Subgroups — #26316096

Order 12032.

642 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20249363	C₂	1	1	yes	maximal, characteristic
2	#20244955	C₂	1	2	yes	maximal
3	#20244631	C₂	1	2	yes	maximal
4	#20244081	C₂	1	2	yes	maximal
5	#20244067	C₂	1	2	yes	maximal
6	#20243403	C₂	1	2	yes	maximal
7	#20243393	C₂	1	2	yes	maximal
8	#20243245	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#2327	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×#13093948	C₂²	1	2	yes	—
12	C₂×#13093948	C₂²	1	2	yes	—
13	C₂×#13093948	C₂²	1	2	yes	—
14	C₂×#13093948	C₂²	1	2	yes	—
15	C₂×#13093935	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	#15689065	—	2	2	no	—
18	#15689057	—	2	2	no	—
19	#15689021	C₂²	1	2	yes	—
20	#15689021	C₂²	1	2	yes	—
21	#15688993	C₂²	1	1	yes	characteristic
22	#15688989	C₂²	1	1	yes	characteristic
23	#15688885	C₂²	1	2	yes	—
24	#15688885	C₂²	1	2	yes	—
25	#15688885	C₂²	1	2	yes	—
26	#15688885	C₂²	1	2	yes	—
27	#15688878	C₂²	1	2	yes	—
28	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
30	C₄₇×#391	C₂²	1	2	yes	—
31	C₄₇×#387	—	2	2	no	—
32	C₄₇×#387	—	2	2	no	—
33	C₄₇×#382	C₂²	1	1	yes	characteristic
34	C₄₇×#376	C₂²	1	1	yes	characteristic
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	C₂×C₄⋊Dic₄₇	—	2	2	no	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
40	C₂×Dic₄₇⋊C₄	—	2	2	no	—
41	C₂×Dic₄₇⋊C₄	—	2	2	no	—
42	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
43	#13093948	—	2	4	no	—
44	#13093948	—	2	4	no	—
45	#13093948	—	4	2	no	—
46	#13093948	—	4	2	no	—
47	#13093935	—	4	1	no	—
48	C₄×C₄₇⋊C₈	—	4	1	no	—
49	C₄₇⋊#95	—	2	2	no	—
50	C₄₇⋊#95	—	2	2	no	—
51	C₄₇⋊#95	—	2	2	no	—
52	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
53	C₄⋊Q₈×C₄₇	—	2	4	no	—
54	C₄×Q₈×C₄₇	—	2	4	no	—
55	C₄×Q₈×C₄₇	—	2	4	no	—
56	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
59	C₂×C₄²×C₄₇	—	2	2	no	—
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	#95×C₄₇	—	2	2	no	—
64	#95×C₄₇	—	2	2	no	—
65	#95×C₄₇	—	2	2	no	—
66	#95×C₄₇	D₄	1	2	yes	—
67	#95×C₄₇	D₄	1	2	yes	—
68	C₂×Q₈×C₄₇	—	2	2	no	—
69	C₂×Q₈×C₄₇	—	2	2	no	—
70	C₂×Q₈×C₄₇	—	2	2	no	—
71	C₂×Q₈×C₄₇	—	2	2	no	—
72	C₂×Q₈×C₄₇	—	2	2	no	—
73	C₂×Q₈×C₄₇	#41	1	4	yes	—
74	C₂²×C₄×C₄₇	—	2	2	no	—
75	C₂²×C₄×C₄₇	—	2	2	no	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₄₇	—	2	2	no	—
82	C₂²×Dic₄₇	—	2	2	no	—
83	C₄⋊C₄×C₄₇	—	2	8	no	—
84	C₄⋊C₄×C₄₇	—	2	2	no	—
85	C₄⋊C₄×C₄₇	—	2	2	no	—
86	C₄⋊C₄×C₄₇	—	4	4	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	—	2	1	no	—
89	C₄²×C₄₇	—	2	1	no	—
90	C₄²×C₄₇	—	4	4	no	—
91	C₄²×C₄₇	#41	1	4	yes	—
92	C₄⋊Dic₄₇	—	2	2	no	—
93	C₄⋊Dic₄₇	—	2	2	no	—
94	C₄⋊Dic₄₇	—	2	2	no	—
95	C₄⋊Dic₄₇	—	2	2	no	—
96	C₄⋊Dic₄₇	—	8	2	no	—
97	Dic₄₇⋊C₄	—	8	2	no	—
98	Dic₄₇⋊C₄	—	8	2	no	—
99	C₂×C₄₇⋊C₈	—	2	2	no	—
100	C₂×C₄₇⋊C₈	—	2	2	no	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	—	2	2	no	—
103	C₂×C₄₇⋊C₈	—	2	2	no	—
104	C₂×C₄₇⋊C₈	—	2	2	no	—
105	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
106	Q₈×C₄₇	—	2	4	no	—
107	Q₈×C₄₇	—	2	4	no	—
108	Q₈×C₄₇	—	4	2	no	—
109	Q₈×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	1	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	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	#124	1	4	yes	—
126	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
128	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	—	4	2	no	—
134	C₂×Dic₄₇	—	4	2	no	—
135	C₂×Dic₄₇	—	4	2	no	—
136	C₄₇⋊C₈	—	4	2	no	—
137	C₄₇⋊C₈	—	4	2	no	—
138	#29138	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	#525	1	1	yes	characteristic
140	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
141	C₂²×C₄₇	#537	1	1	yes	characteristic
142	C₂²×C₄₇	#577	1	2	yes	—
143	C₂²×C₄₇	#578	1	2	yes	—
144	C₄×C₄₇	—	2	4	no	—
145	C₄×C₄₇	—	4	4	no	—
146	C₄×C₄₇	—	4	2	no	—
147	C₄×C₄₇	—	4	1	no	—
148	C₄×C₄₇	—	4	1	no	—
149	C₄×C₄₇	—	8	2	no	—
150	C₄×C₄₇	#552	1	2	yes	—
151	C₄×C₄₇	#553	1	2	yes	—
152	Dic₄₇	—	8	2	no	—
153	Dic₄₇	—	8	2	no	—
154	#2647	—	47	1	no	—
155	#2327	D₄₇	1	1	yes	characteristic
156	#2101	—	47	2	no	—
157	#2043	—	47	2	no	—
158	#1935	—	47	2	no	—
159	#1933	—	47	2	no	—
160	#1805	—	47	2	no	—
161	#1803	—	47	2	no	—
162	#1768	—	47	1	no	—
163	C₂×C₄₇	#2562	1	1	yes	characteristic
164	C₂×C₄₇	#3121	1	2	yes	—
165	C₂×C₄₇	#3137	1	2	yes	—
166	C₂×C₄₇	#3260	1	1	yes	characteristic
167	C₂×C₄₇	#3261	1	1	yes	characteristic
168	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
169	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
170	C₂×C₄⋊C₈	—	47	2	no	—
171	C₂×#103	—	47	2	no	—
172	C₂×#103	—	47	2	no	—
173	C₂×#103	—	47	2	no	—
174	C₂×#103	—	47	2	no	—
175	C₂×#97	—	47	1	no	—
176	C₂×C₄×C₈	—	47	1	no	—
177	#400	—	94	2	no	—
178	#398	—	94	2	no	—
179	#391	C₂×D₄₇	1	2	yes	—
180	#389	—	47	2	no	—
181	#389	—	47	2	no	—
182	#387	—	2	2	no	—
183	#387	—	2	2	no	—
184	#384	—	47	1	no	—
185	#382	—	47	1	no	—
186	#382	C₂×D₄₇	1	1	yes	characteristic
187	#376	C₂×D₄₇	1	1	yes	characteristic
188	#340	—	47	2	no	—
189	#340	—	47	2	no	—
190	#340	—	47	2	no	—
191	#340	—	47	2	no	—
192	#336	—	47	2	no	—
193	C₄₇	#29138	1	1	yes	characteristic, complemented, Sylow
194	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
195	C₂²×C₈	—	47	2	no	—
196	C₂²×C₈	—	47	2	no	—
197	C₄⋊Q₈	—	2	4	no	—
198	C₄×Q₈	—	2	4	no	—
199	C₄×Q₈	—	2	4	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	47	2	no	—
203	C₂×C₄⋊C₄	—	47	2	no	—
204	C₂×C₄⋊C₄	—	94	2	no	—
205	C₂×C₄⋊C₄	—	94	2	no	—
206	C₂×C₄⋊C₄	—	94	2	no	—
207	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
210	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
212	C₄⋊C₈	—	94	4	no	—
213	#103	—	94	4	no	—
214	#103	—	94	4	no	—
215	#103	—	188	2	no	—
216	#103	—	188	2	no	—
217	#97	—	188	1	no	—
218	C₄×C₈	—	188	1	no	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	94	2	no	—
223	#95	—	94	2	no	—
224	#95	—	94	2	no	—
225	#95	C₂²⋊D₄₇	1	2	yes	—
226	#95	C₂²⋊D₄₇	1	2	yes	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	#10625350	1	4	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	94	2	no	—
236	C₂²×C₄	—	94	2	no	—
237	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
240	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
241	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
242	C₂×C₈	—	94	2	no	—
243	C₂×C₈	—	94	2	no	—
244	C₂×C₈	—	94	2	no	—
245	C₂×C₈	—	94	2	no	—
246	C₂×C₈	—	94	2	no	—
247	C₂×C₈	—	94	2	no	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	—	94	2	no	—
253	C₄⋊C₄	—	94	2	no	—
254	C₄⋊C₄	—	94	2	no	—
255	C₄⋊C₄	—	94	2	no	—
256	C₄⋊C₄	—	376	2	no	—
257	C₄⋊C₄	—	376	2	no	—
258	C₄⋊C₄	—	376	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	4	4	no	—
263	C₄²	#10625352	1	4	yes	—
264	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
265	Q₈	—	2	4	no	—
266	Q₈	—	2	4	no	—
267	Q₈	—	4	2	no	—
268	Q₈	—	4	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	188	2	no	—
285	C₂×C₄	—	188	2	no	—
286	C₂×C₄	—	188	2	no	—
287	C₂×C₄	—	188	2	no	—
288	C₂×C₄	—	188	2	no	—
289	C₂×C₄	—	188	2	no	—
290	C₂×C₄	#13094011	1	4	yes	—
291	C₂×C₄	C₂×#10625352	1	2	yes	—
292	C₂×C₄	C₂×#10625350	1	2	yes	—
293	C₂×C₄	#13094070	1	1	yes	characteristic
294	C₂×C₄	#13094071	1	1	yes	characteristic
295	C₈	—	188	2	no	—
296	C₈	—	188	2	no	—
297	C₂²	#15689634	1	1	yes	characteristic
298	C₂²	C₂×#13094011	1	1	yes	characteristic
299	C₂²	#15689701	1	1	yes	characteristic
300	C₂²	#15689931	1	2	yes	—
301	C₂²	#15689938	1	2	yes	—
302	C₄	—	2	4	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	2	no	—
306	C₄	—	4	4	no	—
307	C₄	—	8	2	no	—
308	C₄	—	376	2	no	—
309	C₄	—	376	2	no	—
310	C₄	#15689813	1	2	yes	—
311	C₄	#15689819	1	2	yes	—
312	C₂	#20248554	1	1	yes	characteristic
313	C₂	#20253617	1	2	yes	—
314	C₂	#20253753	1	2	yes	—
315	C₂	#20254829	1	1	yes	characteristic
316	C₂	#20254838	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2327	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	#2327	D₄₇
hypocenter	C₄₇	#29138
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #2327

Complements

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

N	H ≅ G/N	type
193. C₄₇	#29138	semidirect

Maximal subgroups

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

subgroup	index
#20249363	2
#20244955	2
#20244631	2
#20244081	2
#20244067	2
#20243403	2
#20243393	2
#20243245	2
C₄₇×#2327	2
#29138	47