Subgroups — #26316801

Order 12032.

770 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20249367	C₂	1	1	yes	maximal, characteristic, complemented
2	#20246114	C₂	1	1	yes	maximal, characteristic
3	#20244772	C₂	1	2	yes	maximal, complemented
4	#20244758	C₂	1	2	yes	maximal, complemented
5	#20244195	C₂	1	1	yes	maximal, characteristic
6	#20243647	C₂	1	2	yes	maximal, complemented
7	#20243633	C₂	1	2	yes	maximal, complemented
8	#20243550	C₂	1	1	yes	maximal, characteristic, complemented
9	#20243522	C₂	1	1	yes	maximal, characteristic, complemented
10	#20243248	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1961	C₂	1	1	yes	maximal, characteristic
12	C₂×D₄×Dic₄₇	C₂²	1	2	yes	—
13	C₂×#13093957	C₂²	1	2	yes	—
14	C₂×#13093955	C₂²	1	2	yes	—
15	C₂×Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×#13093945	C₂²	1	2	yes	—
18	C₂×#13093945	C₂²	1	2	yes	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	#15689065	—	2	2	no	—
21	#15689055	—	2	2	no	—
22	#15689026	C₂²	1	2	yes	—
23	#15689022	C₂²	1	2	yes	—
24	#15689019	C₂²	1	2	yes	—
25	#15689003	—	2	2	no	—
26	#15689003	—	2	2	no	—
27	#15688993	C₂²	1	1	yes	characteristic
28	#15688989	C₂²	1	1	yes	characteristic
29	#15688969	C₂²	1	1	yes	characteristic
30	#15688886	C₂²	1	2	yes	—
31	#15688886	C₂²	1	2	yes	—
32	#15688877	C₂²	1	1	yes	characteristic
33	#15688877	C₂²	1	1	yes	characteristic
34	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×#102×C₄₇	C₂²	1	2	yes	—
36	C₂×#102×C₄₇	C₂²	1	2	yes	—
37	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#382	C₂²	1	1	yes	characteristic
39	C₂³×Dic₄₇	—	2	2	no	—
40	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	D₄×Dic₄₇	—	4	2	no	—
43	D₄×Dic₄₇	—	4	2	no	—
44	C₂×C₄⋊Dic₄₇	—	2	2	no	—
45	C₂×C₄⋊Dic₄₇	—	2	2	no	—
46	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
48	C₂×Dic₄₇⋊C₄	—	2	2	no	—
49	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
50	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
51	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
52	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
54	#13093957	—	4	2	no	—
55	#13093955	—	4	2	no	—
56	Dic₄₇⋊C₈	—	2	2	no	—
57	Dic₄₇⋊C₈	—	2	2	no	—
58	#13093945	—	2	4	no	—
59	#13093945	—	2	4	no	—
60	C₄×C₄₇⋊C₈	—	2	2	no	—
61	C₄₇⋊#95	—	2	2	no	—
62	C₄₇⋊#95	—	2	2	no	—
63	C₄₇⋊#95	—	2	2	no	—
64	C₄₇⋊#95	D₄	1	2	yes	—
65	C₄₇⋊#95	D₄	1	2	yes	—
66	C₂²×D₄×C₄₇	—	2	2	no	—
67	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
68	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
70	#127×C₄₇	—	2	2	no	—
71	#127×C₄₇	Q₈	1	4	yes	—
72	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
73	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
74	#102×C₄₇	—	4	2	no	—
75	#102×C₄₇	—	4	2	no	—
76	#97×C₄₇	—	2	2	no	—
77	#95×C₄₇	—	2	2	no	—
78	C₂⁴×C₄₇	—	2	2	no	—
79	C₂×D₄×C₄₇	—	2	8	no	—
80	C₂×D₄×C₄₇	—	2	2	no	—
81	C₂×D₄×C₄₇	—	2	2	no	—
82	C₂×D₄×C₄₇	—	2	2	no	—
83	C₂×D₄×C₄₇	—	2	2	no	—
84	C₂×D₄×C₄₇	—	2	2	no	—
85	C₂×D₄×C₄₇	—	8	2	no	—
86	C₂×D₄×C₄₇	—	8	2	no	—
87	C₂×D₄×C₄₇	C₂×Q₈	1	4	yes	—
88	C₂²×C₄×C₄₇	—	2	2	no	—
89	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
91	C₂²×Dic₄₇	—	2	2	no	—
92	C₂²×Dic₄₇	—	2	2	no	—
93	C₂²×Dic₄₇	—	4	4	no	—
94	C₂²×Dic₄₇	—	8	2	no	—
95	C₂²×Dic₄₇	—	8	2	no	—
96	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
98	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₇	—	2	1	no	—
100	C₂×C₈×C₄₇	—	2	1	no	—
101	C₂×C₈×C₄₇	—	2	1	no	—
102	C₂×C₈×C₄₇	—	2	1	no	—
103	C₂×C₈×C₄₇	—	2	1	no	—
104	C₂×C₈×C₄₇	—	2	1	no	—
105	C₄⋊C₄×C₄₇	—	2	2	no	—
106	C₄⋊C₄×C₄₇	—	2	2	no	—
107	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
108	C₄²×C₄₇	#41	1	2	yes	—
109	C₂²⋊Dic₄₇	—	4	4	no	—
110	C₂²⋊Dic₄₇	—	8	2	no	—
111	C₂²⋊Dic₄₇	—	8	2	no	—
112	C₄⋊Dic₄₇	—	2	2	no	—
113	C₄⋊Dic₄₇	—	2	2	no	—
114	C₄⋊Dic₄₇	—	2	2	no	—
115	C₄⋊Dic₄₇	—	2	2	no	—
116	C₄⋊Dic₄₇	—	8	2	no	—
117	C₄⋊Dic₄₇	—	8	2	no	—
118	Dic₄₇⋊C₄	—	8	2	no	—
119	C₄×Dic₄₇	—	2	1	no	—
120	C₄×Dic₄₇	—	2	1	no	—
121	C₄×Dic₄₇	—	2	1	no	—
122	C₄×Dic₄₇	—	2	1	no	—
123	C₂×C₄₇⋊C₈	—	2	1	no	—
124	C₂×C₄₇⋊C₈	—	2	1	no	—
125	C₂×C₄₇⋊C₈	—	2	1	no	—
126	C₂×C₄₇⋊C₈	—	2	1	no	—
127	C₂×C₄₇⋊C₈	#41	1	2	yes	—
128	C₂×C₄₇⋊C₈	#41	1	2	yes	—
129	C₂³×C₄₇	—	2	4	no	—
130	C₂³×C₄₇	—	4	2	no	—
131	C₂³×C₄₇	—	4	2	no	—
132	C₂³×C₄₇	—	4	2	no	—
133	C₂³×C₄₇	—	4	2	no	—
134	C₂³×C₄₇	—	4	2	no	—
135	C₂³×C₄₇	—	4	2	no	—
136	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
137	D₄×C₄₇	—	2	4	no	—
138	D₄×C₄₇	—	2	4	no	—
139	D₄×C₄₇	—	4	2	no	—
140	D₄×C₄₇	—	4	2	no	—
141	D₄×C₄₇	—	8	8	no	—
142	C₂×C₄×C₄₇	—	2	2	no	—
143	C₂×C₄×C₄₇	—	2	2	no	—
144	C₂×C₄×C₄₇	—	4	2	no	—
145	C₂×C₄×C₄₇	—	4	2	no	—
146	C₂×C₄×C₄₇	—	4	2	no	—
147	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
148	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂×Dic₄₇	—	2	1	no	—
155	C₂×Dic₄₇	—	2	1	no	—
156	C₂×Dic₄₇	—	2	2	no	—
157	C₂×Dic₄₇	—	2	2	no	—
158	C₂×Dic₄₇	—	2	2	no	—
159	C₂×Dic₄₇	—	2	2	no	—
160	C₂×Dic₄₇	—	2	2	no	—
161	C₂×Dic₄₇	—	4	2	no	—
162	C₂×Dic₄₇	—	4	2	no	—
163	C₂×Dic₄₇	—	4	2	no	—
164	C₂×Dic₄₇	—	4	2	no	—
165	C₂×Dic₄₇	—	4	2	no	—
166	C₂×Dic₄₇	—	4	2	no	—
167	C₂×Dic₄₇	—	8	8	no	—
168	C₈×C₄₇	—	4	1	no	—
169	C₈×C₄₇	—	4	1	no	—
170	C₄₇⋊C₈	—	2	2	no	—
171	C₄₇⋊C₈	—	2	2	no	—
172	#29201	—	47	1	no	maximal, Sylow
173	C₂²×C₄₇	—	4	4	no	—
174	C₂²×C₄₇	—	4	4	no	—
175	C₂²×C₄₇	—	4	4	no	—
176	C₂²×C₄₇	—	8	2	no	—
177	C₂²×C₄₇	—	8	2	no	—
178	C₂²×C₄₇	—	8	2	no	—
179	C₂²×C₄₇	—	8	2	no	—
180	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
181	C₂²×C₄₇	#532	1	1	yes	characteristic
182	C₂²×C₄₇	#535	1	1	yes	characteristic
183	C₂²×C₄₇	#576	1	2	yes	—
184	C₂²×C₄₇	#577	1	2	yes	—
185	C₄×C₄₇	—	2	4	no	—
186	C₄×C₄₇	—	8	2	no	—
187	C₄×C₄₇	#553	1	1	yes	characteristic
188	C₄×C₄₇	#553	1	1	yes	characteristic
189	C₄×C₄₇	#554	1	1	yes	characteristic
190	C₄×C₄₇	#554	1	1	yes	characteristic
191	Dic₄₇	—	4	1	no	—
192	Dic₄₇	—	4	1	no	—
193	Dic₄₇	—	4	2	no	—
194	Dic₄₇	—	8	2	no	—
195	Dic₄₇	—	8	2	no	—
196	#2647	—	47	1	no	—
197	#2328	—	47	1	no	—
198	#2067	—	47	2	no	—
199	#2065	—	47	2	no	—
200	#1961	D₄₇	1	1	yes	characteristic
201	#1957	—	47	1	no	—
202	#1864	—	47	2	no	—
203	#1862	—	47	2	no	—
204	#1846	—	47	1	no	—
205	#1836	—	47	1	no	—
206	#1768	—	47	1	no	—
207	C₂×C₄₇	—	8	4	no	—
208	C₂×C₄₇	#2566	1	1	yes	characteristic
209	C₂×C₄₇	#3081	1	2	yes	—
210	C₂×C₄₇	#3174	1	1	yes	characteristic
211	C₂×C₄₇	#3176	1	1	yes	characteristic
212	C₂×C₄₇	#3224	1	2	yes	—
213	C₂×#127	C₂×D₄₇	1	1	yes	characteristic
214	C₂×C₄×D₄	—	47	2	no	—
215	C₂×#107	—	47	2	no	—
216	C₂×#106	—	47	2	no	—
217	C₂×C₄⋊C₈	—	47	1	no	—
218	C₂×C₄⋊C₈	—	47	1	no	—
219	C₂×#102	—	47	2	no	—
220	C₂×#102	—	47	2	no	—
221	C₂×#102	C₂×D₄₇	1	2	yes	—
222	C₂×#102	C₂×D₄₇	1	2	yes	—
223	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
224	C₂×C₄×C₈	—	47	1	no	—
225	#400	—	94	2	no	—
226	#398	—	94	2	no	—
227	#390	—	47	2	no	—
228	#389	—	47	2	no	—
229	#389	—	47	2	no	—
230	#386	—	94	2	no	—
231	#386	—	94	2	no	—
232	#384	—	47	1	no	—
233	#382	—	47	1	no	—
234	#382	C₂×D₄₇	1	1	yes	characteristic
235	#376	—	47	1	no	—
236	#340	—	47	2	no	—
237	#340	—	47	2	no	—
238	#336	—	47	1	no	—
239	#336	—	47	1	no	—
240	C₄₇	#29201	1	1	yes	characteristic, complemented, Sylow
241	C₂²×D₄	—	2	2	no	—
242	C₂²×D₄	C₂²×D₄₇	1	2	yes	—
243	C₂³×C₄	—	94	2	no	—
244	C₂²×C₈	—	47	1	no	—
245	C₂²×C₈	—	47	1	no	—
246	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
247	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
248	#127	—	2	2	no	—
249	#127	Dic₉₄	1	4	yes	—
250	C₄×D₄	—	188	2	no	—
251	C₄×D₄	—	188	2	no	—
252	C₂×C₄⋊C₄	—	47	2	no	—
253	C₂×C₄⋊C₄	—	47	2	no	—
254	C₂×C₄⋊C₄	—	94	2	no	—
255	C₂×C₄⋊C₄	—	94	2	no	—
256	C₂×C₄⋊C₄	—	94	2	no	—
257	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
258	C₂×C₂²⋊C₄	—	94	2	no	—
259	C₂×C₂²⋊C₄	—	94	2	no	—
260	C₂×C₂²⋊C₄	—	94	2	no	—
261	C₂×C₄²	—	47	1	no	—
262	C₂×C₄²	—	47	1	no	—
263	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
264	#107	—	188	2	no	—
265	#106	—	188	2	no	—
266	C₄⋊C₈	—	94	2	no	—
267	C₄⋊C₈	—	94	2	no	—
268	#102	—	4	2	no	—
269	#102	—	4	2	no	—
270	#102	—	94	4	no	—
271	#102	—	94	4	no	—
272	#97	—	2	2	no	—
273	C₄×C₈	—	94	2	no	—
274	#95	—	2	2	no	—
275	#95	—	47	2	no	—
276	#95	—	47	2	no	—
277	#95	—	94	2	no	—
278	#95	—	94	2	no	—
279	#95	—	94	2	no	—
280	C₂⁴	—	2	2	no	—
281	C₂×D₄	—	2	8	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	8	2	no	—
288	C₂×D₄	—	8	2	no	—
289	C₂×D₄	C₂×Dic₉₄	1	4	yes	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	47	1	no	—
292	C₂²×C₄	—	47	2	no	—
293	C₂²×C₄	—	47	1	no	—
294	C₂²×C₄	—	94	2	no	—
295	C₂²×C₄	—	94	2	no	—
296	C₂²×C₄	—	376	2	no	—
297	C₂²×C₄	—	376	2	no	—
298	C₂²×C₄	—	188	4	no	—
299	C₂²×C₄	D₄×D₄₇	1	2	yes	—
300	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	47	2	no	—
308	C₂×C₈	—	47	2	no	—
309	C₂×C₈	—	94	1	no	—
310	C₂×C₈	—	94	1	no	—
311	C₂×C₈	—	94	1	no	—
312	C₂×C₈	—	94	1	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	94	2	no	—
316	C₄⋊C₄	—	94	2	no	—
317	C₄⋊C₄	—	94	2	no	—
318	C₄⋊C₄	—	94	2	no	—
319	C₄⋊C₄	—	376	2	no	—
320	C₄⋊C₄	—	376	2	no	—
321	C₄⋊C₄	—	376	2	no	—
322	C₂²⋊C₄	—	376	2	no	—
323	C₂²⋊C₄	—	376	2	no	—
324	C₂²⋊C₄	—	188	4	no	—
325	C₄²	—	94	1	no	—
326	C₄²	—	94	1	no	—
327	C₄²	—	94	1	no	—
328	C₄²	—	94	1	no	—
329	C₄²	C₂×Dic₉₄	1	2	yes	—
330	C₄²	#10625352	1	2	yes	—
331	C₂³	—	2	4	no	—
332	C₂³	—	4	2	no	—
333	C₂³	—	4	2	no	—
334	C₂³	—	4	2	no	—
335	C₂³	—	4	2	no	—
336	C₂³	—	4	2	no	—
337	C₂³	—	4	2	no	—
338	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
339	D₄	—	2	4	no	—
340	D₄	—	2	4	no	—
341	D₄	—	4	2	no	—
342	D₄	—	4	2	no	—
343	D₄	—	8	8	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	94	2	no	—
350	C₂×C₄	—	94	2	no	—
351	C₂×C₄	—	94	1	no	—
352	C₂×C₄	—	94	1	no	—
353	C₂×C₄	—	94	2	no	—
354	C₂×C₄	—	94	2	no	—
355	C₂×C₄	—	94	2	no	—
356	C₂×C₄	—	376	8	no	—
357	C₂×C₄	—	188	2	no	—
358	C₂×C₄	—	188	2	no	—
359	C₂×C₄	—	188	2	no	—
360	C₂×C₄	—	188	2	no	—
361	C₂×C₄	—	188	2	no	—
362	C₂×C₄	—	188	2	no	—
363	C₂×C₄	#13094001	1	4	yes	—
364	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
365	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
366	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
367	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
368	C₂×C₄	#13094069	1	1	yes	characteristic
369	C₂×C₄	#13094069	1	1	yes	characteristic
370	C₈	—	4	1	no	—
371	C₈	—	4	1	no	—
372	C₈	—	94	2	no	—
373	C₈	—	94	2	no	—
374	C₂²	—	4	4	no	—
375	C₂²	—	4	4	no	—
376	C₂²	—	4	4	no	—
377	C₂²	—	8	2	no	—
378	C₂²	—	8	2	no	—
379	C₂²	—	8	2	no	—
380	C₂²	—	8	2	no	—
381	C₂²	C₂×#13094001	1	1	yes	characteristic
382	C₂²	#15689672	1	1	yes	characteristic
383	C₂²	#15689686	1	1	yes	characteristic
384	C₂²	#15689927	1	2	yes	—
385	C₂²	#15689933	1	2	yes	—
386	C₄	—	2	4	no	—
387	C₄	—	8	2	no	—
388	C₄	—	376	2	no	—
389	C₄	—	376	2	no	—
390	C₄	—	188	2	no	—
391	C₄	—	188	1	no	—
392	C₄	—	188	1	no	—
393	C₄	#15689820	1	1	yes	characteristic
394	C₄	#15689820	1	1	yes	characteristic
395	C₄	#15689832	1	1	yes	characteristic
396	C₄	#15689832	1	1	yes	characteristic
397	C₂	—	8	4	no	—
398	C₂	#20248591	1	1	yes	characteristic
399	C₂	#20253221	1	2	yes	—
400	C₂	#20254015	1	1	yes	characteristic
401	C₂	#20254037	1	1	yes	characteristic
402	C₂	#20254477	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1961	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂×#127×C₄₇	C₂²
hypercenter	#1961	D₄₇
hypocenter	C₄₇	#29201
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1961

Complements

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

N	H ≅ G/N	type
1. #20249367	C₂	semidirect
3. #20244772	C₂	semidirect
4. #20244758	C₂	semidirect
6. #20243647	C₂	semidirect
7. #20243633	C₂	semidirect
8. #20243550	C₂	semidirect
9. #20243522	C₂	semidirect
10. #20243248	C₂	semidirect
240. C₄₇	#29201	semidirect

Maximal subgroups

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

subgroup	index
#20249367	2
#20246114	2
#20244772	2
#20244758	2
#20244195	2
#20243647	2
#20243633	2
#20243550	2
#20243522	2
#20243248	2
C₄₇×#1961	2
#29201	47