Subgroups — #26374016

Order 12032.

518 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256362	C₂	1	1	yes	maximal, characteristic
2	#20256281	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245628	C₂	1	2	yes	maximal, complemented
4	#20245613	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245548	C₂	1	2	yes	maximal, complemented
6	#20245429	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245427	C₂	1	2	yes	maximal, complemented
8	#20245293	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245251	C₂	1	2	yes	maximal, complemented
10	C₄×#13093968	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2224	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689848	—	2	1	no	—
13	#15689794	—	2	1	no	—
14	C₂×#13093969	C₂²	1	2	yes	—
15	C₂×#13093968	C₂²	1	1	yes	characteristic
16	#15689524	C₂²	1	2	yes	—
17	#15689500	C₂²	1	1	yes	characteristic
18	#15689498	C₂²	1	2	yes	—
19	#15689429	—	2	2	no	—
20	#15689405	—	2	1	no	—
21	#15689318	—	2	1	no	—
22	#15689317	—	2	2	no	—
23	C₈×Dic₉₄	C₂²	1	1	yes	characteristic
24	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
25	#15689217	C₂²	1	2	yes	—
26	#15689205	C₂²	1	2	yes	—
27	#15688950	C₂²	1	1	yes	characteristic
28	Dic₄₇⋊C₁₆	C₂²	1	2	yes	—
29	#15688926	C₂²	1	1	yes	characteristic
30	#15688925	C₂²	1	2	yes	—
31	#15688924	C₂²	1	1	yes	characteristic
32	#15688923	C₂²	1	2	yes	—
33	#15688923	C₂²	1	2	yes	—
34	#15688898	C₂²	1	2	yes	—
35	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₄₇×#500	C₂²	1	1	yes	characteristic
37	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#366	C₂²	1	2	yes	—
39	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₄₇×#357	C₂²	1	2	yes	—
41	C₂×#10625350	—	4	1	no	—
42	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
43	C₂×D₃₇₆	—	2	1	no	—
44	C₂×D₃₇₆	C₂³	1	2	yes	—
45	C₂×#10625334	—	2	2	no	—
46	#13094022	—	4	1	no	—
47	#13094021	—	2	1	no	—
48	#13094020	—	2	2	no	—
49	#13094015	—	2	2	no	—
50	#13094014	—	4	2	no	—
51	#13093998	—	4	1	no	—
52	#13093997	—	4	2	no	—
53	Q₈×Dic₄₇	—	4	1	no	—
54	C₄×Dic₉₄	—	2	1	no	—
55	C₄×Dic₉₄	D₄	1	1	yes	characteristic
56	#13093989	—	2	2	no	—
57	#13093986	—	2	2	no	—
58	#13093969	—	2	4	no	—
59	#13093968	—	2	2	no	—
60	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
62	#13093958	C₂³	1	1	yes	characteristic
63	#13093955	—	2	2	no	—
64	C₄⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
65	Dic₄₇⋊C₈	D₄	1	2	yes	—
66	#13093949	—	2	1	no	—
67	#13093947	—	2	2	no	—
68	#13093944	—	2	2	no	—
69	C₈×Dic₄₇	C₂³	1	2	yes	—
70	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₄₇	—	2	1	no	—
72	C₄×D₄×C₄₇	—	2	1	no	—
73	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
74	#107×C₄₇	—	2	1	no	—
75	#107×C₄₇	C₂³	1	2	yes	—
76	#107×C₄₇	C₂³	1	1	yes	characteristic
77	#102×C₄₇	—	2	1	no	—
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₄₇	—	4	1	no	—
80	C₂×D₄×C₄₇	—	2	1	no	—
81	C₂²×C₄×C₄₇	—	4	1	no	—
82	C₂×C₂²⋊D₄₇	—	4	2	no	—
83	C₂×C₂²⋊D₄₇	—	4	2	no	—
84	#10625350	—	8	4	no	—
85	C₂×D₁₈₈	—	2	2	no	—
86	C₂×D₁₈₈	—	4	1	no	—
87	C₂×D₁₈₈	—	4	1	no	—
88	C₂×C₄×D₄₇	—	4	2	no	—
89	C₂×Dic₉₄	—	2	1	no	—
90	C₂×Dic₉₄	D₈	1	1	yes	characteristic
91	D₈×C₄₇	—	2	1	no	—
92	D₈×C₄₇	—	2	1	no	—
93	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₄₇	—	2	2	no	—
96	C₄⋊C₄×C₄₇	—	2	1	no	—
97	C₄⋊C₄×C₄₇	—	4	1	no	—
98	C₂²⋊C₄×C₄₇	—	4	1	no	—
99	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
100	#10625334	—	8	2	no	—
101	(C₂×D₄₇)⋊C₄	—	4	2	no	—
102	C₄⋊Dic₄₇	—	2	1	no	—
103	C₄⋊Dic₄₇	—	4	2	no	—
104	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
105	Dic₄₇⋊C₄	—	4	1	no	—
106	Dic₄₇⋊C₄	—	4	2	no	—
107	Dic₄₇⋊C₄	—	4	2	no	—
108	Dic₄₇⋊C₄	D₈	1	2	yes	—
109	C₄×Dic₄₇	—	4	1	no	—
110	C₄×Dic₄₇	—	4	2	no	—
111	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
112	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
113	Dic₁₈₈	—	2	1	no	—
114	Dic₁₈₈	#41	1	2	yes	complemented
115	D₃₇₆	—	2	2	no	—
116	D₃₇₆	—	2	2	no	—
117	D₃₇₆	—	8	1	no	—
118	C₁₆×C₄₇	—	2	2	no	—
119	C₄₇⋊C₁₆	—	2	1	no	—
120	C₄₇⋊C₁₆	—	2	1	no	—
121	C₂³×C₄₇	—	4	1	no	—
122	C₂²×D₄₇	—	4	2	no	—
123	Q₈×C₄₇	—	8	2	no	—
124	D₄×C₄₇	—	4	1	no	—
125	D₄×C₄₇	—	4	1	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	1	no	—
128	C₂×C₄×C₄₇	—	8	2	no	—
129	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
132	C₂²⋊D₄₇	—	8	4	no	—
133	C₂²⋊D₄₇	—	16	2	no	—
134	C₂×Dic₄₇	—	4	1	no	—
135	C₂×Dic₄₇	—	4	1	no	—
136	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
137	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
138	D₁₈₈	—	4	2	no	—
139	D₁₈₈	—	4	2	no	—
140	D₁₈₈	—	8	2	no	—
141	D₁₈₈	—	16	1	no	—
142	C₄×D₄₇	—	8	4	no	—
143	Dic₉₄	—	2	2	no	—
144	Dic₉₄	—	2	2	no	—
145	Dic₉₄	—	4	1	no	—
146	C₈×C₄₇	—	2	1	no	—
147	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
148	C₈×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
149	C₄₇⋊C₈	—	2	2	no	—
150	#33935	—	47	1	no	maximal, Sylow
151	C₂²×C₄₇	—	8	1	no	—
152	C₂²×C₄₇	—	8	1	no	—
153	C₂²×C₄₇	—	8	1	no	—
154	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₄₇	—	8	2	no	—
156	C₂×D₄₇	—	8	2	no	—
157	C₂×D₄₇	—	8	2	no	—
158	C₄×C₄₇	—	2	1	no	—
159	C₄×C₄₇	—	8	1	no	—
160	C₄×C₄₇	—	8	2	no	—
161	C₄×C₄₇	#509	1	2	yes	—
162	C₄×C₄₇	#548	1	1	yes	characteristic
163	C₄×C₄₇	#577	1	1	yes	characteristic
164	Dic₄₇	—	2	2	no	—
165	Dic₄₇	—	2	2	no	—
166	Dic₄₇	—	4	2	no	—
167	Dic₄₇	—	8	1	no	—
168	#3391	—	47	1	no	—
169	#3383	—	47	1	no	—
170	#2244	—	47	2	no	—
171	#2240	—	47	1	no	—
172	#2228	—	47	2	no	—
173	#2224	D₄₇	1	1	yes	characteristic, complemented
174	#2207	—	47	1	no	—
175	#2207	—	47	2	no	—
176	#2183	—	47	1	no	—
177	#2175	—	47	2	no	—
178	C₄×SD₃₂	—	47	1	no	—
179	C₂×C₄₇	—	16	1	no	—
180	C₂×C₄₇	#3358	1	1	yes	characteristic
181	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
182	C₂×C₄₇	#3416	1	1	yes	characteristic
183	D₄₇	—	16	2	no	—
184	#557	—	94	1	no	—
185	#550	—	94	1	no	—
186	C₂×SD₃₂	—	47	1	no	—
187	C₂×SD₃₂	—	47	2	no	—
188	#500	C₂×D₄₇	1	1	yes	characteristic
189	#499	—	47	2	no	—
190	#493	—	47	2	no	—
191	#493	—	47	1	no	—
192	#479	—	94	2	no	—
193	#475	—	94	1	no	—
194	#459	—	94	2	no	—
195	#459	—	94	1	no	—
196	C₈×Q₈	—	47	1	no	—
197	C₄×Q₁₆	—	47	2	no	—
198	C₄×Q₁₆	—	47	1	no	—
199	C₄×D₈	—	47	2	no	—
200	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
201	#367	—	47	1	no	—
202	#366	C₂×D₄₇	1	2	yes	—
203	C₄⋊C₁₆	—	47	2	no	—
204	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
205	#358	—	47	1	no	—
206	#357	—	47	1	no	—
207	#357	—	47	2	no	—
208	#357	—	47	2	no	—
209	#357	—	47	2	no	—
210	#357	C₂×D₄₇	1	2	yes	—
211	#346	—	47	2	no	—
212	C₄×C₁₆	—	47	1	no	—
213	C₄₇	#33935	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	188	1	no	—
215	C₂×Q₁₆	—	47	1	no	—
216	C₂×D₈	—	47	2	no	—
217	C₂×D₈	—	94	2	no	—
218	C₂×D₈	—	94	1	no	—
219	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	188	1	no	—
222	#127	—	94	2	no	—
223	#127	—	94	1	no	—
224	#125	—	94	2	no	—
225	#125	—	188	2	no	—
226	#121	—	188	2	no	—
227	#121	—	188	1	no	—
228	C₄×Q₈	—	47	1	no	—
229	C₄×Q₈	—	94	2	no	—
230	C₄×Q₈	—	94	1	no	—
231	C₄×Q₈	—	188	1	no	—
232	C₄×D₄	—	2	1	no	—
233	C₄×D₄	—	94	2	no	—
234	SD₃₂	—	94	4	no	—
235	SD₃₂	—	94	2	no	—
236	C₂×C₁₆	—	47	1	no	—
237	C₂×C₁₆	—	47	1	no	—
238	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
239	#107	—	2	1	no	—
240	#107	—	47	1	no	—
241	#107	C₂²×D₄₇	1	2	yes	—
242	#107	C₂²×D₄₇	1	1	yes	characteristic
243	#106	—	94	2	no	—
244	C₄⋊C₈	—	47	2	no	—
245	C₄⋊C₈	—	47	1	no	—
246	#103	—	94	2	no	—
247	#103	—	94	1	no	—
248	#102	—	2	1	no	—
249	#102	—	94	2	no	—
250	C₄×C₈	—	47	2	no	—
251	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
252	#41	—	376	4	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×Q₈	—	47	1	no	—
255	C₂×Q₈	—	94	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	94	2	no	—
258	C₂×D₄	—	188	2	no	—
259	C₂×D₄	—	188	2	no	—
260	C₂×D₄	—	188	1	no	—
261	C₂×D₄	—	188	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	188	2	no	—
264	Q₁₆	—	47	2	no	—
265	Q₁₆	—	94	1	no	—
266	D₈	—	2	1	no	—
267	D₈	—	2	1	no	—
268	D₈	—	94	2	no	—
269	D₈	—	94	2	no	—
270	D₈	—	376	1	no	—
271	D₈	—	376	2	no	—
272	C₂×C₈	—	47	2	no	—
273	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	47	2	no	—
279	C₄⋊C₄	—	47	1	no	—
280	C₄⋊C₄	—	94	1	no	—
281	C₄⋊C₄	—	188	2	no	—
282	C₄⋊C₄	—	188	2	no	—
283	C₄⋊C₄	—	188	2	no	—
284	C₄⋊C₄	—	188	1	no	—
285	C₂²⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	188	2	no	—
287	C₄²	—	47	2	no	—
288	C₄²	—	188	2	no	—
289	C₄²	—	188	1	no	—
290	C₄²	D₄×D₄₇	1	1	yes	characteristic
291	C₁₆	—	2	2	no	—
292	C₁₆	—	94	1	no	—
293	C₁₆	—	94	1	no	—
294	C₂³	—	4	1	no	—
295	C₂³	—	188	2	no	—
296	Q₈	—	8	2	no	—
297	Q₈	—	94	2	no	—
298	Q₈	—	94	2	no	—
299	Q₈	—	188	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	4	1	no	—
302	D₄	—	376	4	no	—
303	D₄	—	376	2	no	—
304	D₄	—	188	2	no	—
305	D₄	—	188	2	no	—
306	D₄	—	752	1	no	—
307	D₄	—	752	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	8	2	no	—
311	C₂×C₄	—	47	2	no	—
312	C₂×C₄	—	47	2	no	—
313	C₂×C₄	—	376	4	no	—
314	C₂×C₄	—	188	1	no	—
315	C₂×C₄	—	188	1	no	—
316	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
317	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
318	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	94	2	no	—
321	C₈	#41×D₄₇	1	1	yes	characteristic
322	C₈	#13094072	1	1	yes	characteristic
323	C₂²	—	8	1	no	—
324	C₂²	—	8	1	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	376	2	no	—
327	C₂²	—	376	2	no	—
328	C₂²	—	376	2	no	—
329	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	8	1	no	—
332	C₄	—	8	2	no	—
333	C₄	—	94	2	no	—
334	C₄	—	94	2	no	—
335	C₄	—	376	1	no	—
336	C₄	—	188	2	no	—
337	C₄	#15689557	1	2	yes	—
338	C₄	#15689783	1	1	yes	characteristic
339	C₄	#15689932	1	1	yes	characteristic
340	C₂	—	16	1	no	—
341	C₂	—	752	2	no	—
342	C₂	#20256022	1	1	yes	characteristic
343	C₂	C₂×#15689557	1	1	yes	characteristic
344	C₂	#20256533	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2224	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	#20256362	C₂
hypercenter	#2224	D₄₇
hypocenter	C₄₇	#33935
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2224

Complements

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

N	H ≅ G/N	type
2. #20256281	C₂	semidirect
3. #20245628	C₂	semidirect
4. #20245613	C₂	semidirect
5. #20245548	C₂	semidirect
6. #20245429	C₂	semidirect
7. #20245427	C₂	semidirect
8. #20245293	C₂	semidirect
9. #20245251	C₂	semidirect
10. C₄×#13093968	C₂	semidirect
11. C₄₇×#2224	C₂	semidirect
114. Dic₁₈₈	#41	semidirect
173. #2224	D₄₇	semidirect
213. C₄₇	#33935	semidirect

Maximal subgroups

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

subgroup	index
#20256362	2
#20256281	2
#20245628	2
#20245613	2
#20245548	2
#20245429	2
#20245427	2
#20245293	2
#20245251	2
C₄×#13093968	2
C₄₇×#2224	2
#33935	47