Subgroups — #26125007

Order 12032.

878 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688878	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688877	C₂	1	4	yes	maximal
4	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
6	C₂²×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
7	C₂²×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
8	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
9	C₄×C₄⋊Dic₄₇	—	2	2	no	—
10	C₄×C₄⋊Dic₄₇	—	2	2	no	—
11	C₂×C₄₇⋊#95	C₄	1	2	yes	—
12	#15688878	—	2	2	no	—
13	#15688877	—	2	8	no	—
14	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#336	—	2	2	no	—
17	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
18	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄₇⋊C₈	—	2	4	no	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	4	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	Q₈	1	4	yes	—
24	C₂×C₄⋊Dic₄₇	—	2	2	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	4	no	—
28	C₂×C₄⋊Dic₄₇	C₈	1	4	yes	complemented
29	C₂×C₄×Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	4	no	—
32	C₂×C₄×Dic₄₇	—	2	4	no	—
33	C₄₇⋊#95	—	2	4	no	—
34	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₄₇	—	2	2	no	—
38	C₂²×C₈×C₄₇	—	2	2	no	—
39	C₂²×C₈×C₄₇	—	2	4	no	—
40	C₂²×C₈×C₄₇	—	2	4	no	—
41	C₂²×C₈×C₄₇	D₄	1	4	yes	—
42	C₂×C₄²×C₄₇	—	2	4	no	—
43	C₂×C₄²×C₄₇	D₄	1	2	yes	—
44	C₂×C₄²×C₄₇	D₄	1	2	yes	—
45	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₄₇	#41	1	2	yes	—
60	C₂²×C₄×C₄₇	#41	1	4	yes	—
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂²×Dic₄₇	—	2	2	no	—
63	C₂²×Dic₄₇	—	2	2	no	—
64	C₂²×Dic₄₇	—	2	4	no	—
65	C₂²×Dic₄₇	—	2	4	no	—
66	C₂²×Dic₄₇	—	2	2	no	—
67	C₂²×Dic₄₇	—	2	2	no	—
68	C₂²×Dic₄₇	—	2	4	no	—
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂²×Dic₄₇	C₂×C₈	1	4	yes	—
71	C₂²×Dic₄₇	M₁₆	1	4	yes	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	—	2	8	no	—
74	C₂×C₈×C₄₇	—	4	2	no	—
75	C₂×C₈×C₄₇	—	4	2	no	—
76	C₂×C₈×C₄₇	—	4	4	no	—
77	C₄²×C₄₇	—	2	4	no	—
78	C₄²×C₄₇	—	2	4	no	—
79	C₄⋊Dic₄₇	—	2	8	no	—
80	C₄⋊Dic₄₇	—	2	8	no	—
81	C₄×Dic₄₇	—	4	4	no	—
82	C₄×Dic₄₇	—	4	2	no	—
83	C₄×Dic₄₇	—	4	2	no	—
84	C₂×C₄₇⋊C₈	—	2	8	no	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	4	4	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂³×C₄₇	#117	1	2	yes	—
91	C₂³×C₄₇	C₄×D₄	1	2	yes	—
92	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
93	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
94	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
95	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
96	C₂³×C₄₇	#131	1	2	yes	—
97	C₂³×C₄₇	#131	1	2	yes	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
107	C₂×C₄×C₄₇	#121	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
109	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₄₇	#123	1	2	yes	—
111	C₂×C₄×C₄₇	#123	1	4	yes	—
112	C₂×C₄×C₄₇	#123	1	2	yes	—
113	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
115	C₂×C₄×C₄₇	#125	1	4	yes	—
116	C₂×C₄×C₄₇	#126	1	4	yes	—
117	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
119	C₂×Dic₄₇	—	2	4	no	—
120	C₂×Dic₄₇	—	2	4	no	—
121	C₂×Dic₄₇	—	2	8	no	—
122	C₂×Dic₄₇	—	2	8	no	—
123	C₂×Dic₄₇	—	4	2	no	—
124	C₂×Dic₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	4	4	no	—
128	C₂×Dic₄₇	—	4	4	no	—
129	C₈×C₄₇	—	4	4	no	—
130	C₄₇⋊C₈	—	4	4	no	—
131	#11346	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	#381	1	1	yes	characteristic
133	C₂²×C₄₇	#387	1	4	yes	—
134	C₂²×C₄₇	#390	1	1	yes	characteristic
135	C₂²×C₄₇	#391	1	1	yes	characteristic
136	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
137	C₂²×C₄₇	#407	1	1	yes	characteristic
138	C₂²×C₄₇	#408	1	1	yes	characteristic
139	C₂²×C₄₇	#408	1	1	yes	characteristic
140	C₂²×C₄₇	#431	1	2	yes	—
141	C₂²×C₄₇	#432	1	2	yes	—
142	C₂²×C₄₇	#433	1	4	yes	—
143	C₂²×C₄₇	C₈×D₄	1	2	yes	—
144	C₂²×C₄₇	#435	1	4	yes	—
145	C₂²×C₄₇	#436	1	2	yes	—
146	C₂²×C₄₇	C₈×Q₈	1	2	yes	—
147	C₂²×C₄₇	#446	1	2	yes	—
148	C₂²×C₄₇	#446	1	4	yes	—
149	C₄×C₄₇	—	2	4	no	—
150	C₄×C₄₇	—	4	2	no	—
151	C₄×C₄₇	#397	1	2	yes	—
152	C₄×C₄₇	#398	1	2	yes	—
153	C₄×C₄₇	#399	1	2	yes	—
154	C₄×C₄₇	#400	1	2	yes	—
155	Dic₄₇	—	4	4	no	—
156	Dic₄₇	—	4	4	no	—
157	C₂×C₄×C₄⋊C₄	—	47	1	no	—
158	C₂×#336	—	47	1	no	—
159	C₂×#336	—	47	4	no	—
160	C₂×#336	D₄₇	1	1	yes	characteristic
161	C₂×C₄₇	#1445	1	1	yes	characteristic
162	C₂×C₄₇	#1817	1	1	yes	characteristic
163	C₂×C₄₇	#1818	1	1	yes	characteristic
164	C₂×C₄₇	#1916	1	4	yes	—
165	C₂×C₄₇	#1917	1	4	yes	—
166	C₂×C₄₇	#1954	1	1	yes	characteristic
167	C₂×C₄₇	#1955	1	1	yes	characteristic
168	C₂×C₄₇	#1984	1	1	yes	characteristic
169	C₂×C₄₇	#1985	1	1	yes	characteristic
170	C₂³×C₈	—	47	2	no	—
171	C₂³×C₈	C₂×D₄₇	1	2	yes	—
172	C₂²×C₄⋊C₄	—	47	1	no	—
173	C₂²×C₄⋊C₄	—	47	1	no	—
174	C₂²×C₄²	—	47	2	no	—
175	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
176	C₄×C₄⋊C₄	—	94	2	no	—
177	C₄×C₄⋊C₄	—	94	2	no	—
178	C₂×#95	—	47	2	no	—
179	#336	—	2	2	no	—
180	#336	—	94	8	no	—
181	#336	—	94	2	no	—
182	C₄₇	#11346	1	1	yes	characteristic, complemented, Sylow
183	C₂³×C₄	—	47	2	no	—
184	C₂³×C₄	—	47	2	no	—
185	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
186	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
187	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	—	2	4	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	47	4	no	—
193	C₂²×C₈	—	94	4	no	—
194	C₂²×C₈	—	94	2	no	—
195	C₂²×C₈	—	94	4	no	—
196	C₂²×C₈	—	94	2	no	—
197	C₂²×C₈	D₁₈₈	1	4	yes	—
198	C₂×C₄⋊C₄	—	47	4	no	—
199	C₂×C₄⋊C₄	—	94	4	no	—
200	C₂×C₄⋊C₄	—	94	2	no	—
201	C₂×C₄⋊C₄	—	94	2	no	—
202	C₂×C₄⋊C₄	—	94	2	no	—
203	C₂×C₄²	—	2	4	no	—
204	C₂×C₄²	—	94	4	no	—
205	C₂×C₄²	—	94	2	no	—
206	C₂×C₄²	—	94	4	no	—
207	C₂×C₄²	—	94	2	no	—
208	C₂×C₄²	D₁₈₈	1	2	yes	—
209	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
210	#95	—	94	4	no	—
211	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	47	4	no	—
218	C₂²×C₄	—	47	4	no	—
219	C₂²×C₄	—	94	2	no	—
220	C₂²×C₄	—	94	2	no	—
221	C₂²×C₄	—	94	4	no	—
222	C₂²×C₄	—	94	4	no	—
223	C₂²×C₄	—	94	2	no	—
224	C₂²×C₄	—	94	2	no	—
225	C₂²×C₄	—	94	4	no	—
226	C₂²×C₄	—	94	4	no	—
227	C₂²×C₄	C₈×D₄₇	1	2	yes	—
228	C₂²×C₄	#10625324	1	2	yes	—
229	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
230	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
231	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
232	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
233	C₂²×C₄	#10625350	1	2	yes	—
234	C₂²×C₄	#10625352	1	4	yes	—
235	C₂²×C₄	#10625352	1	2	yes	—
236	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
237	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	8	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	4	4	no	—
243	C₂×C₈	—	94	8	no	—
244	C₂×C₈	—	94	4	no	—
245	C₂×C₈	—	188	4	no	—
246	C₂×C₈	—	188	2	no	—
247	C₂×C₈	—	188	2	no	—
248	C₄⋊C₄	—	94	8	no	—
249	C₄⋊C₄	—	94	8	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	188	4	no	—
253	C₄²	—	188	2	no	—
254	C₄²	—	188	2	no	—
255	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
256	C₂³	#13093980	1	2	yes	—
257	C₂³	C₄×D₁₈₈	1	2	yes	—
258	C₂³	#13093989	1	2	yes	—
259	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
260	C₂³	M₁₆×D₄₇	1	2	yes	—
261	C₂³	C₂×#10625324	1	1	yes	characteristic
262	C₂³	#13094030	1	2	yes	—
263	C₂³	#13094032	1	2	yes	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	94	4	no	—
273	C₂×C₄	—	94	4	no	—
274	C₂×C₄	—	94	8	no	—
275	C₂×C₄	—	94	8	no	—
276	C₂×C₄	—	188	2	no	—
277	C₂×C₄	—	188	2	no	—
278	C₂×C₄	—	188	2	no	—
279	C₂×C₄	—	188	2	no	—
280	C₂×C₄	—	188	4	no	—
281	C₂×C₄	—	188	4	no	—
282	C₂×C₄	(C₂×D₄₇)⋊C₈	1	4	yes	—
283	C₂×C₄	#13093998	1	2	yes	—
284	C₂×C₄	#13094004	1	4	yes	—
285	C₂×C₄	#13094004	1	2	yes	—
286	C₂×C₄	#13094005	1	2	yes	—
287	C₂×C₄	#13094008	1	4	yes	—
288	C₂×C₄	#13094008	1	2	yes	—
289	C₂×C₄	#13094011	1	1	yes	characteristic
290	C₂×C₄	#13094012	1	1	yes	characteristic
291	C₂×C₄	#13094014	1	4	yes	—
292	C₂×C₄	#13094017	1	4	yes	—
293	C₂×C₄	#13094021	1	1	yes	characteristic
294	C₂×C₄	#13094022	1	1	yes	characteristic
295	C₈	—	4	4	no	—
296	C₈	—	188	4	no	—
297	C₂²	#15688984	1	1	yes	characteristic
298	C₂²	#15689008	1	4	yes	—
299	C₂²	#15689025	1	1	yes	characteristic
300	C₂²	#15689028	1	1	yes	characteristic
301	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
302	C₂²	#15689086	1	1	yes	characteristic
303	C₂²	#15689089	1	1	yes	characteristic
304	C₂²	#15689091	1	1	yes	characteristic
305	C₂²	#15689173	1	2	yes	—
306	C₂²	#15689177	1	2	yes	—
307	C₂²	#15689181	1	4	yes	—
308	C₂²	C₈×D₁₈₈	1	2	yes	—
309	C₂²	#15689196	1	4	yes	—
310	C₂²	#15689202	1	2	yes	—
311	C₂²	#15689247	1	2	yes	—
312	C₂²	#15689250	1	2	yes	—
313	C₂²	#15689252	1	4	yes	—
314	C₄	—	2	4	no	—
315	C₄	—	4	2	no	—
316	C₄	—	188	4	no	—
317	C₄	—	188	4	no	—
318	C₄	#15689048	1	2	yes	—
319	C₄	#15689053	1	2	yes	—
320	C₄	#15689060	1	2	yes	—
321	C₄	#15689063	1	2	yes	—
322	C₂	#20241655	1	1	yes	characteristic
323	C₂	#20243454	1	1	yes	characteristic
324	C₂	#20243457	1	1	yes	characteristic
325	C₂	#20243946	1	4	yes	—
326	C₂	#20243953	1	4	yes	—
327	C₂	#20244182	1	1	yes	characteristic
328	C₂	#20244186	1	1	yes	characteristic
329	C₂	#20244323	1	1	yes	characteristic
330	C₂	#20244327	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₄₇
commutator	C₂³×C₄₇	C₂²×C₈
Fitting	C₂×C₄₇×#336	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#336	D₄₇
hypocenter	C₄₇	#11346
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
28. C₂×C₄⋊Dic₄₇	C₈	semidirect
182. C₄₇	#11346	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄⋊Dic₄₇	2
C₂×#15688878	2
C₂×#15688877	2
C₂×C₄₇×#336	2
#11346	47