Subgroups — #26200298

Order 12032.

658 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253181	C₂	1	2	yes	maximal
2	C₂×#15689202	C₂	1	1	yes	maximal, characteristic
3	#20242960	C₂	1	4	yes	maximal, complemented
4	#20242908	C₂	1	2	yes	maximal, complemented
5	#20242874	C₂	1	2	yes	maximal, complemented
6	#20242144	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242078	C₂	1	2	yes	maximal, complemented
8	C₄₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10625324	—	2	1	no	—
10	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
11	#15689420	—	2	4	no	—
12	#15689410	C₂²	1	4	yes	—
13	#15689400	C₂²	1	4	yes	—
14	#15689202	—	2	1	no	—
15	#15689202	—	2	1	no	—
16	#15689202	C₂²	1	2	yes	—
17	#15689202	C₂²	1	2	yes	—
18	#15689166	—	2	2	no	—
19	#15689147	C₂²	1	2	yes	—
20	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
21	#15689127	—	2	2	no	—
22	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
23	#15688874	C₂²	1	2	yes	complemented
24	#15688874	C₂²	1	2	yes	complemented
25	#15688872	C₂²	1	4	yes	complemented
26	#15688869	C₂²	1	4	yes	complemented
27	C₄₇×#533	C₂²	1	2	yes	—
28	C₄₇×#500	C₂²	1	1	yes	characteristic, complemented
29	C₄₇×#494	C₂²	1	1	yes	characteristic, complemented
30	C₄₇×#487	C₂²	1	2	yes	complemented
31	C₄₇×#467	—	2	1	no	—
32	C₄₇×#467	—	2	1	no	—
33	C₄×Q₁₆×C₄₇	—	2	1	no	—
34	C₄×Q₁₆×C₄₇	—	2	1	no	—
35	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₄₇	—	2	1	no	—
38	C₂×#10625329	—	2	2	no	—
39	C₂×#10625324	—	2	2	no	—
40	C₂×#10625324	—	2	2	no	—
41	C₂×#10625324	—	4	4	no	—
42	C₂²×C₄₇⋊C₈	—	2	1	no	—
43	C₄×D₁₈₈	—	2	1	no	—
44	C₄×D₁₈₈	—	2	1	no	—
45	C₄×D₁₈₈	D₄	1	2	yes	—
46	C₄×D₁₈₈	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
48	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
49	#13093957	—	2	4	no	—
50	#13093955	—	2	4	no	—
51	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
52	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
53	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
54	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
55	#13093944	—	2	4	no	—
56	#13093944	—	2	4	no	—
57	(C₂×D₄₇)⋊C₈	—	2	2	no	—
58	(C₂×D₄₇)⋊C₈	—	2	2	no	—
59	C₂×Q₁₆×C₄₇	—	2	1	no	—
60	C₂×Q₁₆×C₄₇	—	2	1	no	—
61	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
63	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
65	#125×C₄₇	—	2	2	no	—
66	C₂²⋊Q₈×C₄₇	—	2	2	no	—
67	C₂²⋊Q₈×C₄₇	—	2	2	no	—
68	C₄×Q₈×C₄₇	—	2	2	no	—
69	C₄×Q₈×C₄₇	—	2	2	no	—
70	#117×C₄₇	—	2	2	no	—
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	#107×C₄₇	—	2	1	no	—
73	#107×C₄₇	—	2	1	no	—
74	#103×C₄₇	—	2	2	no	—
75	#103×C₄₇	—	2	2	no	—
76	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
77	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
80	C₂³×D₄₇	—	2	1	no	—
81	C₂×Q₈×C₄₇	—	2	2	no	—
82	C₂×Q₈×C₄₇	—	2	2	no	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
86	C₂²×Dic₄₇	—	2	1	no	—
87	C₂×D₁₈₈	—	2	4	no	—
88	C₂×D₁₈₈	—	2	1	no	—
89	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
90	C₂×C₄×D₄₇	—	2	2	no	—
91	C₂×C₄×D₄₇	—	2	2	no	—
92	C₂×C₄×D₄₇	—	4	2	no	—
93	C₂×C₄×D₄₇	—	4	2	no	—
94	Q₁₆×C₄₇	—	4	2	no	—
95	Q₁₆×C₄₇	—	4	2	no	—
96	C₂×C₈×C₄₇	—	2	2	no	—
97	C₂×C₈×C₄₇	—	4	1	no	—
98	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₄⋊C₄×C₄₇	—	2	2	no	—
103	C₄⋊C₄×C₄₇	—	2	2	no	—
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	—	2	2	no	—
106	C₄⋊C₄×C₄₇	—	4	2	no	—
107	C₄⋊C₄×C₄₇	—	4	2	no	—
108	C₂²⋊C₄×C₄₇	—	4	2	no	—
109	C₄²×C₄₇	—	4	2	no	—
110	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₄₇	#41	1	1	yes	characteristic
113	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
114	(C₂×D₄₇)⋊C₄	—	2	2	no	—
115	(C₂×D₄₇)⋊C₄	—	2	2	no	—
116	C₄⋊Dic₄₇	—	2	1	no	—
117	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
118	#10625329	—	4	4	no	—
119	C₂×C₄₇⋊C₈	—	2	2	no	—
120	C₂×C₄₇⋊C₈	—	2	4	no	—
121	C₂×C₄₇⋊C₈	—	4	2	no	—
122	#10625324	—	4	4	no	—
123	#10625324	—	4	4	no	—
124	C₂³×C₄₇	#121	1	1	yes	characteristic
125	C₂²×D₄₇	—	2	2	no	—
126	C₂²×D₄₇	—	4	2	no	—
127	C₂²×D₄₇	—	4	2	no	—
128	C₂²×D₄₇	—	4	2	no	—
129	Q₈×C₄₇	—	4	2	no	—
130	Q₈×C₄₇	—	4	2	no	—
131	Q₈×C₄₇	—	8	2	no	—
132	C₂×C₄×C₄₇	—	2	1	no	—
133	C₂×C₄×C₄₇	—	2	1	no	—
134	C₂×C₄×C₄₇	—	4	1	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
138	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
139	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#137	1	4	yes	—
141	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
144	C₂×Dic₄₇	—	2	2	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	D₁₈₈	—	2	4	no	—
147	D₁₈₈	—	4	2	no	—
148	C₄×D₄₇	—	4	4	no	—
149	C₄×D₄₇	—	4	2	no	—
150	C₄×D₄₇	—	4	2	no	—
151	C₈×C₄₇	—	2	1	no	—
152	C₈×C₄₇	—	2	1	no	—
153	C₈×C₄₇	—	2	1	no	—
154	C₈×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
155	C₄₇⋊C₈	—	4	2	no	—
156	C₄₇⋊C₈	—	4	4	no	—
157	#19281	—	47	1	no	maximal, Sylow
158	C₂²×C₄₇	—	2	1	no	—
159	C₂²×C₄₇	—	2	1	no	—
160	C₂²×C₄₇	#470	1	2	yes	—
161	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
162	C₂×D₄₇	—	4	2	no	—
163	C₂×D₄₇	—	4	2	no	—
164	C₂×D₄₇	—	4	2	no	—
165	C₂×D₄₇	—	8	4	no	—
166	C₄×C₄₇	—	2	1	no	—
167	C₄×C₄₇	—	2	1	no	—
168	C₄×C₄₇	—	2	1	no	—
169	C₄×C₄₇	—	8	2	no	—
170	C₄×C₄₇	—	8	2	no	—
171	C₄×C₄₇	#470	1	2	yes	—
172	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	C₄×C₄₇	C₂×#137	1	2	yes	—
174	C₄×C₄₇	#575	1	1	yes	characteristic
175	Dic₄₇	—	4	2	no	—
176	#3170	D₄₇	1	1	yes	characteristic, complemented
177	#3077	—	47	2	no	—
178	C₂×#436	—	47	1	no	—
179	#1703	—	47	4	no	—
180	#1695	—	47	2	no	—
181	#1689	—	47	2	no	—
182	#1566	—	47	1	no	—
183	#1550	—	47	2	no	—
184	C₂×C₄₇	—	2	2	no	—
185	C₂×C₄₇	#3036	1	1	yes	characteristic
186	C₂×C₄₇	C₂×#470	1	1	yes	characteristic
187	C₂×C₄₇	#3067	1	1	yes	characteristic
188	D₄₇	—	8	2	no	—
189	C₂²×M₁₆	—	94	1	no	—
190	#533	C₂×D₄₇	1	2	yes	—
191	C₂×C₄×D₄	—	47	1	no	—
192	#500	C₂×D₄₇	1	1	yes	characteristic, complemented
193	#494	C₂×D₄₇	1	1	yes	characteristic, complemented
194	#487	C₂×D₄₇	1	2	yes	complemented
195	#478	—	94	4	no	—
196	#476	—	47	4	no	—
197	#474	—	47	4	no	—
198	#467	—	2	1	no	—
199	#467	—	2	1	no	—
200	C₄×Q₁₆	—	2	1	no	—
201	C₄×Q₁₆	—	2	1	no	—
202	#436	—	47	2	no	—
203	#436	—	47	2	no	—
204	#436	—	94	1	no	—
205	#436	—	94	1	no	—
206	#428	—	94	2	no	—
207	#423	—	47	2	no	—
208	C₂×C₄⋊C₈	—	47	1	no	—
209	#418	—	94	2	no	—
210	C₂×C₂²⋊C₈	—	94	1	no	—
211	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
212	#334	—	47	2	no	—
213	#334	—	47	2	no	—
214	#333	—	47	4	no	—
215	#332	—	47	4	no	—
216	C₄₇	#19281	1	1	yes	characteristic, complemented, Sylow
217	C₂²×D₄	—	47	1	no	—
218	C₂³×C₄	—	94	1	no	—
219	C₂×Q₁₆	—	2	1	no	—
220	C₂×Q₁₆	—	2	1	no	—
221	C₂×M₁₆	—	94	2	no	—
222	C₂×M₁₆	—	94	2	no	—
223	C₂×M₁₆	—	94	2	no	—
224	C₂×M₁₆	—	188	4	no	—
225	C₂²×C₈	—	94	1	no	—
226	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
227	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
228	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
229	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
230	#125	—	2	2	no	—
231	C₂²⋊Q₈	—	2	2	no	—
232	C₂²⋊Q₈	—	2	2	no	—
233	C₄×Q₈	—	2	2	no	—
234	C₄×Q₈	—	2	2	no	—
235	C₄×D₄	—	47	2	no	—
236	C₄×D₄	—	47	2	no	—
237	C₄×D₄	—	94	1	no	—
238	C₄×D₄	—	94	1	no	—
239	#117	—	2	2	no	—
240	C₂×C₄⋊C₄	—	47	1	no	—
241	C₂×C₂²⋊C₄	—	94	1	no	—
242	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
243	#107	—	2	1	no	—
244	#107	—	2	1	no	—
245	#107	—	94	4	no	—
246	#106	—	94	4	no	—
247	C₄⋊C₈	—	47	2	no	—
248	C₄⋊C₈	—	47	4	no	—
249	C₄⋊C₈	—	94	2	no	—
250	C₄⋊C₈	—	94	1	no	—
251	#103	—	2	2	no	—
252	#103	—	2	2	no	—
253	#102	—	94	4	no	—
254	#102	—	94	4	no	—
255	C₂²⋊C₈	—	94	2	no	—
256	C₂²⋊C₈	—	94	2	no	—
257	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
258	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
259	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
261	C₂⁴	—	94	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×D₄	—	47	2	no	—
265	C₂×D₄	—	94	1	no	—
266	C₂×D₄	—	94	4	no	—
267	C₂²×C₄	—	94	1	no	—
268	C₂²×C₄	—	94	2	no	—
269	C₂²×C₄	—	94	2	no	—
270	C₂²×C₄	—	188	2	no	—
271	C₂²×C₄	—	188	2	no	—
272	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
273	C₂²×C₄	#10625352	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
275	Q₁₆	—	4	2	no	—
276	Q₁₆	—	4	2	no	—
277	M₁₆	—	188	4	no	—
278	M₁₆	—	188	4	no	—
279	M₁₆	—	188	4	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	94	4	no	—
283	C₂×C₈	—	94	2	no	—
284	C₂×C₈	—	188	2	no	—
285	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
286	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
287	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
288	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄⋊C₄	—	47	2	no	—
296	C₄⋊C₄	—	94	1	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	94	2	no	—
299	C₂²⋊C₄	—	94	2	no	—
300	C₄²	—	4	2	no	—
301	C₄²	D₄×D₄₇	1	1	yes	characteristic
302	C₄²	#10625352	1	1	yes	characteristic
303	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
304	C₄²	C₂³×D₄₇	1	1	yes	characteristic
305	C₂³	—	94	2	no	—
306	C₂³	—	188	2	no	—
307	C₂³	—	188	2	no	—
308	C₂³	—	188	2	no	—
309	C₂³	#13093995	1	1	yes	characteristic
310	Q₈	—	4	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	8	2	no	—
313	D₄	—	94	4	no	—
314	D₄	—	188	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	94	2	no	—
321	C₂×C₄	—	188	4	no	—
322	C₂×C₄	—	188	2	no	—
323	C₂×C₄	—	188	2	no	—
324	C₂×C₄	—	188	2	no	—
325	C₂×C₄	#13093995	1	1	yes	characteristic
326	C₂×C₄	#13093995	1	1	yes	characteristic
327	C₂×C₄	#13093995	1	1	yes	characteristic
328	C₂×C₄	#13094056	1	4	yes	—
329	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
331	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	188	2	no	—
336	C₈	—	188	4	no	—
337	C₈	#13094049	1	2	yes	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	376	4	no	—
341	C₂²	—	188	2	no	—
342	C₂²	—	188	2	no	—
343	C₂²	—	188	2	no	—
344	C₂²	#15689378	1	2	yes	—
345	C₂²	C₂×#13093995	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	8	2	no	—
350	C₄	—	8	2	no	—
351	C₄	—	188	2	no	—
352	C₄	#15689378	1	2	yes	—
353	C₄	C₂×#13094049	1	1	yes	characteristic
354	C₄	C₂×#13094056	1	2	yes	—
355	C₄	#15689921	1	1	yes	characteristic
356	C₂	—	2	2	no	—
357	C₂	—	376	2	no	—
358	C₂	#20252848	1	1	yes	characteristic
359	C₂	C₂×#15689378	1	1	yes	characteristic
360	C₂	#20253104	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093995
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3170	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂²×D₁₈₈	D₄
hypercenter	#3170	D₄₇
hypocenter	C₄₇	#19281
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #20242960	C₂	semidirect
4. #20242908	C₂	semidirect
5. #20242874	C₂	semidirect
6. #20242144	C₂	semidirect
7. #20242078	C₂	semidirect
8. C₄₇×#3170	C₂	semidirect
23. #15688874	C₂²	semidirect
24. #15688874	C₂²	semidirect
25. #15688872	C₂²	semidirect
26. #15688869	C₂²	semidirect
28. C₄₇×#500	C₂²	semidirect
29. C₄₇×#494	C₂²	semidirect
30. C₄₇×#487	C₂²	semidirect
176. #3170	D₄₇	semidirect
192. #500	C₂×D₄₇	semidirect
193. #494	C₂×D₄₇	semidirect
194. #487	C₂×D₄₇	semidirect
216. C₄₇	#19281	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×#15689202	2
#20242960	2
#20242908	2
#20242874	2
#20242144	2
#20242078	2
C₄₇×#3170	2
#19281	47