Subgroups — #26216574

Order 12032.

802 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243533	C₂	1	2	yes	maximal, complemented
2	#20243508	C₂	1	2	yes	maximal, complemented
3	#20243456	C₂	1	2	yes	maximal
4	#20243368	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	C₂×C₄₇×#376	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	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#376	C₄	1	2	yes	—
24	C₄₇×#376	C₄	1	2	yes	—
25	C₄₇×#376	C₂²	1	1	yes	characteristic
26	C₄₇×#376	C₂²	1	1	yes	characteristic
27	C₄₇×#376	C₂²	1	1	yes	characteristic
28	C₄₇×#376	C₂²	1	1	yes	characteristic
29	C₂×#95×C₄₇	C₄	1	2	yes	—
30	C₂×#95×C₄₇	C₄	1	2	yes	—
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	1	yes	characteristic
39	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	—	2	4	no	—
44	C₂×C₄²×C₄₇	—	2	4	no	—
45	C₂×C₄²×C₄₇	—	2	4	no	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	#95×C₄₇	C₂×C₄	1	2	yes	—
57	#95×C₄₇	C₂×C₄	1	2	yes	—
58	#95×C₄₇	C₂×C₄	1	2	yes	—
59	#95×C₄₇	C₂×C₄	1	2	yes	—
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	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
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	—	2	4	no	—
73	C₂²×C₄×C₄₇	—	2	4	no	—
74	C₂²×C₄×C₄₇	—	4	2	no	—
75	C₂²×C₄×C₄₇	—	4	2	no	—
76	C₂²×C₄×C₄₇	—	4	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₄₇	#41	1	4	yes	—
87	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
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	—	2	2	no	—
95	C₄²×C₄₇	—	2	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	#117	1	2	yes	—
108	C₂³×C₄₇	#131	1	4	yes	—
109	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	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	2	no	—
116	C₂×C₄×C₄₇	—	2	8	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	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	—	2	2	no	—
131	C₂×C₄×C₄₇	—	4	8	no	—
132	C₂×C₄×C₄₇	—	4	8	no	—
133	C₂×C₄×C₄₇	—	4	4	no	—
134	C₂×C₄×C₄₇	—	4	4	no	—
135	C₂×C₄×C₄₇	#117	1	2	yes	—
136	C₂×C₄×C₄₇	#117	1	4	yes	—
137	C₂×C₄×C₄₇	#117	1	4	yes	—
138	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
139	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
140	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
141	C₄₇⋊C₈	—	4	4	no	—
142	C₄₇⋊C₈	—	4	4	no	—
143	#20779	—	47	1	no	maximal, Sylow
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	4	no	—
146	C₂²×C₄₇	—	2	4	no	—
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	#433	1	4	yes	—
149	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
150	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
152	C₂²×C₄₇	C₂×#131	1	2	yes	—
153	C₂²×C₄₇	#568	1	2	yes	—
154	C₄×C₄₇	—	2	4	no	—
155	C₄×C₄₇	—	2	4	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	4	1	no	—
158	C₄×C₄₇	—	4	1	no	—
159	C₄×C₄₇	—	4	1	no	—
160	C₄×C₄₇	—	4	1	no	—
161	C₄×C₄₇	—	4	2	no	—
162	C₄×C₄₇	—	4	2	no	—
163	C₄×C₄₇	#529	1	4	yes	—
164	C₂×#376	D₄₇	1	1	yes	characteristic
165	#1840	—	47	2	no	—
166	#1832	—	47	2	no	—
167	#1817	—	47	2	no	—
168	#1797	—	47	2	no	—
169	#1768	—	47	2	no	—
170	#1749	—	47	1	no	—
171	C₄×C₂²⋊C₈	—	47	1	no	—
172	#1746	—	47	2	no	—
173	C₂×C₄₇	—	2	4	no	—
174	C₂×C₄₇	#2283	1	1	yes	characteristic
175	C₂×C₄₇	C₂×#433	1	1	yes	characteristic
176	C₂×C₄₇	#2921	1	1	yes	characteristic
177	C₂×C₄₇	#2924	1	2	yes	—
178	C₂×C₄₇	#2924	1	2	yes	—
179	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
180	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
181	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
182	C₂×C₄⋊C₈	—	47	4	no	—
183	C₂×C₂²⋊C₈	—	47	2	no	—
184	C₂×C₂²⋊C₈	—	47	2	no	—
185	C₂×#97	—	47	2	no	—
186	C₂×C₄×C₈	—	47	2	no	—
187	#376	Dic₄₇	1	2	yes	—
188	#376	Dic₄₇	1	2	yes	—
189	#376	C₂×D₄₇	1	1	yes	characteristic
190	#376	C₂×D₄₇	1	1	yes	characteristic
191	#376	C₂×D₄₇	1	1	yes	characteristic
192	#376	C₂×D₄₇	1	1	yes	characteristic
193	C₂×#95	Dic₄₇	1	2	yes	—
194	C₂×#95	Dic₄₇	1	2	yes	—
195	#336	—	47	4	no	—
196	#336	—	47	4	no	—
197	#336	—	47	4	no	—
198	#336	—	47	2	no	—
199	#336	—	47	2	no	—
200	C₄₇	#20779	1	1	yes	characteristic, complemented, Sylow
201	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
202	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
203	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
204	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
205	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
206	C₂²×C₈	—	47	4	no	—
207	C₂²×C₈	—	47	4	no	—
208	C₂×C₄²	—	2	4	no	—
209	C₂×C₄²	—	2	4	no	—
210	C₂×C₄²	—	2	4	no	—
211	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
212	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
213	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
214	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
215	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
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	C₄⋊C₈	—	94	8	no	—
222	C₂²⋊C₈	—	94	4	no	—
223	C₂²⋊C₈	—	94	4	no	—
224	#97	—	94	4	no	—
225	C₄×C₈	—	94	4	no	—
226	#95	C₂×Dic₄₇	1	2	yes	—
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	#95	C₂×Dic₄₇	1	2	yes	—
232	#95	C₂×Dic₄₇	1	2	yes	—
233	#95	C₂×Dic₄₇	1	2	yes	—
234	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	#10625350	1	4	yes	—
248	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
251	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
252	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
255	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
256	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
257	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
258	C₂×C₈	—	94	4	no	—
259	C₂×C₈	—	94	4	no	—
260	C₂×C₈	—	94	4	no	—
261	C₂×C₈	—	94	4	no	—
262	C₂×C₈	—	94	4	no	—
263	C₂×C₈	—	94	4	no	—
264	C₄²	—	2	4	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	#10625350	1	4	yes	—
273	C₄²	#10625350	1	4	yes	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	2	no	—
277	C₂³	#13093983	1	2	yes	—
278	C₂³	#13094031	1	4	yes	—
279	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	8	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	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	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	8	no	—
302	C₂×C₄	—	4	8	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	#13093983	1	2	yes	—
306	C₂×C₄	#13093983	1	4	yes	—
307	C₂×C₄	#13093983	1	4	yes	—
308	C₂×C₄	C₂×#10625350	1	2	yes	—
309	C₂×C₄	C₂×#10625350	1	2	yes	—
310	C₂×C₄	C₂×#10625350	1	2	yes	—
311	C₈	—	188	4	no	—
312	C₈	—	188	4	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	4	no	—
316	C₂²	—	2	4	no	—
317	C₂²	#15689184	1	4	yes	—
318	C₂²	C₂×#13093983	1	1	yes	characteristic
319	C₂²	C₂×#13093983	1	1	yes	characteristic
320	C₂²	C₂×#13093983	1	1	yes	characteristic
321	C₂²	C₂×#13094031	1	2	yes	—
322	C₂²	#15689890	1	2	yes	—
323	C₄	—	2	2	no	—
324	C₄	—	2	4	no	—
325	C₄	—	2	4	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	2	no	—
332	C₄	#15689657	1	4	yes	—
333	C₂	—	2	4	no	—
334	C₂	#20245786	1	1	yes	characteristic
335	C₂	C₂×#15689184	1	1	yes	characteristic
336	C₂	#20252026	1	1	yes	characteristic
337	C₂	#20252046	1	2	yes	—
338	C₂	#20252046	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#376	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#376	D₄₇
hypocenter	C₄₇	#20779
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
1. #20243533	C₂	semidirect
2. #20243508	C₂	semidirect
5. #20243245	C₂	semidirect
8. #20243140	C₂	semidirect
200. C₄₇	#20779	semidirect

Maximal subgroups

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

subgroup	index
#20243533	2
#20243508	2
#20243456	2
#20243368	2
#20243245	2
#20243150	2
C₄×C₂²⋊(C₄₇⋊C₈)	2
#20243140	2
C₂×C₄₇×#376	2
#20779	47