Subgroups — #26216617

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243531	C₂	1	2	yes	maximal, complemented
2	#20243505	C₂	1	2	yes	maximal, complemented
3	#20243459	C₂	1	2	yes	maximal
4	#20243372	C₂	1	2	yes	maximal
5	#20243245	C₂	1	2	yes	maximal, complemented
6	#20243150	C₂	1	1	yes	maximal, characteristic
7	C₄×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
8	#20243140	C₂	1	2	yes	maximal, complemented
9	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×#13093935	C₂²	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	#15688878	C₂²	1	4	yes	—
16	#15688878	C₂²	1	4	yes	—
17	#15688878	C₂²	1	2	yes	—
18	#15688878	C₂²	1	2	yes	—
19	#15688878	C₂²	1	4	yes	—
20	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
24	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
25	C₄×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
26	C₄×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
27	C₄₇×#376	C₂²	1	1	yes	characteristic
28	C₄₇×#376	C₂²	1	1	yes	characteristic
29	C₄³×C₄₇	C₂²	1	1	yes	characteristic
30	C₄³×C₄₇	C₂²	1	1	yes	characteristic
31	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
33	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
34	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
35	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
36	#13093935	—	2	4	no	—
37	C₄×C₄₇⋊C₈	—	2	4	no	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	#117×C₄₇	—	2	4	no	—
41	#117×C₄₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₇	—	2	4	no	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	#95×C₄₇	C₂×C₄	1	2	yes	—
61	#95×C₄₇	C₂×C₄	1	2	yes	—
62	#95×C₄₇	C₂×C₄	1	2	yes	—
63	#95×C₄₇	C₂×C₄	1	2	yes	—
64	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	—	2	4	no	—
68	C₂²×C₄×C₄₇	—	2	4	no	—
69	C₂²×C₄×C₄₇	—	4	2	no	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂²×C₄	1	2	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	2	yes	—
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	#41	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₄₇	—	2	4	no	—
82	C₄⋊C₄×C₄₇	—	2	2	no	—
83	C₄⋊C₄×C₄₇	—	2	2	no	—
84	C₂²⋊C₄×C₄₇	—	2	2	no	—
85	C₂²⋊C₄×C₄₇	—	2	2	no	—
86	C₂²⋊C₄×C₄₇	—	2	4	no	—
87	C₄²×C₄₇	—	2	4	no	—
88	C₄²×C₄₇	—	2	4	no	—
89	C₄²×C₄₇	—	2	4	no	—
90	C₄²×C₄₇	—	2	2	no	—
91	C₄²×C₄₇	—	2	2	no	—
92	C₄²×C₄₇	—	2	2	no	—
93	C₄²×C₄₇	—	2	2	no	—
94	C₄²×C₄₇	—	4	2	no	—
95	C₄²×C₄₇	—	4	2	no	—
96	C₄²×C₄₇	#41	1	4	yes	—
97	C₄²×C₄₇	#41	1	4	yes	—
98	C₂×C₄₇⋊C₈	—	2	4	no	—
99	C₂×C₄₇⋊C₈	—	2	4	no	—
100	C₂×C₄₇⋊C₈	—	2	4	no	—
101	C₂×C₄₇⋊C₈	—	2	4	no	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	C₂×C₄₇⋊C₈	—	2	4	no	—
104	C₂³×C₄₇	—	2	1	no	—
105	C₂³×C₄₇	—	2	1	no	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	—	2	2	no	—
108	C₂³×C₄₇	#117	1	2	yes	—
109	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	8	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	1	no	—
122	C₂×C₄×C₄₇	—	2	1	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	2	2	no	—
127	C₂×C₄×C₄₇	—	2	2	no	—
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	2	2	no	—
130	C₂×C₄×C₄₇	—	4	8	no	—
131	C₂×C₄×C₄₇	#117	1	4	yes	—
132	C₂×C₄×C₄₇	#117	1	4	yes	—
133	C₂×C₄×C₄₇	#117	1	2	yes	—
134	C₂×C₄×C₄₇	#131	1	4	yes	—
135	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
136	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
137	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
138	C₄₇⋊C₈	—	4	4	no	—
139	C₄₇⋊C₈	—	4	4	no	—
140	#20784	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	4	2	no	—
145	C₂²×C₄₇	—	4	2	no	—
146	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×#131	1	2	yes	—
150	C₂²×C₄₇	#568	1	2	yes	—
151	C₄×C₄₇	—	2	4	no	—
152	C₄×C₄₇	—	2	4	no	—
153	C₄×C₄₇	—	2	4	no	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	—	4	1	no	—
156	C₄×C₄₇	—	4	1	no	—
157	C₄×C₄₇	—	4	1	no	—
158	C₄×C₄₇	—	4	1	no	—
159	C₄×C₄₇	—	4	2	no	—
160	C₄×C₄₇	#529	1	4	yes	—
161	C₄×#117	D₄₇	1	1	yes	characteristic
162	#1839	—	47	2	no	—
163	#1831	—	47	2	no	—
164	#1818	—	47	2	no	—
165	#1798	—	47	2	no	—
166	#1768	—	47	2	no	—
167	#1749	—	47	1	no	—
168	C₄×C₂²⋊C₈	—	47	1	no	—
169	#1746	—	47	2	no	—
170	C₂×C₄₇	—	4	2	no	—
171	C₂×C₄₇	#2283	1	1	yes	characteristic
172	C₂×C₄₇	#2922	1	1	yes	characteristic
173	C₂×C₄₇	#2923	1	1	yes	characteristic
174	C₂×C₄₇	#2924	1	2	yes	—
175	C₂×C₄₇	#2925	1	2	yes	—
176	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
177	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
178	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
179	C₂×C₄⋊C₈	—	47	4	no	—
180	C₂×C₂²⋊C₈	—	47	2	no	—
181	C₂×C₂²⋊C₈	—	47	2	no	—
182	C₂×#97	—	47	2	no	—
183	C₂×C₄×C₈	—	47	2	no	—
184	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
185	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
186	C₄×C₂²⋊C₄	Dic₄₇	1	2	yes	—
187	C₄×C₂²⋊C₄	Dic₄₇	1	2	yes	—
188	#376	C₂×D₄₇	1	1	yes	characteristic
189	#376	C₂×D₄₇	1	1	yes	characteristic
190	C₄³	C₂×D₄₇	1	1	yes	characteristic
191	C₄³	C₂×D₄₇	1	1	yes	characteristic
192	#336	—	47	4	no	—
193	#336	—	47	4	no	—
194	#336	—	47	4	no	—
195	#336	—	47	2	no	—
196	#336	—	47	2	no	—
197	C₄₇	#20784	1	1	yes	characteristic, complemented, Sylow
198	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
199	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
200	C₂²×C₈	—	47	4	no	—
201	C₂²×C₈	—	47	4	no	—
202	#117	—	2	4	no	—
203	#117	—	2	4	no	—
204	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
205	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
206	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
207	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
208	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
210	C₂×C₄²	—	2	4	no	—
211	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
212	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
213	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
214	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
215	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
216	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
217	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
218	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
219	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
220	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
222	C₄⋊C₈	—	94	8	no	—
223	C₂²⋊C₈	—	94	4	no	—
224	C₂²⋊C₈	—	94	4	no	—
225	#97	—	94	4	no	—
226	C₄×C₈	—	94	4	no	—
227	#95	C₂×Dic₄₇	1	2	yes	—
228	#95	C₂×Dic₄₇	1	2	yes	—
229	#95	C₂×Dic₄₇	1	2	yes	—
230	#95	C₂×Dic₄₇	1	2	yes	—
231	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	#10625350	1	4	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	2	yes	—
241	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
244	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
245	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
246	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
247	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
248	C₂×C₈	—	94	4	no	—
249	C₂×C₈	—	94	4	no	—
250	C₂×C₈	—	94	4	no	—
251	C₂×C₈	—	94	4	no	—
252	C₂×C₈	—	94	4	no	—
253	C₂×C₈	—	94	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	2	4	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	4	2	no	—
269	C₄²	#10625350	1	4	yes	—
270	C₄²	#10625350	1	4	yes	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	#13093983	1	2	yes	—
276	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	8	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	8	no	—
298	C₂×C₄	#13093983	1	4	yes	—
299	C₂×C₄	#13093983	1	4	yes	—
300	C₂×C₄	#13093983	1	2	yes	—
301	C₂×C₄	#13094031	1	4	yes	—
302	C₂×C₄	C₂×#10625350	1	2	yes	—
303	C₂×C₄	C₂×#10625350	1	2	yes	—
304	C₂×C₄	C₂×#10625350	1	2	yes	—
305	C₈	—	188	4	no	—
306	C₈	—	188	4	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	C₂×#13093983	1	1	yes	characteristic
313	C₂²	C₂×#13093983	1	1	yes	characteristic
314	C₂²	C₂×#13093983	1	1	yes	characteristic
315	C₂²	C₂×#13094031	1	2	yes	—
316	C₂²	#15689890	1	2	yes	—
317	C₄	—	2	2	no	—
318	C₄	—	2	4	no	—
319	C₄	—	2	4	no	—
320	C₄	—	2	4	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	2	no	—
326	C₄	#15689657	1	4	yes	—
327	C₂	—	4	2	no	—
328	C₂	#20245786	1	1	yes	characteristic
329	C₂	#20252032	1	1	yes	characteristic
330	C₂	#20252039	1	1	yes	characteristic
331	C₂	#20252046	1	2	yes	—
332	C₂	#20252054	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄×#117×C₄₇	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#20784
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
1. #20243531	C₂	semidirect
2. #20243505	C₂	semidirect
5. #20243245	C₂	semidirect
8. #20243140	C₂	semidirect
197. C₄₇	#20784	semidirect

Maximal subgroups

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

subgroup	index
#20243531	2
#20243505	2
#20243459	2
#20243372	2
#20243245	2
#20243150	2
C₄×C₂²⋊(C₄₇⋊C₈)	2
#20243140	2
C₄×#117×C₄₇	2
#20784	47