Subgroups — #26214529

Order 12032.

990 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689180	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20243257	C₂	1	2	yes	maximal, complemented
4	C₄⋊C₄×C₄₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	#20243166	C₂	1	2	yes	maximal
6	C₂²⋊C₄×C₄₇⋊C₈	C₂	1	2	yes	maximal
7	#20243140	C₂	1	2	yes	maximal, complemented
8	#20243075	C₂	1	2	yes	maximal, complemented
9	C₄₇×#2283	C₂	1	1	yes	maximal, characteristic
10	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	#15689180	C₄	1	8	yes	complemented
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×#13093935	C₂²	1	4	yes	—
16	C₂×#13093935	C₂²	1	2	yes	—
17	C₂×C₄×C₄₇⋊C₈	C₄	1	8	yes	complemented
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
23	#15688878	C₂²	1	4	yes	—
24	#15688878	C₂²	1	4	yes	—
25	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
26	C₄₇×#383	C₄	1	2	yes	—
27	C₄₇×#382	C₄	1	2	yes	—
28	C₄₇×#381	C₄	1	2	yes	—
29	C₄₇×#380	C₄	1	2	yes	—
30	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
31	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
32	C₄₇×#376	C₂²	1	2	yes	—
33	C₂²×C₄₇⋊C₈	—	2	4	no	—
34	C₂²×C₄₇⋊C₈	—	2	4	no	—
35	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
36	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
37	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
40	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
43	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
44	#13093935	—	2	8	no	—
45	#13093935	C₂×C₄	1	8	yes	—
46	C₄×C₄₇⋊C₈	—	2	8	no	—
47	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
48	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
49	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
50	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
51	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
55	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
56	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
59	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
60	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	#95×C₄₇	C₂×C₄	1	4	yes	—
64	#95×C₄₇	C₂×C₄	1	4	yes	—
65	#95×C₄₇	C₂×C₄	1	4	yes	—
66	#95×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	2	no	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
73	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
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	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄×C₄₇	—	2	8	no	—
84	C₂²⋊C₄×C₄₇	—	2	8	no	—
85	C₄²×C₄₇	—	2	4	no	—
86	C₄²×C₄₇	—	2	4	no	—
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	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	8	no	—
98	C₂×C₄₇⋊C₈	—	2	8	no	—
99	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
101	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
102	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
103	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
104	C₂×C₄₇⋊C₈	#41	1	4	yes	—
105	C₂×C₄₇⋊C₈	#41	1	4	yes	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	—	2	2	no	—
108	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
109	C₂³×C₄₇	#131	1	4	yes	—
110	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	—	2	2	no	—
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	8	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	8	no	—
120	C₂×C₄×C₄₇	—	2	8	no	—
121	C₂×C₄×C₄₇	—	2	8	no	—
122	C₂×C₄×C₄₇	—	2	8	no	—
123	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₄₇	#131	1	4	yes	—
129	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
133	C₄₇⋊C₈	—	2	2	no	—
134	C₄₇⋊C₈	—	2	2	no	—
135	C₄₇⋊C₈	—	2	8	no	—
136	C₄₇⋊C₈	#117	1	4	yes	—
137	C₄₇⋊C₈	#117	1	4	yes	—
138	#20476	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	8	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	#405	1	4	yes	—
143	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
144	C₂²×C₄₇	C₄×#41	1	2	yes	—
145	C₂²×C₄₇	C₂×#131	1	2	yes	—
146	C₂²×C₄₇	C₂×#131	1	2	yes	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	—	4	8	no	—
152	C₄×C₄₇	#405	1	4	yes	—
153	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
154	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
155	C₄×C₄₇	C₄×#41	1	2	yes	—
156	C₂×#432	—	47	1	no	—
157	C₂²×C₄×C₈	—	47	1	no	—
158	#2283	D₄₇	1	1	yes	characteristic
159	#1770	—	47	2	no	—
160	C₈×C₄⋊C₄	—	47	2	no	—
161	#1752	—	47	2	no	—
162	C₈×C₂²⋊C₄	—	47	2	no	—
163	#1746	—	47	2	no	—
164	#1726	—	47	2	no	—
165	C₂×C₄₇	—	2	4	no	—
166	C₂×C₄₇	C₄×#117	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
168	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
169	C₂×C₄₇	#2966	1	4	yes	—
170	C₂³×C₈	—	47	1	no	—
171	C₂³×C₈	—	47	1	no	—
172	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
173	#432	—	47	8	no	—
174	C₂×C₄⋊C₈	—	47	2	no	—
175	C₂×C₂²⋊C₈	—	47	2	no	—
176	C₂×#97	—	47	2	no	—
177	C₂×#97	—	47	4	no	—
178	C₂×C₄×C₈	—	47	8	no	—
179	C₂×C₄×C₈	—	47	1	no	—
180	C₂×C₄×C₈	—	47	1	no	—
181	C₂×C₄×C₈	—	47	1	no	—
182	C₂×C₄×C₈	—	47	1	no	—
183	C₂×C₄×C₈	—	47	4	no	—
184	#383	Dic₄₇	1	2	yes	—
185	#382	Dic₄₇	1	2	yes	—
186	#381	Dic₄₇	1	2	yes	—
187	#380	Dic₄₇	1	2	yes	—
188	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
189	C₄×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
190	#376	C₂×D₄₇	1	2	yes	—
191	#336	—	47	4	no	—
192	#336	—	47	4	no	—
193	C₄₇	#20476	1	1	yes	characteristic, complemented, Sylow
194	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
195	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
196	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
197	C₂²×C₈	—	47	1	no	—
198	C₂²×C₈	—	47	1	no	—
199	C₂²×C₈	—	47	4	no	—
200	C₂²×C₈	—	47	4	no	—
201	C₂²×C₈	—	47	1	no	—
202	C₂²×C₈	—	47	1	no	—
203	C₂²×C₈	—	47	4	no	—
204	C₂²×C₈	—	94	4	no	—
205	C₂²×C₈	—	94	4	no	—
206	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
207	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
208	C₂×C₄²	C₄×D₄₇	1	4	yes	—
209	C₂×C₄²	C₄×D₄₇	1	4	yes	—
210	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
212	C₂×C₄²	C₂²×D₄₇	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	C₂²⋊C₈	—	47	8	no	—
217	#97	—	47	8	no	—
218	#97	—	94	8	no	—
219	C₄×C₈	—	47	4	no	—
220	C₄×C₈	—	47	4	no	—
221	C₄×C₈	—	47	4	no	—
222	C₄×C₈	—	47	4	no	—
223	C₄×C₈	—	94	8	no	—
224	#95	C₂×Dic₄₇	1	4	yes	—
225	#95	C₂×Dic₄₇	1	4	yes	—
226	#95	C₂×Dic₄₇	1	4	yes	—
227	#95	C₂×Dic₄₇	1	4	yes	—
228	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
234	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
235	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
236	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
237	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
238	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
244	C₂×C₈	—	47	2	no	—
245	C₂×C₈	—	47	2	no	—
246	C₂×C₈	—	47	4	no	—
247	C₂×C₈	—	47	4	no	—
248	C₂×C₈	—	47	2	no	—
249	C₂×C₈	—	47	2	no	—
250	C₂×C₈	—	47	8	no	—
251	C₂×C₈	—	94	8	no	—
252	C₂×C₈	—	94	8	no	—
253	C₂×C₈	—	94	4	no	—
254	C₂×C₈	—	94	4	no	—
255	C₂×C₈	—	94	2	no	—
256	C₂×C₈	—	94	2	no	—
257	C₄⋊C₄	—	2	8	no	—
258	C₂²⋊C₄	—	2	8	no	—
259	C₄²	—	2	4	no	—
260	C₄²	—	2	4	no	—
261	C₄²	C₄×Dic₄₇	1	4	yes	—
262	C₄²	C₄×Dic₄₇	1	4	yes	—
263	C₄²	C₂×C₄×D₄₇	1	2	yes	—
264	C₄²	C₂×C₄×D₄₇	1	2	yes	—
265	C₄²	C₂×C₄×D₄₇	1	2	yes	—
266	C₄²	C₂×C₄×D₄₇	1	2	yes	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
270	C₂³	#13094031	1	4	yes	—
271	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	8	no	—
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	C₂×C₄×Dic₄₇	1	2	yes	—
289	C₂×C₄	#13094031	1	4	yes	—
290	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
292	C₂×C₄	#41×D₄₇	1	2	yes	—
293	C₂×C₄	#41×D₄₇	1	2	yes	—
294	C₈	—	47	4	no	—
295	C₈	—	47	4	no	—
296	C₈	—	94	2	no	—
297	C₈	—	94	2	no	—
298	C₈	—	94	8	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	8	no	—
301	C₂²	—	2	2	no	—
302	C₂²	#15689079	1	4	yes	—
303	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
304	C₂²	#41×Dic₄₇	1	2	yes	—
305	C₂²	C₂×#13094031	1	2	yes	—
306	C₂²	C₂×#13094031	1	2	yes	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	8	no	—
312	C₄	#15689079	1	4	yes	—
313	C₄	#117×D₄₇	1	1	yes	characteristic
314	C₄	#117×D₄₇	1	1	yes	characteristic
315	C₄	#15689591	1	2	yes	—
316	C₂	—	2	4	no	—
317	C₂	#117×Dic₄₇	1	1	yes	characteristic
318	C₂	C₂×#15689079	1	1	yes	characteristic
319	C₂	C₂×#15689079	1	1	yes	characteristic
320	C₂	#20252364	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#2283	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2283	D₄₇
hypocenter	C₄₇	#20476
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2283

Complements

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

N	H ≅ G/N	type
3. #20243257	C₂	semidirect
4. C₄⋊C₄×C₄₇⋊C₈	C₂	semidirect
7. #20243140	C₂	semidirect
8. #20243075	C₂	semidirect
12. #15689180	C₄	semidirect
17. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
193. C₄₇	#20476	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689180	2
C₂²×C₄×C₄₇⋊C₈	2
#20243257	2
C₄⋊C₄×C₄₇⋊C₈	2
#20243166	2
C₂²⋊C₄×C₄₇⋊C₈	2
#20243140	2
#20243075	2
C₄₇×#2283	2
#20476	47