Subgroups — #26207879

Order 12032.

638 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256362	C₂	1	1	yes	maximal, characteristic
2	#20255720	C₂	1	2	yes	maximal, complemented
3	#20242981	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242923	C₂	1	2	yes	maximal, complemented
5	#20242789	C₂	1	1	yes	maximal, characteristic, complemented
6	#20242748	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242514	C₂	1	2	yes	maximal, complemented
8	#20242460	C₂	1	2	yes	maximal, complemented
9	#20242273	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#10625337	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#3380	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689794	C₂²	1	2	yes	—
13	#15689500	C₂²	1	1	yes	characteristic
14	#15689498	—	2	1	no	—
15	#15689497	C₂²	1	2	yes	—
16	#15689471	C₂²	1	2	yes	—
17	#15689436	—	2	2	no	—
18	#15689415	C₂²	1	2	yes	—
19	#15689371	—	2	2	no	—
20	#15689323	C₂²	1	2	yes	—
21	#15689318	C₂²	1	2	yes	—
22	#15689317	—	2	2	no	—
23	#15689283	—	2	2	no	—
24	C₈×Dic₉₄	C₂²	1	1	yes	characteristic
25	C₄×#10625337	C₂²	1	1	yes	characteristic
26	#15689205	—	2	1	no	—
27	#15689201	C₂²	1	2	yes	—
28	#15689122	—	2	2	no	—
29	#15688876	C₂²	1	1	yes	characteristic
30	#15688875	C₂²	1	1	yes	characteristic
31	#15688872	C₂²	1	2	yes	—
32	#15688866	C₂²	1	2	yes	—
33	Q₈⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
34	#15688853	C₂²	1	1	yes	characteristic
35	D₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
36	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
37	C₄₇×#554	C₂²	1	2	yes	—
38	C₄₇×#500	C₂²	1	1	yes	characteristic
39	C₄₇×#499	—	2	1	no	—
40	C₄₇×#478	—	2	2	no	—
41	C₄₇×#476	C₂²	1	2	yes	—
42	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
43	C₄×D₈×C₄₇	—	2	1	no	—
44	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
45	C₂×#10625350	—	2	2	no	—
46	C₂×#10625337	D₄	1	1	yes	characteristic
47	C₂×#10625336	—	2	2	no	—
48	C₂×#10625336	—	4	2	no	—
49	C₂×D₃₇₆	—	2	1	no	—
50	C₂×D₃₇₆	—	2	1	no	—
51	C₂×#10625334	—	2	2	no	—
52	C₂×#10625329	—	2	2	no	—
53	#13094021	C₂³	1	2	yes	—
54	#13094020	—	2	2	no	—
55	#13093998	—	2	2	no	—
56	#13093997	—	4	2	no	—
57	C₄×Dic₉₄	C₂³	1	1	yes	characteristic
58	#13093986	—	2	2	no	—
59	#13093957	—	2	2	no	—
60	#13093957	D₄	1	1	yes	characteristic
61	C₄⋊(C₄₇⋊C₈)	C₂³	1	1	yes	characteristic
62	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
63	Dic₄₇⋊C₈	—	2	1	no	—
64	#13093948	D₄	1	1	yes	characteristic
65	#13093947	—	2	2	no	—
66	#13093947	D₄	1	1	yes	characteristic
67	#13093944	—	2	2	no	—
68	#13093944	—	2	2	no	—
69	#13093944	—	2	2	no	—
70	#13093944	—	2	2	no	—
71	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
72	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
74	C₈×Dic₄₇	—	2	1	no	—
75	C₂×D₈×C₄₇	D₄	1	1	yes	characteristic
76	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
77	#125×C₄₇	—	2	2	no	—
78	C₂²⋊Q₈×C₄₇	—	2	2	no	—
79	C₂²⋊Q₈×C₄₇	—	4	2	no	—
80	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
81	C₄×D₄×C₄₇	—	2	2	no	—
82	C₄×D₄×C₄₇	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
84	#107×C₄₇	—	2	1	no	—
85	#107×C₄₇	—	2	1	no	—
86	#107×C₄₇	D₄	1	1	yes	characteristic
87	#106×C₄₇	—	2	2	no	—
88	C₄⋊C₈×C₄₇	—	2	1	no	—
89	C₄⋊C₈×C₄₇	C₂³	1	1	yes	characteristic
90	#102×C₄₇	—	2	2	no	—
91	#102×C₄₇	D₄	1	2	yes	—
92	C₄×C₈×C₄₇	—	2	1	no	—
93	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₄₇	—	2	2	no	—
95	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₄₇	—	2	2	no	—
98	C₂²×C₄×C₄₇	—	4	2	no	—
99	C₂×C₂²⋊D₄₇	—	4	2	no	—
100	C₂×C₂²⋊D₄₇	—	4	2	no	—
101	#10625350	—	4	4	no	—
102	#10625350	—	8	2	no	—
103	C₂×D₁₈₈	—	2	2	no	—
104	C₂×D₁₈₈	—	2	2	no	—
105	C₂×D₁₈₈	—	2	2	no	—
106	C₂×C₄×D₄₇	—	4	2	no	—
107	C₂×Dic₉₄	C₂×D₄	1	1	yes	characteristic
108	D₈×C₄₇	—	2	2	no	—
109	C₂×C₈×C₄₇	—	2	1	no	—
110	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
112	C₄⋊C₄×C₄₇	—	2	1	no	—
113	C₄⋊C₄×C₄₇	—	2	2	no	—
114	C₄⋊C₄×C₄₇	—	2	2	no	—
115	C₄⋊C₄×C₄₇	—	4	2	no	—
116	C₄⋊C₄×C₄₇	—	4	2	no	—
117	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
118	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
119	C₂²⋊C₄×C₄₇	—	2	2	no	—
120	C₂²⋊C₄×C₄₇	—	4	2	no	—
121	C₄²×C₄₇	—	2	1	no	—
122	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
123	#10625337	—	2	2	no	—
124	#10625336	—	8	4	no	—
125	#10625336	—	8	2	no	—
126	#10625334	—	8	2	no	—
127	(C₂×D₄₇)⋊C₄	—	4	2	no	—
128	C₄⋊Dic₄₇	C₂×D₄	1	1	yes	characteristic
129	Dic₄₇⋊C₄	—	2	1	no	—
130	C₄×Dic₄₇	—	2	1	no	—
131	#10625329	—	8	2	no	—
132	C₂×C₄₇⋊C₈	—	2	1	no	—
133	C₂×C₄₇⋊C₈	—	2	1	no	—
134	C₂×C₄₇⋊C₈	—	2	2	no	—
135	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
137	D₃₇₆	—	4	2	no	—
138	D₃₇₆	—	8	1	no	—
139	C₂³×C₄₇	—	2	2	no	—
140	C₂²×D₄₇	—	4	2	no	—
141	Q₈×C₄₇	—	2	1	no	—
142	Q₈×C₄₇	—	2	1	no	—
143	Q₈×C₄₇	—	8	2	no	—
144	D₄×C₄₇	—	2	2	no	—
145	D₄×C₄₇	—	2	2	no	—
146	C₂×C₄×C₄₇	—	2	2	no	—
147	C₂×C₄×C₄₇	—	2	1	no	—
148	C₂×C₄×C₄₇	—	2	1	no	—
149	C₂×C₄×C₄₇	—	4	2	no	—
150	C₂×C₄×C₄₇	—	4	2	no	—
151	C₂×C₄×C₄₇	—	4	2	no	—
152	C₂×C₄×C₄₇	—	16	2	no	—
153	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²⋊D₄₇	—	8	4	no	—
157	C₂²⋊D₄₇	—	16	2	no	—
158	C₂×Dic₄₇	—	2	1	no	—
159	C₂×Dic₄₇	—	2	1	no	—
160	D₁₈₈	—	4	2	no	—
161	D₁₈₈	—	4	2	no	—
162	D₁₈₈	—	4	2	no	—
163	D₁₈₈	—	4	2	no	—
164	D₁₈₈	—	8	2	no	—
165	C₄×D₄₇	—	8	4	no	—
166	Dic₉₄	—	2	1	no	—
167	Dic₉₄	#135	1	2	yes	—
168	C₈×C₄₇	—	2	1	no	—
169	C₈×C₄₇	—	4	1	no	—
170	C₈×C₄₇	#137	1	2	yes	—
171	C₄₇⋊C₈	—	2	2	no	—
172	C₄₇⋊C₈	—	2	1	no	—
173	C₄₇⋊C₈	—	2	1	no	—
174	C₄₇⋊C₈	—	4	2	no	—
175	C₄₇⋊C₈	—	4	1	no	—
176	#19909	—	47	1	no	maximal, Sylow
177	C₂²×C₄₇	—	4	2	no	—
178	C₂²×C₄₇	—	4	2	no	—
179	C₂²×C₄₇	—	4	2	no	—
180	C₂²×C₄₇	D₄²	1	1	yes	characteristic
181	C₂×D₄₇	—	8	2	no	—
182	C₂×D₄₇	—	8	2	no	—
183	C₂×D₄₇	—	8	2	no	—
184	C₄×C₄₇	—	2	1	no	—
185	C₄×C₄₇	—	4	2	no	—
186	C₄×C₄₇	—	4	1	no	—
187	C₄×C₄₇	—	4	1	no	—
188	C₄×C₄₇	—	8	2	no	—
189	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
190	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
191	C₄×C₄₇	#576	1	1	yes	characteristic
192	C₄×C₄₇	#577	1	1	yes	characteristic
193	Dic₄₇	—	2	2	no	—
194	Dic₄₇	—	4	1	no	—
195	#3391	—	47	1	no	—
196	#3380	D₄₇	1	1	yes	characteristic, complemented
197	#3333	—	47	2	no	—
198	#1706	—	47	1	no	—
199	#1697	—	47	2	no	—
200	#1676	—	47	1	no	—
201	#1668	—	47	1	no	—
202	#1631	—	47	2	no	—
203	#1622	—	47	2	no	—
204	#1589	—	47	1	no	—
205	C₈×Q₁₆	—	47	1	no	—
206	C₂×C₄₇	—	8	2	no	—
207	C₂×C₄₇	#3278	1	1	yes	characteristic
208	C₂×C₄₇	#3280	1	1	yes	characteristic
209	C₂×C₄₇	#3292	1	1	yes	characteristic
210	D₄₇	—	16	2	no	—
211	#554	C₂×D₄₇	1	2	yes	—
212	#550	—	47	2	no	—
213	#500	C₂×D₄₇	1	1	yes	characteristic
214	#499	—	2	1	no	—
215	#493	—	47	1	no	—
216	#493	—	94	1	no	—
217	#492	—	47	2	no	—
218	#486	—	47	2	no	—
219	#480	—	94	2	no	—
220	#478	—	2	2	no	—
221	#477	—	47	2	no	—
222	#476	C₂×D₄₇	1	2	yes	—
223	#468	—	94	2	no	—
224	#460	—	47	2	no	—
225	#459	—	47	2	no	—
226	#459	—	94	2	no	—
227	#452	—	94	2	no	—
228	C₈×Q₈	—	47	1	no	—
229	C₈×Q₈	C₂×D₄₇	1	1	yes	characteristic
230	C₄×Q₁₆	—	47	1	no	—
231	C₄×D₈	—	2	1	no	—
232	C₄×D₈	—	94	1	no	—
233	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
234	#436	—	47	2	no	—
235	#417	—	94	2	no	—
236	#335	—	47	1	no	—
237	#335	—	47	1	no	—
238	#333	—	47	2	no	—
239	#331	—	47	2	no	—
240	Q₈⋊C₈	—	47	1	no	—
241	Q₈⋊C₈	—	47	1	no	—
242	D₄⋊C₈	—	47	2	no	—
243	C₈²	—	47	1	no	—
244	C₄₇	#19909	1	1	yes	characteristic, complemented, Sylow
245	C₂×#41	—	94	2	no	—
246	C₂×Q₁₆	—	47	1	no	—
247	C₂×SD₁₆	—	94	2	no	—
248	C₂×SD₁₆	—	188	2	no	—
249	C₂×D₈	—	94	2	no	—
250	C₂×D₈	—	94	1	no	—
251	C₂×D₈	—	94	1	no	—
252	C₂×D₈	C₂²⋊D₄₇	1	1	yes	characteristic
253	C₂×M₁₆	—	94	2	no	—
254	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
255	#127	—	47	2	no	—
256	#127	—	94	2	no	—
257	#125	—	2	2	no	—
258	C₂²⋊Q₈	—	2	2	no	—
259	C₂²⋊Q₈	—	4	2	no	—
260	#121	—	94	2	no	—
261	#121	—	188	2	no	—
262	C₄×Q₈	—	47	1	no	—
263	C₄×Q₈	C₂²×D₄₇	1	1	yes	characteristic
264	C₄×D₄	—	2	2	no	—
265	C₄×D₄	—	94	2	no	—
266	C₄×D₄	C₂²×D₄₇	1	2	yes	—
267	C₂×C₄⋊C₄	—	2	2	no	—
268	#107	—	2	1	no	—
269	#107	—	2	1	no	—
270	#107	—	47	1	no	—
271	#107	—	94	2	no	—
272	#107	C₂²⋊D₄₇	1	1	yes	characteristic
273	#106	—	2	2	no	—
274	C₄⋊C₈	—	2	1	no	—
275	C₄⋊C₈	—	47	2	no	—
276	C₄⋊C₈	—	47	1	no	—
277	C₄⋊C₈	—	94	1	no	—
278	C₄⋊C₈	C₂²×D₄₇	1	1	yes	characteristic
279	#103	—	47	1	no	—
280	#103	—	47	1	no	—
281	#103	—	94	2	no	—
282	#102	—	2	2	no	—
283	#102	—	94	2	no	—
284	#102	—	94	2	no	—
285	#102	—	94	2	no	—
286	#102	—	94	2	no	—
287	#102	C₂²⋊D₄₇	1	2	yes	—
288	C₂²⋊C₈	—	94	2	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	—	47	1	no	—
291	C₄×C₈	—	47	1	no	—
292	C₄×C₈	—	94	1	no	—
293	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
294	#41	—	376	2	no	—
295	#41	—	188	4	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	47	1	no	—
298	C₂×Q₈	D₄×D₄₇	1	1	yes	characteristic
299	C₂×D₄	—	94	2	no	—
300	C₂×D₄	—	94	2	no	—
301	C₂×D₄	—	94	2	no	—
302	C₂×D₄	—	188	2	no	—
303	C₂×D₄	—	188	2	no	—
304	C₂×D₄	C₂×C₂²⋊D₄₇	1	2	yes	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	4	2	no	—
307	C₂²×C₄	—	188	2	no	—
308	Q₁₆	—	94	2	no	—
309	SD₁₆	—	376	4	no	—
310	SD₁₆	—	376	2	no	—
311	D₈	—	2	2	no	—
312	D₈	—	376	1	no	—
313	D₈	—	376	2	no	—
314	D₈	—	188	2	no	—
315	M₁₆	—	376	2	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	47	1	no	—
318	C₂×C₈	—	47	1	no	—
319	C₂×C₈	—	94	1	no	—
320	C₂×C₈	—	94	2	no	—
321	C₂×C₈	—	94	1	no	—
322	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
323	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	4	2	no	—
328	C₄⋊C₄	—	4	2	no	—
329	C₄⋊C₄	—	47	1	no	—
330	C₄⋊C₄	—	94	1	no	—
331	C₄⋊C₄	D₄×D₄₇	1	1	yes	characteristic
332	C₄⋊C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
333	C₂²⋊C₄	—	2	2	no	—
334	C₂²⋊C₄	—	4	2	no	—
335	C₂²⋊C₄	—	188	2	no	—
336	C₄²	—	2	1	no	—
337	C₄²	—	94	1	no	—
338	C₄²	C₂³×D₄₇	1	1	yes	characteristic
339	C₂³	—	2	2	no	—
340	C₂³	—	188	2	no	—
341	Q₈	—	2	1	no	—
342	Q₈	—	2	1	no	—
343	Q₈	—	8	2	no	—
344	Q₈	—	47	2	no	—
345	Q₈	—	94	1	no	—
346	D₄	—	2	2	no	—
347	D₄	—	2	2	no	—
348	D₄	—	376	2	no	—
349	D₄	—	376	4	no	—
350	D₄	—	188	2	no	—
351	D₄	—	188	2	no	—
352	D₄	—	188	2	no	—
353	D₄	—	188	2	no	—
354	D₄	—	752	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	16	2	no	—
362	C₂×C₄	—	94	1	no	—
363	C₂×C₄	—	94	1	no	—
364	C₂×C₄	—	376	4	no	—
365	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
366	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
367	C₂×C₄	#13094069	1	1	yes	characteristic
368	C₈	—	2	1	no	—
369	C₈	—	4	1	no	—
370	C₈	—	94	2	no	—
371	C₈	—	94	1	no	—
372	C₈	—	94	1	no	—
373	C₈	—	188	2	no	—
374	C₈	—	188	1	no	—
375	C₈	#13094055	1	2	yes	—
376	C₂²	—	4	2	no	—
377	C₂²	—	4	2	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	376	2	no	—
380	C₂²	—	376	2	no	—
381	C₂²	—	376	2	no	—
382	C₂²	D₄×C₂²⋊D₄₇	1	1	yes	characteristic
383	C₄	—	2	1	no	—
384	C₄	—	4	2	no	—
385	C₄	—	4	1	no	—
386	C₄	—	4	1	no	—
387	C₄	—	8	2	no	—
388	C₄	—	94	2	no	—
389	C₄	—	188	1	no	—
390	C₄	#135×D₄₇	1	1	yes	characteristic
391	C₄	C₂×#13094055	1	1	yes	characteristic
392	C₄	#15689925	1	1	yes	characteristic
393	C₄	#15689931	1	1	yes	characteristic
394	C₂	—	8	2	no	—
395	C₂	—	752	2	no	—
396	C₂	#20255050	1	1	yes	characteristic
397	C₂	#20255067	1	1	yes	characteristic
398	C₂	#20255185	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₄₇
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3380	C₂
socle	C₂²×C₄₇	D₄²
prime radical	#20256362	C₂
hypercenter	#3380	D₄₇
hypocenter	C₄₇	#19909
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #3380

Complements

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

N	H ≅ G/N	type
2. #20255720	C₂	semidirect
3. #20242981	C₂	semidirect
4. #20242923	C₂	semidirect
5. #20242789	C₂	semidirect
6. #20242748	C₂	semidirect
7. #20242514	C₂	semidirect
8. #20242460	C₂	semidirect
9. #20242273	C₂	semidirect
10. C₈×#10625337	C₂	semidirect
11. C₄₇×#3380	C₂	semidirect
196. #3380	D₄₇	semidirect
244. C₄₇	#19909	semidirect

Maximal subgroups

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

subgroup	index
#20256362	2
#20255720	2
#20242981	2
#20242923	2
#20242789	2
#20242748	2
#20242514	2
#20242460	2
#20242273	2
C₈×#10625337	2
C₄₇×#3380	2
#19909	47