Subgroups — #26275179

Order 12032.

690 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253916	C₂	1	1	yes	maximal, characteristic, complemented
2	#20253255	C₂	1	2	yes	maximal
3	#20251979	C₂	1	1	yes	maximal, characteristic
4	#20244899	C₂	1	2	yes	maximal, complemented
5	#20244733	C₂	1	2	yes	maximal, complemented
6	#20244166	C₂	1	1	yes	maximal, characteristic, complemented
7	#20244136	C₂	1	1	yes	maximal, characteristic, complemented
8	#20243613	C₂	1	2	yes	maximal
9	#20243448	C₂	1	1	yes	maximal, characteristic
10	#20243225	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094030	C₂²	1	1	yes	characteristic
13	#15689754	C₂²	1	2	yes	—
14	#15689604	C₂²	1	2	yes	—
15	#15689530	—	2	2	no	—
16	#15689523	—	2	1	no	—
17	#15689520	—	2	1	no	—
18	#15689433	—	2	2	no	—
19	#15689422	—	2	2	no	—
20	#15689412	—	2	2	no	—
21	#15689407	—	2	2	no	—
22	C₂×#13093960	C₂²	1	1	yes	characteristic
23	#15689163	C₂²	1	2	yes	—
24	#15689163	C₂²	1	2	yes	—
25	#15689163	C₂²	1	2	yes	—
26	#15689162	—	2	2	no	—
27	#15689161	C₂²	1	1	yes	characteristic
28	#15689158	—	2	1	no	—
29	#106×D₄₇	—	2	1	no	—
30	#15689149	C₂²	1	2	yes	—
31	C₂×#13093952	C₂²	1	1	yes	characteristic
32	#15689122	C₂²	1	2	yes	—
33	#15689120	C₂²	1	1	yes	characteristic
34	#15689078	C₂²	1	1	yes	characteristic
35	#15688892	C₂²	1	2	yes	—
36	#15688892	C₂²	1	2	yes	—
37	#15688888	C₂²	1	1	yes	characteristic
38	#15688883	C₂²	1	1	yes	characteristic
39	#15688883	C₂²	1	2	yes	—
40	C₂×#106×C₄₇	C₂²	1	2	yes	—
41	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
42	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
44	C₄₇×#384	C₂²	1	2	yes	—
45	C₂×#10625350	C₂³	1	1	yes	characteristic
46	#13094030	—	2	2	no	—
47	#13094030	Q₈	1	4	yes	complemented
48	C₂×#10625329	C₂³	1	2	yes	—
49	C₂×#10625329	C₂³	1	1	yes	characteristic
50	C₂×#10625329	C₂³	1	1	yes	characteristic
51	C₂×#10625324	—	2	1	no	—
52	C₂×C₈×D₄₇	—	2	1	no	—
53	#13094023	—	2	2	no	—
54	#13094023	—	2	2	no	—
55	#13094015	—	2	2	no	—
56	#13094015	—	2	2	no	—
57	#13094003	—	4	2	no	—
58	#13093989	—	2	2	no	—
59	#13093986	—	2	2	no	—
60	C₄×C₂²⋊D₄₇	—	4	2	no	—
61	#13093983	C₂³	1	1	yes	characteristic
62	#13093983	C₂³	1	2	yes	—
63	#13093982	—	2	2	no	—
64	C₄⋊C₄×D₄₇	—	2	2	no	—
65	#13093960	—	2	2	no	—
66	#13093958	D₄	1	2	yes	—
67	#13093957	—	2	2	no	—
68	#13093957	—	2	2	no	—
69	#13093957	—	2	2	no	—
70	#13093956	D₄	1	1	yes	characteristic
71	#13093956	D₄	1	1	yes	characteristic
72	#13093955	—	2	2	no	—
73	#13093955	—	2	2	no	—
74	#13093955	—	2	2	no	—
75	Dic₄₇⋊C₈	—	2	2	no	—
76	Dic₄₇⋊C₈	—	2	2	no	—
77	#13093952	Q₈	1	4	yes	complemented
78	#13093949	D₄	1	2	yes	—
79	#13093947	—	2	2	no	—
80	#13093946	D₄	1	2	yes	—
81	#13093944	—	2	2	no	—
82	#13093934	—	2	1	no	—
83	C₈×Dic₄₇	—	2	1	no	—
84	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
87	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
89	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
90	#106×C₄₇	—	2	1	no	—
91	#106×C₄₇	—	2	1	no	—
92	#106×C₄₇	—	2	2	no	—
93	#106×C₄₇	—	2	1	no	—
94	#106×C₄₇	—	2	2	no	—
95	#106×C₄₇	—	2	1	no	—
96	C₄×C₈×C₄₇	—	2	2	no	—
97	C₂²×C₄×C₄₇	—	2	1	no	—
98	C₂²×C₄×C₄₇	—	2	2	no	—
99	C₂²×C₄×C₄₇	—	2	2	no	—
100	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
101	C₂×C₂²⋊D₄₇	—	2	1	no	—
102	#10625350	—	2	1	no	—
103	#10625350	—	2	1	no	—
104	#10625350	C₂×Q₈	1	2	yes	—
105	#10625350	C₂×Q₈	1	2	yes	—
106	C₂×D₁₈₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×D₄₇	—	2	1	no	—
108	C₂×C₄×D₄₇	—	4	2	no	—
109	C₂×Dic₉₄	—	4	2	no	—
110	C₂×Dic₉₄	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₄₇	—	2	1	no	—
112	C₂×C₈×C₄₇	—	2	1	no	—
113	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
117	C₂×C₈×C₄₇	#41	1	2	yes	—
118	C₄⋊C₄×C₄₇	—	2	2	no	—
119	C₄⋊C₄×C₄₇	—	2	2	no	—
120	C₄⋊C₄×C₄₇	—	2	2	no	—
121	C₄⋊C₄×C₄₇	—	2	2	no	—
122	C₄⋊C₄×C₄₇	—	8	4	no	—
123	C₄²×C₄₇	—	2	1	no	—
124	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
125	C₂²⋊Dic₄₇	—	2	2	no	—
126	C₂²⋊Dic₄₇	—	2	1	no	—
127	(C₂×D₄₇)⋊C₄	—	4	2	no	—
128	C₄⋊Dic₄₇	—	4	2	no	—
129	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
130	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
131	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
132	Dic₄₇⋊C₄	—	4	2	no	—
133	Dic₄₇⋊C₄	—	4	2	no	—
134	Dic₄₇⋊C₄	—	4	2	no	—
135	C₄×Dic₄₇	—	2	1	no	—
136	C₄×Dic₄₇	—	2	2	no	—
137	C₄×Dic₄₇	—	4	2	no	—
138	#10625329	—	2	1	no	—
139	#10625329	—	2	1	no	—
140	#10625329	—	2	2	no	—
141	#10625329	—	2	2	no	—
142	#10625329	C₂×Q₈	1	2	yes	—
143	#10625329	C₂×Q₈	1	2	yes	—
144	C₂×C₄₇⋊C₈	—	2	1	no	—
145	C₂×C₄₇⋊C₈	—	2	1	no	—
146	C₂×C₄₇⋊C₈	—	2	2	no	—
147	#10625324	—	2	4	no	—
148	C₈×D₄₇	—	2	4	no	—
149	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂²×D₄₇	—	2	1	no	—
151	C₂×C₄×C₄₇	—	2	4	no	—
152	C₂×C₄×C₄₇	—	2	2	no	—
153	C₂×C₄×C₄₇	—	4	2	no	—
154	C₂×C₄×C₄₇	—	4	2	no	—
155	C₂×C₄×C₄₇	—	8	2	no	—
156	C₂×C₄×C₄₇	—	8	2	no	—
157	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
161	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
162	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
163	C₂²⋊D₄₇	—	2	2	no	—
164	C₂²⋊D₄₇	—	2	2	no	—
165	C₂×Dic₄₇	—	2	1	no	—
166	C₂×Dic₄₇	—	2	2	no	—
167	C₂×Dic₄₇	—	2	2	no	—
168	C₂×Dic₄₇	—	2	2	no	—
169	D₁₈₈	—	2	1	no	—
170	D₁₈₈	C₄⋊Q₈	1	2	yes	—
171	C₄×D₄₇	—	2	2	no	—
172	C₄×D₄₇	—	2	2	no	—
173	C₄×D₄₇	—	16	2	no	—
174	Dic₉₄	—	2	1	no	—
175	Dic₉₄	—	16	2	no	—
176	Dic₉₄	C₄⋊Q₈	1	2	yes	—
177	C₈×C₄₇	—	2	2	no	—
178	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
179	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
180	C₄₇⋊C₈	—	2	2	no	—
181	C₄₇⋊C₈	—	2	2	no	—
182	C₄₇⋊C₈	—	4	2	no	—
183	#26081	—	47	1	no	maximal, Sylow
184	C₂²×C₄₇	—	2	2	no	—
185	C₂²×C₄₇	#544	1	1	yes	characteristic
186	C₂²×C₄₇	D₄²	1	1	yes	characteristic
187	C₂²×C₄₇	#554	1	1	yes	characteristic
188	C₂×D₄₇	—	2	2	no	—
189	C₂×D₄₇	—	4	2	no	—
190	C₄×C₄₇	—	2	4	no	—
191	C₄×C₄₇	—	8	2	no	—
192	C₄×C₄₇	—	8	2	no	—
193	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
194	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
195	C₄×C₄₇	#547	1	1	yes	characteristic
196	C₄×C₄₇	D₄×Q₈	1	1	yes	characteristic
197	Dic₄₇	—	2	2	no	—
198	Dic₄₇	—	4	2	no	—
199	Dic₄₇	—	4	2	no	—
200	Dic₄₇	—	4	2	no	—
201	#3163	—	47	1	no	—
202	#3084	—	47	2	no	—
203	#2912	—	47	1	no	—
204	#2090	—	47	2	no	—
205	#2060	—	47	2	no	—
206	#1951	—	47	1	no	—
207	#1945	—	47	1	no	—
208	#1859	—	47	2	no	—
209	#1844	D₄₇	1	1	yes	characteristic, complemented
210	#1815	—	47	1	no	—
211	#1764	—	47	1	no	—
212	C₂×C₄₇	—	2	1	no	—
213	C₂×C₄₇	—	2	1	no	—
214	C₂×C₄₇	#2440	1	1	yes	characteristic
215	C₂×C₄₇	#3291	1	2	yes	—
216	D₄₇	—	4	2	no	—
217	C₂×#131	—	47	1	no	—
218	#544	—	47	2	no	—
219	#520	—	47	2	no	—
220	#501	—	94	2	no	—
221	#499	—	94	1	no	—
222	#498	—	94	1	no	—
223	#479	—	94	2	no	—
224	#478	—	94	2	no	—
225	#476	—	94	2	no	—
226	#475	—	94	2	no	—
227	C₂×#108	—	47	1	no	—
228	#428	—	47	2	no	—
229	#428	—	47	2	no	—
230	#428	—	47	2	no	—
231	#428	—	94	2	no	—
232	#427	—	47	1	no	—
233	#427	—	94	1	no	—
234	C₂×#106	—	94	1	no	—
235	C₂×#106	C₂×D₄₇	1	2	yes	—
236	C₂×#106	C₂×D₄₇	1	1	yes	characteristic
237	C₂×#106	C₂×D₄₇	1	1	yes	characteristic
238	#424	—	47	2	no	—
239	C₂×#104	—	47	1	no	—
240	#417	—	47	2	no	—
241	#416	—	47	1	no	—
242	#405	—	47	1	no	—
243	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
244	#384	C₂×D₄₇	1	2	yes	—
245	#343	—	47	2	no	—
246	#343	—	47	2	no	—
247	#341	—	47	1	no	—
248	#339	—	47	1	no	—
249	#339	—	47	2	no	—
250	C₄₇	#26081	1	1	yes	characteristic, complemented, Sylow
251	C₂×#41	—	47	1	no	—
252	#131	—	47	4	no	—
253	#131	—	94	2	no	—
254	C₂×M₁₆	—	47	1	no	—
255	C₂×M₁₆	—	47	1	no	—
256	C₂×M₁₆	—	47	2	no	—
257	C₂×M₁₆	—	94	1	no	—
258	C₂²×C₈	—	94	1	no	—
259	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
260	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
261	C₄⋊Q₈	—	94	2	no	—
262	C₄⋊Q₈	—	94	2	no	—
263	#125	—	94	2	no	—
264	#125	—	94	2	no	—
265	C₂²⋊Q₈	—	188	2	no	—
266	C₄×Q₈	—	94	2	no	—
267	C₄×D₄	—	94	2	no	—
268	C₄×D₄	—	188	2	no	—
269	#117	—	47	2	no	—
270	#117	—	47	1	no	—
271	#117	—	94	2	no	—
272	C₂×C₄⋊C₄	—	4	2	no	—
273	C₂×C₄⋊C₄	—	94	2	no	—
274	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
275	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
276	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
277	#108	—	94	2	no	—
278	#107	—	47	2	no	—
279	#107	—	94	2	no	—
280	#107	—	94	2	no	—
281	#107	—	94	2	no	—
282	#106	—	2	1	no	—
283	#106	—	2	1	no	—
284	#106	—	2	2	no	—
285	#106	—	2	2	no	—
286	#106	—	2	1	no	—
287	#106	—	2	1	no	—
288	#106	—	47	1	no	—
289	#106	—	47	1	no	—
290	#106	—	94	2	no	—
291	#106	—	94	2	no	—
292	#106	—	94	2	no	—
293	C₄⋊C₈	—	94	2	no	—
294	C₄⋊C₈	—	94	2	no	—
295	#104	—	47	4	no	—
296	#103	—	47	2	no	—
297	#103	—	94	2	no	—
298	#102	—	47	2	no	—
299	#102	—	94	2	no	—
300	#97	—	94	1	no	—
301	C₄×C₈	—	2	2	no	—
302	C₄×C₈	—	94	1	no	—
303	#41	—	47	2	no	—
304	#41	—	47	2	no	—
305	#41	—	94	1	no	—
306	#41	—	94	1	no	—
307	C₂×Q₈	—	47	1	no	—
308	C₂×Q₈	—	188	2	no	—
309	C₂×D₄	—	47	1	no	—
310	C₂×D₄	—	94	1	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	94	1	no	—
315	C₂²×C₄	—	188	2	no	—
316	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
317	M₁₆	—	47	2	no	—
318	M₁₆	—	47	2	no	—
319	M₁₆	—	94	1	no	—
320	M₁₆	—	94	1	no	—
321	M₁₆	—	94	4	no	—
322	M₁₆	—	94	2	no	—
323	M₁₆	—	94	2	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	94	1	no	—
327	C₂×C₈	—	94	1	no	—
328	C₂×C₈	—	94	4	no	—
329	C₂×C₈	—	94	2	no	—
330	C₂×C₈	#10625350	1	2	yes	—
331	C₂×C₈	D₄×D₄₇	1	2	yes	—
332	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
333	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
334	C₂×C₈	Q₈×D₄₇	1	2	yes	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	8	4	no	—
340	C₄⋊C₄	—	47	2	no	—
341	C₄⋊C₄	—	47	2	no	—
342	C₄⋊C₄	—	47	2	no	—
343	C₄⋊C₄	—	188	2	no	—
344	C₄⋊C₄	—	188	2	no	—
345	C₄⋊C₄	—	188	2	no	—
346	C₄⋊C₄	—	188	2	no	—
347	C₂²⋊C₄	—	94	2	no	—
348	C₂²⋊C₄	—	94	1	no	—
349	C₂²⋊C₄	—	188	2	no	—
350	C₄²	—	2	1	no	—
351	C₄²	—	94	1	no	—
352	C₄²	—	94	2	no	—
353	C₄²	—	188	2	no	—
354	C₄²	Q₈×D₄₇	1	2	yes	—
355	C₂³	—	94	1	no	—
356	C₂³	#13094069	1	1	yes	characteristic
357	Q₈	—	47	2	no	—
358	Q₈	—	94	1	no	—
359	Q₈	—	752	2	no	—
360	D₄	—	47	2	no	—
361	D₄	—	94	2	no	—
362	D₄	—	94	2	no	—
363	D₄	—	94	1	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	8	2	no	—
369	C₂×C₄	—	8	2	no	—
370	C₂×C₄	—	94	1	no	—
371	C₂×C₄	—	94	2	no	—
372	C₂×C₄	—	94	2	no	—
373	C₂×C₄	—	94	2	no	—
374	C₂×C₄	—	94	2	no	—
375	C₂×C₄	—	94	2	no	—
376	C₂×C₄	—	752	2	no	—
377	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
378	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
379	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
380	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
381	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
382	C₂×C₄	#13094072	1	1	yes	characteristic
383	C₈	—	2	2	no	—
384	C₈	—	94	2	no	—
385	C₈	—	94	2	no	—
386	C₈	—	188	2	no	—
387	C₈	#13094002	1	2	yes	—
388	C₈	#13094002	1	2	yes	—
389	C₂²	—	2	2	no	—
390	C₂²	—	94	2	no	—
391	C₂²	—	188	2	no	—
392	C₂²	#15689750	1	1	yes	characteristic
393	C₂²	#15689758	1	1	yes	characteristic
394	C₂²	#15689828	1	1	yes	characteristic
395	C₄	—	2	4	no	—
396	C₄	—	8	2	no	—
397	C₄	—	8	2	no	—
398	C₄	—	94	2	no	—
399	C₄	—	188	2	no	—
400	C₄	—	188	2	no	—
401	C₄	—	188	2	no	—
402	C₄	C₂×#13094002	1	1	yes	characteristic
403	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
404	C₄	#15689774	1	1	yes	characteristic
405	C₄	#15689788	1	1	yes	characteristic
406	C₂	—	2	1	no	—
407	C₂	—	2	1	no	—
408	C₂	—	188	2	no	—
409	C₂	#20247093	1	1	yes	characteristic
410	C₂	#20255181	1	2	yes	—

	subgroup	G/H
center	C₂²	#15689758
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1844	C₂
socle	C₂²×C₄₇	D₄²
prime radical	C₂×#10625350	C₂³
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#26081
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1844

Complements

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

N	H ≅ G/N	type
1. #20253916	C₂	semidirect
4. #20244899	C₂	semidirect
5. #20244733	C₂	semidirect
6. #20244166	C₂	semidirect
7. #20244136	C₂	semidirect
11. C₄₇×#1844	C₂	semidirect
47. #13094030	Q₈	semidirect
77. #13093952	Q₈	semidirect
209. #1844	D₄₇	semidirect
250. C₄₇	#26081	semidirect

Maximal subgroups

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

subgroup	index
#20253916	2
#20253255	2
#20251979	2
#20244899	2
#20244733	2
#20244166	2
#20244136	2
#20243613	2
#20243448	2
#20243225	2
C₄₇×#1844	2
#26081	47