Subgroups — #26371116

Order 12032.

588 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	#20251988	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245576	C₂	1	2	yes	maximal, complemented
3	#20245576	C₂	1	2	yes	maximal, complemented
4	#20245571	C₂	1	2	yes	maximal, complemented
5	#20245566	C₂	1	2	yes	maximal, complemented
6	#20245177	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245170	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245104	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15688924	C₂	1	1	yes	maximal, characteristic, complemented
10	#20245002	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#3058	C₂	1	1	yes	maximal, characteristic
12	#15689606	—	2	1	no	—
13	C₂×#13093964	C₂²	1	1	yes	characteristic
14	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	#15689165	—	2	2	no	—
16	#15689165	—	2	1	no	—
17	#15689161	C₂²	1	1	yes	characteristic
18	#15689161	C₂²	1	2	yes	complemented
19	C₂×#13093958	C₂²	1	1	yes	characteristic
20	C₂×#13093958	C₂²	1	2	yes	complemented
21	#15688950	C₂²	1	1	yes	characteristic, complemented
22	#15688950	C₂²	1	2	yes	complemented
23	#15688950	C₂²	1	1	yes	characteristic, complemented
24	#15688948	C₂²	1	1	yes	characteristic, complemented
25	#15688948	C₂²	1	2	yes	complemented
26	#15688948	C₂²	1	1	yes	characteristic, complemented
27	#15688924	C₂²	1	2	yes	—
28	#15688924	C₂²	1	2	yes	—
29	#15688924	C₂²	1	2	yes	—
30	#15688924	C₂²	1	2	yes	—
31	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	complemented
32	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	complemented
33	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
35	C₄₇×#535	—	2	1	no	—
36	C₄₇×#469	—	2	2	no	—
37	C₄₇×#466	C₂²	1	2	yes	—
38	C₄₇×#466	C₂²	1	2	yes	—
39	C₄₇×#427	C₂²	1	1	yes	characteristic
40	C₄₇×#418	—	2	1	no	—
41	Q₈×Dic₄₇	—	4	1	no	—
42	D₄×Dic₄₇	—	4	2	no	—
43	D₄×Dic₄₇	—	4	1	no	—
44	#13093983	—	2	1	no	—
45	#13093983	—	2	2	no	—
46	C₂×C₄⋊Dic₄₇	—	2	1	no	—
47	C₂×C₄⋊Dic₄₇	—	2	2	no	—
48	#13093964	—	2	2	no	—
49	C₂×C₄₇⋊C₁₆	—	2	2	no	—
50	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
53	#13093958	—	2	1	no	—
54	#13093958	—	2	2	no	—
55	#13093958	C₂³	1	2	yes	—
56	#13093958	C₂³	1	1	yes	characteristic
57	#13093958	C₂³	1	1	yes	characteristic
58	#13093956	—	2	1	no	—
59	#13093956	—	2	2	no	—
60	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
61	C₂²×D₄×C₄₇	—	2	1	no	—
62	C₂×D₈×C₄₇	—	2	1	no	—
63	C₂×D₈×C₄₇	—	2	2	no	—
64	C₂×D₈×C₄₇	—	2	2	no	—
65	C₂×D₈×C₄₇	C₂³	1	2	yes	—
66	#131×C₄₇	—	2	2	no	—
67	#131×C₄₇	—	2	2	no	—
68	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
69	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
70	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
71	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	#127×C₄₇	—	4	1	no	—
73	#124×C₄₇	—	4	1	no	—
74	#121×C₄₇	—	2	2	no	—
75	#121×C₄₇	—	2	2	no	—
76	#120×C₄₇	—	4	2	no	—
77	#117×C₄₇	—	2	1	no	—
78	#107×C₄₇	C₂³	1	1	yes	characteristic
79	#107×C₄₇	C₂³	1	1	yes	characteristic
80	#106×C₄₇	—	2	1	no	—
81	#102×C₄₇	—	2	2	no	—
82	#102×C₄₇	—	2	2	no	—
83	C₂⁴×C₄₇	—	4	1	no	—
84	#41×C₄₇	—	2	2	no	—
85	#41×C₄₇	—	2	2	no	—
86	C₂×Q₈×C₄₇	D₈	1	1	yes	characteristic
87	C₂×D₄×C₄₇	—	2	2	no	—
88	C₂×D₄×C₄₇	—	4	2	no	—
89	C₂×D₄×C₄₇	—	4	2	no	—
90	C₂×D₄×C₄₇	—	4	2	no	—
91	C₂×D₄×C₄₇	—	4	1	no	—
92	C₂×D₄×C₄₇	—	4	2	no	—
93	C₂×D₄×C₄₇	D₈	1	2	yes	—
94	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
95	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×Dic₄₇	—	4	2	no	—
98	C₂²×Dic₄₇	—	4	1	no	—
99	D₈×C₄₇	—	2	2	no	—
100	D₈×C₄₇	—	2	2	no	—
101	D₈×C₄₇	—	4	2	no	—
102	M₁₆×C₄₇	—	2	2	no	—
103	M₁₆×C₄₇	—	2	4	no	—
104	C₂×C₈×C₄₇	—	2	2	no	—
105	C₂×C₈×C₄₇	—	2	2	no	—
106	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₇	#41	1	2	yes	—
109	C₂×C₈×C₄₇	#41	1	2	yes	—
110	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₄₇	—	2	1	no	—
112	C₄⋊C₄×C₄₇	—	2	1	no	—
113	C₂²⋊C₄×C₄₇	—	4	1	no	—
114	C₂²⋊C₄×C₄₇	—	4	2	no	—
115	C₂²⋊C₄×C₄₇	—	4	2	no	—
116	C₄²×C₄₇	—	4	1	no	—
117	C₂²⋊Dic₄₇	—	4	2	no	—
118	C₂²⋊Dic₄₇	—	4	1	no	—
119	C₄⋊Dic₄₇	—	2	1	no	—
120	C₄⋊Dic₄₇	—	2	1	no	—
121	C₄⋊Dic₄₇	—	2	2	no	—
122	C₄⋊Dic₄₇	—	4	2	no	—
123	C₄⋊Dic₄₇	—	4	1	no	—
124	C₄×Dic₄₇	—	4	2	no	—
125	C₄×Dic₄₇	—	4	1	no	—
126	C₄₇⋊C₁₆	—	2	1	no	—
127	C₄₇⋊C₁₆	—	2	1	no	—
128	C₄₇⋊C₁₆	—	2	2	no	—
129	C₂³×C₄₇	—	4	2	no	—
130	C₂³×C₄₇	—	8	2	no	—
131	C₂³×C₄₇	—	16	1	no	—
132	C₂³×C₄₇	—	16	1	no	—
133	C₂³×C₄₇	C₂×D₈	1	2	yes	—
134	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
135	Q₈×C₄₇	—	2	2	no	—
136	D₄×C₄₇	—	2	4	no	—
137	D₄×C₄₇	—	2	2	no	—
138	D₄×C₄₇	—	4	2	no	—
139	D₄×C₄₇	—	4	2	no	—
140	D₄×C₄₇	—	8	2	no	—
141	D₄×C₄₇	—	16	2	no	—
142	D₄×C₄₇	—	16	2	no	—
143	C₂×C₄×C₄₇	—	2	2	no	—
144	C₂×C₄×C₄₇	—	2	2	no	—
145	C₂×C₄×C₄₇	—	4	1	no	—
146	C₂×C₄×C₄₇	—	4	1	no	—
147	C₂×C₄×C₄₇	#123	1	2	yes	—
148	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
149	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₄₇	#137	1	2	yes	—
151	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×Dic₄₇	—	4	1	no	—
153	C₂×Dic₄₇	—	4	2	no	—
154	C₂×Dic₄₇	—	4	1	no	—
155	C₂×Dic₄₇	—	16	1	no	—
156	C₂×Dic₄₇	—	16	2	no	—
157	C₈×C₄₇	—	2	2	no	—
158	C₈×C₄₇	#123	1	2	yes	—
159	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
160	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
161	#33650	—	47	1	no	maximal, Sylow
162	C₂²×C₄₇	—	2	1	no	—
163	C₂²×C₄₇	—	2	1	no	—
164	C₂²×C₄₇	—	2	2	no	—
165	C₂²×C₄₇	—	2	2	no	—
166	C₂²×C₄₇	—	2	2	no	—
167	C₂²×C₄₇	—	8	2	no	—
168	C₂²×C₄₇	—	8	2	no	—
169	C₂²×C₄₇	—	8	2	no	—
170	C₂²×C₄₇	—	16	4	no	—
171	C₂²×C₄₇	#480	1	2	yes	—
172	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
173	C₄×C₄₇	—	2	2	no	—
174	C₄×C₄₇	—	8	1	no	—
175	C₄×C₄₇	—	8	1	no	—
176	C₄×C₄₇	#480	1	2	yes	—
177	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
178	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
179	Dic₄₇	—	8	2	no	—
180	Dic₄₇	—	8	1	no	—
181	Dic₄₇	—	8	1	no	—
182	#3058	D₄₇	1	1	yes	characteristic
183	#2913	—	47	1	no	—
184	#2234	—	47	2	no	—
185	#2234	—	47	2	no	—
186	#2233	—	47	2	no	—
187	#2232	—	47	2	no	—
188	#2159	—	47	1	no	—
189	#2158	—	47	1	no	—
190	#2140	—	47	1	no	—
191	C₂×#357	—	47	1	no	—
192	#2113	—	47	1	no	—
193	C₂×C₄₇	—	2	2	no	—
194	C₂×C₄₇	—	4	2	no	—
195	C₂×C₄₇	—	16	2	no	—
196	C₂×C₄₇	C₂×#480	1	1	yes	characteristic
197	C₂×C₄₇	#3415	1	1	yes	characteristic
198	C₂×C₄₇	#3416	1	1	yes	characteristic
199	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
200	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
201	#535	—	2	1	no	—
202	#520	—	94	1	no	—
203	C₂×M₃₂	—	47	1	no	—
204	C₂²×C₁₆	—	47	1	no	—
205	#469	—	2	2	no	—
206	#466	C₂×D₄₇	1	2	yes	—
207	#466	C₂×D₄₇	1	2	yes	—
208	#428	—	94	1	no	—
209	#428	—	94	2	no	—
210	#427	—	47	1	no	—
211	#427	—	47	2	no	—
212	#427	C₂×D₄₇	1	1	yes	characteristic
213	C₂×#107	—	47	1	no	—
214	C₂×#107	—	47	2	no	—
215	#418	—	2	1	no	—
216	#367	—	47	1	no	—
217	#367	—	47	1	no	—
218	#367	—	47	2	no	—
219	#366	—	47	1	no	—
220	#366	—	47	1	no	—
221	#366	—	47	2	no	—
222	#357	—	47	2	no	—
223	#357	—	47	2	no	—
224	#357	—	47	2	no	—
225	#357	—	47	2	no	—
226	C₂²⋊C₁₆	—	47	2	no	—
227	C₂²⋊C₁₆	—	47	2	no	—
228	C₄₇	#33650	1	1	yes	characteristic, complemented, Sylow
229	C₂×#41	C₂²⋊D₄₇	1	1	yes	characteristic
230	C₂²×D₄	—	2	1	no	—
231	C₂×D₈	—	2	2	no	—
232	C₂×D₈	—	2	2	no	—
233	C₂×D₈	—	2	1	no	—
234	C₂×D₈	C₂²×D₄₇	1	2	yes	—
235	#131	—	2	2	no	—
236	#131	—	2	2	no	—
237	C₂×M₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
238	C₂×M₁₆	C₂²⋊D₄₇	1	2	yes	—
239	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
240	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
241	#127	—	4	1	no	—
242	#124	—	4	1	no	—
243	#121	—	2	2	no	—
244	#121	—	2	2	no	—
245	#120	—	4	2	no	—
246	C₄×Q₈	—	188	1	no	—
247	C₄×D₄	—	188	2	no	—
248	C₄×D₄	—	188	1	no	—
249	#117	—	2	1	no	—
250	#117	—	94	2	no	—
251	#117	—	94	1	no	—
252	C₂×C₄⋊C₄	—	94	2	no	—
253	C₂×C₄⋊C₄	—	94	1	no	—
254	M₃₂	—	94	2	no	—
255	C₂×C₁₆	—	47	1	no	—
256	C₂×C₁₆	—	47	2	no	—
257	C₂×C₁₆	—	47	1	no	—
258	C₂×C₁₆	—	94	2	no	—
259	#107	—	47	2	no	—
260	#107	—	47	1	no	—
261	#107	—	47	1	no	—
262	#107	—	94	2	no	—
263	#107	—	94	1	no	—
264	#107	C₂²×D₄₇	1	1	yes	characteristic
265	#107	C₂²×D₄₇	1	1	yes	characteristic
266	#106	—	2	1	no	—
267	#106	—	94	2	no	—
268	#106	—	94	1	no	—
269	#102	—	2	2	no	—
270	#102	—	2	2	no	—
271	C₂⁴	—	4	1	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	C₂×Q₈	#10625334	1	1	yes	characteristic
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	4	2	no	—
277	C₂×D₄	—	4	2	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	—	4	1	no	—
280	C₂×D₄	—	4	2	no	—
281	C₂×D₄	#10625334	1	2	yes	—
282	C₂×D₄	#10625334	1	1	yes	characteristic
283	C₂²×C₄	—	188	2	no	—
284	C₂²×C₄	—	188	1	no	—
285	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
286	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
287	D₈	—	2	2	no	—
288	D₈	—	2	2	no	—
289	D₈	—	4	2	no	—
290	M₁₆	—	2	2	no	—
291	M₁₆	—	2	4	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	#10625352	1	2	yes	—
295	C₂×C₈	#10625352	1	2	yes	—
296	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
297	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
298	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	94	1	no	—
302	C₄⋊C₄	—	94	1	no	—
303	C₄⋊C₄	—	94	2	no	—
304	C₄⋊C₄	—	188	2	no	—
305	C₄⋊C₄	—	188	1	no	—
306	C₂²⋊C₄	—	4	1	no	—
307	C₂²⋊C₄	—	4	2	no	—
308	C₂²⋊C₄	—	4	2	no	—
309	C₂²⋊C₄	—	188	2	no	—
310	C₂²⋊C₄	—	188	1	no	—
311	C₄²	—	4	1	no	—
312	C₄²	—	188	2	no	—
313	C₄²	—	188	1	no	—
314	C₁₆	—	94	1	no	—
315	C₁₆	—	94	1	no	—
316	C₁₆	—	94	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	8	2	no	—
319	C₂³	—	16	1	no	—
320	C₂³	—	16	1	no	—
321	C₂³	C₂×#10625334	1	2	yes	—
322	C₂³	C₂×#10625334	1	1	yes	characteristic
323	Q₈	—	2	2	no	—
324	D₄	—	2	4	no	—
325	D₄	—	2	2	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	8	2	no	—
329	D₄	—	16	2	no	—
330	D₄	—	16	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	188	1	no	—
336	C₂×C₄	—	188	2	no	—
337	C₂×C₄	—	188	1	no	—
338	C₂×C₄	—	752	1	no	—
339	C₂×C₄	—	752	2	no	—
340	C₂×C₄	#13094005	1	2	yes	—
341	C₂×C₄	C₂×#10625334	1	2	yes	—
342	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
343	C₂×C₄	#13094054	1	2	yes	—
344	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	#13094005	1	2	yes	—
347	C₈	C₂×#10625352	1	1	yes	characteristic
348	C₈	C₂×#10625352	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	16	4	no	—
358	C₂²	#15689434	1	2	yes	—
359	C₂²	C₂²×#10625334	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	8	1	no	—
362	C₄	—	8	1	no	—
363	C₄	—	376	2	no	—
364	C₄	—	376	1	no	—
365	C₄	—	376	1	no	—
366	C₄	#15689434	1	2	yes	—
367	C₄	C₂×#13094005	1	1	yes	characteristic
368	C₄	C₂×#13094054	1	1	yes	characteristic
369	C₂	—	2	2	no	—
370	C₂	—	4	2	no	—
371	C₂	—	16	2	no	—
372	C₂	C₂×#15689434	1	1	yes	characteristic
373	C₂	#20256527	1	1	yes	characteristic
374	C₂	#20256535	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting, prime radical	C₄₇×#3058	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#3058	D₄₇
hypocenter	C₄₇	#33650
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ #3058

Complements

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

N	H ≅ G/N	type
1. #20251988	C₂	semidirect
2. #20245576	C₂	semidirect
3. #20245576	C₂	semidirect
4. #20245571	C₂	semidirect
5. #20245566	C₂	semidirect
6. #20245177	C₂	semidirect
7. #20245170	C₂	semidirect
8. #20245104	C₂	semidirect
9. C₂×#15688924	C₂	semidirect
10. #20245002	C₂	semidirect
18. #15689161	C₂²	semidirect
20. C₂×#13093958	C₂²	semidirect
21. #15688950	C₂²	semidirect
22. #15688950	C₂²	semidirect
23. #15688950	C₂²	semidirect
24. #15688948	C₂²	semidirect
25. #15688948	C₂²	semidirect
26. #15688948	C₂²	semidirect
31. C₂²⋊(C₄₇⋊C₁₆)	C₂²	semidirect
32. C₂²⋊(C₄₇⋊C₁₆)	C₂²	semidirect
228. C₄₇	#33650	semidirect

Maximal subgroups

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

subgroup	index
#20251988	2
#20245576	2
#20245571	2
#20245566	2
#20245177	2
#20245170	2
#20245104	2
C₂×#15688924	2
#20245002	2
C₄₇×#3058	2
#33650	47