Subgroups — #26224652

Order 12032.

638 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246693	C₂	1	1	yes	maximal, characteristic
2	#20244111	C₂	1	1	yes	maximal, characteristic
3	#20244111	C₂	1	1	yes	maximal, characteristic
4	#20243945	C₂	1	2	yes	maximal
5	#20243945	C₂	1	2	yes	maximal
6	#20243508	C₂	1	1	yes	maximal, characteristic
7	#20243264	C₂	1	2	yes	maximal
8	#20243256	C₂	1	2	yes	maximal
9	#20243255	C₂	1	1	yes	maximal, characteristic
10	C₈×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1726	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
16	C₂×#13093934	C₂²	1	2	yes	—
17	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
18	#15689065	C₄	1	2	yes	—
19	#15689065	C₄	1	2	yes	—
20	#15689057	C₄	1	2	yes	—
21	#15689057	C₄	1	2	yes	—
22	C₄×Dic₄₇⋊C₄	C₂²	1	2	yes	—
23	C₄×Dic₄₇⋊C₄	C₂²	1	2	yes	—
24	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
25	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
26	#15688878	C₂²	1	2	yes	—
27	#15688878	C₂²	1	2	yes	—
28	#15688877	C₂²	1	2	yes	—
29	#15688877	C₂²	1	2	yes	—
30	#15688877	C₂²	1	2	yes	—
31	#15688877	C₂²	1	2	yes	—
32	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#376	C₂²	1	1	yes	characteristic
39	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
40	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
43	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
48	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
49	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
50	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
51	Dic₄₇⋊C₈	—	2	4	no	—
52	Dic₄₇⋊C₈	—	2	4	no	—
53	#13093934	—	2	4	no	—
54	C₈×Dic₄₇	—	2	4	no	—
55	C₄₇⋊#95	C₂×C₄	1	1	yes	characteristic
56	C₄₇⋊#95	C₂×C₄	1	1	yes	characteristic
57	C₄₇⋊#95	C₂×C₄	1	1	yes	characteristic
58	C₄₇⋊#95	C₂×C₄	1	1	yes	characteristic
59	C₄₇⋊#95	C₂×C₄	1	2	yes	—
60	C₄₇⋊#95	C₂×C₄	1	2	yes	—
61	C₄₇⋊#95	C₂×C₄	1	2	yes	—
62	C₄₇⋊#95	C₂×C₄	1	2	yes	—
63	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
70	#97×C₄₇	—	2	2	no	—
71	#97×C₄₇	—	2	2	no	—
72	#97×C₄₇	—	2	2	no	—
73	C₄×C₈×C₄₇	—	2	2	no	—
74	C₄×C₈×C₄₇	—	2	2	no	—
75	C₄×C₈×C₄₇	Q₈	1	4	yes	—
76	#95×C₄₇	C₂×C₄	1	2	yes	—
77	#95×C₄₇	C₂×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
83	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
84	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
85	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
86	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₇	—	2	1	no	—
88	C₂×C₈×C₄₇	—	2	1	no	—
89	C₂×C₈×C₄₇	—	2	1	no	—
90	C₂×C₈×C₄₇	—	2	1	no	—
91	C₂×C₈×C₄₇	—	2	1	no	—
92	C₂×C₈×C₄₇	—	2	1	no	—
93	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₄₇	#41	1	2	yes	—
96	C₂×C₈×C₄₇	#41	1	2	yes	—
97	C₂×C₈×C₄₇	#41	1	2	yes	—
98	C₂×C₈×C₄₇	#41	1	2	yes	—
99	C₄²×C₄₇	—	2	1	no	—
100	C₄²×C₄₇	—	2	1	no	—
101	C₄²×C₄₇	—	2	1	no	—
102	C₄²×C₄₇	—	2	1	no	—
103	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
104	C₄²×C₄₇	#41	1	2	yes	—
105	C₄⋊Dic₄₇	—	2	4	no	—
106	C₄⋊Dic₄₇	—	4	2	no	—
107	Dic₄₇⋊C₄	—	2	4	no	—
108	Dic₄₇⋊C₄	—	2	4	no	—
109	Dic₄₇⋊C₄	—	4	2	no	—
110	Dic₄₇⋊C₄	—	4	2	no	—
111	C₄×Dic₄₇	—	2	2	no	—
112	C₄×Dic₄₇	—	2	2	no	—
113	C₄×Dic₄₇	—	2	2	no	—
114	C₄×Dic₄₇	—	2	2	no	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	—	2	2	no	—
117	C₂×C₄₇⋊C₈	—	2	2	no	—
118	C₂×C₄₇⋊C₈	—	2	2	no	—
119	C₂×C₄₇⋊C₈	—	2	2	no	—
120	C₂×C₄₇⋊C₈	—	2	2	no	—
121	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
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	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
132	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₄₇	—	2	2	no	—
139	C₂×Dic₄₇	—	2	2	no	—
140	C₂×Dic₄₇	—	2	2	no	—
141	C₂×Dic₄₇	—	2	2	no	—
142	C₂×Dic₄₇	—	2	2	no	—
143	C₂×Dic₄₇	—	2	2	no	—
144	C₂×Dic₄₇	—	2	2	no	—
145	C₂×Dic₄₇	—	2	2	no	—
146	C₂×Dic₄₇	—	2	2	no	—
147	C₂×Dic₄₇	—	2	2	no	—
148	C₂×Dic₄₇	—	2	2	no	—
149	C₂×Dic₄₇	#131	1	4	yes	—
150	C₈×C₄₇	—	2	2	no	—
151	C₈×C₄₇	—	2	2	no	—
152	C₈×C₄₇	—	2	2	no	—
153	C₈×C₄₇	#125	1	4	yes	—
154	C₄₇⋊C₈	—	4	2	no	—
155	C₄₇⋊C₈	—	4	2	no	—
156	#21679	—	47	1	no	maximal, Sylow
157	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
158	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
159	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
160	C₂²×C₄₇	C₂×#131	1	2	yes	—
161	C₂²×C₄₇	#568	1	2	yes	—
162	C₄×C₄₇	—	2	2	no	—
163	C₄×C₄₇	—	2	2	no	—
164	C₄×C₄₇	—	4	2	no	—
165	C₄×C₄₇	—	4	2	no	—
166	C₄×C₄₇	C₂×#125	1	1	yes	characteristic
167	C₄×C₄₇	#529	1	1	yes	characteristic
168	C₄×C₄₇	#529	1	1	yes	characteristic
169	C₄×C₄₇	#533	1	1	yes	characteristic
170	Dic₄₇	—	2	4	no	—
171	Dic₄₇	—	4	2	no	—
172	Dic₄₇	—	4	2	no	—
173	Dic₄₇	—	4	2	no	—
174	#2402	—	47	1	no	—
175	#1940	—	47	1	no	—
176	#1940	—	47	1	no	—
177	#1916	—	47	2	no	—
178	#1916	—	47	2	no	—
179	#1832	—	47	1	no	—
180	#1771	—	47	2	no	—
181	#1770	—	47	2	no	—
182	#1770	—	47	1	no	—
183	C₈×C₄⋊C₄	—	47	1	no	—
184	#1726	D₄₇	1	1	yes	characteristic
185	C₂×C₄₇	C₄×#125	1	1	yes	characteristic
186	C₂×C₄₇	#2960	1	1	yes	characteristic
187	C₂×C₄₇	#2961	1	1	yes	characteristic
188	C₂×C₄₇	#2968	1	2	yes	—
189	C₂×C₄₇	#2969	1	2	yes	—
190	C₂×C₄⋊C₈	—	47	2	no	—
191	C₂×C₄⋊C₈	—	47	2	no	—
192	C₂×C₄⋊C₈	—	47	1	no	—
193	C₂×C₄⋊C₈	—	47	1	no	—
194	C₂×#97	—	47	2	no	—
195	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
196	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
197	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
198	C₂×C₄×C₈	—	47	2	no	—
199	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
200	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
201	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
202	#400	—	47	2	no	—
203	#400	—	47	2	no	—
204	#398	—	47	2	no	—
205	#398	—	47	2	no	—
206	C₄×C₄⋊C₄	—	47	1	no	—
207	C₄×C₄⋊C₄	—	47	1	no	—
208	C₄×C₄⋊C₄	—	47	2	no	—
209	C₄×C₄⋊C₄	—	47	2	no	—
210	#376	C₂×D₄₇	1	1	yes	characteristic
211	#336	—	47	2	no	—
212	#336	—	47	2	no	—
213	#336	—	47	2	no	—
214	#336	—	47	2	no	—
215	#336	—	47	2	no	—
216	#336	—	47	2	no	—
217	C₄₇	#21679	1	1	yes	characteristic, complemented, Sylow
218	C₂²×C₈	—	47	2	no	—
219	C₂²×C₈	—	47	2	no	—
220	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
221	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
222	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
223	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
224	C₂×C₄⋊C₄	—	47	2	no	—
225	C₂×C₄⋊C₄	—	47	2	no	—
226	C₂×C₄⋊C₄	—	47	2	no	—
227	C₂×C₄⋊C₄	—	47	2	no	—
228	C₂×C₄⋊C₄	—	47	2	no	—
229	C₂×C₄⋊C₄	—	47	2	no	—
230	C₂×C₄²	—	47	2	no	—
231	C₂×C₄²	—	47	2	no	—
232	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
233	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
234	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
235	C₄⋊C₈	—	94	4	no	—
236	C₄⋊C₈	—	94	4	no	—
237	C₄⋊C₈	—	94	2	no	—
238	C₄⋊C₈	—	94	2	no	—
239	#97	—	2	2	no	—
240	#97	—	2	2	no	—
241	#97	—	2	2	no	—
242	#97	—	94	4	no	—
243	C₄×C₈	—	2	2	no	—
244	C₄×C₈	—	2	2	no	—
245	C₄×C₈	—	94	4	no	—
246	C₄×C₈	Dic₉₄	1	4	yes	—
247	#95	—	47	1	no	—
248	#95	—	47	1	no	—
249	#95	—	47	2	no	—
250	#95	—	47	2	no	—
251	#95	—	47	2	no	—
252	#95	—	47	2	no	—
253	#95	—	47	1	no	—
254	#95	—	47	1	no	—
255	#95	C₄×D₄₇	1	2	yes	—
256	#95	C₄×D₄₇	1	2	yes	—
257	C₂²×C₄	—	47	2	no	—
258	C₂²×C₄	—	47	2	no	—
259	C₂²×C₄	—	47	2	no	—
260	C₂²×C₄	—	47	2	no	—
261	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
262	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
263	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
265	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	94	2	no	—
273	C₂×C₈	—	94	2	no	—
274	C₂×C₈	—	94	2	no	—
275	C₂×C₈	—	94	2	no	—
276	C₂×C₈	—	94	2	no	—
277	C₂×C₈	—	94	2	no	—
278	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
279	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
280	C₂×C₈	#10625350	1	2	yes	—
281	C₂×C₈	#10625350	1	2	yes	—
282	C₂×C₈	#10625350	1	2	yes	—
283	C₂×C₈	#10625350	1	2	yes	—
284	C₄⋊C₄	—	94	4	no	—
285	C₄⋊C₄	—	94	4	no	—
286	C₄⋊C₄	—	94	4	no	—
287	C₄⋊C₄	—	188	2	no	—
288	C₄⋊C₄	—	188	2	no	—
289	C₄⋊C₄	—	188	2	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	94	2	no	—
295	C₄²	—	94	2	no	—
296	C₄²	—	94	2	no	—
297	C₄²	—	94	2	no	—
298	C₄²	C₂×Dic₉₄	1	2	yes	—
299	C₄²	#10625350	1	2	yes	—
300	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	47	4	no	—
310	C₂×C₄	—	94	2	no	—
311	C₂×C₄	—	94	2	no	—
312	C₂×C₄	—	94	2	no	—
313	C₂×C₄	—	94	2	no	—
314	C₂×C₄	—	94	2	no	—
315	C₂×C₄	—	94	2	no	—
316	C₂×C₄	—	94	2	no	—
317	C₂×C₄	—	94	2	no	—
318	C₂×C₄	—	94	2	no	—
319	C₂×C₄	—	94	2	no	—
320	C₂×C₄	—	94	2	no	—
321	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
322	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
323	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
324	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
325	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
326	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
327	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
328	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	2	2	no	—
331	C₈	—	2	2	no	—
332	C₈	—	188	2	no	—
333	C₈	—	188	2	no	—
334	C₈	#13094016	1	4	yes	—
335	C₂²	C₂×C₄×Dic₉₄	1	1	yes	characteristic
336	C₂²	C₄×#10625350	1	1	yes	characteristic
337	C₂²	C₄×#10625350	1	1	yes	characteristic
338	C₂²	#131×D₄₇	1	2	yes	—
339	C₂²	#15689888	1	2	yes	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	4	2	no	—
343	C₄	—	4	2	no	—
344	C₄	—	94	4	no	—
345	C₄	—	188	2	no	—
346	C₄	—	188	2	no	—
347	C₄	—	188	2	no	—
348	C₄	C₂×#13094016	1	1	yes	characteristic
349	C₄	#15689657	1	1	yes	characteristic
350	C₄	#15689657	1	1	yes	characteristic
351	C₄	#15689680	1	1	yes	characteristic
352	C₂	C₄×#13094016	1	1	yes	characteristic
353	C₂	#20252320	1	1	yes	characteristic
354	C₂	#20252326	1	1	yes	characteristic
355	C₂	#20252378	1	2	yes	—
356	C₂	#20252388	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#1726	C₂
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1726	D₄₇
hypocenter	C₄₇	#21679
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1726

Complements

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

N	H ≅ G/N	type
217. C₄₇	#21679	semidirect

Maximal subgroups

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

subgroup	index
#20246693	2
#20244111	2
#20243945	2
#20243508	2
#20243264	2
#20243256	2
#20243255	2
C₈×C₄⋊Dic₄₇	2
C₄₇×#1726	2
#21679	47