Subgroups — #26201464

Order 12032.

654 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254336	C₂	1	1	yes	maximal, characteristic, complemented
2	#20253187	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689400	C₂	1	1	yes	maximal, characteristic
4	#20252411	C₂	1	1	yes	maximal, characteristic
5	#20242838	C₂	1	4	yes	maximal, complemented
6	#20242752	C₂	1	4	yes	maximal, complemented
7	#20242164	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242043	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1549	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689680	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
12	#15689502	C₂²	1	2	yes	complemented
13	#15689473	C₂²	1	2	yes	complemented
14	#15689424	—	2	2	no	—
15	#15689414	C₂²	1	2	yes	—
16	#15689404	C₂²	1	2	yes	—
17	#15689400	—	2	2	no	—
18	#15689400	C₂²	1	4	yes	—
19	#15689330	—	2	2	no	—
20	#15689279	—	2	2	no	—
21	C₈×D₁₈₈	—	2	2	no	—
22	#15689188	C₂²	1	4	yes	—
23	#15689176	C₂²	1	1	yes	characteristic
24	#15689165	—	2	1	no	—
25	C₂×#13093957	—	2	1	no	—
26	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
27	#15689129	—	2	1	no	—
28	C₂×#13093944	—	2	1	no	—
29	C₂²⋊C₈×D₄₇	—	2	2	no	—
30	#15688875	C₂²	1	4	yes	complemented
31	#15688871	C₂²	1	2	yes	complemented
32	#15688870	C₂²	1	2	yes	complemented
33	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₄₇×#431	C₂²	1	1	yes	characteristic
35	C₄₇×#423	C₂²	1	1	yes	characteristic
36	C₄₇×#332	C₂²	1	4	yes	complemented
37	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
38	C₂²×C₄×D₄₇	—	2	1	no	—
39	C₂×#10625337	—	2	4	no	—
40	C₂²×C₄₇⋊C₈	—	2	1	no	—
41	C₂×C₈×D₄₇	—	2	4	no	—
42	#13094024	C₂³	1	1	yes	characteristic
43	#13094024	C₂³	1	1	yes	characteristic
44	#13094016	—	2	1	no	—
45	#13094003	—	2	1	no	—
46	#13094003	—	2	1	no	—
47	C₄×Dic₉₄	—	2	1	no	—
48	C₄×Dic₉₄	—	2	1	no	—
49	C₄×D₁₈₈	—	2	1	no	—
50	C₄×D₁₈₈	—	2	1	no	—
51	C₄×D₁₈₈	D₄	1	2	yes	—
52	C₄×D₁₈₈	C₂³	1	2	yes	—
53	#13093983	—	2	1	no	—
54	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
55	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
56	#13093958	—	2	2	no	—
57	#13093957	—	2	4	no	—
58	#13093956	—	2	2	no	—
59	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
60	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
61	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
62	#13093948	—	2	4	no	—
63	#13093946	—	2	2	no	—
64	#13093946	—	2	2	no	—
65	#13093944	—	2	4	no	—
66	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
67	(C₂×D₄₇)⋊C₈	—	2	4	no	—
68	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
69	C₂²×Q₈×C₄₇	—	2	1	no	—
70	C₂×M₁₆×C₄₇	—	2	1	no	—
71	C₄⋊Q₈×C₄₇	—	2	1	no	—
72	C₄⋊Q₈×C₄₇	—	2	1	no	—
73	C₄⋊Q₈×C₄₇	—	2	1	no	—
74	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
76	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
77	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₄₇	—	2	1	no	—
79	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
80	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
81	C₂²⋊C₈×C₄₇	—	2	1	no	—
82	C₄×C₈×C₄₇	—	2	1	no	—
83	C₂³×D₄₇	—	2	1	no	—
84	C₂×Q₈×C₄₇	—	2	2	no	—
85	C₂×Q₈×C₄₇	—	4	1	no	—
86	C₂×Q₈×C₄₇	—	8	2	no	—
87	C₂²×C₄×C₄₇	—	4	1	no	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
91	C₂²×Dic₄₇	—	2	1	no	—
92	C₂×D₁₈₈	—	2	1	no	—
93	C₂×D₁₈₈	—	2	2	no	—
94	C₂×D₁₈₈	Q₁₆	1	4	yes	complemented
95	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
96	C₂×C₄×D₄₇	—	2	2	no	—
97	C₂×C₄×D₄₇	—	2	2	no	—
98	C₂×C₄×D₄₇	—	8	2	no	—
99	C₂×Dic₉₄	—	2	1	no	—
100	C₂×Dic₉₄	—	2	1	no	—
101	M₁₆×C₄₇	—	4	2	no	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	—	2	2	no	—
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	—	2	2	no	—
106	C₄⋊C₄×C₄₇	—	4	1	no	—
107	C₄⋊C₄×C₄₇	—	4	1	no	—
108	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₄₇	#41	1	1	yes	characteristic
111	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
112	C₂²⋊Dic₄₇	—	4	1	no	—
113	#10625337	—	8	4	no	—
114	(C₂×D₄₇)⋊C₄	—	2	2	no	—
115	(C₂×D₄₇)⋊C₄	—	2	2	no	—
116	C₄⋊Dic₄₇	—	2	1	no	—
117	C₄⋊Dic₄₇	—	2	1	no	—
118	C₄⋊Dic₄₇	—	2	1	no	—
119	C₄⋊Dic₄₇	—	2	1	no	—
120	C₄⋊Dic₄₇	—	2	1	no	—
121	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
122	Dic₄₇⋊C₄	—	4	1	no	—
123	Dic₄₇⋊C₄	—	4	1	no	—
124	C₄×Dic₄₇	—	4	1	no	—
125	C₂×C₄₇⋊C₈	—	2	2	no	—
126	C₂×C₄₇⋊C₈	—	4	2	no	—
127	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
128	C₈×D₄₇	—	8	4	no	—
129	C₂³×C₄₇	#121	1	1	yes	characteristic
130	C₂²×D₄₇	—	2	2	no	—
131	C₂²×D₄₇	—	2	4	no	—
132	C₂²×D₄₇	—	8	1	no	—
133	C₂²×D₄₇	—	8	1	no	—
134	Q₈×C₄₇	—	4	4	no	—
135	Q₈×C₄₇	—	8	2	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	4	1	no	—
138	C₂×C₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	—	8	2	no	—
140	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
142	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
144	C₂×C₄×C₄₇	#135	1	2	yes	—
145	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₄₇	—	2	2	no	—
149	C₂×Dic₄₇	—	4	1	no	—
150	C₂×Dic₄₇	—	4	1	no	—
151	C₂×Dic₄₇	—	8	1	no	—
152	D₁₈₈	—	2	2	no	—
153	D₁₈₈	—	2	4	no	—
154	D₁₈₈	C₂×Q₁₆	1	4	yes	—
155	C₄×D₄₇	—	4	4	no	—
156	C₄×D₄₇	—	8	2	no	—
157	Dic₉₄	—	4	2	no	—
158	Dic₉₄	—	8	1	no	—
159	C₈×C₄₇	—	4	2	no	—
160	C₈×C₄₇	—	4	2	no	—
161	C₄₇⋊C₈	—	2	4	no	—
162	C₄₇⋊C₈	—	4	2	no	—
163	#19387	—	47	1	no	maximal, Sylow
164	C₂²×C₄₇	—	2	1	no	—
165	C₂²×C₄₇	—	2	1	no	—
166	C₂²×C₄₇	#467	1	2	yes	—
167	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
168	C₂×D₄₇	—	2	4	no	—
169	C₂×D₄₇	—	4	2	no	—
170	C₂×D₄₇	—	4	2	no	—
171	C₂×D₄₇	—	8	4	no	—
172	C₄×C₄₇	—	2	1	no	—
173	C₄×C₄₇	—	2	1	no	—
174	C₄×C₄₇	—	2	1	no	—
175	C₄×C₄₇	—	8	2	no	—
176	C₄×C₄₇	#467	1	2	yes	—
177	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
178	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
179	C₄×C₄₇	#576	1	2	yes	—
180	Dic₄₇	—	4	2	no	—
181	Dic₄₇	—	8	1	no	—
182	Dic₄₇	—	8	1	no	—
183	#3209	—	47	1	no	—
184	#3077	—	47	1	no	—
185	C₂×#474	—	47	1	no	—
186	#2972	—	47	1	no	—
187	#1683	—	47	4	no	—
188	#1669	—	47	4	no	—
189	#1571	—	47	1	no	—
190	#1549	D₄₇	1	1	yes	characteristic, complemented
191	#1542	—	47	1	no	—
192	C₂×C₄₇	—	2	2	no	—
193	C₂×C₄₇	#3041	1	1	yes	characteristic
194	C₂×C₄₇	C₂×#467	1	1	yes	characteristic
195	C₂×C₄₇	#3069	1	1	yes	characteristic
196	D₄₇	—	4	4	no	—
197	#533	—	47	1	no	—
198	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
199	C₂×C₄×D₄	—	47	1	no	—
200	#494	—	47	2	no	—
201	#487	—	47	2	no	—
202	#478	—	94	2	no	—
203	#476	—	47	2	no	—
204	#474	—	47	4	no	—
205	#474	—	47	2	no	—
206	#474	—	94	2	no	—
207	#462	—	94	2	no	—
208	#451	—	94	2	no	—
209	C₈×D₄	—	47	4	no	—
210	C₈×D₄	—	94	2	no	—
211	#431	—	47	1	no	—
212	#431	C₂×D₄₇	1	1	yes	characteristic
213	#428	—	94	1	no	—
214	C₂×#107	—	94	1	no	—
215	#423	C₂×D₄₇	1	1	yes	characteristic
216	C₂×C₄⋊C₈	—	47	1	no	—
217	#418	—	94	1	no	—
218	C₂×#102	—	94	1	no	—
219	C₂×C₂²⋊C₈	—	94	2	no	—
220	#335	—	47	4	no	—
221	#333	—	47	2	no	—
222	#332	—	47	2	no	—
223	#332	C₂×D₄₇	1	4	yes	complemented
224	C₄₇	#19387	1	1	yes	characteristic, complemented, Sylow
225	C₂²×Q₈	—	2	1	no	—
226	C₂²×D₄	—	47	1	no	—
227	C₂³×C₄	—	94	1	no	—
228	C₂×Q₁₆	—	94	4	no	—
229	C₂×M₁₆	—	2	1	no	—
230	C₂²×C₈	—	94	4	no	—
231	C₂²×C₈	—	94	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	—	47	1	no	—
236	C₄⋊Q₈	—	47	1	no	—
237	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
238	#125	—	94	1	no	—
239	C₂²⋊Q₈	—	94	1	no	—
240	C₂²⋊Q₈	—	94	1	no	—
241	C₄×Q₈	—	94	1	no	—
242	C₄×Q₈	—	94	1	no	—
243	C₄×D₄	—	47	2	no	—
244	C₄×D₄	—	47	2	no	—
245	C₄×D₄	—	94	1	no	—
246	C₄×D₄	—	94	1	no	—
247	#117	—	94	1	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄⋊C₄	—	47	1	no	—
251	C₂×C₂²⋊C₄	—	94	1	no	—
252	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
253	#107	—	94	4	no	—
254	#107	—	94	2	no	—
255	#106	—	94	2	no	—
256	C₄⋊C₈	—	2	1	no	—
257	C₄⋊C₈	—	47	2	no	—
258	C₄⋊C₈	—	47	2	no	—
259	C₄⋊C₈	—	94	1	no	—
260	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
261	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
262	#103	—	94	4	no	—
263	#102	—	94	4	no	—
264	#102	—	94	2	no	—
265	#102	—	94	2	no	—
266	C₂²⋊C₈	—	2	1	no	—
267	C₂²⋊C₈	—	47	2	no	—
268	C₂²⋊C₈	—	94	4	no	—
269	C₄×C₈	—	2	1	no	—
270	C₄×C₈	—	47	2	no	—
271	C₂⁴	—	94	1	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×Q₈	—	4	1	no	—
274	C₂×Q₈	—	8	2	no	—
275	C₂×Q₈	—	94	1	no	—
276	C₂×Q₈	—	94	1	no	—
277	C₂×D₄	—	47	4	no	—
278	C₂×D₄	—	47	2	no	—
279	C₂×D₄	—	94	1	no	—
280	C₂×D₄	—	94	2	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	94	1	no	—
283	C₂²×C₄	—	94	2	no	—
284	C₂²×C₄	—	94	2	no	—
285	C₂²×C₄	—	376	2	no	—
286	C₂²×C₄	#10625350	1	1	yes	characteristic
287	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
288	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
289	Q₁₆	—	376	4	no	—
290	M₁₆	—	4	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	47	4	no	—
294	C₂×C₈	—	94	2	no	—
295	C₂×C₈	—	376	4	no	—
296	C₂×C₈	—	188	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	47	2	no	—
302	C₄⋊C₄	—	94	1	no	—
303	C₄⋊C₄	—	94	1	no	—
304	C₄⋊C₄	—	94	1	no	—
305	C₄⋊C₄	—	94	1	no	—
306	C₄⋊C₄	—	94	1	no	—
307	C₄⋊C₄	—	188	1	no	—
308	C₄⋊C₄	—	188	1	no	—
309	C₂²⋊C₄	—	94	2	no	—
310	C₂²⋊C₄	—	94	2	no	—
311	C₂²⋊C₄	—	188	1	no	—
312	C₄²	—	188	1	no	—
313	C₄²	#10625350	1	1	yes	characteristic
314	C₄²	D₄×D₄₇	1	1	yes	characteristic
315	C₄²	D₄×D₄₇	1	1	yes	characteristic
316	C₄²	C₂³×D₄₇	1	1	yes	characteristic
317	C₂³	—	94	4	no	—
318	C₂³	—	94	2	no	—
319	C₂³	—	376	1	no	—
320	C₂³	—	376	1	no	—
321	C₂³	#13093997	1	1	yes	characteristic
322	Q₈	—	4	4	no	—
323	Q₈	—	8	2	no	—
324	Q₈	—	376	1	no	—
325	Q₈	—	188	2	no	—
326	D₄	—	47	4	no	—
327	D₄	—	94	4	no	—
328	D₄	—	94	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	8	2	no	—
333	C₂×C₄	—	94	2	no	—
334	C₂×C₄	—	376	2	no	—
335	C₂×C₄	—	376	1	no	—
336	C₂×C₄	—	188	1	no	—
337	C₂×C₄	—	188	1	no	—
338	C₂×C₄	—	188	4	no	—
339	C₂×C₄	#13093997	1	1	yes	characteristic
340	C₂×C₄	#13093997	1	1	yes	characteristic
341	C₂×C₄	#13093997	1	1	yes	characteristic
342	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
343	C₂×C₄	#13094045	1	2	yes	—
344	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
345	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
346	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
347	C₈	—	4	2	no	—
348	C₈	—	4	2	no	—
349	C₈	—	94	4	no	—
350	C₈	—	188	2	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	94	4	no	—
354	C₂²	—	376	4	no	—
355	C₂²	—	188	2	no	—
356	C₂²	—	188	2	no	—
357	C₂²	#15689363	1	2	yes	—
358	C₂²	C₂×#13093997	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	8	2	no	—
363	C₄	—	376	1	no	—
364	C₄	—	376	1	no	—
365	C₄	—	188	2	no	—
366	C₄	#15689363	1	2	yes	—
367	C₄	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
368	C₄	C₂×#13094045	1	1	yes	characteristic
369	C₄	#15689927	1	2	yes	—
370	C₂	—	2	2	no	—
371	C₂	—	188	4	no	—
372	C₂	#20252890	1	1	yes	characteristic
373	C₂	C₂×#15689363	1	1	yes	characteristic
374	C₂	#20253124	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093997
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1549	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂²×D₁₈₈	D₄
hypercenter	#1549	D₄₇
hypocenter	C₄₇	#19387
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1549

Complements

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

N	H ≅ G/N	type
1. #20254336	C₂	semidirect
5. #20242838	C₂	semidirect
6. #20242752	C₂	semidirect
7. #20242164	C₂	semidirect
8. #20242043	C₂	semidirect
9. C₄₇×#1549	C₂	semidirect
12. #15689502	C₂²	semidirect
13. #15689473	C₂²	semidirect
30. #15688875	C₂²	semidirect
31. #15688871	C₂²	semidirect
32. #15688870	C₂²	semidirect
36. C₄₇×#332	C₂²	semidirect
94. C₂×D₁₈₈	Q₁₆	semidirect
190. #1549	D₄₇	semidirect
223. #332	C₂×D₄₇	semidirect
224. C₄₇	#19387	semidirect

Maximal subgroups

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

subgroup	index
#20254336	2
#20253187	2
C₂×#15689400	2
#20252411	2
#20242838	2
#20242752	2
#20242164	2
#20242043	2
C₄₇×#1549	2
#19387	47