Subgroups — #26132765

Order 12032.

384 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689539	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688944	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245146	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15688906	C₂	1	1	yes	maximal, characteristic
5	#20243225	C₂	1	1	yes	maximal, characteristic
6	#20241476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094030	C₂²	1	1	yes	characteristic
9	#15689539	—	2	2	no	—
10	#15689539	—	2	2	no	—
11	C₂×#13093964	C₂²	1	1	yes	characteristic
12	C₂×#13093964	C₂²	1	1	yes	characteristic
13	C₂×#13093963	—	2	1	no	—
14	C₂×C₁₆×D₄₇	—	2	1	no	—
15	C₂×#13093952	C₄	1	1	yes	characteristic
16	#15689120	C₄	1	1	yes	characteristic
17	#15689078	C₂²	1	1	yes	characteristic
18	#15688944	—	2	2	no	—
19	Dic₄₇⋊C₁₆	—	2	1	no	—
20	Dic₄₇⋊C₁₆	—	2	1	no	—
21	#15688906	—	2	2	no	—
22	#15688888	C₄	1	1	yes	characteristic
23	#15688883	C₄	1	1	yes	characteristic
24	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#10625350	C₂×C₄	1	1	yes	characteristic
28	#13094030	—	2	1	no	—
29	#13094030	—	2	1	no	—
30	#13094030	D₄	1	2	yes	—
31	#13094030	Q₈	1	2	yes	—
32	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
33	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
34	C₂×#10625324	—	2	1	no	—
35	C₂×C₈×D₄₇	—	2	1	no	—
36	#13093983	C₂×C₄	1	1	yes	characteristic
37	#13093964	—	2	1	no	—
38	#13093964	—	2	1	no	—
39	#13093964	—	2	1	no	—
40	#13093964	D₄	1	2	yes	—
41	#13093963	—	2	4	no	—
42	C₂×C₄₇⋊C₁₆	—	2	1	no	—
43	C₂×C₄₇⋊C₁₆	—	2	1	no	—
44	C₁₆×D₄₇	—	2	4	no	—
45	#13093952	—	2	2	no	—
46	#13093949	C₈	1	1	yes	characteristic
47	#13093949	C₈	1	1	yes	characteristic
48	#13093946	C₈	1	1	yes	characteristic
49	#13093946	C₈	1	1	yes	characteristic
50	#13093934	—	2	1	no	—
51	C₈×Dic₄₇	—	2	1	no	—
52	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₁₆×C₄₇	—	2	1	no	—
56	C₂×C₁₆×C₄₇	—	2	1	no	—
57	C₂×C₁₆×C₄₇	—	2	1	no	—
58	C₂×C₁₆×C₄₇	—	2	1	no	—
59	C₂×C₁₆×C₄₇	—	2	1	no	—
60	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
61	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
62	C₄×C₈×C₄₇	—	2	1	no	—
63	C₄×C₈×C₄₇	—	2	1	no	—
64	C₂²×C₄×C₄₇	—	2	1	no	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊D₄₇	—	2	1	no	—
67	#10625350	—	2	1	no	—
68	#10625350	—	2	1	no	—
69	#10625350	C₄⋊C₄	1	2	yes	—
70	#10625350	M₁₆	1	2	yes	—
71	C₂×D₁₈₈	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₄₇	—	2	1	no	—
73	C₂×Dic₉₄	C₂×C₈	1	1	yes	characteristic
74	C₂×C₈×C₄₇	—	2	1	no	—
75	C₂×C₈×C₄₇	—	2	1	no	—
76	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
77	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
83	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
84	C₄²×C₄₇	—	2	1	no	—
85	C₄²×C₄₇	—	2	1	no	—
86	C₂²⋊Dic₄₇	—	2	1	no	—
87	C₄⋊Dic₄₇	C₂×C₈	1	1	yes	characteristic
88	C₄⋊Dic₄₇	C₂×C₈	1	1	yes	characteristic
89	C₄×Dic₄₇	—	2	1	no	—
90	#10625329	—	2	1	no	—
91	#10625329	—	2	1	no	—
92	#10625329	—	2	1	no	—
93	#10625329	C₄⋊C₄	1	2	yes	—
94	C₂×C₄₇⋊C₈	—	2	1	no	—
95	C₂×C₄₇⋊C₈	—	2	1	no	—
96	#10625324	—	2	2	no	—
97	#10625324	—	2	2	no	—
98	C₈×D₄₇	—	2	2	no	—
99	C₈×D₄₇	—	2	2	no	—
100	C₁₆×C₄₇	—	2	1	no	—
101	C₁₆×C₄₇	—	2	1	no	—
102	C₁₆×C₄₇	—	4	1	no	—
103	C₄₇⋊C₁₆	—	2	2	no	—
104	C₄₇⋊C₁₆	—	4	1	no	—
105	C₂³×C₄₇	#131	1	1	yes	characteristic
106	C₂²×D₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	4	1	no	—
109	C₂×C₄×C₄₇	—	4	1	no	—
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
116	C₂²⋊D₄₇	—	2	2	no	—
117	C₂²⋊D₄₇	—	2	2	no	—
118	C₂×Dic₄₇	—	2	1	no	—
119	C₂×Dic₄₇	—	2	1	no	—
120	C₂×Dic₄₇	—	2	1	no	—
121	D₁₈₈	—	2	1	no	—
122	D₁₈₈	C₄⋊C₈	1	2	yes	—
123	C₄×D₄₇	—	2	2	no	—
124	C₄×D₄₇	—	2	2	no	—
125	Dic₉₄	—	2	1	no	—
126	Dic₉₄	C₄⋊C₈	1	2	yes	—
127	C₈×C₄₇	—	2	1	no	—
128	C₈×C₄₇	—	2	1	no	—
129	C₈×C₄₇	#120	1	1	yes	characteristic
130	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
131	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
132	C₈×C₄₇	#123	1	1	yes	characteristic
133	C₄₇⋊C₈	—	2	2	no	—
134	C₄₇⋊C₈	—	4	1	no	—
135	#12917	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	1	no	—
137	C₂²×C₄₇	—	2	1	no	—
138	C₂²×C₄₇	#424	1	1	yes	characteristic
139	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
140	C₂²×C₄₇	#435	1	1	yes	characteristic
141	C₂×D₄₇	—	2	2	no	—
142	C₂×D₄₇	—	4	1	no	—
143	C₂×D₄₇	—	4	1	no	—
144	C₄×C₄₇	—	4	2	no	—
145	C₄×C₄₇	#385	1	1	yes	characteristic
146	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
147	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
148	C₄×C₄₇	#435	1	1	yes	characteristic
149	Dic₄₇	—	2	2	no	—
150	Dic₄₇	—	4	1	no	—
151	Dic₄₇	—	4	1	no	—
152	C₂×#504	—	47	1	no	—
153	C₂×#364	—	47	1	no	—
154	#2151	—	47	1	no	—
155	C₂×#350	—	47	1	no	—
156	#1764	—	47	1	no	—
157	#1379	—	47	1	no	—
158	#1374	D₄₇	1	1	yes	characteristic, complemented
159	C₂×C₄₇	—	2	1	no	—
160	C₂×C₄₇	—	2	1	no	—
161	C₂×C₄₇	#1850	1	1	yes	characteristic
162	C₂×C₄₇	#2170	1	1	yes	characteristic
163	C₂×C₄₇	#2171	1	1	yes	characteristic
164	D₄₇	—	4	2	no	—
165	C₂×#131	—	47	1	no	—
166	#504	—	94	2	no	—
167	#504	—	94	2	no	—
168	C₂×M₃₂	—	47	1	no	—
169	C₂×M₃₂	—	47	1	no	—
170	C₂×M₃₂	—	94	1	no	—
171	C₂²×C₁₆	—	94	1	no	—
172	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
173	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
174	C₂×#104	—	47	1	no	—
175	#416	—	47	1	no	—
176	#405	—	47	1	no	—
177	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
178	#364	—	94	2	no	—
179	C₄⋊C₁₆	—	94	1	no	—
180	C₄⋊C₁₆	—	94	1	no	—
181	#350	—	94	2	no	—
182	#341	—	47	1	no	—
183	#339	—	47	1	no	—
184	C₄₇	#12917	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	47	1	no	—
186	#131	—	47	2	no	—
187	#131	—	47	2	no	—
188	#131	—	94	1	no	—
189	#131	—	94	1	no	—
190	C₂×M₁₆	—	47	1	no	—
191	C₂×M₁₆	—	47	1	no	—
192	C₂×M₁₆	—	94	1	no	—
193	C₂²×C₈	—	94	1	no	—
194	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
196	#117	—	47	1	no	—
197	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
198	M₃₂	—	47	2	no	—
199	M₃₂	—	94	4	no	—
200	M₃₂	—	94	1	no	—
201	M₃₂	—	94	1	no	—
202	M₃₂	—	94	1	no	—
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	94	4	no	—
209	C₂×C₁₆	—	94	1	no	—
210	C₂×C₁₆	—	94	1	no	—
211	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
212	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
213	#104	—	94	2	no	—
214	#103	—	47	1	no	—
215	#103	—	47	1	no	—
216	#102	—	47	1	no	—
217	#102	—	47	1	no	—
218	#97	—	94	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	94	1	no	—
222	#41	—	47	2	no	—
223	#41	—	47	2	no	—
224	#41	—	94	1	no	—
225	#41	—	94	1	no	—
226	C₂×Q₈	—	47	1	no	—
227	C₂×D₄	—	47	1	no	—
228	C₂×D₄	—	94	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	94	1	no	—
231	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
232	M₁₆	—	47	2	no	—
233	M₁₆	—	94	1	no	—
234	M₁₆	—	94	2	no	—
235	M₁₆	—	94	2	no	—
236	M₁₆	—	94	1	no	—
237	M₁₆	—	94	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	94	1	no	—
241	C₂×C₈	—	94	2	no	—
242	C₂×C₈	—	94	2	no	—
243	C₂×C₈	—	94	1	no	—
244	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
245	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
246	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
247	C₂×C₈	#10625350	1	1	yes	characteristic
248	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
249	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
250	C₂×C₈	#10625352	1	1	yes	characteristic
251	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
252	C₄⋊C₄	—	47	1	no	—
253	C₄⋊C₄	—	47	1	no	—
254	C₂²⋊C₄	—	94	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	94	1	no	—
258	C₁₆	—	2	1	no	—
259	C₁₆	—	2	1	no	—
260	C₁₆	—	4	1	no	—
261	C₁₆	—	94	2	no	—
262	C₁₆	—	188	1	no	—
263	C₂³	—	94	1	no	—
264	C₂³	#13094030	1	1	yes	characteristic
265	Q₈	—	47	2	no	—
266	Q₈	—	94	1	no	—
267	D₄	—	47	2	no	—
268	D₄	—	94	2	no	—
269	D₄	—	94	2	no	—
270	D₄	—	94	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	94	1	no	—
275	C₂×C₄	—	94	2	no	—
276	C₂×C₄	—	94	2	no	—
277	C₂×C₄	—	94	1	no	—
278	C₂×C₄	—	94	1	no	—
279	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
280	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
282	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
283	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
284	C₂×C₄	#13094032	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	94	2	no	—
288	C₈	—	188	1	no	—
289	C₈	#13093994	1	1	yes	characteristic
290	C₈	#13094002	1	1	yes	characteristic
291	C₈	#13094004	1	1	yes	characteristic
292	C₈	#13094006	1	1	yes	characteristic
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	94	2	no	—
296	C₂²	—	188	1	no	—
297	C₂²	—	188	1	no	—
298	C₂²	#15689148	1	1	yes	characteristic
299	C₂²	C₈×D₁₈₈	1	1	yes	characteristic
300	C₂²	#15689197	1	1	yes	characteristic
301	C₄	—	4	2	no	—
302	C₄	—	94	2	no	—
303	C₄	—	188	1	no	—
304	C₄	—	188	1	no	—
305	C₄	#15688997	1	1	yes	characteristic
306	C₄	C₄⋊C₈×D₄₇	1	1	yes	characteristic
307	C₄	(C₂²⋊D₄₇)⋊C₈	1	1	yes	characteristic
308	C₄	#15689196	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	188	2	no	—
312	C₂	#20243567	1	1	yes	characteristic
313	C₂	#20245226	1	1	yes	characteristic
314	C₂	#20245231	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1374	C₂
socle	C₂²×C₄₇	C₈×D₄
prime radical	C₂×#10625350	C₂×C₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1374	D₄₇
hypocenter	C₄₇	#12917
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ #1374

Complements

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

N	H ≅ G/N	type
2. C₂×#15688944	C₂	semidirect
3. #20245146	C₂	semidirect
6. #20241476	C₂	semidirect
7. C₄₇×#1374	C₂	semidirect
158. #1374	D₄₇	semidirect
184. C₄₇	#12917	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689539	2
C₂×#15688944	2
#20245146	2
C₂×#15688906	2
#20243225	2
#20241476	2
C₄₇×#1374	2
#12917	47