Subgroups — #26214411

Order 12032.

986 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20243247	C₂	1	4	yes	maximal, complemented
4	#20243156	C₂	1	4	yes	maximal
5	#20243133	C₂	1	1	yes	maximal, characteristic, complemented
6	#20243133	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#13093934	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄₇×#431	C₂	1	1	yes	maximal, characteristic
10	C₂×#13093983	C₂²	1	2	yes	—
11	#15689176	C₄	1	8	yes	complemented
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×#13093934	C₂²	1	4	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
18	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
19	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
20	C₄×C₂²⋊Dic₄₇	C₄	1	2	yes	—
21	C₄×C₂²⋊Dic₄₇	C₄	1	2	yes	—
22	#15688969	C₂²	1	2	yes	—
23	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
24	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
25	#15688877	C₂²	1	4	yes	—
26	#15688877	C₂²	1	4	yes	—
27	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#431	C₄	1	8	yes	complemented
29	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	#13093983	—	2	8	no	—
36	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
38	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
43	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
44	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
45	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
46	#13093934	—	2	8	no	—
47	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
48	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
49	C₈×Dic₄₇	—	2	8	no	—
50	C₄₇⋊#95	C₂×C₄	1	4	yes	—
51	C₄₇⋊#95	C₂×C₄	1	4	yes	—
52	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
54	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
60	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
64	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
65	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
66	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
67	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	—	2	4	no	—
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	—	2	4	no	—
71	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
72	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
82	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₄₇	—	2	4	no	—
84	C₂×C₈×C₄₇	—	2	4	no	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
87	C₄²×C₄₇	C₄²	1	4	yes	—
88	C₄²×C₄₇	C₄²	1	4	yes	—
89	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
90	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
91	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
93	C₂²⋊Dic₄₇	—	2	8	no	—
94	C₄⋊Dic₄₇	—	2	8	no	—
95	C₄×Dic₄₇	—	2	4	no	—
96	C₄×Dic₄₇	—	2	4	no	—
97	C₄×Dic₄₇	—	2	4	no	—
98	C₄×Dic₄₇	—	2	4	no	—
99	C₄×Dic₄₇	—	2	4	no	—
100	C₄×Dic₄₇	#41	1	8	yes	—
101	C₂×C₄₇⋊C₈	—	2	4	no	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
104	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
105	C₂³×C₄₇	—	2	2	no	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	—	2	2	no	—
108	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
109	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
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	4	no	—
115	C₂×C₄×C₄₇	—	2	8	no	—
116	C₂×C₄×C₄₇	—	2	8	no	—
117	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₄₇	#131	1	4	yes	—
123	C₂×C₄×C₄₇	#131	1	4	yes	—
124	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	2	4	no	—
129	C₂×Dic₄₇	—	2	4	no	—
130	C₂×Dic₄₇	—	2	4	no	—
131	C₂×Dic₄₇	—	2	4	no	—
132	C₂×Dic₄₇	#117	1	8	yes	—
133	C₈×C₄₇	—	2	4	no	—
134	C₈×C₄₇	—	2	4	no	—
135	C₄₇⋊C₈	—	2	4	no	—
136	C₄₇⋊C₈	—	2	4	no	—
137	#20462	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	4	4	no	—
142	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×#117	1	2	yes	—
144	C₂²×C₄₇	C₂×#131	1	2	yes	—
145	C₂²×C₄₇	C₂×#131	1	2	yes	—
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	#405	1	4	yes	—
150	C₄×C₄₇	#405	1	4	yes	—
151	C₄×C₄₇	C₄×#41	1	2	yes	—
152	C₄×C₄₇	C₄×#41	1	2	yes	—
153	Dic₄₇	—	2	8	no	—
154	Dic₄₇	—	4	4	no	—
155	C₂×#431	—	47	1	no	—
156	C₂×#431	D₄₇	1	1	yes	characteristic
157	C₄×#117	—	47	1	no	—
158	#1768	—	47	4	no	—
159	#1750	—	47	4	no	—
160	#1744	—	47	1	no	—
161	#1744	—	47	1	no	—
162	C₄×#97	—	47	1	no	—
163	C₄²×C₈	—	47	1	no	—
164	C₂×C₄₇	—	4	2	no	—
165	C₂×C₄₇	C₄×#117	1	1	yes	characteristic
166	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
168	C₂×C₄₇	#2922	1	4	yes	—
169	C₂×#117	—	47	2	no	—
170	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
171	#431	—	47	8	no	—
172	#431	Dic₄₇	1	8	yes	complemented
173	C₂×C₄⋊C₈	—	47	2	no	—
174	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
175	C₂×C₂²⋊C₈	—	47	2	no	—
176	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
177	C₂×#97	—	47	4	no	—
178	C₂×C₄×C₈	—	47	1	no	—
179	C₂×C₄×C₈	—	47	1	no	—
180	C₂×C₄×C₈	—	47	4	no	—
181	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
182	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
183	C₄×C₄⋊C₄	—	47	2	no	—
184	C₄×C₄⋊C₄	—	47	2	no	—
185	C₄×C₂²⋊C₄	—	47	2	no	—
186	C₄×C₂²⋊C₄	—	47	2	no	—
187	#376	—	47	2	no	—
188	C₄³	—	47	1	no	—
189	C₄³	—	47	1	no	—
190	#336	—	47	4	no	—
191	#336	—	47	4	no	—
192	C₄₇	#20462	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
194	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
195	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
196	C₂²×C₈	—	47	2	no	—
197	C₂²×C₈	—	47	2	no	—
198	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
199	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
200	#117	—	94	8	no	—
201	C₂×C₄⋊C₄	—	47	4	no	—
202	C₂×C₂²⋊C₄	—	47	4	no	—
203	C₂×C₄²	—	47	2	no	—
204	C₂×C₄²	—	47	2	no	—
205	C₂×C₄²	—	47	2	no	—
206	C₂×C₄²	—	47	2	no	—
207	C₂×C₄²	—	47	2	no	—
208	C₂×C₄²	—	47	2	no	—
209	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₄×D₄₇	1	4	yes	—
212	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
213	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
214	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
215	C₄⋊C₈	—	47	8	no	—
216	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
217	C₂²⋊C₈	—	47	8	no	—
218	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
219	#97	—	94	8	no	—
220	C₄×C₈	—	47	4	no	—
221	C₄×C₈	—	47	4	no	—
222	C₄×C₈	—	94	8	no	—
223	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
224	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
225	#95	—	47	4	no	—
226	#95	—	47	4	no	—
227	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	47	4	no	—
232	C₂²×C₄	—	47	4	no	—
233	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
234	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
235	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
239	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
242	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	47	4	no	—
246	C₂×C₈	—	47	4	no	—
247	C₂×C₈	—	94	4	no	—
248	C₂×C₈	—	94	4	no	—
249	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
250	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
251	C₄⋊C₄	—	94	8	no	—
252	C₂²⋊C₄	—	94	8	no	—
253	C₄²	—	47	8	no	—
254	C₄²	—	94	4	no	—
255	C₄²	—	94	4	no	—
256	C₄²	—	94	4	no	—
257	C₄²	—	94	4	no	—
258	C₄²	—	94	4	no	—
259	C₄²	C₄×Dic₄₇	1	4	yes	—
260	C₄²	C₄×Dic₄₇	1	4	yes	—
261	C₄²	C₂×C₄×D₄₇	1	2	yes	—
262	C₄²	C₂×C₄×D₄₇	1	2	yes	—
263	C₄²	C₂²×Dic₄₇	1	2	yes	—
264	C₄²	C₂²×Dic₄₇	1	2	yes	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
269	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	47	8	no	—
278	C₂×C₄	—	94	4	no	—
279	C₂×C₄	—	94	4	no	—
280	C₂×C₄	—	94	4	no	—
281	C₂×C₄	—	94	4	no	—
282	C₂×C₄	—	94	4	no	—
283	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
284	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
285	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
286	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
287	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
288	C₂×C₄	#13094030	1	4	yes	—
289	C₂×C₄	#13094032	1	4	yes	—
290	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
291	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
292	C₂×C₄	#41×D₄₇	1	4	yes	—
293	C₈	—	2	4	no	—
294	C₈	—	2	4	no	—
295	C₈	—	94	4	no	—
296	C₈	—	94	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	4	4	no	—
301	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
302	C₂²	#117×D₄₇	1	2	yes	—
303	C₂²	C₂×#13094030	1	2	yes	—
304	C₂²	C₂×#13094032	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	94	8	no	—
309	C₄	—	188	4	no	—
310	C₄	#15689078	1	4	yes	—
311	C₄	#15689081	1	4	yes	—
312	C₄	#15689591	1	2	yes	—
313	C₄	#41×Dic₄₇	1	2	yes	—
314	C₂	—	4	2	no	—
315	C₂	#117×Dic₄₇	1	1	yes	characteristic
316	C₂	C₂×#15689078	1	1	yes	characteristic
317	C₂	C₂×#15689081	1	1	yes	characteristic
318	C₂	#20252027	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#431	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#431	D₄₇
hypocenter	C₄₇	#20462
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20243247	C₂	semidirect
5. #20243133	C₂	semidirect
6. #20243133	C₂	semidirect
7. C₄×#13093934	C₂	semidirect
8. C₄×C₈×Dic₄₇	C₂	semidirect
11. #15689176	C₄	semidirect
28. C₄₇×#431	C₄	semidirect
172. #431	Dic₄₇	semidirect
192. C₄₇	#20462	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
C₄×#13093983	2
#20243247	2
#20243156	2
#20243133	2
C₄×#13093934	2
C₄×C₈×Dic₄₇	2
C₂×C₄₇×#431	2
#20462	47