Subgroups — #26109151

Order 12032.

462 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13094032	C₂	1	1	yes	maximal, characteristic
2	#20242134	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15688873	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241776	C₂	1	1	yes	maximal, characteristic
5	#20241773	C₂	1	1	yes	maximal, characteristic
6	#20241239	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094032	C₄	1	1	yes	characteristic
9	C₂×#13094032	C₄	1	1	yes	characteristic
10	C₄×#10625350	C₂²	1	1	yes	characteristic
11	#15689176	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	#15689081	C₄	1	1	yes	characteristic
14	#15689081	C₄	1	1	yes	characteristic
15	#15689080	—	2	1	no	—
16	#15689080	—	2	1	no	—
17	C₄×#10625324	—	2	1	no	—
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₄₇	—	2	1	no	—
20	#15688875	—	2	1	no	—
21	#15688873	—	2	1	no	—
22	#15688873	—	2	1	no	—
23	#15688873	—	2	1	no	—
24	#15688855	—	2	1	no	—
25	#15688853	—	2	1	no	—
26	#15688852	—	2	1	no	—
27	#15688850	—	2	1	no	—
28	C₄₇×#431	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×#10625350	C₂×C₄	1	1	yes	characteristic
32	#13094032	—	2	2	no	—
33	#13094032	—	2	2	no	—
34	#13094032	—	2	2	no	—
35	#13094032	C₈	1	4	yes	complemented
36	C₂×#10625324	—	2	1	no	—
37	C₂×#10625324	—	2	1	no	—
38	C₂²×C₄₇⋊C₈	—	2	1	no	—
39	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×D₄₇	—	2	1	no	—
42	C₂×C₈×D₄₇	—	2	1	no	—
43	C₄×Dic₉₄	D₄	1	1	yes	characteristic
44	C₄×Dic₉₄	Q₈	1	1	yes	characteristic
45	C₄×D₁₈₈	D₄	1	1	yes	characteristic
46	C₄×D₁₈₈	Q₈	1	1	yes	characteristic
47	C₄×C₂²⋊D₄₇	—	2	1	no	—
48	C₄×C₂²⋊D₄₇	—	2	1	no	—
49	#13093983	C₂×C₄	1	1	yes	characteristic
50	#13093981	—	2	1	no	—
51	C₄²×D₄₇	—	2	1	no	—
52	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
53	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
54	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
55	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
56	#13093934	—	2	1	no	—
57	#13093934	—	2	1	no	—
58	C₄×C₄₇⋊C₈	—	2	1	no	—
59	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
60	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
61	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
62	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
63	C₈×Dic₄₇	—	2	1	no	—
64	C₈×Dic₄₇	—	2	1	no	—
65	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
66	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₄₇	—	2	1	no	—
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₄₇	—	2	1	no	—
70	C₄⋊C₈×C₄₇	—	2	1	no	—
71	C₄⋊C₈×C₄₇	—	2	1	no	—
72	C₂²⋊C₈×C₄₇	—	2	1	no	—
73	#97×C₄₇	—	2	1	no	—
74	C₄×C₈×C₄₇	—	2	1	no	—
75	C₄×C₈×C₄₇	—	2	1	no	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
79	C₂×C₂²⋊D₄₇	—	2	1	no	—
80	#10625350	—	2	1	no	—
81	#10625350	—	2	1	no	—
82	#10625350	C₂×C₈	1	2	yes	—
83	#10625350	M₁₆	1	2	yes	—
84	C₂×D₁₈₈	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×D₄₇	—	2	1	no	—
86	C₂×C₄×D₄₇	—	2	1	no	—
87	C₂×C₄×D₄₇	—	2	1	no	—
88	C₂×Dic₉₄	C₄⋊C₄	1	1	yes	characteristic
89	M₁₆×C₄₇	—	2	1	no	—
90	M₁₆×C₄₇	—	2	1	no	—
91	M₁₆×C₄₇	C₂×C₈	1	2	yes	—
92	M₁₆×C₄₇	M₁₆	1	2	yes	—
93	C₂×C₈×C₄₇	—	2	1	no	—
94	C₂×C₈×C₄₇	—	2	1	no	—
95	C₂×C₈×C₄₇	—	4	1	no	—
96	C₂×C₈×C₄₇	—	4	1	no	—
97	C₂×C₈×C₄₇	—	8	1	no	—
98	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
102	C₂²⋊Dic₄₇	—	2	1	no	—
103	(C₂×D₄₇)⋊C₄	—	2	1	no	—
104	(C₂×D₄₇)⋊C₄	—	2	1	no	—
105	C₄⋊Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
106	C₄⋊Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
107	Dic₄₇⋊C₄	—	2	1	no	—
108	Dic₄₇⋊C₄	—	2	1	no	—
109	C₄×Dic₄₇	—	2	1	no	—
110	C₄×Dic₄₇	—	2	1	no	—
111	C₄×Dic₄₇	—	2	1	no	—
112	C₂×C₄₇⋊C₈	—	2	1	no	—
113	C₂×C₄₇⋊C₈	—	2	1	no	—
114	C₂×C₄₇⋊C₈	—	4	1	no	—
115	C₂×C₄₇⋊C₈	—	4	1	no	—
116	C₂×C₄₇⋊C₈	—	8	1	no	—
117	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
122	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
123	#10625324	—	2	4	no	—
124	#10625324	—	2	4	no	—
125	C₈×D₄₇	—	2	4	no	—
126	C₈×D₄₇	—	2	4	no	—
127	C₂³×C₄₇	#117	1	1	yes	characteristic
128	C₂²×D₄₇	—	2	1	no	—
129	C₂×C₄×C₄₇	—	2	1	no	—
130	C₂×C₄×C₄₇	—	2	1	no	—
131	C₂×C₄×C₄₇	—	4	1	no	—
132	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
135	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
142	C₂²⋊D₄₇	—	2	2	no	—
143	C₂²⋊D₄₇	—	2	2	no	—
144	C₂×Dic₄₇	—	2	1	no	—
145	C₂×Dic₄₇	—	2	1	no	—
146	C₂×Dic₄₇	—	2	1	no	—
147	D₁₈₈	—	2	1	no	—
148	D₁₈₈	C₄⋊C₈	1	2	yes	—
149	C₄×D₄₇	—	2	2	no	—
150	C₄×D₄₇	—	2	2	no	—
151	C₄×D₄₇	—	4	2	no	—
152	C₄×D₄₇	—	4	1	no	—
153	C₄×D₄₇	—	4	1	no	—
154	Dic₉₄	—	2	1	no	—
155	Dic₉₄	C₄⋊C₈	1	2	yes	—
156	C₈×C₄₇	—	2	2	no	—
157	C₈×C₄₇	—	2	2	no	—
158	C₈×C₄₇	—	8	1	no	—
159	C₈×C₄₇	—	8	1	no	—
160	C₄₇⋊C₈	—	2	1	no	—
161	C₄₇⋊C₈	—	2	1	no	—
162	C₄₇⋊C₈	—	8	1	no	—
163	C₄₇⋊C₈	—	8	1	no	—
164	C₄₇⋊C₈	C₂²⋊C₈	1	2	yes	—
165	C₄₇⋊C₈	C₂²⋊C₈	1	2	yes	—
166	#8604	—	47	1	no	maximal, Sylow
167	C₂²×C₄₇	—	2	1	no	—
168	C₂²×C₄₇	—	2	1	no	—
169	C₂²×C₄₇	#380	1	1	yes	characteristic
170	C₂²×C₄₇	#431	1	1	yes	characteristic
171	C₂²×C₄₇	#432	1	1	yes	characteristic
172	C₂×D₄₇	—	2	2	no	—
173	C₂×D₄₇	—	4	1	no	—
174	C₂×D₄₇	—	4	1	no	—
175	C₄×C₄₇	—	2	1	no	—
176	C₄×C₄₇	—	2	1	no	—
177	C₄×C₄₇	—	2	1	no	—
178	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
179	C₄×C₄₇	#407	1	1	yes	characteristic
180	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
181	C₄×C₄₇	#423	1	1	yes	characteristic
182	C₄×C₄₇	#471	1	1	yes	characteristic
183	C₄×C₄₇	#471	1	1	yes	characteristic
184	Dic₄₇	—	2	2	no	—
185	Dic₄₇	—	4	1	no	—
186	Dic₄₇	—	4	1	no	—
187	C₄×#131	—	47	1	no	—
188	#1564	—	47	1	no	—
189	C₂×#334	—	47	1	no	—
190	#1477	—	47	1	no	—
191	#1477	—	47	1	no	—
192	#1283	—	47	1	no	—
193	#1283	D₄₇	1	1	yes	characteristic, complemented
194	C₂×C₄₇	—	2	1	no	—
195	C₂×C₄₇	—	2	1	no	—
196	C₂×C₄₇	#1797	1	1	yes	characteristic
197	C₂×C₄₇	#1814	1	1	yes	characteristic
198	C₂×C₄₇	#1815	1	1	yes	characteristic
199	D₄₇	—	4	2	no	—
200	C₂×#131	—	47	1	no	—
201	C₂×#131	—	47	1	no	—
202	C₄×#41	—	47	1	no	—
203	#431	—	47	1	no	—
204	#431	C₂×D₄₇	1	1	yes	characteristic
205	C₂×C₄⋊C₈	—	47	1	no	—
206	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
207	#405	—	47	1	no	—
208	#405	—	47	1	no	—
209	#405	—	94	1	no	—
210	#405	—	94	1	no	—
211	C₄×M₁₆	—	94	1	no	—
212	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
213	C₂×C₄×C₈	—	47	1	no	—
214	C₂×C₄×C₈	—	94	1	no	—
215	#335	—	94	1	no	—
216	#334	—	94	1	no	—
217	#334	—	94	1	no	—
218	#334	—	94	1	no	—
219	Q₈⋊C₈	—	94	1	no	—
220	Q₈⋊C₈	—	94	1	no	—
221	D₄⋊C₈	—	94	1	no	—
222	D₄⋊C₈	—	94	1	no	—
223	C₄₇	#8604	1	1	yes	characteristic, complemented, Sylow
224	C₂×#41	—	47	1	no	—
225	#131	—	47	4	no	—
226	#131	—	94	2	no	—
227	#131	—	94	2	no	—
228	#131	—	94	2	no	—
229	C₂×M₁₆	—	94	1	no	—
230	C₂×M₁₆	—	94	1	no	—
231	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
232	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
233	C₂²×C₈	—	2	1	no	—
234	C₂²×C₈	—	47	1	no	—
235	C₂²×C₈	—	47	1	no	—
236	C₂²×C₈	—	94	1	no	—
237	C₂²×C₈	—	94	1	no	—
238	C₂²×C₈	—	94	1	no	—
239	C₄×Q₈	—	47	1	no	—
240	C₄×Q₈	—	47	1	no	—
241	C₄×D₄	—	47	1	no	—
242	C₄×D₄	—	47	1	no	—
243	C₄×D₄	—	94	1	no	—
244	C₄×D₄	—	94	1	no	—
245	#117	—	47	1	no	—
246	#117	—	94	1	no	—
247	C₂×C₄²	—	94	1	no	—
248	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	2	1	no	—
252	C₄⋊C₈	—	94	1	no	—
253	C₄⋊C₈	—	94	1	no	—
254	C₄⋊C₈	—	94	1	no	—
255	C₂²⋊C₈	—	2	1	no	—
256	C₂²⋊C₈	—	94	1	no	—
257	#97	—	2	1	no	—
258	#97	—	94	1	no	—
259	#97	—	94	1	no	—
260	C₄×C₈	—	2	1	no	—
261	C₄×C₈	—	2	1	no	—
262	C₄×C₈	—	47	1	no	—
263	C₄×C₈	—	47	1	no	—
264	C₄×C₈	—	47	1	no	—
265	C₄×C₈	—	47	1	no	—
266	C₄×C₈	—	94	1	no	—
267	C₄×C₈	—	94	1	no	—
268	C₄×C₈	—	94	1	no	—
269	#41	—	47	2	no	—
270	#41	—	47	2	no	—
271	#41	—	94	1	no	—
272	#41	—	94	1	no	—
273	C₂×Q₈	—	47	1	no	—
274	C₂×D₄	—	47	1	no	—
275	C₂×D₄	—	94	1	no	—
276	C₂²×C₄	—	94	1	no	—
277	C₂²×C₄	—	94	1	no	—
278	C₂²×C₄	—	94	1	no	—
279	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
280	C₂²×C₄	#10625350	1	1	yes	characteristic
281	C₂²×C₄	#10625350	1	1	yes	characteristic
282	M₁₆	—	2	1	no	—
283	M₁₆	—	2	1	no	—
284	M₁₆	—	94	4	no	—
285	M₁₆	—	94	4	no	—
286	M₁₆	C₈×D₄₇	1	2	yes	—
287	M₁₆	#10625324	1	2	yes	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	4	1	no	—
291	C₂×C₈	—	4	1	no	—
292	C₂×C₈	—	8	1	no	—
293	C₂×C₈	—	47	1	no	—
294	C₂×C₈	—	47	1	no	—
295	C₂×C₈	—	47	2	no	—
296	C₂×C₈	—	47	2	no	—
297	C₂×C₈	—	47	1	no	—
298	C₂×C₈	—	47	1	no	—
299	C₂×C₈	—	94	4	no	—
300	C₂×C₈	—	94	4	no	—
301	C₂×C₈	—	94	1	no	—
302	C₂×C₈	—	94	1	no	—
303	C₂×C₈	—	376	1	no	—
304	C₂×C₈	—	188	1	no	—
305	C₂×C₈	—	188	1	no	—
306	C₄⋊C₄	—	47	1	no	—
307	C₄⋊C₄	—	47	1	no	—
308	C₄⋊C₄	—	94	1	no	—
309	C₄⋊C₄	—	94	1	no	—
310	C₂²⋊C₄	—	94	1	no	—
311	C₂²⋊C₄	—	94	1	no	—
312	C₂²⋊C₄	—	94	1	no	—
313	C₄²	—	94	1	no	—
314	C₄²	—	94	1	no	—
315	C₄²	—	94	1	no	—
316	C₄²	D₄×D₄₇	1	1	yes	characteristic
317	C₄²	D₄×D₄₇	1	1	yes	characteristic
318	C₄²	D₄×D₄₇	1	1	yes	characteristic
319	C₄²	Q₈×D₄₇	1	1	yes	characteristic
320	C₂³	—	94	1	no	—
321	C₂³	#13093981	1	1	yes	characteristic
322	Q₈	—	47	2	no	—
323	Q₈	—	94	1	no	—
324	D₄	—	47	2	no	—
325	D₄	—	94	2	no	—
326	D₄	—	94	2	no	—
327	D₄	—	94	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	94	1	no	—
332	C₂×C₄	—	94	2	no	—
333	C₂×C₄	—	94	2	no	—
334	C₂×C₄	—	94	1	no	—
335	C₂×C₄	—	94	1	no	—
336	C₂×C₄	—	188	2	no	—
337	C₂×C₄	—	188	1	no	—
338	C₂×C₄	—	188	1	no	—
339	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
340	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
341	C₂×C₄	#13093997	1	1	yes	characteristic
342	C₂×C₄	#13093997	1	1	yes	characteristic
343	C₂×C₄	#13094002	1	1	yes	characteristic
344	C₂×C₄	#13094002	1	1	yes	characteristic
345	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
346	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
347	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
348	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
349	C₈	—	2	2	no	—
350	C₈	—	2	2	no	—
351	C₈	—	8	1	no	—
352	C₈	—	8	1	no	—
353	C₈	—	47	2	no	—
354	C₈	—	47	2	no	—
355	C₈	—	94	1	no	—
356	C₈	—	94	1	no	—
357	C₈	—	376	1	no	—
358	C₈	—	376	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	94	2	no	—
362	C₂²	—	188	1	no	—
363	C₂²	—	188	1	no	—
364	C₂²	#15688982	1	1	yes	characteristic
365	C₂²	#15689174	1	1	yes	characteristic
366	C₂²	#15689178	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	94	2	no	—
371	C₄	—	188	1	no	—
372	C₄	—	188	1	no	—
373	C₄	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
374	C₄	#15689085	1	1	yes	characteristic
375	C₄	C₄⋊C₈×D₄₇	1	1	yes	characteristic
376	C₄	#15689145	1	1	yes	characteristic
377	C₄	#15689385	1	1	yes	characteristic
378	C₄	#15689389	1	1	yes	characteristic
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	—	188	2	no	—
382	C₂	#20243367	1	1	yes	characteristic
383	C₂	#20243441	1	1	yes	characteristic
384	C₂	#20243446	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093997
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1283	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂×#10625350	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1283	D₄₇
hypocenter	C₄₇	#8604
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1283

Complements

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

N	H ≅ G/N	type
2. #20242134	C₂	semidirect
3. C₂×#15688873	C₂	semidirect
6. #20241239	C₂	semidirect
7. C₄₇×#1283	C₂	semidirect
35. #13094032	C₈	semidirect
193. #1283	D₄₇	semidirect
223. C₄₇	#8604	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13094032	2
#20242134	2
C₂×#15688873	2
#20241776	2
#20241773	2
#20241239	2
C₄₇×#1283	2
#8604	47