Subgroups — #26201519

Order 12032.

642 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	#20253181	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689410	C₂	1	1	yes	maximal, characteristic
3	#20252411	C₂	1	1	yes	maximal, characteristic
4	#20242836	C₂	1	4	yes	maximal, complemented
5	#20242784	C₂	1	4	yes	maximal, complemented
6	#20242163	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242069	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242045	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#3230	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
11	#15689420	—	2	2	no	—
12	#15689410	—	2	2	no	—
13	#15689410	C₂²	1	2	yes	—
14	#15689410	C₂²	1	4	yes	—
15	#15689400	C₂²	1	2	yes	—
16	C₈×D₁₈₈	C₂²	1	4	yes	—
17	#15689188	—	2	2	no	—
18	#15689176	C₂²	1	1	yes	characteristic
19	#15689166	—	2	1	no	—
20	C₂×#13093955	—	2	1	no	—
21	#15689147	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
23	#15689127	—	2	1	no	—
24	C₂×#13093944	—	2	1	no	—
25	C₂²⋊C₈×D₄₇	—	2	2	no	—
26	#15688876	C₂²	1	4	yes	complemented
27	#15688872	C₂²	1	2	yes	complemented
28	#15688872	C₂²	1	4	yes	complemented
29	#15688869	C₂²	1	2	yes	complemented
30	C₄₇×#533	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#500	C₂²	1	2	yes	complemented
33	C₄₇×#487	C₂²	1	2	yes	complemented
34	C₄₇×#484	—	2	2	no	—
35	C₄₇×#477	—	2	2	no	—
36	C₄₇×#431	C₂²	1	1	yes	characteristic
37	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
38	C₂²×C₄×D₄₇	—	2	1	no	—
39	C₂×#10625329	—	2	1	no	—
40	C₂²×C₄₇⋊C₈	—	2	1	no	—
41	C₂×C₈×D₄₇	—	2	4	no	—
42	C₄×D₁₈₈	—	2	1	no	—
43	C₄×D₁₈₈	—	2	1	no	—
44	C₄×D₁₈₈	D₄	1	2	yes	—
45	C₄×D₁₈₈	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
47	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
48	#13093957	—	2	2	no	—
49	#13093955	—	2	4	no	—
50	#13093955	—	2	2	no	—
51	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
52	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
53	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
54	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
55	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
56	#13093944	—	2	4	no	—
57	#13093944	—	2	2	no	—
58	#13093944	—	2	2	no	—
59	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
60	(C₂×D₄₇)⋊C₈	—	2	4	no	—
61	C₄×C₄₇⋊C₈	—	2	1	no	—
62	C₂²×Q₈×C₄₇	—	2	1	no	—
63	C₄⋊Q₈×C₄₇	—	2	1	no	—
64	C₄⋊Q₈×C₄₇	—	2	1	no	—
65	C₄⋊Q₈×C₄₇	—	2	1	no	—
66	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
68	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
69	#125×C₄₇	—	2	1	no	—
70	C₂²⋊Q₈×C₄₇	—	2	1	no	—
71	C₂²⋊Q₈×C₄₇	—	2	1	no	—
72	C₄×Q₈×C₄₇	—	2	1	no	—
73	C₄×Q₈×C₄₇	—	2	1	no	—
74	#117×C₄₇	—	2	1	no	—
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	#107×C₄₇	—	2	4	no	—
79	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
80	#103×C₄₇	—	2	4	no	—
81	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
82	C₄×C₈×C₄₇	C₂³	1	2	yes	—
83	C₂³×D₄₇	—	2	1	no	—
84	C₂×Q₈×C₄₇	—	2	1	no	—
85	C₂×Q₈×C₄₇	—	2	1	no	—
86	C₂×Q₈×C₄₇	—	2	2	no	—
87	C₂×Q₈×C₄₇	—	4	1	no	—
88	C₂×Q₈×C₄₇	—	8	2	no	—
89	C₂²×C₄×C₄₇	—	4	1	no	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
93	C₂²×Dic₄₇	—	2	1	no	—
94	C₂×D₁₈₈	—	2	1	no	—
95	C₂×D₁₈₈	—	2	2	no	—
96	C₂×D₁₈₈	Q₁₆	1	4	yes	—
97	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
98	C₂×C₄×D₄₇	—	2	2	no	—
99	C₂×C₄×D₄₇	—	2	2	no	—
100	C₂×C₄×D₄₇	—	8	2	no	—
101	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
102	C₄⋊C₄×C₄₇	—	2	2	no	—
103	C₄⋊C₄×C₄₇	—	2	1	no	—
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	—	2	1	no	—
106	C₄⋊C₄×C₄₇	—	2	1	no	—
107	C₄⋊C₄×C₄₇	—	2	1	no	—
108	C₄⋊C₄×C₄₇	—	4	1	no	—
109	C₄⋊C₄×C₄₇	—	4	1	no	—
110	C₄⋊C₄×C₄₇	—	4	1	no	—
111	C₄⋊C₄×C₄₇	—	4	1	no	—
112	C₂²⋊C₄×C₄₇	—	4	1	no	—
113	C₄²×C₄₇	—	4	1	no	—
114	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₄₇	#41	1	1	yes	characteristic
117	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
118	(C₂×D₄₇)⋊C₄	—	2	2	no	—
119	(C₂×D₄₇)⋊C₄	—	2	2	no	—
120	C₄⋊Dic₄₇	—	2	1	no	—
121	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
122	#10625329	—	4	2	no	—
123	C₂×C₄₇⋊C₈	—	2	2	no	—
124	C₂×C₄₇⋊C₈	—	2	2	no	—
125	C₂×C₄₇⋊C₈	—	2	2	no	—
126	C₂×C₄₇⋊C₈	—	4	2	no	—
127	C₈×D₄₇	—	8	4	no	—
128	C₂³×C₄₇	#121	1	1	yes	characteristic
129	C₂²×D₄₇	—	2	2	no	—
130	C₂²×D₄₇	—	2	4	no	—
131	C₂²×D₄₇	—	8	1	no	—
132	C₂²×D₄₇	—	8	1	no	—
133	Q₈×C₄₇	—	4	2	no	—
134	Q₈×C₄₇	—	8	1	no	—
135	Q₈×C₄₇	—	8	2	no	—
136	Q₈×C₄₇	—	8	2	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	4	1	no	—
139	C₂×C₄×C₄₇	—	4	1	no	—
140	C₂×C₄×C₄₇	—	4	1	no	—
141	C₂×C₄×C₄₇	—	4	2	no	—
142	C₂×C₄×C₄₇	—	8	2	no	—
143	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
144	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
145	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
147	C₂×C₄×C₄₇	#135	1	2	yes	—
148	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
151	C₂×Dic₄₇	—	2	2	no	—
152	C₂×Dic₄₇	—	8	1	no	—
153	D₁₈₈	—	2	2	no	—
154	D₁₈₈	—	2	4	no	—
155	D₁₈₈	C₂×Q₁₆	1	4	yes	—
156	C₄×D₄₇	—	4	4	no	—
157	C₄×D₄₇	—	8	2	no	—
158	C₈×C₄₇	—	2	4	no	—
159	C₄₇⋊C₈	—	4	2	no	—
160	C₄₇⋊C₈	—	4	2	no	—
161	C₄₇⋊C₈	—	4	2	no	—
162	#19393	—	47	1	no	maximal, Sylow
163	C₂²×C₄₇	—	2	1	no	—
164	C₂²×C₄₇	—	2	1	no	—
165	C₂²×C₄₇	#467	1	2	yes	—
166	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
167	C₂×D₄₇	—	2	4	no	—
168	C₂×D₄₇	—	4	2	no	—
169	C₂×D₄₇	—	4	2	no	—
170	C₂×D₄₇	—	8	4	no	—
171	C₄×C₄₇	—	2	1	no	—
172	C₄×C₄₇	—	2	1	no	—
173	C₄×C₄₇	—	2	1	no	—
174	C₄×C₄₇	—	8	1	no	—
175	C₄×C₄₇	—	8	2	no	—
176	C₄×C₄₇	—	8	1	no	—
177	C₄×C₄₇	#467	1	2	yes	—
178	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
179	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
180	C₄×C₄₇	#577	1	2	yes	—
181	Dic₄₇	—	4	2	no	—
182	#3230	D₄₇	1	1	yes	characteristic, complemented
183	#3077	—	47	1	no	—
184	C₂×#476	—	47	1	no	—
185	#2972	—	47	1	no	—
186	#1683	—	47	4	no	—
187	#1675	—	47	4	no	—
188	#1571	—	47	1	no	—
189	#1548	—	47	1	no	—
190	#1542	—	47	1	no	—
191	C₂×C₄₇	—	2	2	no	—
192	C₂×C₄₇	#3042	1	1	yes	characteristic
193	C₂×C₄₇	C₂×#467	1	1	yes	characteristic
194	C₂×C₄₇	#3068	1	1	yes	characteristic
195	D₄₇	—	4	4	no	—
196	#533	C₂×D₄₇	1	1	yes	characteristic
197	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
198	C₂×C₄×D₄	—	47	1	no	—
199	#500	C₂×D₄₇	1	2	yes	complemented
200	#487	C₂×D₄₇	1	2	yes	complemented
201	#484	—	2	2	no	—
202	#478	—	94	2	no	—
203	#477	—	2	2	no	—
204	#476	—	47	2	no	—
205	#476	—	47	4	no	—
206	#476	—	94	2	no	—
207	#474	—	47	2	no	—
208	C₈×D₄	—	47	4	no	—
209	C₈×D₄	—	94	2	no	—
210	#431	—	47	1	no	—
211	#431	C₂×D₄₇	1	1	yes	characteristic
212	#428	—	94	1	no	—
213	C₂×#106	—	94	1	no	—
214	#423	—	47	1	no	—
215	C₂×C₄⋊C₈	—	47	1	no	—
216	#418	—	94	1	no	—
217	C₂×#102	—	94	1	no	—
218	C₂×C₂²⋊C₈	—	94	2	no	—
219	#335	—	47	4	no	—
220	#333	—	47	2	no	—
221	#333	—	47	4	no	—
222	#332	—	47	2	no	—
223	C₄₇	#19393	1	1	yes	characteristic, complemented, Sylow
224	C₂²×Q₈	—	2	1	no	—
225	C₂²×D₄	—	47	1	no	—
226	C₂³×C₄	—	94	1	no	—
227	C₂×M₁₆	—	94	1	no	—
228	C₂²×C₈	—	94	1	no	—
229	C₂²×C₈	—	94	4	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
234	C₄⋊Q₈	C₂²×D₄₇	1	1	yes	characteristic
235	C₄⋊Q₈	C₂²×D₄₇	1	1	yes	characteristic
236	#125	—	2	1	no	—
237	C₂²⋊Q₈	—	2	1	no	—
238	C₂²⋊Q₈	—	2	1	no	—
239	C₄×Q₈	—	2	1	no	—
240	C₄×Q₈	—	2	1	no	—
241	C₄×D₄	—	47	2	no	—
242	C₄×D₄	—	47	2	no	—
243	C₄×D₄	—	94	1	no	—
244	C₄×D₄	—	94	1	no	—
245	#117	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄⋊C₄	—	47	1	no	—
249	C₂×C₂²⋊C₄	—	94	1	no	—
250	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
251	#107	—	2	4	no	—
252	#107	—	94	2	no	—
253	#106	—	94	4	no	—
254	#106	—	94	2	no	—
255	C₄⋊C₈	—	47	2	no	—
256	C₄⋊C₈	—	47	2	no	—
257	C₄⋊C₈	—	47	2	no	—
258	C₄⋊C₈	—	94	1	no	—
259	C₄⋊C₈	—	94	1	no	—
260	C₄⋊C₈	C₂²⋊D₄₇	1	2	yes	—
261	#103	—	2	4	no	—
262	#102	—	94	4	no	—
263	#102	—	94	2	no	—
264	#102	—	94	2	no	—
265	C₂²⋊C₈	—	94	1	no	—
266	C₂²⋊C₈	—	94	4	no	—
267	C₂²⋊C₈	C₂²⋊D₄₇	1	2	yes	—
268	C₄×C₈	—	94	1	no	—
269	C₄×C₈	C₂²×D₄₇	1	2	yes	—
270	C₂⁴	—	94	1	no	—
271	C₂×Q₈	—	2	1	no	—
272	C₂×Q₈	—	2	1	no	—
273	C₂×Q₈	—	2	2	no	—
274	C₂×Q₈	—	4	1	no	—
275	C₂×Q₈	—	8	2	no	—
276	C₂×D₄	—	47	4	no	—
277	C₂×D₄	—	47	2	no	—
278	C₂×D₄	—	94	1	no	—
279	C₂×D₄	—	94	2	no	—
280	C₂²×C₄	—	4	1	no	—
281	C₂²×C₄	—	94	1	no	—
282	C₂²×C₄	—	94	2	no	—
283	C₂²×C₄	—	94	2	no	—
284	C₂²×C₄	—	376	2	no	—
285	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
286	C₂²×C₄	#10625352	1	1	yes	characteristic
287	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
288	M₁₆	—	188	2	no	—
289	C₂×C₈	—	94	2	no	—
290	C₂×C₈	—	94	2	no	—
291	C₂×C₈	—	94	2	no	—
292	C₂×C₈	—	376	4	no	—
293	C₂×C₈	—	188	2	no	—
294	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	47	2	no	—
306	C₄⋊C₄	—	94	1	no	—
307	C₂²⋊C₄	—	4	1	no	—
308	C₂²⋊C₄	—	94	2	no	—
309	C₂²⋊C₄	—	94	2	no	—
310	C₄²	—	4	1	no	—
311	C₄²	D₄×D₄₇	1	1	yes	characteristic
312	C₄²	#10625352	1	1	yes	characteristic
313	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
314	C₄²	C₂³×D₄₇	1	1	yes	characteristic
315	C₂³	—	94	4	no	—
316	C₂³	—	94	2	no	—
317	C₂³	—	376	1	no	—
318	C₂³	—	376	1	no	—
319	C₂³	#13093995	1	1	yes	characteristic
320	Q₈	—	4	2	no	—
321	Q₈	—	8	1	no	—
322	Q₈	—	8	2	no	—
323	Q₈	—	8	2	no	—
324	D₄	—	47	4	no	—
325	D₄	—	94	4	no	—
326	D₄	—	94	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	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	4	no	—
337	C₂×C₄	#13093995	1	1	yes	characteristic
338	C₂×C₄	#13093995	1	1	yes	characteristic
339	C₂×C₄	#13093995	1	1	yes	characteristic
340	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
341	C₂×C₄	#13094047	1	2	yes	—
342	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
343	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
344	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
345	C₈	—	2	4	no	—
346	C₈	—	188	2	no	—
347	C₈	—	188	2	no	—
348	C₈	—	188	2	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	94	4	no	—
352	C₂²	—	376	4	no	—
353	C₂²	—	188	2	no	—
354	C₂²	—	188	2	no	—
355	C₂²	#15689361	1	2	yes	—
356	C₂²	C₂×#13093995	1	1	yes	characteristic
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	8	2	no	—
361	C₄	—	8	1	no	—
362	C₄	—	8	1	no	—
363	C₄	—	188	2	no	—
364	C₄	#15689361	1	2	yes	—
365	C₄	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
366	C₄	C₂×#13094047	1	1	yes	characteristic
367	C₄	#15689931	1	2	yes	—
368	C₂	—	2	2	no	—
369	C₂	—	188	4	no	—
370	C₂	#20252896	1	1	yes	characteristic
371	C₂	C₂×#15689361	1	1	yes	characteristic
372	C₂	#20253112	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093995
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3230	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂²×D₁₈₈	D₄
hypercenter	#3230	D₄₇
hypocenter	C₄₇	#19393
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3230

Complements

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

N	H ≅ G/N	type
4. #20242836	C₂	semidirect
5. #20242784	C₂	semidirect
6. #20242163	C₂	semidirect
7. #20242069	C₂	semidirect
8. #20242045	C₂	semidirect
9. C₄₇×#3230	C₂	semidirect
26. #15688876	C₂²	semidirect
27. #15688872	C₂²	semidirect
28. #15688872	C₂²	semidirect
29. #15688869	C₂²	semidirect
32. C₄₇×#500	C₂²	semidirect
33. C₄₇×#487	C₂²	semidirect
182. #3230	D₄₇	semidirect
199. #500	C₂×D₄₇	semidirect
200. #487	C₂×D₄₇	semidirect
223. C₄₇	#19393	semidirect

Maximal subgroups

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

subgroup	index
#20253181	2
C₂×#15689410	2
#20252411	2
#20242836	2
#20242784	2
#20242163	2
#20242069	2
#20242045	2
C₄₇×#3230	2
#19393	47