Subgroups — #26186433

Order 12032.

586 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253428	C₂	1	1	yes	maximal, characteristic
2	#20252482	C₂	1	1	yes	maximal, characteristic
3	#20252468	C₂	1	1	yes	maximal, characteristic
4	#20242261	C₂	1	2	yes	maximal, complemented
5	#20242240	C₂	1	2	yes	maximal, complemented
6	#20242208	C₂	1	2	yes	maximal, complemented
7	C₈×Dic₁₈₈	C₂	1	2	yes	maximal, complemented
8	#20242168	C₂	1	1	yes	maximal, characteristic
9	#20241836	C₂	1	1	yes	maximal, characteristic
10	#20241810	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1453	C₂	1	1	yes	maximal, characteristic
12	#15689680	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
14	#15689459	C₄	1	2	yes	—
15	#15689419	C₄	1	2	yes	—
16	#15689334	C₄	1	2	yes	—
17	#15689281	C₄	1	2	yes	—
18	#15689253	C₂²	1	2	yes	—
19	#15689252	—	2	2	no	—
20	C₈×Dic₉₄	C₂²	1	2	yes	—
21	C₈×Dic₉₄	C₂²	1	2	yes	—
22	C₈×Dic₉₄	C₂²	1	2	yes	—
23	#15689247	—	2	2	no	—
24	C₄×Dic₁₈₈	C₂²	1	2	yes	—
25	C₄×Dic₁₈₈	C₂²	1	2	yes	—
26	#15689181	—	2	2	no	—
27	#15689180	C₂²	1	1	yes	characteristic
28	#15689176	C₂²	1	1	yes	characteristic
29	#15689173	—	2	2	no	—
30	#15688876	C₂²	1	2	yes	—
31	#15688874	C₂²	1	2	yes	—
32	#15688855	C₂²	1	2	yes	—
33	#15688855	C₂²	1	2	yes	—
34	#15688855	C₂²	1	2	yes	—
35	#15688855	C₂²	1	2	yes	—
36	C₄₇×#431	C₂²	1	1	yes	characteristic
37	C₄₇×#431	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×#322	C₂²	1	1	yes	characteristic
40	C₄₇×#322	C₂²	1	1	yes	characteristic
41	C₄₇×#322	C₂²	1	1	yes	characteristic
42	C₈²×C₄₇	C₂²	1	1	yes	characteristic
43	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
44	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
45	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
46	#13094024	C₂×C₄	1	1	yes	characteristic
47	#13094024	C₂×C₄	1	1	yes	characteristic
48	#13094016	—	2	1	no	—
49	#13094003	—	2	1	no	—
50	#13094003	C₂×C₄	1	2	yes	—
51	C₄×Dic₉₄	—	2	1	no	—
52	C₄×Dic₉₄	—	2	1	no	—
53	C₄×Dic₉₄	—	2	1	no	—
54	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
55	C₄×Dic₉₄	C₂³	1	2	yes	—
56	C₄×Dic₉₄	C₂³	1	2	yes	—
57	#13093983	—	2	1	no	—
58	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂×Dic₄₇⋊C₄	—	2	1	no	—
60	C₂×C₄×Dic₄₇	—	2	1	no	—
61	#13093958	C₂×C₄	1	2	yes	—
62	#13093958	C₂×C₄	1	2	yes	—
63	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
64	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
65	Dic₄₇⋊C₈	—	2	2	no	—
66	Dic₄₇⋊C₈	—	2	2	no	—
67	Dic₄₇⋊C₈	—	2	2	no	—
68	Dic₄₇⋊C₈	—	2	2	no	—
69	#13093949	C₂×C₄	1	2	yes	—
70	#13093949	C₂×C₄	1	2	yes	—
71	#13093949	C₂×C₄	1	2	yes	—
72	#13093949	C₂×C₄	1	2	yes	—
73	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
74	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
75	#13093935	—	2	1	no	—
76	#13093934	—	2	2	no	—
77	C₄×C₄₇⋊C₈	—	2	1	no	—
78	C₈×Dic₄₇	—	2	2	no	—
79	C₈×Dic₄₇	—	2	2	no	—
80	C₈×Dic₄₇	—	2	2	no	—
81	C₂×M₁₆×C₄₇	—	2	1	no	—
82	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
84	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
85	C₂²⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
86	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
87	#97×C₄₇	—	2	1	no	—
88	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
97	C₂²×Dic₄₇	—	2	1	no	—
98	C₂²×Dic₄₇	—	2	1	no	—
99	C₂×Dic₉₄	—	2	1	no	—
100	C₂×Dic₉₄	—	2	4	no	—
101	C₂×Dic₉₄	—	2	1	no	—
102	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
103	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
104	M₁₆×C₄₇	—	4	2	no	—
105	C₂×C₈×C₄₇	—	2	1	no	—
106	C₂×C₈×C₄₇	—	2	1	no	—
107	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
111	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₄₇	#41	1	1	yes	characteristic
114	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
115	C₂²⋊Dic₄₇	—	2	2	no	—
116	C₄⋊Dic₄₇	—	2	1	no	—
117	C₄⋊Dic₄₇	—	2	1	no	—
118	C₄⋊Dic₄₇	—	2	1	no	—
119	C₄⋊Dic₄₇	—	2	1	no	—
120	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
121	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
122	Dic₄₇⋊C₄	—	2	2	no	—
123	Dic₄₇⋊C₄	—	2	2	no	—
124	Dic₄₇⋊C₄	—	2	2	no	—
125	Dic₄₇⋊C₄	—	2	2	no	—
126	C₄×Dic₄₇	—	2	2	no	—
127	C₄×Dic₄₇	—	2	2	no	—
128	C₄×Dic₄₇	—	2	2	no	—
129	C₂×C₄₇⋊C₈	—	2	2	no	—
130	C₂×C₄₇⋊C₈	—	2	2	no	—
131	Dic₁₈₈	—	2	4	no	—
132	Dic₁₈₈	—	2	4	no	—
133	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	4	1	no	—
137	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
143	C₂×Dic₄₇	—	2	4	no	—
144	C₂×Dic₄₇	—	2	2	no	—
145	C₂×Dic₄₇	—	2	2	no	—
146	C₂×Dic₄₇	—	2	2	no	—
147	C₂×Dic₄₇	—	2	2	no	—
148	C₂×Dic₄₇	—	8	1	no	—
149	Dic₉₄	—	2	4	no	—
150	Dic₉₄	—	2	2	no	—
151	Dic₉₄	—	2	2	no	—
152	Dic₉₄	—	2	2	no	—
153	Dic₉₄	—	8	1	no	—
154	Dic₉₄	#131	1	4	yes	—
155	C₈×C₄₇	—	2	2	no	—
156	C₈×C₄₇	—	2	2	no	—
157	C₈×C₄₇	—	2	2	no	—
158	C₈×C₄₇	—	2	2	no	—
159	C₈×C₄₇	—	2	2	no	—
160	C₈×C₄₇	—	4	1	no	—
161	C₄₇⋊C₈	—	4	2	no	—
162	C₄₇⋊C₈	—	4	2	no	—
163	#17925	—	47	1	no	maximal, Sylow
164	C₂²×C₄₇	—	2	1	no	—
165	C₂²×C₄₇	—	2	1	no	—
166	C₂²×C₄₇	—	2	1	no	—
167	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
168	C₄×C₄₇	—	2	1	no	—
169	C₄×C₄₇	—	2	1	no	—
170	C₄×C₄₇	—	2	1	no	—
171	C₄×C₄₇	—	2	1	no	—
172	C₄×C₄₇	C₂×#131	1	2	yes	—
173	C₄×C₄₇	#578	1	2	yes	—
174	Dic₄₇	—	2	4	no	—
175	Dic₄₇	—	4	2	no	—
176	Dic₄₇	—	4	2	no	—
177	Dic₄₇	—	4	2	no	—
178	#3104	—	47	1	no	—
179	#2985	—	47	1	no	—
180	#2982	—	47	1	no	—
181	#1587	—	47	2	no	—
182	#1584	—	47	2	no	—
183	#1579	—	47	2	no	—
184	C₈×Q₁₆	—	47	2	no	—
185	#1572	—	47	1	no	—
186	#1491	—	47	1	no	—
187	#1485	—	47	1	no	—
188	#1453	D₄₇	1	1	yes	characteristic
189	C₂×C₄₇	—	4	1	no	—
190	C₂×C₄₇	#2929	1	1	yes	characteristic
191	C₂×C₄₇	#2950	1	1	yes	characteristic
192	C₂×C₄₇	#2956	1	1	yes	characteristic
193	#533	—	47	1	no	—
194	C₂×C₄×Q₈	—	47	1	no	—
195	#484	—	47	2	no	—
196	#477	—	47	2	no	—
197	#462	—	47	2	no	—
198	#451	—	47	2	no	—
199	#446	—	47	2	no	—
200	#446	—	94	2	no	—
201	C₈×Q₈	—	47	2	no	—
202	C₈×Q₈	—	47	2	no	—
203	C₈×Q₈	—	47	2	no	—
204	C₈×Q₈	—	94	2	no	—
205	C₄×Q₁₆	—	47	2	no	—
206	C₄×Q₁₆	—	47	2	no	—
207	#433	—	94	2	no	—
208	#432	—	47	1	no	—
209	#431	—	47	1	no	—
210	#431	—	94	2	no	—
211	#431	C₂×D₄₇	1	1	yes	characteristic
212	#431	C₂×D₄₇	1	1	yes	characteristic
213	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
214	#335	—	47	2	no	—
215	#334	—	47	2	no	—
216	Q₈⋊C₈	—	47	2	no	—
217	Q₈⋊C₈	—	47	2	no	—
218	Q₈⋊C₈	—	47	2	no	—
219	Q₈⋊C₈	—	47	2	no	—
220	#322	C₂×D₄₇	1	1	yes	characteristic
221	#322	C₂×D₄₇	1	1	yes	characteristic
222	#322	C₂×D₄₇	1	1	yes	characteristic
223	C₈²	C₂×D₄₇	1	1	yes	characteristic
224	C₄₇	#17925	1	1	yes	characteristic, complemented, Sylow
225	C₂²×Q₈	—	47	1	no	—
226	C₂×Q₁₆	—	47	2	no	—
227	C₂×Q₁₆	—	47	2	no	—
228	C₂×M₁₆	—	2	1	no	—
229	C₄⋊Q₈	—	47	1	no	—
230	C₄⋊Q₈	—	47	1	no	—
231	#125	—	94	1	no	—
232	C₂²⋊Q₈	—	47	2	no	—
233	C₂²⋊Q₈	—	94	1	no	—
234	C₄×Q₈	—	47	2	no	—
235	C₄×Q₈	—	47	2	no	—
236	C₄×Q₈	—	47	2	no	—
237	C₄×Q₈	—	94	1	no	—
238	C₄×Q₈	—	94	1	no	—
239	C₄×Q₈	—	94	1	no	—
240	#117	—	94	1	no	—
241	C₂×C₄⋊C₄	—	47	1	no	—
242	C₂×C₄⋊C₄	—	94	1	no	—
243	C₂×C₄²	—	94	1	no	—
244	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
245	#107	—	47	2	no	—
246	#107	—	47	2	no	—
247	C₄⋊C₈	—	47	2	no	—
248	C₄⋊C₈	—	47	2	no	—
249	C₄⋊C₈	—	94	2	no	—
250	C₄⋊C₈	—	94	2	no	—
251	C₄⋊C₈	—	94	2	no	—
252	C₄⋊C₈	—	94	2	no	—
253	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
254	C₄⋊C₈	D₁₈₈	1	2	yes	—
255	#103	—	47	2	no	—
256	#103	—	47	2	no	—
257	#103	—	47	2	no	—
258	#103	—	47	2	no	—
259	C₂²⋊C₈	—	94	1	no	—
260	C₂²⋊C₈	—	94	1	no	—
261	C₂²⋊C₈	C₄×D₄₇	1	2	yes	—
262	C₂²⋊C₈	D₁₈₈	1	2	yes	—
263	#97	—	2	1	no	—
264	#97	—	94	2	no	—
265	#97	—	94	1	no	—
266	C₄×C₈	—	94	2	no	—
267	C₄×C₈	—	94	2	no	—
268	C₄×C₈	—	94	2	no	—
269	C₄×C₈	—	94	1	no	—
270	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
271	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
272	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
276	C₂×Q₈	—	47	2	no	—
277	C₂×Q₈	—	47	2	no	—
278	C₂×Q₈	—	94	4	no	—
279	C₂×Q₈	—	94	1	no	—
280	C₂×Q₈	—	94	1	no	—
281	C₂²×C₄	—	94	1	no	—
282	C₂²×C₄	—	94	1	no	—
283	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
284	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
285	C₂²×C₄	#10625350	1	1	yes	characteristic
286	Q₁₆	—	94	4	no	—
287	Q₁₆	—	94	4	no	—
288	M₁₆	—	4	2	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	94	2	no	—
292	C₂×C₈	—	94	2	no	—
293	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
294	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
295	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
296	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
297	C₄⋊C₄	—	47	2	no	—
298	C₄⋊C₄	—	47	2	no	—
299	C₄⋊C₄	—	94	2	no	—
300	C₄⋊C₄	—	94	2	no	—
301	C₄⋊C₄	—	94	1	no	—
302	C₄⋊C₄	—	94	1	no	—
303	C₄⋊C₄	—	94	1	no	—
304	C₄⋊C₄	—	94	1	no	—
305	C₄⋊C₄	—	94	2	no	—
306	C₄⋊C₄	—	94	2	no	—
307	C₂²⋊C₄	—	94	2	no	—
308	C₄²	—	94	2	no	—
309	C₄²	—	94	2	no	—
310	C₄²	—	94	2	no	—
311	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
312	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
313	C₄²	#10625350	1	1	yes	characteristic
314	C₄²	C₂³×D₄₇	1	1	yes	characteristic
315	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
316	Q₈	—	47	4	no	—
317	Q₈	—	94	4	no	—
318	Q₈	—	94	2	no	—
319	Q₈	—	94	2	no	—
320	Q₈	—	94	2	no	—
321	Q₈	—	376	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	94	4	no	—
326	C₂×C₄	—	94	2	no	—
327	C₂×C₄	—	94	2	no	—
328	C₂×C₄	—	94	2	no	—
329	C₂×C₄	—	94	2	no	—
330	C₂×C₄	—	376	1	no	—
331	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
332	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
333	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
335	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
336	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
337	C₈	—	2	2	no	—
338	C₈	—	2	2	no	—
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	2	2	no	—
342	C₈	—	4	1	no	—
343	C₈	—	188	2	no	—
344	C₈	—	188	2	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	94	4	no	—
354	C₄	—	188	2	no	—
355	C₄	—	188	2	no	—
356	C₄	—	188	2	no	—
357	C₄	#131×D₄₇	1	2	yes	—
358	C₄	#15689940	1	2	yes	—
359	C₂	—	4	1	no	—
360	C₂	#20252084	1	1	yes	characteristic
361	C₂	#20252257	1	1	yes	characteristic
362	C₂	#20252299	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1453	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1453	D₄₇
hypocenter	C₄₇	#17925
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1453

Complements

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

N	H ≅ G/N	type
4. #20242261	C₂	semidirect
5. #20242240	C₂	semidirect
6. #20242208	C₂	semidirect
7. C₈×Dic₁₈₈	C₂	semidirect
224. C₄₇	#17925	semidirect

Maximal subgroups

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

subgroup	index
#20253428	2
#20252482	2
#20252468	2
#20242261	2
#20242240	2
#20242208	2
C₈×Dic₁₈₈	2
#20242168	2
#20241836	2
#20241810	2
C₄₇×#1453	2
#17925	47