Subgroups — #26235157

Order 12032.

730 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689253	C₂	1	1	yes	maximal, characteristic
2	#20246340	C₂	1	2	yes	maximal
3	#20244157	C₂	1	2	yes	maximal
4	#20244154	C₂	1	2	yes	maximal
5	#20244148	C₂	1	2	yes	maximal
6	#20243555	C₂	1	2	yes	maximal
7	#20243287	C₂	1	2	yes	maximal
8	C₄×#13093956	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1846	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
11	#15689253	C₄	1	8	yes	complemented
12	C₂×#13093957	C₂²	1	2	yes	—
13	C₂×#13093956	C₂²	1	1	yes	characteristic
14	C₂×#13093956	C₂²	1	1	yes	characteristic
15	C₂×#13093955	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
18	C₂×#13093934	C₂²	1	2	yes	—
19	#15689027	—	2	2	no	—
20	#15689008	—	2	2	no	—
21	#15689005	—	2	2	no	—
22	#15688992	C₂²	1	2	yes	—
23	#15688988	C₂²	1	2	yes	—
24	C₄⋊C₄×Dic₄₇	C₂²	1	2	yes	—
25	C₄⋊C₄×Dic₄₇	C₂²	1	2	yes	—
26	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
27	#15688974	—	2	2	no	—
28	#15688886	C₂²	1	2	yes	—
29	#15688885	C₂²	1	2	yes	—
30	#15688885	C₂²	1	2	yes	—
31	C₂×#107×C₄₇	C₂²	1	2	yes	—
32	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
35	C₄₇×#382	C₂²	1	2	yes	—
36	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
37	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
38	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
39	C₄×Dic₉₄	Q₈	1	4	yes	—
40	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
42	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
43	C₂×Dic₄₇⋊C₄	—	2	2	no	—
44	C₂×Dic₄₇⋊C₄	—	2	2	no	—
45	C₂×Dic₄₇⋊C₄	D₄	1	2	yes	—
46	C₂×C₄×Dic₄₇	—	2	2	no	—
47	C₂×C₄×Dic₄₇	—	2	2	no	—
48	C₂×C₄×Dic₄₇	—	2	2	no	—
49	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
50	#13093957	—	4	2	no	—
51	#13093956	—	2	2	no	—
52	#13093956	Q₈	1	4	yes	complemented
53	#13093955	—	4	2	no	—
54	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
55	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	—
56	#13093934	C₂×C₄	1	8	yes	—
57	C₄₇⋊#95	—	2	2	no	—
58	C₄₇⋊#95	—	2	2	no	—
59	C₄₇⋊#95	—	2	2	no	—
60	C₄₇⋊#95	—	2	2	no	—
61	C₄₇⋊#95	—	2	2	no	—
62	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
66	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	#107×C₄₇	—	2	4	no	—
68	#106×C₄₇	—	4	1	no	—
69	#106×C₄₇	—	4	1	no	—
70	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
71	#95×C₄₇	—	2	2	no	—
72	#95×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	—	2	2	no	—
74	C₂²×C₄×C₄₇	—	2	2	no	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
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	2	yes	—
82	C₂×Dic₉₄	—	4	2	no	—
83	C₂×Dic₉₄	C₄⋊C₄	1	4	yes	—
84	C₂×C₈×C₄₇	—	2	1	no	—
85	C₂×C₈×C₄₇	—	2	1	no	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₄₇	#41	1	2	yes	—
90	C₄⋊C₄×C₄₇	—	2	2	no	—
91	C₄⋊C₄×C₄₇	—	2	2	no	—
92	C₄⋊C₄×C₄₇	—	2	2	no	—
93	C₄⋊C₄×C₄₇	—	2	2	no	—
94	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
95	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
96	C₄⋊Dic₄₇	—	2	2	no	—
97	C₄⋊Dic₄₇	—	2	2	no	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
100	C₄⋊Dic₄₇	C₂×Q₈	1	4	yes	—
101	Dic₄₇⋊C₄	—	4	4	no	—
102	Dic₄₇⋊C₄	—	8	2	no	—
103	Dic₄₇⋊C₄	C₄⋊C₄	1	8	yes	—
104	C₄×Dic₄₇	—	4	4	no	—
105	C₄×Dic₄₇	—	8	2	no	—
106	C₄×Dic₄₇	—	8	2	no	—
107	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
108	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
109	C₂×C₄₇⋊C₈	—	2	2	no	—
110	C₂×C₄₇⋊C₈	—	2	2	no	—
111	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
112	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
129	C₂×Dic₄₇	—	2	2	no	—
130	C₂×Dic₄₇	—	2	2	no	—
131	C₂×Dic₄₇	—	2	2	no	—
132	C₂×Dic₄₇	—	2	2	no	—
133	C₂×Dic₄₇	—	2	4	no	—
134	C₂×Dic₄₇	—	4	2	no	—
135	C₂×Dic₄₇	—	4	2	no	—
136	C₂×Dic₄₇	—	4	2	no	—
137	C₂×Dic₄₇	—	4	2	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
140	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
141	Dic₉₄	—	4	4	no	—
142	C₈×C₄₇	—	2	2	no	—
143	C₈×C₄₇	C₄×Q₈	1	4	yes	—
144	C₄₇⋊C₈	—	2	4	no	—
145	#22730	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₄₇	D₄×Q₈	1	1	yes	characteristic
148	C₂²×C₄₇	#550	1	1	yes	characteristic
149	C₂²×C₄₇	#575	1	2	yes	—
150	C₂²×C₄₇	#577	1	2	yes	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	8	2	no	—
154	C₄×C₄₇	—	8	2	no	—
155	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
156	C₄×C₄₇	#520	1	1	yes	characteristic
157	C₄×C₄₇	#557	1	1	yes	characteristic
158	C₄×C₄₇	Q₈²	1	1	yes	characteristic
159	Dic₄₇	—	2	4	no	—
160	Dic₄₇	—	2	4	no	—
161	Dic₄₇	—	4	4	no	—
162	Dic₄₇	—	8	2	no	—
163	C₂×#446	—	47	1	no	—
164	#2355	—	47	2	no	—
165	#1949	—	47	2	no	—
166	#1949	—	47	2	no	—
167	#1948	—	47	2	no	—
168	#1847	—	47	2	no	—
169	#1846	D₄₇	1	1	yes	characteristic
170	#1776	—	47	2	no	—
171	C₄×#106	—	47	1	no	—
172	C₂×C₄₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄₇	#2911	1	1	yes	characteristic
174	C₂×C₄₇	#2912	1	1	yes	characteristic
175	C₂×C₄₇	#3388	1	2	yes	—
176	C₂×C₄₇	#3401	1	2	yes	—
177	C₂×C₄×Q₈	—	47	1	no	—
178	#446	—	47	8	no	—
179	C₂×#107	—	47	2	no	—
180	C₂×#107	C₂×D₄₇	1	2	yes	—
181	C₂×#106	—	47	1	no	—
182	C₂×#106	—	47	1	no	—
183	C₂×#106	—	47	2	no	—
184	C₂×#106	C₂×D₄₇	1	1	yes	characteristic
185	C₂×#106	C₂×D₄₇	1	1	yes	characteristic
186	C₂×C₄⋊C₈	—	47	2	no	—
187	C₂×C₄⋊C₈	—	47	1	no	—
188	C₂×#97	—	47	2	no	—
189	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
190	#391	—	94	2	no	—
191	#387	—	94	2	no	—
192	#387	—	94	2	no	—
193	#384	—	47	2	no	—
194	#382	—	47	2	no	—
195	#382	C₂×D₄₇	1	2	yes	—
196	C₄×C₄⋊C₄	—	47	2	no	—
197	C₄×C₄⋊C₄	—	47	2	no	—
198	C₄×C₄⋊C₄	—	47	2	no	—
199	C₄×C₄⋊C₄	—	94	2	no	—
200	#340	—	47	2	no	—
201	#340	—	47	2	no	—
202	#340	—	47	2	no	—
203	C₄₇	#22730	1	1	yes	characteristic, complemented, Sylow
204	C₂²×Q₈	—	47	1	no	—
205	C₂²×C₈	—	47	2	no	—
206	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
208	C₄×Q₈	—	47	4	no	—
209	C₄×Q₈	—	47	4	no	—
210	C₂×C₄⋊C₄	—	47	2	no	—
211	C₂×C₄⋊C₄	—	47	1	no	—
212	C₂×C₄⋊C₄	—	47	2	no	—
213	C₂×C₄⋊C₄	—	47	2	no	—
214	C₂×C₄⋊C₄	—	94	2	no	—
215	C₂×C₄⋊C₄	—	94	2	no	—
216	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
217	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
218	C₂×C₄²	—	47	2	no	—
219	C₂×C₄²	—	94	2	no	—
220	C₂×C₄²	—	94	2	no	—
221	C₂×C₄²	—	94	2	no	—
222	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
223	#107	—	2	4	no	—
224	#107	—	188	2	no	—
225	#106	—	4	1	no	—
226	#106	—	4	1	no	—
227	#106	—	47	4	no	—
228	#106	—	94	2	no	—
229	#106	—	188	2	no	—
230	C₄⋊C₈	—	47	8	no	—
231	C₄⋊C₈	—	47	4	no	—
232	#97	—	47	8	no	—
233	C₄×C₈	C₄×D₄₇	1	4	yes	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	#95	—	94	2	no	—
237	#95	—	94	2	no	—
238	#95	—	94	2	no	—
239	#95	—	94	2	no	—
240	#95	—	94	2	no	—
241	C₂×Q₈	—	47	4	no	—
242	C₂×Q₈	—	188	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	47	2	no	—
246	C₂²×C₄	—	47	2	no	—
247	C₂²×C₄	—	94	2	no	—
248	C₂²×C₄	—	94	2	no	—
249	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
250	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
251	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	47	4	no	—
255	C₂×C₈	—	94	2	no	—
256	C₂×C₈	—	94	2	no	—
257	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
258	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
259	C₂×C₈	#10625352	1	2	yes	—
260	C₂×C₈	Q₈×D₄₇	1	2	yes	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	47	8	no	—
266	C₄⋊C₄	—	47	4	no	—
267	C₄⋊C₄	—	47	4	no	—
268	C₄⋊C₄	—	94	2	no	—
269	C₄⋊C₄	—	94	2	no	—
270	C₄⋊C₄	—	94	2	no	—
271	C₄⋊C₄	—	376	2	no	—
272	C₄⋊C₄	—	188	4	no	—
273	C₄²	—	47	4	no	—
274	C₄²	—	47	4	no	—
275	C₄²	—	376	2	no	—
276	C₄²	—	376	2	no	—
277	C₄²	—	188	4	no	—
278	C₄²	C₂×C₄×D₄₇	1	2	yes	—
279	C₄²	Q₈×D₄₇	1	2	yes	—
280	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
281	Q₈	—	188	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	47	4	no	—
291	C₂×C₄	—	47	4	no	—
292	C₂×C₄	—	94	2	no	—
293	C₂×C₄	—	94	2	no	—
294	C₂×C₄	—	94	2	no	—
295	C₂×C₄	—	94	2	no	—
296	C₂×C₄	—	94	4	no	—
297	C₂×C₄	—	188	2	no	—
298	C₂×C₄	—	188	2	no	—
299	C₂×C₄	—	188	2	no	—
300	C₂×C₄	—	188	2	no	—
301	C₂×C₄	—	188	2	no	—
302	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
303	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
304	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
305	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
306	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
307	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
308	C₂×C₄	#13094069	1	1	yes	characteristic
309	C₂×C₄	#13094072	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	94	4	no	—
312	C₈	#13093989	1	4	yes	—
313	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
314	C₂²	#15689788	1	1	yes	characteristic
315	C₂²	#15689793	1	1	yes	characteristic
316	C₂²	#15689920	1	2	yes	—
317	C₂²	#15689930	1	2	yes	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	94	4	no	—
323	C₄	—	94	4	no	—
324	C₄	—	376	2	no	—
325	C₄	—	188	4	no	—
326	C₄	C₂×#13093989	1	1	yes	characteristic
327	C₄	#15689604	1	1	yes	characteristic
328	C₄	#15689848	1	1	yes	characteristic
329	C₄	#15689853	1	1	yes	characteristic
330	C₂	#20246300	1	1	yes	characteristic
331	C₂	#20251976	1	1	yes	characteristic
332	C₂	#20251980	1	1	yes	characteristic
333	C₂	#20256339	1	2	yes	—
334	C₂	#20256460	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1846	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1846	D₄₇
hypocenter	C₄₇	#22730
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
11. #15689253	C₄	semidirect
52. #13093956	Q₈	semidirect
203. C₄₇	#22730	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689253	2
#20246340	2
#20244157	2
#20244154	2
#20244148	2
#20243555	2
#20243287	2
C₄×#13093956	2
C₄₇×#1846	2
#22730	47