Subgroups — #26121896

Order 12032.

870 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688878	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688877	C₂	1	2	yes	maximal
3	C₂×#15688877	C₂	1	2	yes	maximal
4	C₄×C₄₇⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂²×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
8	C₂×C₄₇⋊#95	C₄	1	4	yes	—
9	#15688878	C₄	1	4	yes	complemented
10	#15688877	—	2	4	no	—
11	#15688877	—	2	4	no	—
12	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#97×C₄₇	—	2	4	no	—
16	C₂×#97×C₄₇	C₄	1	4	yes	complemented
17	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
18	C₂²×C₄₇⋊C₈	—	2	2	no	—
19	C₂²×C₄₇⋊C₈	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
25	C₂×C₄×Dic₄₇	—	2	2	no	—
26	C₂×C₄×Dic₄₇	—	2	2	no	—
27	C₂×C₄×Dic₄₇	—	2	2	no	—
28	C₂×C₄×Dic₄₇	—	2	2	no	—
29	C₂×C₄×Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₄₇⋊#95	—	2	8	no	—
32	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	—	2	2	no	—
35	C₂²×C₈×C₄₇	—	2	2	no	—
36	C₂²×C₈×C₄₇	—	2	2	no	—
37	C₂²×C₈×C₄₇	—	2	1	no	—
38	C₂²×C₈×C₄₇	—	2	1	no	—
39	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₄₇	D₄	1	4	yes	—
42	C₂²×C₈×C₄₇	D₄	1	2	yes	—
43	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
44	C₂×C₄²×C₄₇	—	2	2	no	—
45	C₂×C₄²×C₄₇	—	2	2	no	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
48	#97×C₄₇	—	2	8	no	—
49	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	#41	1	2	yes	—
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	#41	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
65	C₂²×Dic₄₇	—	2	4	no	—
66	C₂²×Dic₄₇	—	2	4	no	—
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂²×Dic₄₇	—	2	4	no	—
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂²×Dic₄₇	M₁₆	1	8	yes	—
72	C₂×C₈×C₄₇	—	2	2	no	—
73	C₂×C₈×C₄₇	—	2	2	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	—	2	4	no	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
81	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	—	2	4	no	—
85	C₄×Dic₄₇	—	4	2	no	—
86	C₄×Dic₄₇	—	4	2	no	—
87	C₄×Dic₄₇	—	4	2	no	—
88	C₄×Dic₄₇	—	4	2	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₂×C₄₇⋊C₈	—	4	2	no	—
93	C₂×C₄₇⋊C₈	—	4	2	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	C₂×C₄₇⋊C₈	—	4	2	no	—
96	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
97	C₂³×C₄₇	#117	1	2	yes	—
98	C₂³×C₄₇	C₄×D₄	1	2	yes	—
99	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
100	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
101	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
102	C₂³×C₄₇	#131	1	4	yes	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	4	4	no	—
107	C₂×C₄×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#117	1	2	yes	—
111	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#121	1	2	yes	—
119	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₄₇	#123	1	2	yes	—
121	C₂×C₄×C₄₇	#123	1	2	yes	—
122	C₂×C₄×C₄₇	#124	1	2	yes	—
123	C₂×C₄×C₄₇	#125	1	2	yes	—
124	C₂×C₄×C₄₇	#126	1	2	yes	—
125	C₂×C₄×C₄₇	#126	1	2	yes	—
126	C₂×Dic₄₇	—	2	8	no	—
127	C₂×Dic₄₇	—	2	8	no	—
128	C₂×Dic₄₇	—	2	8	no	—
129	C₂×Dic₄₇	—	4	4	no	—
130	C₂×Dic₄₇	—	4	4	no	—
131	C₂×Dic₄₇	—	4	4	no	—
132	C₂×Dic₄₇	—	4	4	no	—
133	C₈×C₄₇	—	2	4	no	—
134	C₈×C₄₇	—	2	4	no	—
135	C₄₇⋊C₈	—	4	4	no	—
136	#10798	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	#376	1	1	yes	characteristic
138	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₄₇	#387	1	2	yes	—
141	C₂²×C₄₇	#388	1	2	yes	—
142	C₂²×C₄₇	C₄×M₁₆	1	2	yes	—
143	C₂²×C₄₇	#405	1	2	yes	—
144	C₂²×C₄₇	#432	1	4	yes	—
145	C₂²×C₄₇	#433	1	4	yes	—
146	C₂²×C₄₇	#435	1	4	yes	—
147	C₂²×C₄₇	#435	1	4	yes	—
148	C₂²×C₄₇	#436	1	4	yes	—
149	C₂²×C₄₇	#446	1	4	yes	—
150	C₄×C₄₇	—	4	4	no	—
151	C₄×C₄₇	#380	1	1	yes	characteristic
152	C₄×C₄₇	#381	1	1	yes	characteristic
153	C₄×C₄₇	#382	1	1	yes	characteristic
154	C₄×C₄₇	#383	1	1	yes	characteristic
155	C₄×C₄₇	#387	1	2	yes	—
156	C₄×C₄₇	#388	1	2	yes	—
157	Dic₄₇	—	4	8	no	—
158	C₂²×#97	D₄₇	1	1	yes	characteristic
159	C₂×#336	—	47	1	no	—
160	C₂×#336	—	47	2	no	—
161	C₂×#336	—	47	2	no	—
162	C₄×#95	—	47	1	no	—
163	C₂×C₄₇	#1434	1	1	yes	characteristic
164	C₂×C₄₇	#1745	1	1	yes	characteristic
165	C₂×C₄₇	#1746	1	1	yes	characteristic
166	C₂×C₄₇	#1753	1	2	yes	—
167	C₂×C₄₇	#1771	1	2	yes	—
168	C₂×C₄₇	#1917	1	4	yes	—
169	C₂×C₄₇	#1924	1	4	yes	—
170	C₂³×C₈	—	47	2	no	—
171	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
172	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
173	C₂²×C₄²	—	47	2	no	—
174	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
175	C₂×#97	—	2	4	no	—
176	C₂×#97	Dic₄₇	1	4	yes	complemented
177	C₂×#95	—	47	4	no	—
178	#336	—	47	4	no	—
179	#336	—	94	4	no	—
180	#336	—	94	4	no	—
181	C₄₇	#10798	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₄	—	47	4	no	—
183	C₂³×C₄	C₄×D₄₇	1	2	yes	—
184	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	47	4	no	—
191	C₂²×C₈	—	94	2	no	—
192	C₂²×C₈	—	94	2	no	—
193	C₂²×C₈	—	94	2	no	—
194	C₂²×C₈	—	94	2	no	—
195	C₂²×C₈	—	94	2	no	—
196	C₂²×C₈	—	94	2	no	—
197	C₂²×C₈	Dic₉₄	1	2	yes	—
198	C₂²×C₈	D₁₈₈	1	2	yes	—
199	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
200	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
201	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	94	2	no	—
205	C₂×C₄²	—	94	2	no	—
206	C₂×C₄²	—	94	2	no	—
207	C₂×C₄²	—	94	2	no	—
208	C₂×C₄²	—	94	2	no	—
209	C₂×C₄²	—	94	2	no	—
210	C₂×C₄²	C₄×D₄₇	1	2	yes	—
211	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
212	#97	—	2	8	no	—
213	#95	—	94	8	no	—
214	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	47	8	no	—
220	C₂²×C₄	—	94	4	no	—
221	C₂²×C₄	—	94	4	no	—
222	C₂²×C₄	—	94	4	no	—
223	C₂²×C₄	—	94	4	no	—
224	C₂²×C₄	—	94	4	no	—
225	C₂²×C₄	—	94	4	no	—
226	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
227	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
230	C₂²×C₄	#10625350	1	2	yes	—
231	C₂²×C₄	#10625350	1	2	yes	—
232	C₂²×C₄	#10625350	1	2	yes	—
233	C₂²×C₄	#10625350	1	1	yes	characteristic
234	C₂²×C₄	#10625350	1	1	yes	characteristic
235	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	94	4	no	—
246	C₂×C₈	—	94	4	no	—
247	C₂×C₈	—	94	4	no	—
248	C₂×C₈	—	188	2	no	—
249	C₂×C₈	—	188	2	no	—
250	C₂×C₈	—	188	2	no	—
251	C₂×C₈	—	188	2	no	—
252	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
253	C₂×C₈	C₂²⋊Dic₄₇	1	4	yes	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	188	2	no	—
258	C₄²	—	188	2	no	—
259	C₄²	—	188	2	no	—
260	C₄²	—	188	2	no	—
261	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
262	C₂³	#13093981	1	2	yes	—
263	C₂³	C₄×C₂²⋊D₄₇	1	2	yes	—
264	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
265	C₂³	C₄×Dic₉₄	1	1	yes	characteristic
266	C₂³	M₁₆×D₄₇	1	4	yes	—
267	C₂³	#13094032	1	4	yes	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	94	8	no	—
274	C₂×C₄	—	94	8	no	—
275	C₂×C₄	—	94	8	no	—
276	C₂×C₄	—	188	4	no	—
277	C₂×C₄	—	188	4	no	—
278	C₂×C₄	—	188	4	no	—
279	C₂×C₄	—	188	4	no	—
280	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
281	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
282	C₂×C₄	#13093981	1	2	yes	—
283	C₂×C₄	#13093983	1	1	yes	characteristic
284	C₂×C₄	#13093983	1	1	yes	characteristic
285	C₂×C₄	#13093983	1	1	yes	characteristic
286	C₂×C₄	#13093983	1	1	yes	characteristic
287	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
288	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
289	C₂×C₄	C₄×Dic₉₄	1	1	yes	characteristic
290	C₂×C₄	#13093998	1	2	yes	—
291	C₂×C₄	#13094003	1	2	yes	—
292	C₂×C₄	#13094007	1	2	yes	—
293	C₂×C₄	#13094007	1	2	yes	—
294	C₂×C₄	#13094012	1	2	yes	—
295	C₂×C₄	#13094016	1	2	yes	—
296	C₂×C₄	#13094018	1	2	yes	—
297	C₂×C₄	#13094018	1	2	yes	—
298	C₈	—	2	4	no	—
299	C₈	—	2	4	no	—
300	C₈	—	188	4	no	—
301	C₂²	#15688969	1	1	yes	characteristic
302	C₂²	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
303	C₂²	C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
304	C₂²	#15689007	1	2	yes	—
305	C₂²	#15689014	1	2	yes	—
306	C₂²	M₁₆×Dic₄₇	1	2	yes	—
307	C₂²	#15689081	1	2	yes	—
308	C₂²	#15689178	1	4	yes	—
309	C₂²	#15689185	1	4	yes	—
310	C₂²	#15689192	1	4	yes	—
311	C₂²	#15689196	1	4	yes	—
312	C₂²	#15689199	1	4	yes	—
313	C₂²	#15689251	1	4	yes	—
314	C₄	—	4	4	no	—
315	C₄	—	188	8	no	—
316	C₄	#15688983	1	1	yes	characteristic
317	C₄	#15688986	1	1	yes	characteristic
318	C₄	#15688989	1	1	yes	characteristic
319	C₄	#15688991	1	1	yes	characteristic
320	C₄	#15689007	1	2	yes	—
321	C₄	#15689014	1	2	yes	—
322	C₂	#20241615	1	1	yes	characteristic
323	C₂	#20243136	1	1	yes	characteristic
324	C₂	#20243141	1	1	yes	characteristic
325	C₂	#20243169	1	2	yes	—
326	C₂	#20243260	1	2	yes	—
327	C₂	#20243952	1	4	yes	—
328	C₂	#20244001	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₄₇
commutator	C₂³×C₄₇	C₂×C₄²
Fitting	C₂²×#97×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×#97	D₄₇
hypocenter	C₄₇	#10798
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
9. #15688878	C₄	semidirect
16. C₂×#97×C₄₇	C₄	semidirect
176. C₂×#97	Dic₄₇	semidirect
181. C₄₇	#10798	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688878	2
C₂×#15688877	2
C₄×C₄₇⋊#95	2
C₂²×#97×C₄₇	2
#10798	47