Subgroups — #26186434

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	#20253423	C₂	1	1	yes	maximal, characteristic
2	#20252483	C₂	1	1	yes	maximal, characteristic
3	#20252468	C₂	1	1	yes	maximal, characteristic
4	#20242262	C₂	1	2	yes	maximal, complemented
5	#20242241	C₂	1	2	yes	maximal, complemented
6	#20242204	C₂	1	2	yes	maximal, complemented
7	#20242192	C₂	1	2	yes	maximal, complemented
8	#20242166	C₂	1	1	yes	maximal, characteristic
9	#20241836	C₂	1	1	yes	maximal, characteristic
10	#20241685	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1485	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
13	#15689456	C₄	1	2	yes	—
14	#15689415	C₄	1	2	yes	—
15	#15689330	C₄	1	2	yes	—
16	#15689278	C₄	1	2	yes	—
17	#15689254	—	2	2	no	—
18	#15689254	C₂²	1	2	yes	—
19	Q₈×C₄₇⋊C₈	C₂²	1	2	yes	—
20	C₈×Dic₉₄	—	2	2	no	—
21	#15689247	C₂²	1	2	yes	—
22	#15689247	C₂²	1	2	yes	—
23	C₄×#10625337	C₂²	1	2	yes	—
24	C₄×#10625337	C₂²	1	2	yes	—
25	#15689184	—	2	2	no	—
26	#15689180	C₂²	1	1	yes	characteristic
27	#15689176	C₂²	1	1	yes	characteristic
28	#15689176	C₂²	1	1	yes	characteristic
29	#15689173	—	2	2	no	—
30	#15688875	C₂²	1	2	yes	—
31	#15688873	C₂²	1	2	yes	—
32	#15688855	C₂²	1	2	yes	—
33	#15688855	C₂²	1	2	yes	—
34	#15688845	C₂²	1	1	yes	characteristic
35	#15688845	C₂²	1	1	yes	characteristic
36	#15688844	C₂²	1	1	yes	characteristic
37	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
38	C₄₇×#533	C₂²	1	1	yes	characteristic
39	C₄₇×#431	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
41	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	—
42	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	—
43	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
44	C₂×#10625337	C₂×C₄	1	2	yes	—
45	C₂×#10625337	C₂×C₄	1	2	yes	—
46	C₄×Dic₉₄	—	2	1	no	—
47	C₄×Dic₉₄	—	2	1	no	—
48	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
49	C₄×Dic₉₄	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×Dic₄₇⋊C₄	—	2	1	no	—
52	C₂×C₄×Dic₄₇	—	2	1	no	—
53	#13093957	C₂×C₄	1	2	yes	—
54	#13093957	C₂×C₄	1	2	yes	—
55	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
56	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
57	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
58	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
59	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
60	Dic₄₇⋊C₈	—	2	2	no	—
61	Dic₄₇⋊C₈	—	2	2	no	—
62	#13093948	C₂×C₄	1	2	yes	—
63	#13093948	C₂×C₄	1	2	yes	—
64	#13093947	C₂×C₄	1	2	yes	—
65	#13093947	C₂×C₄	1	2	yes	—
66	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
67	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
68	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
69	#13093935	—	2	1	no	—
70	#13093935	—	2	2	no	—
71	C₄×C₄₇⋊C₈	—	2	2	no	—
72	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₈×Dic₄₇	—	2	2	no	—
77	C₈×Dic₄₇	—	2	2	no	—
78	C₂×M₁₆×C₄₇	—	2	1	no	—
79	C₄⋊Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
80	C₄⋊Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
81	#125×C₄₇	—	2	1	no	—
82	C₂²⋊Q₈×C₄₇	—	2	1	no	—
83	C₂²⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
84	C₄×Q₈×C₄₇	—	2	1	no	—
85	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
86	#117×C₄₇	—	2	1	no	—
87	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
89	C₂²⋊C₈×C₄₇	—	2	1	no	—
90	#97×C₄₇	—	2	1	no	—
91	C₄×C₈×C₄₇	—	2	1	no	—
92	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₄₇	—	2	1	no	—
95	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
99	C₂²×Dic₄₇	—	2	1	no	—
100	C₂²×Dic₄₇	—	2	1	no	—
101	C₂×Dic₉₄	—	2	1	no	—
102	C₂×Dic₉₄	—	2	4	no	—
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	2	no	—
107	C₂×C₈×C₄₇	—	2	1	no	—
108	C₄⋊C₄×C₄₇	—	2	1	no	—
109	C₄⋊C₄×C₄₇	—	2	1	no	—
110	C₄⋊C₄×C₄₇	—	2	1	no	—
111	C₄⋊C₄×C₄₇	—	2	2	no	—
112	C₄⋊C₄×C₄₇	—	2	2	no	—
113	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
114	C₂²⋊C₄×C₄₇	—	2	2	no	—
115	C₄²×C₄₇	—	2	2	no	—
116	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₄₇	#41	1	1	yes	characteristic
119	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
120	#10625337	—	2	4	no	—
121	#10625337	—	2	4	no	—
122	C₄⋊Dic₄₇	—	2	1	no	—
123	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
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	C₂×C₄₇⋊C₈	—	2	2	no	—
129	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
130	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
131	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
132	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
133	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
134	Q₈×C₄₇	—	2	2	no	—
135	Q₈×C₄₇	—	2	2	no	—
136	Q₈×C₄₇	—	8	1	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	2	2	no	—
139	C₂×C₄×C₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	—	2	2	no	—
141	C₂×C₄×C₄₇	—	4	1	no	—
142	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₄₇	—	2	4	no	—
149	C₂×Dic₄₇	—	2	2	no	—
150	C₂×Dic₄₇	—	2	2	no	—
151	C₂×Dic₄₇	—	8	1	no	—
152	Dic₉₄	—	2	4	no	—
153	Dic₉₄	—	2	2	no	—
154	Dic₉₄	#131	1	4	yes	—
155	C₈×C₄₇	—	2	2	no	—
156	C₈×C₄₇	—	4	1	no	—
157	C₈×C₄₇	—	4	2	no	—
158	C₄₇⋊C₈	—	2	2	no	—
159	C₄₇⋊C₈	—	2	2	no	—
160	C₄₇⋊C₈	—	2	2	no	—
161	C₄₇⋊C₈	—	2	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	—	4	2	no	—
173	C₄×C₄₇	—	4	2	no	—
174	C₄×C₄₇	C₂×#131	1	2	yes	—
175	C₄×C₄₇	#578	1	2	yes	—
176	Dic₄₇	—	2	4	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	D₄₇	1	1	yes	characteristic
188	#1453	—	47	1	no	—
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	C₂×D₄₇	1	1	yes	characteristic
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	—	47	1	no	—
211	#431	—	94	2	no	—
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	C₂×D₄₇	1	2	yes	—
219	Q₈⋊C₈	C₂×D₄₇	1	2	yes	—
220	#322	—	47	1	no	—
221	#322	—	47	1	no	—
222	#322	—	47	1	no	—
223	C₈²	—	47	1	no	—
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	C₄×D₄₇	1	1	yes	characteristic
230	C₄⋊Q₈	C₄×D₄₇	1	1	yes	characteristic
231	#125	—	2	1	no	—
232	C₂²⋊Q₈	—	2	1	no	—
233	C₂²⋊Q₈	C₄×D₄₇	1	2	yes	—
234	C₄×Q₈	—	2	1	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	C₂²×D₄₇	1	2	yes	—
240	#117	—	2	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	—	47	2	no	—
250	C₄⋊C₈	—	94	2	no	—
251	C₄⋊C₈	—	94	2	no	—
252	C₄⋊C₈	—	94	2	no	—
253	C₄⋊C₈	—	94	2	no	—
254	C₄⋊C₈	C₂²×D₄₇	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	—	2	1	no	—
260	C₂²⋊C₈	—	47	2	no	—
261	C₂²⋊C₈	—	47	2	no	—
262	C₂²⋊C₈	—	94	1	no	—
263	#97	—	2	1	no	—
264	#97	—	94	2	no	—
265	#97	—	94	1	no	—
266	C₄×C₈	—	2	1	no	—
267	C₄×C₈	—	47	1	no	—
268	C₄×C₈	—	47	1	no	—
269	C₄×C₈	—	47	1	no	—
270	C₄×C₈	—	47	1	no	—
271	C₄×C₈	—	94	2	no	—
272	C₄×C₈	—	94	2	no	—
273	C₄×C₈	—	94	2	no	—
274	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	47	2	no	—
278	C₂×Q₈	—	94	4	no	—
279	C₂×Q₈	—	94	1	no	—
280	C₂×Q₈	C₂×C₄×D₄₇	1	2	yes	—
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	D₄×D₄₇	1	1	yes	characteristic
285	C₂²×C₄	#10625352	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	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	47	2	no	—
293	C₂×C₈	—	47	2	no	—
294	C₂×C₈	—	47	2	no	—
295	C₂×C₈	—	47	2	no	—
296	C₂×C₈	—	94	2	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	47	2	no	—
303	C₄⋊C₄	—	94	2	no	—
304	C₄⋊C₄	—	94	1	no	—
305	C₄⋊C₄	—	94	2	no	—
306	C₄⋊C₄	C₂×C₄×D₄₇	1	2	yes	—
307	C₂²⋊C₄	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	94	2	no	—
310	C₄²	—	94	2	no	—
311	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
312	C₄²	D₄×D₄₇	1	1	yes	characteristic
313	C₄²	#10625352	1	1	yes	characteristic
314	C₄²	C₂³×D₄₇	1	1	yes	characteristic
315	C₂³	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
316	Q₈	—	2	2	no	—
317	Q₈	—	2	2	no	—
318	Q₈	—	8	1	no	—
319	Q₈	—	47	4	no	—
320	Q₈	—	94	4	no	—
321	Q₈	—	94	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	94	4	no	—
328	C₂×C₄	—	94	2	no	—
329	C₂×C₄	—	94	2	no	—
330	C₂×C₄	—	376	1	no	—
331	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
332	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
333	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
335	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
336	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
337	C₈	—	2	2	no	—
338	C₈	—	4	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	94	2	no	—
341	C₈	—	94	2	no	—
342	C₈	—	94	2	no	—
343	C₈	—	94	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×(C₂²⋊D₄₇)⋊C₄	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	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	94	4	no	—
356	C₄	—	188	2	no	—
357	C₄	#131×D₄₇	1	2	yes	—
358	C₄	#15689939	1	2	yes	—
359	C₂	—	4	1	no	—
360	C₂	#20252085	1	1	yes	characteristic
361	C₂	#20252252	1	1	yes	characteristic
362	C₂	#20252294	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1485	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1485	D₄₇
hypocenter	C₄₇	#17925
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1485

Complements

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

N	H ≅ G/N	type
4. #20242262	C₂	semidirect
5. #20242241	C₂	semidirect
6. #20242204	C₂	semidirect
7. #20242192	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
#20253423	2
#20252483	2
#20252468	2
#20242262	2
#20242241	2
#20242204	2
#20242192	2
#20242166	2
#20241836	2
#20241685	2
C₄₇×#1485	2
#17925	47