Subgroups — #26214493

Order 12032.

1050 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20243265	C₂	1	4	yes	maximal, complemented
4	#20243174	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	C₂²×#10625329	C₂²	1	2	yes	—
11	C₂×#13093983	C₄	1	2	yes	—
12	C₄×#10625329	C₄	1	8	yes	complemented
13	C₂×#13093935	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	C₂²	1	2	yes	—
20	C₄×C₂²⋊Dic₄₇	C₄	1	2	yes	—
21	C₄×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
22	#15688969	C₄	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	C₂×#10625329	—	2	8	no	—
34	C₂×#10625329	C₂×C₄	1	8	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	#13093983	—	2	8	no	—
38	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
45	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
46	#13093935	C₂×C₄	1	8	yes	—
47	#13093934	—	2	8	no	—
48	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
49	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
50	C₈×Dic₄₇	—	2	8	no	—
51	C₄₇⋊#95	C₂×C₄	1	4	yes	—
52	C₄₇⋊#95	C₂×C₄	1	4	yes	—
53	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
54	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
55	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
57	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
61	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
65	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
66	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
67	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
68	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	—	2	4	no	—
71	C₂²×C₄×C₄₇	—	2	4	no	—
72	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
73	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
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	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
83	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₄₇	—	2	4	no	—
85	C₂×C₈×C₄₇	—	2	4	no	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
88	C₄²×C₄₇	C₄²	1	4	yes	—
89	C₄²×C₄₇	C₄²	1	4	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	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
94	C₂²⋊Dic₄₇	—	2	8	no	—
95	C₄⋊Dic₄₇	—	2	8	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	—	2	4	no	—
101	C₄×Dic₄₇	M₁₆	1	8	yes	complemented
102	#10625329	—	2	16	no	—
103	C₂×C₄₇⋊C₈	—	2	4	no	—
104	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
105	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
106	C₂×C₄₇⋊C₈	#41	1	8	yes	—
107	C₂³×C₄₇	—	2	2	no	—
108	C₂³×C₄₇	—	2	2	no	—
109	C₂³×C₄₇	—	2	2	no	—
110	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
111	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	8	no	—
118	C₂×C₄×C₄₇	—	2	8	no	—
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	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
125	C₂×C₄×C₄₇	#131	1	4	yes	—
126	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
129	C₂×Dic₄₇	—	2	4	no	—
130	C₂×Dic₄₇	—	2	4	no	—
131	C₂×Dic₄₇	—	2	4	no	—
132	C₂×Dic₄₇	—	2	4	no	—
133	C₂×Dic₄₇	—	2	4	no	—
134	C₂×Dic₄₇	C₂×M₁₆	1	8	yes	—
135	C₈×C₄₇	—	2	4	no	—
136	C₈×C₄₇	—	2	4	no	—
137	C₄₇⋊C₈	—	2	4	no	—
138	C₄₇⋊C₈	#117	1	8	yes	complemented
139	#20472	—	47	1	no	maximal, Sylow
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	4	4	no	—
144	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
145	C₂²×C₄₇	C₄×#41	1	2	yes	—
146	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
147	C₂²×C₄₇	C₂×#131	1	2	yes	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	C₄×M₁₆	1	4	yes	—
152	C₄×C₄₇	#405	1	4	yes	—
153	C₄×C₄₇	C₂×#117	1	2	yes	—
154	C₄×C₄₇	C₄×#41	1	2	yes	—
155	Dic₄₇	—	2	8	no	—
156	Dic₄₇	—	4	4	no	—
157	C₂×#431	D₄₇	1	1	yes	characteristic
158	C₂×C₄×M₁₆	—	47	1	no	—
159	C₄×#117	—	47	1	no	—
160	#1771	—	47	4	no	—
161	#1753	—	47	4	no	—
162	#1744	—	47	1	no	—
163	#1744	—	47	1	no	—
164	C₄×#97	—	47	1	no	—
165	C₄²×C₈	—	47	1	no	—
166	C₂×C₄₇	—	4	2	no	—
167	C₂×C₄₇	C₄×#117	1	1	yes	characteristic
168	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
169	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
170	C₂×C₄₇	#2965	1	4	yes	—
171	C₂²×M₁₆	—	47	2	no	—
172	C₂×#117	—	47	2	no	—
173	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
174	#431	Dic₄₇	1	8	yes	complemented
175	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
176	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
177	C₄×M₁₆	—	47	8	no	—
178	C₂×#97	—	47	2	no	—
179	C₂×#97	—	47	4	no	—
180	C₂×C₄×C₈	—	47	1	no	—
181	C₂×C₄×C₈	—	47	1	no	—
182	C₂×C₄×C₈	—	47	4	no	—
183	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
185	C₄×C₄⋊C₄	—	47	2	no	—
186	C₄×C₄⋊C₄	—	47	2	no	—
187	C₄×C₂²⋊C₄	—	47	2	no	—
188	C₄×C₂²⋊C₄	—	47	2	no	—
189	#376	—	47	2	no	—
190	C₄³	—	47	1	no	—
191	C₄³	—	47	1	no	—
192	#336	—	47	4	no	—
193	#336	—	47	4	no	—
194	C₄₇	#20472	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
196	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
197	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
198	C₂×M₁₆	—	47	8	no	—
199	C₂×M₁₆	—	94	8	no	—
200	C₂²×C₈	—	47	2	no	—
201	C₂²×C₈	—	47	2	no	—
202	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
203	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
204	#117	—	94	8	no	—
205	C₂×C₄⋊C₄	—	47	4	no	—
206	C₂×C₂²⋊C₄	—	47	4	no	—
207	C₂×C₄²	—	47	2	no	—
208	C₂×C₄²	—	47	2	no	—
209	C₂×C₄²	—	47	2	no	—
210	C₂×C₄²	—	47	2	no	—
211	C₂×C₄²	—	47	2	no	—
212	C₂×C₄²	—	47	2	no	—
213	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
214	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
215	C₂×C₄²	C₄×D₄₇	1	4	yes	—
216	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
217	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
218	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
219	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
220	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
221	#97	—	47	8	no	—
222	#97	—	94	8	no	—
223	C₄×C₈	—	47	4	no	—
224	C₄×C₈	—	47	4	no	—
225	C₄×C₈	—	94	8	no	—
226	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
227	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
228	#95	—	47	4	no	—
229	#95	—	47	4	no	—
230	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	47	4	no	—
235	C₂²×C₄	—	47	4	no	—
236	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
237	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
238	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
242	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
244	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
245	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
246	M₁₆	—	94	16	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	47	4	no	—
250	C₂×C₈	—	47	8	no	—
251	C₂×C₈	—	47	4	no	—
252	C₂×C₈	—	94	4	no	—
253	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
254	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
255	C₄⋊C₄	—	94	8	no	—
256	C₂²⋊C₄	—	94	8	no	—
257	C₄²	—	47	8	no	—
258	C₄²	—	94	4	no	—
259	C₄²	—	94	4	no	—
260	C₄²	—	94	4	no	—
261	C₄²	—	94	4	no	—
262	C₄²	—	94	4	no	—
263	C₄²	C₄×Dic₄₇	1	4	yes	—
264	C₄²	C₄×Dic₄₇	1	4	yes	—
265	C₄²	C₂×C₄×D₄₇	1	2	yes	—
266	C₄²	C₂×C₄×D₄₇	1	2	yes	—
267	C₄²	C₂²×Dic₄₇	1	2	yes	—
268	C₄²	C₂²×Dic₄₇	1	2	yes	—
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
273	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	47	8	no	—
282	C₂×C₄	—	94	4	no	—
283	C₂×C₄	—	94	4	no	—
284	C₂×C₄	—	94	4	no	—
285	C₂×C₄	—	94	4	no	—
286	C₂×C₄	—	94	4	no	—
287	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
288	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
289	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
290	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
291	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
292	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
293	C₂×C₄	#13094030	1	4	yes	—
294	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
295	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
296	C₂×C₄	#41×D₄₇	1	4	yes	—
297	C₈	—	2	4	no	—
298	C₈	—	2	4	no	—
299	C₈	—	47	8	no	—
300	C₈	—	94	4	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	4	no	—
305	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
306	C₂²	#15689591	1	2	yes	—
307	C₂²	C₂×M₁₆×D₄₇	1	2	yes	—
308	C₂²	C₂×#13094030	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	94	8	no	—
313	C₄	—	188	4	no	—
314	C₄	M₁₆×Dic₄₇	1	4	yes	—
315	C₄	#15689078	1	4	yes	—
316	C₄	#117×D₄₇	1	2	yes	—
317	C₄	#41×Dic₄₇	1	2	yes	—
318	C₂	—	4	2	no	—
319	C₂	#117×Dic₄₇	1	1	yes	characteristic
320	C₂	C₂×M₁₆×Dic₄₇	1	1	yes	characteristic
321	C₂	C₂×#15689078	1	1	yes	characteristic
322	C₂	#20252352	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	#20472
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. #20243265	C₂	semidirect
5. #20243133	C₂	semidirect
6. #20243133	C₂	semidirect
7. C₄×#13093934	C₂	semidirect
8. C₄×C₈×Dic₄₇	C₂	semidirect
12. C₄×#10625329	C₄	semidirect
28. C₄₇×#431	C₄	semidirect
101. C₄×Dic₄₇	M₁₆	semidirect
138. C₄₇⋊C₈	#117	semidirect
174. #431	Dic₄₇	semidirect
194. C₄₇	#20472	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10625329	2
C₄×#13093983	2
#20243265	2
#20243174	2
#20243133	2
C₄×#13093934	2
C₄×C₈×Dic₄₇	2
C₂×C₄₇×#431	2
#20472	47