Subgroups — #26115800

Order 12032.

606 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244225	C₂	1	2	yes	maximal, complemented
2	#20243417	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241956	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×#466	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093958	—	2	1	no	—
7	C₂×#13093939	—	2	4	no	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₄₇	—	2	1	no	—
10	#15688983	C₂²	1	2	yes	—
11	C₄×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
12	#15688886	—	2	1	no	—
13	#15688877	—	2	1	no	—
14	#15688857	—	2	4	no	—
15	C₂²×D₈×C₄₇	C₄	1	1	yes	characteristic, complemented
16	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#121×C₄₇	C₂²	1	2	yes	—
18	C₄₇×#466	—	2	1	no	—
19	C₄₇×#466	—	2	1	no	—
20	C₄₇×#466	—	2	1	no	—
21	C₄₇×#466	—	2	1	no	—
22	C₂×#107×C₄₇	C₄	1	1	yes	characteristic, complemented
23	C₂×#102×C₄₇	C₄	1	2	yes	complemented
24	C₂²×C₄₇⋊C₈	—	2	1	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
27	C₂×C₂²⋊Dic₄₇	D₄	1	2	yes	—
28	C₂×C₄×Dic₄₇	—	2	1	no	—
29	#13093958	—	4	2	no	—
30	#13093939	—	4	8	no	—
31	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
32	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
33	C₈×Dic₄₇	—	4	2	no	—
34	C₄₇⋊#95	—	2	1	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₂²×D₄×C₄₇	—	2	2	no	—
37	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×D₈×C₄₇	—	2	1	no	—
40	C₂×D₈×C₄₇	—	2	1	no	—
41	C₂×D₈×C₄₇	—	4	2	no	—
42	C₂²×C₈×C₄₇	—	4	2	no	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
49	#121×C₄₇	—	2	2	no	—
50	#121×C₄₇	—	2	2	no	—
51	#121×C₄₇	—	2	2	no	—
52	#121×C₄₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
54	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
55	#107×C₄₇	—	2	1	no	—
56	#107×C₄₇	—	2	1	no	—
57	#102×C₄₇	—	2	2	no	—
58	#102×C₄₇	—	2	2	no	—
59	C₂⁴×C₄₇	—	2	2	no	—
60	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂×D₄×C₄₇	—	2	2	no	—
62	C₂×D₄×C₄₇	—	2	2	no	—
63	C₂×D₄×C₄₇	—	4	2	no	—
64	C₂×D₄×C₄₇	—	4	2	no	—
65	C₂×D₄×C₄₇	—	4	2	no	—
66	C₂×D₄×C₄₇	—	4	2	no	—
67	C₂×D₄×C₄₇	—	4	4	no	—
68	C₂×D₄×C₄₇	—	4	4	no	—
69	C₂²×C₄×C₄₇	—	2	2	no	—
70	C₂²×C₄×C₄₇	—	4	1	no	—
71	C₂²×C₄×C₄₇	—	4	1	no	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
78	C₂²×Dic₄₇	—	2	2	no	—
79	C₂²×Dic₄₇	—	4	2	no	—
80	D₈×C₄₇	—	4	2	no	—
81	D₈×C₄₇	—	4	2	no	—
82	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	—	4	1	no	—
86	C₂×C₈×C₄₇	—	4	1	no	—
87	C₂×C₈×C₄₇	—	4	1	no	—
88	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₄⋊C₄×C₄₇	—	2	2	no	—
93	C₄⋊C₄×C₄₇	—	2	2	no	—
94	C₂²⋊C₄×C₄₇	—	4	2	no	—
95	C₂²⋊C₄×C₄₇	—	4	2	no	—
96	C₂²⋊Dic₄₇	—	2	2	no	—
97	C₂²⋊Dic₄₇	—	4	4	no	—
98	C₂²⋊Dic₄₇	D₈	1	4	yes	complemented
99	C₄⋊Dic₄₇	—	4	2	no	—
100	C₄⋊Dic₄₇	—	4	2	no	—
101	C₄×Dic₄₇	—	4	1	no	—
102	C₄×Dic₄₇	—	4	1	no	—
103	C₂×C₄₇⋊C₈	—	2	2	no	—
104	C₂×C₄₇⋊C₈	—	4	1	no	—
105	C₂×C₄₇⋊C₈	—	4	1	no	—
106	C₂³×C₄₇	—	4	1	no	—
107	C₂³×C₄₇	—	4	1	no	—
108	C₂³×C₄₇	—	4	2	no	—
109	C₂³×C₄₇	—	4	4	no	—
110	C₂³×C₄₇	—	8	2	no	—
111	C₂³×C₄₇	—	8	2	no	—
112	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
113	C₂³×C₄₇	#121	1	1	yes	characteristic
114	C₂³×C₄₇	#127	1	1	yes	characteristic
115	C₂³×C₄₇	C₂×D₈	1	2	yes	—
116	C₂³×C₄₇	#135	1	2	yes	—
117	D₄×C₄₇	—	4	2	no	—
118	D₄×C₄₇	—	4	2	no	—
119	D₄×C₄₇	—	8	2	no	—
120	D₄×C₄₇	—	8	4	no	—
121	D₄×C₄₇	—	8	2	no	—
122	D₄×C₄₇	—	8	2	no	—
123	C₂×C₄×C₄₇	—	2	1	no	—
124	C₂×C₄×C₄₇	—	2	1	no	—
125	C₂×C₄×C₄₇	—	2	1	no	—
126	C₂×C₄×C₄₇	—	2	1	no	—
127	C₂×C₄×C₄₇	—	2	1	no	—
128	C₂×C₄×C₄₇	—	2	1	no	—
129	C₂×C₄×C₄₇	—	4	1	no	—
130	C₂×C₄×C₄₇	—	4	1	no	—
131	C₂×C₄×C₄₇	—	4	1	no	—
132	C₂×C₄×C₄₇	—	4	2	no	—
133	C₂×C₄×C₄₇	—	4	4	no	—
134	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
137	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
138	C₂×Dic₄₇	—	2	4	no	—
139	C₂×Dic₄₇	—	4	2	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	8	2	no	—
142	C₂×Dic₄₇	—	8	2	no	—
143	C₂×Dic₄₇	—	8	2	no	—
144	C₈×C₄₇	—	2	1	no	—
145	C₈×C₄₇	—	2	1	no	—
146	C₈×C₄₇	—	2	1	no	—
147	C₈×C₄₇	#99	1	2	yes	complemented
148	C₄₇⋊C₈	—	4	2	no	—
149	#9783	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	—	2	1	no	—
152	C₂²×C₄₇	—	2	1	no	—
153	C₂²×C₄₇	—	2	1	no	—
154	C₂²×C₄₇	—	2	1	no	—
155	C₂²×C₄₇	—	2	1	no	—
156	C₂²×C₄₇	—	4	2	no	—
157	C₂²×C₄₇	—	4	2	no	—
158	C₂²×C₄₇	—	8	2	no	—
159	C₂²×C₄₇	—	8	2	no	—
160	C₂²×C₄₇	—	8	2	no	—
161	C₂²×C₄₇	—	16	4	no	—
162	C₂²×C₄₇	#390	1	1	yes	characteristic
163	C₂²×C₄₇	C₄×D₈	1	2	yes	—
164	C₂²×C₄₇	#466	1	2	yes	—
165	C₂²×C₄₇	#493	1	1	yes	characteristic
166	C₂²×C₄₇	#495	1	1	yes	characteristic
167	C₄×C₄₇	—	2	1	no	—
168	C₄×C₄₇	—	2	1	no	—
169	C₄×C₄₇	—	2	1	no	—
170	C₄×C₄₇	—	8	2	no	—
171	C₄×C₄₇	C₂×#99	1	1	yes	characteristic
172	C₄×C₄₇	#410	1	1	yes	characteristic
173	Dic₄₇	—	4	4	no	—
174	Dic₄₇	—	16	2	no	—
175	C₂×#466	D₄₇	1	1	yes	characteristic
176	#1964	—	47	2	no	—
177	#1808	—	47	1	no	—
178	C₈×C₂²⋊C₄	—	47	1	no	—
179	#1519	—	47	2	no	—
180	C₂×C₄₇	—	2	2	no	—
181	C₂×C₄₇	—	2	1	no	—
182	C₂×C₄₇	—	2	2	no	—
183	C₂×C₄₇	—	2	1	no	—
184	C₂×C₄₇	—	16	2	no	—
185	C₂×C₄₇	#1964	1	1	yes	characteristic
186	C₂×C₄₇	#1970	1	1	yes	characteristic
187	C₂×C₄₇	#1976	1	1	yes	characteristic
188	C₂²×D₈	Dic₄₇	1	1	yes	characteristic, complemented
189	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
190	C₂×#121	C₂×D₄₇	1	2	yes	—
191	#466	—	2	1	no	—
192	#466	—	2	1	no	—
193	#466	—	2	1	no	—
194	#466	—	2	1	no	—
195	C₂×#107	—	94	1	no	—
196	C₂×#107	Dic₄₇	1	1	yes	characteristic, complemented
197	C₂×#102	Dic₄₇	1	2	yes	complemented
198	C₂×#99	—	94	4	no	—
199	C₂×C₂²⋊C₈	—	47	1	no	—
200	C₂×C₄×C₈	—	94	1	no	—
201	#380	—	47	2	no	—
202	C₄×C₂²⋊C₄	—	47	1	no	—
203	#340	—	94	1	no	—
204	#336	—	94	1	no	—
205	#327	—	94	4	no	—
206	C₄₇	#9783	1	1	yes	characteristic, complemented, Sylow
207	C₂²×D₄	—	2	2	no	—
208	C₂²×D₄	C₂×Dic₄₇	1	2	yes	—
209	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
210	C₂×D₈	—	2	1	no	—
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	—	4	2	no	—
213	C₂²×C₈	—	4	2	no	—
214	C₂²×C₈	—	94	1	no	—
215	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
216	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
217	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
218	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
221	#121	—	2	2	no	—
222	#121	—	2	2	no	—
223	#121	—	2	2	no	—
224	#121	—	2	2	no	—
225	C₂×C₄⋊C₄	—	94	2	no	—
226	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	—	47	2	no	—
229	C₂×C₂²⋊C₄	—	94	2	no	—
230	C₂×C₄²	—	94	1	no	—
231	#107	—	2	1	no	—
232	#107	—	2	1	no	—
233	#107	—	188	2	no	—
234	#102	—	2	2	no	—
235	#102	—	2	2	no	—
236	#99	—	188	8	no	—
237	C₂²⋊C₈	—	47	2	no	—
238	C₂²⋊C₈	—	94	1	no	—
239	C₄×C₈	—	188	2	no	—
240	#95	—	94	1	no	—
241	#95	—	94	2	no	—
242	C₂⁴	—	2	2	no	—
243	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	4	no	—
251	C₂×D₄	—	4	4	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	94	2	no	—
256	C₂²×C₄	—	188	2	no	—
257	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
258	C₂²×C₄	#10625352	1	1	yes	characteristic
259	C₂²×C₄	#10625352	1	1	yes	characteristic
260	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
263	D₈	—	4	2	no	—
264	D₈	—	4	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	94	2	no	—
272	C₂×C₈	—	188	1	no	—
273	C₂×C₈	—	188	1	no	—
274	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
275	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
276	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
277	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	188	2	no	—
281	C₄⋊C₄	—	188	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	47	4	no	—
285	C₂²⋊C₄	—	94	2	no	—
286	C₂²⋊C₄	—	188	4	no	—
287	C₄²	—	188	1	no	—
288	C₄²	—	188	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	4	4	no	—
293	C₂³	—	8	2	no	—
294	C₂³	—	8	2	no	—
295	C₂³	D₄×Dic₄₇	1	1	yes	characteristic
296	C₂³	#13093995	1	1	yes	characteristic
297	C₂³	#13094020	1	1	yes	characteristic
298	C₂³	D₈×D₄₇	1	2	yes	—
299	C₂³	#13094043	1	2	yes	—
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	8	2	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	2	no	—
305	D₄	—	8	4	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	94	4	no	—
318	C₂×C₄	—	376	2	no	—
319	C₂×C₄	—	376	2	no	—
320	C₂×C₄	—	376	2	no	—
321	C₂×C₄	—	188	2	no	—
322	C₂×C₄	—	188	2	no	—
323	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
324	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
325	C₂×C₄	#13093995	1	1	yes	characteristic
326	C₂×C₄	#13094011	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	188	2	no	—
331	C₈	#13093939	1	2	yes	complemented
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	16	4	no	—
344	C₂²	#15689026	1	1	yes	characteristic
345	C₂²	D₈×Dic₄₇	1	2	yes	—
346	C₂²	#15689356	1	2	yes	—
347	C₂²	#15689498	1	1	yes	characteristic
348	C₂²	#15689506	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	8	2	no	—
353	C₄	—	188	4	no	—
354	C₄	—	752	2	no	—
355	C₄	C₂×#13093939	1	1	yes	characteristic
356	C₄	#15689097	1	1	yes	characteristic
357	C₂	—	2	2	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	2	no	—
360	C₂	—	2	1	no	—
361	C₂	—	16	2	no	—
362	C₂	#20244225	1	1	yes	characteristic
363	C₂	#20244255	1	1	yes	characteristic
364	C₂	#20244287	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689026
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting, prime radical	C₂×C₄₇×#466	C₂
socle	C₂²×C₄₇	#390
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#466	D₄₇
hypocenter	C₄₇	#9783
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. #20244225	C₂	semidirect
2. #20243417	C₂	semidirect
3. C₈×C₂²⋊Dic₄₇	C₂	semidirect
4. #20241956	C₂	semidirect
15. C₂²×D₈×C₄₇	C₄	semidirect
22. C₂×#107×C₄₇	C₄	semidirect
23. C₂×#102×C₄₇	C₄	semidirect
98. C₂²⋊Dic₄₇	D₈	semidirect
147. C₈×C₄₇	#99	semidirect
188. C₂²×D₈	Dic₄₇	semidirect
196. C₂×#107	Dic₄₇	semidirect
197. C₂×#102	Dic₄₇	semidirect
206. C₄₇	#9783	semidirect
331. C₈	#13093939	semidirect

Maximal subgroups

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

subgroup	index
#20244225	2
#20243417	2
C₈×C₂²⋊Dic₄₇	2
#20241956	2
C₂×C₄₇×#466	2
#9783	47