄	—	376	4	no	—
324	C₄	#15689477	1	4	yes	—
325	C₄	#15689830	1	2	yes	—
326	C₄	#15689838	1	2	yes	—
327	C₂	—	8	2	no	—
328	C₂	#20248561	1	1	yes	characteristic
329	C₂	#20253176	1	2	yes	—
330	C₂	C₂×#15689477	1	1	yes	characteristic
331	C₂	#20253788	1	1	yes	characteristic
332	C₂	#20254360	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#124×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₄×C₄₇	C₂³
hypercenter	C₄×#124	D₄₇
hypocenter	C₄₇	#29140
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#124

Complements

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

N	H ≅ G/N	type
1. #20251751	C₂	semidirect
2. #20244539	C₂	semidirect
5. #20243518	C₂	semidirect
6. #20243393	C₂	semidirect
8. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
197. C₄₇	#29140	semidirect

Maximal subgroups

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

subgroup	index
#20251751	2
#20244539	2
#20244534	2
#20244200	2
#20243518	2
#20243393	2
#20243388	2
C₄×C₄⋊(C₄₇⋊C₈)	2
C₄×#124×C₄₇	2
#29140	47