Subgroups — #26188333

Order 12032.

598 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253958	C₂	1	1	yes	maximal, characteristic
2	#20252544	C₂	1	2	yes	maximal, complemented
3	#20242301	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242296	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242230	C₂	1	2	yes	maximal, complemented
6	#20242223	C₂	1	2	yes	maximal, complemented
7	#20242211	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242206	C₂	1	1	yes	maximal, characteristic, complemented
9	#20241799	C₂	1	2	yes	maximal, complemented
10	#20241677	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄₇×#334	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689657	C₂²	1	2	yes	—
13	#15689525	C₄	1	1	yes	characteristic
14	#15689507	C₄	1	1	yes	characteristic
15	#15689478	C₄	1	2	yes	—
16	#15689364	C₄	1	1	yes	characteristic
17	#15689359	C₄	1	1	yes	characteristic
18	#15689352	C₄	1	2	yes	—
19	#15689250	C₂²	1	2	yes	—
20	#15689247	C₂²	1	2	yes	—
21	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
22	C₄×#10625325	C₂²	1	2	yes	—
23	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
24	#15689200	C₂²	1	2	yes	—
25	#15689194	—	2	2	no	—
26	#15689188	C₂²	1	2	yes	—
27	C₈×C₂²⋊D₄₇	—	2	2	no	—
28	#15689183	—	2	2	no	—
29	#15689176	C₂²	1	1	yes	characteristic
30	#15689148	—	2	2	no	—
31	C₄×#10625329	C₂²	1	1	yes	characteristic
32	#15688853	C₂²	1	2	yes	—
33	#15688853	C₂²	1	2	yes	—
34	#15688850	C₂²	1	2	yes	—
35	#15688850	C₂²	1	2	yes	—
36	#15688845	C₂²	1	1	yes	characteristic
37	#15688845	C₂²	1	1	yes	characteristic
38	#15688844	C₂²	1	1	yes	characteristic
39	#15688844	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
41	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
42	C₄₇×#334	C₂²	1	1	yes	characteristic
43	C₄₇×#334	C₂²	1	2	yes	—
44	C₄₇×#334	C₂²	1	1	yes	characteristic
45	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
46	C₂×#10625325	C₂×C₄	1	2	yes	—
47	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
48	C₂×#10625329	—	2	1	no	—
49	C₂×#10625324	—	2	2	no	—
50	C₂×C₈×D₄₇	—	2	2	no	—
51	#13094018	—	2	2	no	—
52	#13094016	C₂×C₄	1	2	yes	—
53	#13094012	C₂×C₄	1	2	yes	—
54	#13094007	—	2	2	no	—
55	#13094003	C₂×C₄	1	2	yes	—
56	#13093998	C₂×C₄	1	2	yes	—
57	C₄×Dic₉₄	C₂³	1	2	yes	—
58	C₄×D₁₈₈	C₂³	1	2	yes	—
59	C₄×C₂²⋊D₄₇	—	2	2	no	—
60	#13093981	—	2	2	no	—
61	#13093958	C₂×C₄	1	1	yes	characteristic
62	#13093958	C₂×C₄	1	1	yes	characteristic
63	#13093956	C₂×C₄	1	2	yes	—
64	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
65	Dic₄₇⋊C₈	—	2	2	no	—
66	Dic₄₇⋊C₈	—	2	2	no	—
67	#13093949	C₂×C₄	1	2	yes	—
68	#13093949	C₂×C₄	1	2	yes	—
69	#13093946	C₂×C₄	1	2	yes	—
70	#13093946	C₂×C₄	1	2	yes	—
71	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
72	(C₂×D₄₇)⋊C₈	—	2	2	no	—
73	(C₂×D₄₇)⋊C₈	—	2	2	no	—
74	#13093935	—	2	1	no	—
75	#13093934	—	2	2	no	—
76	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
80	C₈×Dic₄₇	—	2	2	no	—
81	C₂²×C₈×C₄₇	—	2	2	no	—
82	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
83	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
85	C₄⋊C₈×C₄₇	—	2	2	no	—
86	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
87	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
88	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
89	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
90	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
95	C₂×C₂²⋊D₄₇	—	2	2	no	—
96	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
97	C₂×C₄×D₄₇	—	2	2	no	—
98	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	—	2	2	no	—
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	—	4	1	no	—
103	C₂×C₈×C₄₇	—	8	2	no	—
104	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₄₇	#41	1	1	yes	characteristic
111	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
112	C₂²⋊Dic₄₇	—	2	2	no	—
113	(C₂×D₄₇)⋊C₄	—	2	2	no	—
114	(C₂×D₄₇)⋊C₄	—	2	2	no	—
115	(C₂×D₄₇)⋊C₄	—	2	2	no	—
116	(C₂×D₄₇)⋊C₄	—	2	2	no	—
117	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
118	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
119	Dic₄₇⋊C₄	—	2	2	no	—
120	Dic₄₇⋊C₄	—	2	2	no	—
121	Dic₄₇⋊C₄	—	2	2	no	—
122	Dic₄₇⋊C₄	—	2	2	no	—
123	C₄×Dic₄₇	—	2	2	no	—
124	#10625329	—	4	2	no	—
125	C₂×C₄₇⋊C₈	—	2	1	no	—
126	C₂×C₄₇⋊C₈	—	2	1	no	—
127	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
128	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
129	Dic₁₈₈	—	2	2	no	—
130	D₃₇₆	—	2	2	no	—
131	#10625325	—	4	2	no	—
132	#10625324	—	8	2	no	—
133	C₈×D₄₇	—	8	2	no	—
134	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
135	C₂²×D₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	2	1	no	—
138	C₂×C₄×C₄₇	—	2	1	no	—
139	C₂×C₄×C₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
146	C₂²⋊D₄₇	—	4	4	no	—
147	C₂×Dic₄₇	—	2	2	no	—
148	C₂×Dic₄₇	—	2	2	no	—
149	C₂×Dic₄₇	—	2	2	no	—
150	D₁₈₈	—	2	2	no	—
151	D₁₈₈	—	2	2	no	—
152	C₄×D₄₇	—	4	2	no	—
153	C₄×D₄₇	—	4	2	no	—
154	Dic₉₄	—	2	2	no	—
155	Dic₉₄	—	2	2	no	—
156	C₈×C₄₇	—	2	1	no	—
157	C₈×C₄₇	—	2	1	no	—
158	C₈×C₄₇	—	2	1	no	—
159	C₈×C₄₇	—	4	2	no	—
160	C₈×C₄₇	—	4	2	no	—
161	C₈×C₄₇	#131	1	2	yes	—
162	C₄₇⋊C₈	—	2	2	no	—
163	C₄₇⋊C₈	—	2	2	no	—
164	C₄₇⋊C₈	—	4	1	no	—
165	C₄₇⋊C₈	—	4	1	no	—
166	#18093	—	47	1	no	maximal, Sylow
167	C₂²×C₄₇	—	2	1	no	—
168	C₂²×C₄₇	—	2	1	no	—
169	C₂²×C₄₇	#444	1	2	yes	—
170	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×D₄₇	—	4	2	no	—
172	C₂×D₄₇	—	4	2	no	—
173	C₂×D₄₇	—	4	2	no	—
174	C₄×C₄₇	—	2	1	no	—
175	C₄×C₄₇	—	2	1	no	—
176	C₄×C₄₇	—	2	1	no	—
177	C₄×C₄₇	—	2	1	no	—
178	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
179	C₄×C₄₇	#568	1	1	yes	characteristic
180	C₄×C₄₇	#577	1	2	yes	—
181	Dic₄₇	—	4	2	no	—
182	Dic₄₇	—	4	2	no	—
183	Dic₄₇	—	4	2	no	—
184	#3168	—	47	1	no	—
185	#2994	—	47	2	no	—
186	#1594	—	47	1	no	—
187	#1593	—	47	1	no	—
188	#1583	—	47	2	no	—
189	#1582	—	47	2	no	—
190	#1580	—	47	1	no	—
191	#1579	—	47	1	no	—
192	C₂×#334	D₄₇	1	1	yes	characteristic, complemented
193	#1483	—	47	2	no	—
194	#1451	—	47	1	no	—
195	C₂×C₄₇	—	2	2	no	—
196	C₂×C₄₇	#2935	1	1	yes	characteristic
197	C₂×C₄₇	#2952	1	1	yes	characteristic
198	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
199	D₄₇	—	8	2	no	—
200	#529	—	47	2	no	—
201	#499	—	47	1	no	—
202	#495	—	47	1	no	—
203	#488	—	47	2	no	—
204	#467	—	47	1	no	—
205	#466	—	47	1	no	—
206	#465	—	47	2	no	—
207	#446	—	47	2	no	—
208	C₈×Q₈	—	47	2	no	—
209	C₄×Q₁₆	—	47	1	no	—
210	C₄×SD₁₆	—	47	2	no	—
211	C₄×D₈	—	47	1	no	—
212	#436	—	47	2	no	—
213	#435	—	94	2	no	—
214	C₈×D₄	—	47	2	no	—
215	C₈×D₄	—	94	2	no	—
216	#433	—	94	2	no	—
217	#431	—	47	1	no	—
218	#424	—	94	2	no	—
219	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
220	C₄×M₁₆	—	47	1	no	—
221	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
222	#334	C₂×D₄₇	1	1	yes	characteristic
223	#334	C₂×D₄₇	1	2	yes	—
224	#334	C₂×D₄₇	1	1	yes	characteristic
225	Q₈⋊C₈	—	47	2	no	—
226	Q₈⋊C₈	—	47	2	no	—
227	D₄⋊C₈	—	47	2	no	—
228	D₄⋊C₈	—	47	2	no	—
229	#322	—	47	1	no	—
230	#322	—	47	1	no	—
231	#322	—	47	1	no	—
232	#322	—	47	1	no	—
233	C₄₇	#18093	1	1	yes	characteristic, complemented, Sylow
234	C₂×Q₁₆	—	47	1	no	—
235	C₂×SD₁₆	—	47	2	no	—
236	C₂×D₈	—	47	1	no	—
237	C₂×M₁₆	—	94	2	no	—
238	C₂×M₁₆	—	94	1	no	—
239	C₂²×C₈	—	2	2	no	—
240	C₂²×C₈	—	94	2	no	—
241	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
242	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
243	#126	—	94	2	no	—
244	#125	—	47	2	no	—
245	#124	—	47	2	no	—
246	#123	—	94	2	no	—
247	C₂²⋊Q₈	—	47	2	no	—
248	#121	—	47	2	no	—
249	C₄×Q₈	—	47	2	no	—
250	C₄×D₄	—	47	2	no	—
251	C₄×D₄	—	94	2	no	—
252	#117	—	94	2	no	—
253	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
254	#107	—	47	1	no	—
255	#107	—	47	1	no	—
256	#106	—	47	2	no	—
257	C₄⋊C₈	—	2	2	no	—
258	C₄⋊C₈	—	47	2	no	—
259	C₄⋊C₈	—	94	2	no	—
260	C₄⋊C₈	—	94	2	no	—
261	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
262	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
263	#103	—	47	2	no	—
264	#103	—	47	2	no	—
265	#102	—	47	2	no	—
266	#102	—	47	2	no	—
267	C₂²⋊C₈	—	47	2	no	—
268	C₂²⋊C₈	—	94	2	no	—
269	C₂²⋊C₈	—	94	2	no	—
270	#97	—	94	2	no	—
271	#97	—	94	1	no	—
272	C₄×C₈	—	47	1	no	—
273	C₄×C₈	—	47	1	no	—
274	C₄×C₈	—	47	1	no	—
275	C₄×C₈	—	47	1	no	—
276	C₄×C₈	—	94	2	no	—
277	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
278	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
281	C₂×Q₈	—	47	2	no	—
282	C₂×D₄	—	47	2	no	—
283	C₂×D₄	—	94	2	no	—
284	C₂²×C₄	—	94	2	no	—
285	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
286	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
287	C₂²×C₄	#10625354	1	1	yes	characteristic
288	Q₁₆	—	94	2	no	—
289	SD₁₆	—	188	2	no	—
290	D₈	—	94	2	no	—
291	M₁₆	—	376	2	no	—
292	M₁₆	—	188	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	4	1	no	—
297	C₂×C₈	—	8	2	no	—
298	C₂×C₈	—	47	2	no	—
299	C₂×C₈	—	47	2	no	—
300	C₂×C₈	—	94	1	no	—
301	C₂×C₈	—	94	1	no	—
302	C₂×C₈	—	376	2	no	—
303	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
304	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
305	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
306	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
307	C₄⋊C₄	—	47	2	no	—
308	C₄⋊C₄	—	47	2	no	—
309	C₄⋊C₄	—	94	2	no	—
310	C₄⋊C₄	—	94	2	no	—
311	C₄⋊C₄	—	94	2	no	—
312	C₄⋊C₄	—	94	2	no	—
313	C₂²⋊C₄	—	94	2	no	—
314	C₂²⋊C₄	—	94	2	no	—
315	C₂²⋊C₄	—	94	2	no	—
316	C₂²⋊C₄	—	94	2	no	—
317	C₂²⋊C₄	—	94	2	no	—
318	C₄²	—	94	2	no	—
319	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
320	C₄²	D₄×D₄₇	1	1	yes	characteristic
321	C₄²	#10625354	1	1	yes	characteristic
322	C₄²	C₂³×D₄₇	1	1	yes	characteristic
323	C₂³	—	94	2	no	—
324	C₂³	#13093986	1	1	yes	characteristic
325	Q₈	—	94	2	no	—
326	Q₈	—	94	2	no	—
327	D₄	—	94	2	no	—
328	D₄	—	94	2	no	—
329	D₄	—	188	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	94	2	no	—
335	C₂×C₄	—	94	2	no	—
336	C₂×C₄	—	94	2	no	—
337	C₂×C₄	—	188	2	no	—
338	C₂×C₄	—	188	2	no	—
339	C₂×C₄	#13093986	1	1	yes	characteristic
340	C₂×C₄	#13093986	1	1	yes	characteristic
341	C₂×C₄	#13093986	1	1	yes	characteristic
342	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
344	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	4	2	no	—
349	C₈	—	4	2	no	—
350	C₈	—	94	2	no	—
351	C₈	—	94	2	no	—
352	C₈	—	188	1	no	—
353	C₈	—	188	1	no	—
354	C₈	#13094030	1	2	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	188	2	no	—
358	C₂²	—	188	2	no	—
359	C₂²	—	188	2	no	—
360	C₂²	#15689244	1	2	yes	—
361	C₂²	C₂×#13093986	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	188	2	no	—
367	C₄	—	188	2	no	—
368	C₄	—	188	2	no	—
369	C₄	C₂×#13094030	1	1	yes	characteristic
370	C₄	#15689889	1	1	yes	characteristic
371	C₄	#15689929	1	2	yes	—
372	C₂	—	2	2	no	—
373	C₂	—	376	2	no	—
374	C₂	#20252131	1	1	yes	characteristic
375	C₂	#20252271	1	1	yes	characteristic
376	C₂	C₂×#15689244	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#334	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#15689359	C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×#334	D₄₇
hypocenter	C₄₇	#18093
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
2. #20252544	C₂	semidirect
3. #20242301	C₂	semidirect
4. #20242296	C₂	semidirect
5. #20242230	C₂	semidirect
6. #20242223	C₂	semidirect
7. #20242211	C₂	semidirect
8. #20242206	C₂	semidirect
9. #20241799	C₂	semidirect
10. #20241677	C₂	semidirect
11. C₂×C₄₇×#334	C₂	semidirect
192. C₂×#334	D₄₇	semidirect
233. C₄₇	#18093	semidirect

Maximal subgroups

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

subgroup	index
#20253958	2
#20252544	2
#20242301	2
#20242296	2
#20242230	2
#20242223	2
#20242211	2
#20242206	2
#20241799	2
#20241677	2
C₂×C₄₇×#334	2
#18093	47