Subgroups — #26110158

Order 12032.

466 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	#20252496	C₂	1	1	yes	maximal, characteristic
2	#20243539	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13093950	C₂	1	1	yes	maximal, characteristic
4	#20241773	C₂	1	1	yes	maximal, characteristic
5	#20241239	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241220	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1565	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#10625350	C₂²	1	1	yes	characteristic
9	C₈×Dic₉₄	—	2	1	no	—
10	C₈×D₁₈₈	—	2	1	no	—
11	(C₂²⋊D₄₇)⋊C₈	—	2	2	no	—
12	#15689176	C₂²	1	1	yes	characteristic
13	#15689175	—	2	1	no	—
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₇⋊C₈	—	2	2	no	—
16	C₄⋊C₈×D₄₇	—	2	1	no	—
17	C₂×#13093950	C₄	1	1	yes	characteristic
18	C₂×#13093950	C₄	1	1	yes	characteristic
19	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
20	#15688884	C₄	1	1	yes	characteristic
21	#15688884	C₄	1	1	yes	characteristic
22	#15688853	—	2	1	no	—
23	#15688850	—	2	1	no	—
24	C₄₇×#431	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
27	C₄₇×#335	—	2	1	no	—
28	C₄₇×#334	—	2	1	no	—
29	C₂×#10625350	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
31	C₂×C₈×D₄₇	—	2	2	no	—
32	C₄×Dic₉₄	D₄	1	1	yes	characteristic
33	C₄×Dic₉₄	Q₈	1	1	yes	characteristic
34	C₄×D₁₈₈	D₄	1	1	yes	characteristic
35	C₄×D₁₈₈	Q₈	1	1	yes	characteristic
36	C₄×C₂²⋊D₄₇	—	2	1	no	—
37	C₄×C₂²⋊D₄₇	—	2	1	no	—
38	#13093983	C₂×C₄	1	1	yes	characteristic
39	#13093981	—	2	1	no	—
40	C₂×C₄×Dic₄₇	—	2	1	no	—
41	C₂×C₄×Dic₄₇	—	2	1	no	—
42	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
44	C₄²×D₄₇	—	2	1	no	—
45	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
46	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
47	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
48	Dic₄₇⋊C₈	—	2	2	no	—
49	Dic₄₇⋊C₈	—	4	4	no	—
50	#13093950	—	2	2	no	—
51	#13093950	C₈	1	4	yes	complemented
52	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
53	(C₂×D₄₇)⋊C₈	—	2	2	no	—
54	C₄×C₄₇⋊C₈	—	2	1	no	—
55	C₈×Dic₄₇	—	2	2	no	—
56	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₄₇	—	2	1	no	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₄₇	—	2	1	no	—
61	C₄⋊C₈×C₄₇	—	2	1	no	—
62	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
63	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
64	C₂²⋊C₈×C₄₇	—	2	1	no	—
65	#97×C₄₇	—	2	1	no	—
66	C₄×C₈×C₄₇	—	2	1	no	—
67	C₄×C₈×C₄₇	—	2	1	no	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
71	C₂×C₂²⋊D₄₇	—	2	1	no	—
72	C₂²×Dic₄₇	—	2	1	no	—
73	C₂²×Dic₄₇	—	2	1	no	—
74	#10625350	—	2	1	no	—
75	#10625350	—	2	1	no	—
76	#10625350	C₂×C₈	1	2	yes	—
77	#10625350	M₁₆	1	2	yes	—
78	C₂×D₁₈₈	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×D₄₇	—	2	1	no	—
80	C₂×C₄×D₄₇	—	2	1	no	—
81	C₂×C₄×D₄₇	—	2	1	no	—
82	C₂×Dic₉₄	C₄⋊C₄	1	1	yes	characteristic
83	M₁₆×C₄₇	—	2	1	no	—
84	M₁₆×C₄₇	—	2	1	no	—
85	M₁₆×C₄₇	—	2	1	no	—
86	M₁₆×C₄₇	C₂×C₈	1	2	yes	complemented
87	C₂×C₈×C₄₇	—	2	1	no	—
88	C₂×C₈×C₄₇	—	2	1	no	—
89	C₂×C₈×C₄₇	—	2	1	no	—
90	C₂×C₈×C₄₇	—	2	1	no	—
91	C₂×C₈×C₄₇	—	4	2	no	—
92	C₂×C₈×C₄₇	—	4	1	no	—
93	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
96	C₄²×C₄₇	#41	1	1	yes	characteristic
97	C₂²⋊Dic₄₇	—	2	1	no	—
98	(C₂×D₄₇)⋊C₄	—	2	1	no	—
99	(C₂×D₄₇)⋊C₄	—	2	1	no	—
100	C₄⋊Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
101	C₄⋊Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
102	Dic₄₇⋊C₄	—	2	1	no	—
103	Dic₄₇⋊C₄	—	2	1	no	—
104	C₄×Dic₄₇	—	2	2	no	—
105	C₄×Dic₄₇	—	2	2	no	—
106	C₄×Dic₄₇	—	2	1	no	—
107	C₄×Dic₄₇	—	2	1	no	—
108	C₄×Dic₄₇	—	2	1	no	—
109	C₄×Dic₄₇	—	2	1	no	—
110	C₄×Dic₄₇	—	2	1	no	—
111	C₄×Dic₄₇	—	2	1	no	—
112	C₄×Dic₄₇	—	4	2	no	—
113	C₄×Dic₄₇	C₂×C₈	1	2	yes	complemented
114	C₂×C₄₇⋊C₈	—	2	2	no	—
115	C₂×C₄₇⋊C₈	—	4	2	no	—
116	C₂×C₄₇⋊C₈	—	4	1	no	—
117	C₈×D₄₇	—	8	2	no	—
118	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
119	C₂²×D₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	1	no	—
122	C₂×C₄×C₄₇	—	4	1	no	—
123	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
132	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
133	C₂²⋊D₄₇	—	2	2	no	—
134	C₂²⋊D₄₇	—	2	2	no	—
135	C₂×Dic₄₇	—	2	2	no	—
136	C₂×Dic₄₇	—	2	2	no	—
137	C₂×Dic₄₇	—	2	1	no	—
138	C₂×Dic₄₇	—	2	2	no	—
139	C₂×Dic₄₇	—	2	2	no	—
140	C₂×Dic₄₇	—	2	1	no	—
141	C₂×Dic₄₇	—	2	1	no	—
142	C₂×Dic₄₇	—	4	1	no	—
143	C₂×Dic₄₇	—	4	1	no	—
144	D₁₈₈	—	2	1	no	—
145	D₁₈₈	C₄⋊C₈	1	2	yes	complemented
146	C₄×D₄₇	—	2	2	no	—
147	C₄×D₄₇	—	2	2	no	—
148	C₄×D₄₇	—	4	1	no	—
149	C₄×D₄₇	—	4	1	no	—
150	C₄×D₄₇	—	8	1	no	—
151	Dic₉₄	—	2	1	no	—
152	Dic₉₄	C₄⋊C₈	1	2	yes	complemented
153	C₈×C₄₇	—	2	2	no	—
154	C₈×C₄₇	—	4	1	no	—
155	C₈×C₄₇	—	4	1	no	—
156	C₈×C₄₇	—	4	1	no	—
157	C₈×C₄₇	—	8	1	no	—
158	C₄₇⋊C₈	—	4	2	no	—
159	C₄₇⋊C₈	—	8	1	no	—
160	#8839	—	47	1	no	maximal, Sylow
161	C₂²×C₄₇	—	2	1	no	—
162	C₂²×C₄₇	—	2	1	no	—
163	C₂²×C₄₇	#385	1	1	yes	characteristic
164	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
165	C₂²×C₄₇	#435	1	1	yes	characteristic
166	C₂×D₄₇	—	2	2	no	—
167	C₂×D₄₇	—	4	1	no	—
168	C₂×D₄₇	—	4	1	no	—
169	C₄×C₄₇	—	2	1	no	—
170	C₄×C₄₇	—	2	1	no	—
171	C₄×C₄₇	—	2	1	no	—
172	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
173	C₄×C₄₇	#424	1	1	yes	characteristic
174	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
175	C₄×C₄₇	#435	1	1	yes	characteristic
176	C₄×C₄₇	#453	1	1	yes	characteristic
177	C₄×C₄₇	#456	1	1	yes	characteristic
178	Dic₄₇	—	2	4	no	—
179	Dic₄₇	—	2	2	no	—
180	Dic₄₇	—	4	2	no	—
181	Dic₄₇	—	4	1	no	—
182	Dic₄₇	—	4	1	no	—
183	#2988	—	47	1	no	—
184	#1842	—	47	1	no	—
185	C₄×#104	—	47	1	no	—
186	#1565	D₄₇	1	1	yes	characteristic, complemented
187	#1477	—	47	1	no	—
188	#1283	—	47	1	no	—
189	#1276	—	47	1	no	—
190	C₂×C₄₇	—	2	1	no	—
191	C₂×C₄₇	—	2	1	no	—
192	C₂×C₄₇	#1850	1	1	yes	characteristic
193	C₂×C₄₇	#1859	1	1	yes	characteristic
194	C₂×C₄₇	#1861	1	1	yes	characteristic
195	D₄₇	—	4	2	no	—
196	C₄×#41	—	47	1	no	—
197	C₈×Q₈	—	94	1	no	—
198	C₈×D₄	—	94	1	no	—
199	C₈×D₄	—	94	2	no	—
200	#431	—	47	1	no	—
201	#431	—	94	1	no	—
202	#431	C₂×D₄₇	1	1	yes	characteristic
203	C₂×C₄⋊C₈	—	47	1	no	—
204	C₂×C₄⋊C₈	—	94	1	no	—
205	C₂×C₄⋊C₈	—	94	2	no	—
206	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
207	C₂×#104	—	47	1	no	—
208	C₂×#104	—	47	1	no	—
209	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
210	C₄³	—	47	1	no	—
211	#339	—	47	1	no	—
212	#339	—	47	1	no	—
213	#335	—	2	1	no	—
214	#334	—	2	1	no	—
215	Q₈⋊C₈	—	94	1	no	—
216	D₄⋊C₈	—	94	1	no	—
217	C₄₇	#8839	1	1	yes	characteristic, complemented, Sylow
218	C₂×#41	—	47	1	no	—
219	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
220	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
221	C₂²×C₈	—	2	1	no	—
222	C₂²×C₈	—	47	2	no	—
223	C₂²×C₈	—	94	2	no	—
224	C₄×Q₈	—	47	1	no	—
225	C₄×Q₈	—	47	1	no	—
226	C₄×D₄	—	47	1	no	—
227	C₄×D₄	—	47	1	no	—
228	C₄×D₄	—	94	1	no	—
229	C₄×D₄	—	94	1	no	—
230	#117	—	47	1	no	—
231	#117	—	94	1	no	—
232	C₂×C₄²	—	47	1	no	—
233	C₂×C₄²	—	47	1	no	—
234	C₂×C₄²	—	94	1	no	—
235	C₂×C₄²	—	94	1	no	—
236	C₂×C₄²	—	94	1	no	—
237	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	94	2	no	—
241	C₄⋊C₈	—	94	1	no	—
242	C₄⋊C₈	—	94	1	no	—
243	C₄⋊C₈	—	94	1	no	—
244	C₄⋊C₈	—	188	4	no	—
245	C₄⋊C₈	C₂²⋊D₄₇	1	1	yes	characteristic
246	C₄⋊C₈	C₂²⋊D₄₇	1	1	yes	characteristic
247	#104	—	47	4	no	—
248	#104	—	94	2	no	—
249	C₂²⋊C₈	—	2	1	no	—
250	C₂²⋊C₈	—	94	2	no	—
251	C₂²⋊C₈	—	94	1	no	—
252	#97	—	2	1	no	—
253	C₄×C₈	—	2	1	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	—	94	2	no	—
256	C₄×C₈	—	94	1	no	—
257	#41	—	47	2	no	—
258	#41	—	47	2	no	—
259	#41	—	94	1	no	—
260	#41	—	94	1	no	—
261	C₂×Q₈	—	47	1	no	—
262	C₂×D₄	—	47	1	no	—
263	C₂×D₄	—	94	1	no	—
264	C₂²×C₄	—	94	1	no	—
265	C₂²×C₄	—	94	1	no	—
266	C₂²×C₄	—	94	1	no	—
267	C₂²×C₄	—	94	1	no	—
268	C₂²×C₄	—	94	1	no	—
269	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
270	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
271	C₂²×C₄	#10625354	1	1	yes	characteristic
272	M₁₆	—	2	1	no	—
273	M₁₆	—	2	1	no	—
274	M₁₆	—	2	1	no	—
275	M₁₆	C₈×D₄₇	1	2	yes	complemented
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	4	2	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	94	2	no	—
283	C₂×C₈	—	376	2	no	—
284	C₂×C₈	—	188	1	no	—
285	C₂×C₈	—	188	2	no	—
286	C₄⋊C₄	—	47	1	no	—
287	C₄⋊C₄	—	47	1	no	—
288	C₄⋊C₄	—	94	1	no	—
289	C₄⋊C₄	—	94	1	no	—
290	C₂²⋊C₄	—	94	1	no	—
291	C₂²⋊C₄	—	94	1	no	—
292	C₂²⋊C₄	—	94	1	no	—
293	C₄²	—	47	2	no	—
294	C₄²	—	94	2	no	—
295	C₄²	—	94	2	no	—
296	C₄²	—	94	1	no	—
297	C₄²	—	94	1	no	—
298	C₄²	—	94	1	no	—
299	C₄²	—	94	1	no	—
300	C₄²	—	94	1	no	—
301	C₄²	—	94	1	no	—
302	C₄²	—	188	2	no	—
303	C₄²	#10625350	1	1	yes	characteristic
304	C₄²	D₄×D₄₇	1	1	yes	characteristic
305	C₄²	Q₈×D₄₇	1	1	yes	characteristic
306	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
307	C₂³	—	94	1	no	—
308	C₂³	#13093986	1	1	yes	characteristic
309	Q₈	—	47	2	no	—
310	Q₈	—	94	1	no	—
311	D₄	—	47	2	no	—
312	D₄	—	94	2	no	—
313	D₄	—	94	2	no	—
314	D₄	—	94	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	94	2	no	—
319	C₂×C₄	—	94	2	no	—
320	C₂×C₄	—	94	1	no	—
321	C₂×C₄	—	94	2	no	—
322	C₂×C₄	—	94	2	no	—
323	C₂×C₄	—	94	2	no	—
324	C₂×C₄	—	94	2	no	—
325	C₂×C₄	—	94	1	no	—
326	C₂×C₄	—	94	1	no	—
327	C₂×C₄	—	376	1	no	—
328	C₂×C₄	—	188	1	no	—
329	C₂×C₄	—	188	1	no	—
330	C₂×C₄	—	188	1	no	—
331	C₂×C₄	—	188	1	no	—
332	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
333	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
334	C₂×C₄	#13093992	1	1	yes	characteristic
335	C₂×C₄	#13094000	1	1	yes	characteristic
336	C₂×C₄	#13094002	1	1	yes	characteristic
337	C₂×C₄	#13094009	1	1	yes	characteristic
338	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
339	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
340	C₂×C₄	#13094030	1	1	yes	characteristic
341	C₂×C₄	#13094032	1	1	yes	characteristic
342	C₈	—	2	2	no	—
343	C₈	—	4	1	no	—
344	C₈	—	4	1	no	—
345	C₈	—	4	1	no	—
346	C₈	—	8	1	no	—
347	C₈	—	376	1	no	—
348	C₈	—	188	2	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	94	2	no	—
352	C₂²	—	188	1	no	—
353	C₂²	—	188	1	no	—
354	C₂²	#15688994	1	1	yes	characteristic
355	C₂²	#15689188	1	1	yes	characteristic
356	C₂²	#15689193	1	1	yes	characteristic
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	94	4	no	—
361	C₄	—	94	2	no	—
362	C₄	—	188	2	no	—
363	C₄	—	188	1	no	—
364	C₄	—	188	1	no	—
365	C₄	C₄⋊C₈×D₄₇	1	1	yes	characteristic
366	C₄	#15689148	1	1	yes	characteristic
367	C₄	C₈×C₂²⋊D₄₇	1	1	yes	characteristic
368	C₄	#15689192	1	1	yes	characteristic
369	C₄	#15689289	1	1	yes	characteristic
370	C₄	#15689304	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	188	2	no	—
374	C₂	#20243564	1	1	yes	characteristic
375	C₂	#20243611	1	1	yes	characteristic
376	C₂	#20243625	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093992
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1565	C₂
socle	C₂²×C₄₇	#385
prime radical	C₂×#10625350	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1565	D₄₇
hypocenter	C₄₇	#8839
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1565

Complements

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

N	H ≅ G/N	type
2. #20243539	C₂	semidirect
5. #20241239	C₂	semidirect
6. #20241220	C₂	semidirect
7. C₄₇×#1565	C₂	semidirect
51. #13093950	C₈	semidirect
86. M₁₆×C₄₇	C₂×C₈	semidirect
113. C₄×Dic₄₇	C₂×C₈	semidirect
145. D₁₈₈	C₄⋊C₈	semidirect
152. Dic₉₄	C₄⋊C₈	semidirect
186. #1565	D₄₇	semidirect
217. C₄₇	#8839	semidirect
275. M₁₆	C₈×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20252496	2
#20243539	2
C₄×#13093950	2
#20241773	2
#20241239	2
#20241220	2
C₄₇×#1565	2
#8839	47