Subgroups — #26224439

Order 12032.

722 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253711	C₂	1	1	yes	maximal, characteristic
2	#20252325	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252105	C₂	1	1	yes	maximal, characteristic
4	#20244368	C₂	1	1	yes	maximal, characteristic, complemented
5	#20244219	C₂	1	1	yes	maximal, characteristic
6	#20244097	C₂	1	2	yes	maximal, complemented
7	#20244090	C₂	1	2	yes	maximal, complemented
8	#20243231	C₂	1	2	yes	maximal, complemented
9	#20243224	C₂	1	2	yes	maximal, complemented
10	#20243074	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1837	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094032	C₂²	1	2	yes	—
13	#15689680	C₂²	1	1	yes	characteristic
14	#15689673	C₂²	1	1	yes	characteristic
15	#15689478	C₄	1	4	yes	complemented
16	#15689474	C₄	1	2	yes	complemented
17	#15689470	C₄	1	2	yes	complemented
18	#15689252	—	2	4	no	—
19	#15689193	—	2	4	no	—
20	#15689151	C₂²	1	2	yes	—
21	C₂×#13093950	C₂²	1	2	yes	—
22	C₂×#13093950	C₂²	1	2	yes	—
23	#15689120	C₂²	1	2	yes	—
24	#15689120	C₂²	1	2	yes	—
25	#15689091	C₂²	1	2	yes	—
26	#15689081	C₂²	1	2	yes	—
27	C₂×#13093935	C₂²	1	1	yes	characteristic
28	C₂×#13093934	C₂²	1	1	yes	characteristic
29	C₂×#13093934	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
32	#15688969	C₂²	1	1	yes	characteristic
33	#15688884	C₂²	1	2	yes	—
34	#15688884	C₂²	1	2	yes	—
35	C₄₇×#423	C₂²	1	1	yes	characteristic
36	C₄₇×#423	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#341	C₂²	1	2	yes	—
39	C₄₇×#341	C₂²	1	2	yes	—
40	C₂×#10625350	C₂³	1	2	yes	—
41	#13094032	—	2	4	no	—
42	#13094032	—	2	4	no	—
43	C₂×#10625324	—	2	2	no	—
44	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
46	C₂×C₈×D₄₇	—	2	2	no	—
47	#13094024	C₂×C₄	1	1	yes	characteristic
48	#13094024	C₂×C₄	1	2	yes	—
49	#13094021	C₂×C₄	1	1	yes	characteristic
50	#13094016	C₂×C₄	1	2	yes	—
51	#13094012	C₂×C₄	1	2	yes	—
52	#13094003	—	2	2	no	—
53	#13093998	—	2	2	no	—
54	C₄×Dic₉₄	—	2	2	no	—
55	C₄×D₁₈₈	—	2	2	no	—
56	#13093983	C₂³	1	2	yes	—
57	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
58	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
59	Dic₄₇⋊C₈	—	2	4	no	—
60	#13093950	—	2	4	no	—
61	#13093950	—	2	4	no	—
62	#13093949	C₂×C₄	1	4	yes	—
63	#13093949	C₂×C₄	1	4	yes	—
64	#13093946	C₂×C₄	1	4	yes	—
65	#13093946	C₂×C₄	1	4	yes	—
66	(C₂×D₄₇)⋊C₈	—	2	4	no	—
67	#13093935	D₄	1	4	yes	—
68	#13093934	—	2	2	no	—
69	#13093934	—	2	2	no	—
70	#13093934	—	2	2	no	—
71	C₈×Dic₄₇	—	2	2	no	—
72	C₈×Dic₄₇	—	2	2	no	—
73	C₈×Dic₄₇	—	2	2	no	—
74	C₄₇⋊#95	—	2	2	no	—
75	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
76	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
77	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₄₇	—	2	2	no	—
81	C₄⋊C₈×C₄₇	—	2	2	no	—
82	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
84	C₂²×C₄×C₄₇	#41	1	2	yes	—
85	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₂×C₂²⋊D₄₇	—	2	2	no	—
87	C₂²×Dic₄₇	—	2	1	no	—
88	C₂²×Dic₄₇	—	2	1	no	—
89	#10625350	—	2	2	no	—
90	#10625350	—	2	2	no	—
91	#10625350	—	2	2	no	—
92	#10625350	—	2	2	no	—
93	C₂×D₁₈₈	—	2	2	no	—
94	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₄₇	—	2	2	no	—
96	C₂×Dic₉₄	—	2	2	no	—
97	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
98	M₁₆×C₄₇	—	2	2	no	—
99	M₁₆×C₄₇	—	2	2	no	—
100	M₁₆×C₄₇	—	2	2	no	—
101	M₁₆×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	—	2	1	no	—
103	C₂×C₈×C₄₇	—	2	1	no	—
104	C₂×C₈×C₄₇	—	2	1	no	—
105	C₂×C₈×C₄₇	—	2	1	no	—
106	C₂×C₈×C₄₇	—	2	2	no	—
107	C₂×C₈×C₄₇	—	2	2	no	—
108	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
110	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₄₇	C₂×D₄	1	2	yes	—
113	C₂²⋊Dic₄₇	—	2	2	no	—
114	(C₂×D₄₇)⋊C₄	—	2	4	no	—
115	C₄⋊Dic₄₇	—	2	2	no	—
116	C₄⋊Dic₄₇	—	2	2	no	—
117	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
118	Dic₄₇⋊C₄	—	2	4	no	—
119	C₄×Dic₄₇	—	2	2	no	—
120	C₄×Dic₄₇	—	2	1	no	—
121	C₄×Dic₄₇	—	2	1	no	—
122	C₄×Dic₄₇	#41	1	2	yes	—
123	C₄×Dic₄₇	#41	1	2	yes	—
124	C₂×C₄₇⋊C₈	—	2	1	no	—
125	C₂×C₄₇⋊C₈	—	2	1	no	—
126	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
127	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
128	C₂×C₄₇⋊C₈	#41	1	2	yes	—
129	C₂×C₄₇⋊C₈	#41	1	2	yes	—
130	#10625324	—	4	4	no	—
131	C₈×D₄₇	—	4	4	no	—
132	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
133	C₂²×D₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	4	2	no	—
135	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
143	C₂²⋊D₄₇	—	4	2	no	—
144	C₂²⋊D₄₇	—	4	2	no	—
145	C₂×Dic₄₇	—	2	4	no	—
146	C₂×Dic₄₇	—	2	2	no	—
147	C₂×Dic₄₇	—	2	2	no	—
148	C₂×Dic₄₇	—	2	2	no	—
149	C₂×Dic₄₇	—	4	2	no	—
150	C₂×Dic₄₇	—	4	2	no	—
151	D₁₈₈	—	2	2	no	—
152	D₁₈₈	—	2	2	no	—
153	D₁₈₈	—	8	2	no	—
154	C₄×D₄₇	—	4	2	no	—
155	C₄×D₄₇	—	4	2	no	—
156	Dic₉₄	—	2	2	no	—
157	Dic₉₄	—	2	2	no	—
158	Dic₉₄	—	8	2	no	—
159	C₈×C₄₇	—	2	2	no	—
160	C₈×C₄₇	—	2	2	no	—
161	C₈×C₄₇	—	4	2	no	—
162	C₈×C₄₇	—	4	2	no	—
163	C₄₇⋊C₈	—	2	2	no	—
164	C₄₇⋊C₈	#124	1	4	yes	—
165	#21659	—	47	1	no	maximal, Sylow
166	C₂²×C₄₇	—	2	2	no	—
167	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂²×C₄₇	#529	1	1	yes	characteristic
169	C₂²×C₄₇	#529	1	1	yes	characteristic
170	C₂×D₄₇	—	4	4	no	—
171	C₂×D₄₇	—	4	2	no	—
172	C₄×C₄₇	—	2	2	no	—
173	C₄×C₄₇	—	2	2	no	—
174	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
175	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
176	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
177	C₄×C₄₇	#532	1	1	yes	characteristic
178	Dic₄₇	—	4	2	no	—
179	Dic₄₇	—	4	2	no	—
180	Dic₄₇	—	4	4	no	—
181	Dic₄₇	—	4	2	no	—
182	#3131	—	47	1	no	—
183	#2961	—	47	1	no	—
184	#2932	—	47	1	no	—
185	#1995	—	47	1	no	—
186	#1963	—	47	1	no	—
187	#1938	—	47	2	no	—
188	#1937	—	47	2	no	—
189	#1837	D₄₇	1	1	yes	characteristic, complemented
190	#1765	—	47	2	no	—
191	#1764	—	47	2	no	—
192	#1726	—	47	1	no	—
193	C₂×C₄₇	—	2	1	no	—
194	C₂×C₄₇	—	2	1	no	—
195	C₂×C₄₇	C₄×#124	1	1	yes	characteristic
196	C₂×C₄₇	#2957	1	2	yes	—
197	D₄₇	—	8	2	no	—
198	C₂×#131	—	47	2	no	—
199	#533	—	47	1	no	—
200	#532	—	47	1	no	—
201	#488	—	47	4	no	—
202	#487	—	47	2	no	—
203	#486	—	47	2	no	—
204	#446	—	94	4	no	—
205	#435	—	94	4	no	—
206	#424	—	47	2	no	—
207	#423	C₂×D₄₇	1	1	yes	characteristic
208	#423	C₂×D₄₇	1	1	yes	characteristic
209	C₂×#104	—	47	2	no	—
210	C₂×#104	—	47	2	no	—
211	#416	—	47	2	no	—
212	#416	—	47	2	no	—
213	#408	—	47	2	no	—
214	#405	—	47	2	no	—
215	C₂×#97	—	47	1	no	—
216	C₂×#97	—	47	1	no	—
217	C₂×#97	—	47	1	no	—
218	C₂×C₄×C₈	—	47	1	no	—
219	C₂×C₄×C₈	—	47	1	no	—
220	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
221	#376	—	47	1	no	—
222	#341	C₂×D₄₇	1	2	yes	—
223	#341	C₂×D₄₇	1	2	yes	—
224	#339	—	47	2	no	—
225	#339	—	47	2	no	—
226	C₄₇	#21659	1	1	yes	characteristic, complemented, Sylow
227	C₂×#41	—	47	2	no	—
228	#131	—	94	4	no	—
229	#131	—	94	4	no	—
230	C₂×M₁₆	—	94	2	no	—
231	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
232	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
233	C₂²×C₈	—	47	1	no	—
234	C₂²×C₈	—	47	1	no	—
235	C₂²×C₈	—	94	2	no	—
236	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
237	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
238	C₄⋊Q₈	—	47	2	no	—
239	C₄⋊Q₈	—	47	1	no	—
240	#127	—	47	1	no	—
241	#125	—	47	2	no	—
242	#124	—	47	2	no	—
243	C₂²⋊Q₈	—	94	2	no	—
244	#121	—	94	2	no	—
245	C₄×Q₈	—	94	2	no	—
246	C₄×D₄	—	94	2	no	—
247	#117	—	47	2	no	—
248	C₂×C₄²	—	47	1	no	—
249	C₂×C₄²	—	47	1	no	—
250	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
251	C₄⋊C₈	—	2	2	no	—
252	C₄⋊C₈	—	2	2	no	—
253	C₄⋊C₈	—	94	4	no	—
254	#104	—	94	4	no	—
255	#104	—	94	4	no	—
256	#103	—	47	4	no	—
257	#103	—	47	4	no	—
258	#102	—	47	4	no	—
259	#102	—	47	4	no	—
260	C₂²⋊C₈	—	94	4	no	—
261	#97	—	47	4	no	—
262	#97	—	94	2	no	—
263	#97	—	94	2	no	—
264	#97	—	94	2	no	—
265	C₄×C₈	—	94	2	no	—
266	C₄×C₈	—	94	2	no	—
267	C₄×C₈	—	94	2	no	—
268	C₄×C₈	C₄×D₄₇	1	2	yes	—
269	C₄×C₈	C₄×D₄₇	1	2	yes	—
270	#95	—	94	2	no	—
271	#41	—	94	2	no	—
272	#41	—	94	2	no	—
273	#41	—	94	2	no	—
274	#41	—	94	2	no	—
275	C₂×Q₈	—	47	2	no	—
276	C₂×Q₈	—	94	2	no	—
277	C₂×D₄	—	47	2	no	—
278	C₂×D₄	—	94	2	no	—
279	C₂×D₄	—	94	2	no	—
280	C₂²×C₄	—	94	1	no	—
281	C₂²×C₄	—	94	1	no	—
282	C₂²×C₄	—	94	2	no	—
283	C₂²×C₄	#10625354	1	2	yes	—
284	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
285	M₁₆	—	2	2	no	—
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	188	4	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	47	2	no	—
297	C₂×C₈	—	47	2	no	—
298	C₂×C₈	—	47	2	no	—
299	C₂×C₈	—	47	2	no	—
300	C₂×C₈	—	94	1	no	—
301	C₂×C₈	—	94	1	no	—
302	C₂×C₈	—	188	4	no	—
303	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
304	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
305	C₄⋊C₄	—	47	4	no	—
306	C₄⋊C₄	—	94	4	no	—
307	C₄⋊C₄	—	94	2	no	—
308	C₄⋊C₄	—	94	2	no	—
309	C₂²⋊C₄	—	94	2	no	—
310	C₂²⋊C₄	—	94	4	no	—
311	C₄²	—	47	2	no	—
312	C₄²	—	47	2	no	—
313	C₄²	—	94	2	no	—
314	C₄²	—	94	1	no	—
315	C₄²	—	94	1	no	—
316	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
317	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
318	C₄²	D₄×D₄₇	1	2	yes	—
319	C₂³	—	94	2	no	—
320	C₂³	C₂×#10625354	1	1	yes	characteristic
321	Q₈	—	94	2	no	—
322	Q₈	—	94	2	no	—
323	Q₈	—	376	2	no	—
324	D₄	—	94	2	no	—
325	D₄	—	94	2	no	—
326	D₄	—	376	2	no	—
327	D₄	—	188	2	no	—
328	D₄	—	188	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	94	4	no	—
331	C₂×C₄	—	94	2	no	—
332	C₂×C₄	—	94	2	no	—
333	C₂×C₄	—	94	2	no	—
334	C₂×C₄	—	188	2	no	—
335	C₂×C₄	—	188	2	no	—
336	C₂×C₄	—	188	2	no	—
337	C₂×C₄	—	188	2	no	—
338	C₂×C₄	#13093986	1	2	yes	—
339	C₂×C₄	#13093986	1	2	yes	—
340	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
342	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
343	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
344	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
345	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
346	C₈	—	2	2	no	—
347	C₈	—	2	2	no	—
348	C₈	—	4	2	no	—
349	C₈	—	4	2	no	—
350	C₈	—	47	4	no	—
351	C₈	—	94	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	188	4	no	—
354	C₂²	—	188	2	no	—
355	C₂²	C₂×#13093986	1	1	yes	characteristic
356	C₂²	#15689658	1	1	yes	characteristic
357	C₂²	#15689658	1	1	yes	characteristic
358	C₄	—	2	2	no	—
359	C₄	—	2	2	no	—
360	C₄	—	188	2	no	—
361	C₄	—	188	2	no	—
362	C₄	—	188	4	no	—
363	C₄	—	188	2	no	—
364	C₄	#15689591	1	1	yes	characteristic
365	C₄	#15689591	1	1	yes	characteristic
366	C₄	#124×D₄₇	1	1	yes	characteristic
367	C₄	#15689672	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	376	2	no	—
371	C₂	#20245927	1	1	yes	characteristic
372	C₂	#20252304	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1837	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#15689673	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1837	D₄₇
hypocenter	C₄₇	#21659
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

Complements

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

N	H ≅ G/N	type
2. #20252325	C₂	semidirect
4. #20244368	C₂	semidirect
6. #20244097	C₂	semidirect
7. #20244090	C₂	semidirect
8. #20243231	C₂	semidirect
9. #20243224	C₂	semidirect
10. #20243074	C₂	semidirect
11. C₄₇×#1837	C₂	semidirect
15. #15689478	C₄	semidirect
16. #15689474	C₄	semidirect
17. #15689470	C₄	semidirect
189. #1837	D₄₇	semidirect
226. C₄₇	#21659	semidirect

Maximal subgroups

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

subgroup	index
#20253711	2
#20252325	2
#20252105	2
#20244368	2
#20244219	2
#20244097	2
#20244090	2
#20243231	2
#20243224	2
#20243074	2
C₄₇×#1837	2
#21659	47