Subgroups — #20257786

Order 6016.

1114 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689867	C₂	1	4	yes	maximal, complemented
2	#15689834	C₂	1	2	yes	maximal, complemented
3	#15689830	C₂	1	2	yes	maximal, complemented
4	#15689794	C₂	1	2	yes	maximal, complemented
5	#15689793	C₂	1	2	yes	maximal, complemented
6	#15689778	C₂	1	2	yes	maximal, complemented
7	#15689777	C₂	1	2	yes	maximal, complemented
8	#15689726	C₂	1	4	yes	maximal, complemented
9	#15689702	C₂	1	4	yes	maximal, complemented
10	#15689701	C₂	1	4	yes	maximal, complemented
11	#15689680	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689673	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄₇×#532	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#10625350	C₂²	1	4	yes	—
15	C₂×#10625350	C₂²	1	2	yes	—
16	C₂×#10625352	C₂²	1	4	yes	—
17	#13094024	C₂²	1	1	yes	characteristic
18	#13094024	C₂²	1	1	yes	characteristic
19	#13094024	C₂²	1	1	yes	characteristic
20	#13094021	C₂²	1	1	yes	characteristic
21	#13094020	C₂²	1	4	yes	—
22	#13094016	C₂²	1	2	yes	—
23	#13094015	C₂²	1	4	yes	—
24	#13094012	C₂²	1	2	yes	—
25	#13094008	C₂²	1	8	yes	—
26	#13094006	C₂²	1	8	yes	—
27	#13094004	C₂²	1	4	yes	—
28	#13094003	C₂²	1	4	yes	—
29	#13094003	C₂²	1	2	yes	—
30	#13094003	C₂²	1	2	yes	—
31	#13094001	C₂²	1	8	yes	—
32	#13094000	C₂²	1	8	yes	—
33	#13093998	C₂²	1	4	yes	—
34	#13093998	C₂²	1	2	yes	—
35	#13093998	C₂²	1	2	yes	—
36	#13093997	C₂²	1	8	yes	—
37	#13093996	C₂²	1	8	yes	—
38	#13093996	C₂²	1	8	yes	—
39	#13093995	C₂²	1	4	yes	—
40	#13093995	C₂²	1	8	yes	—
41	C₄×Dic₉₄	C₂²	1	2	yes	—
42	C₄×Dic₉₄	C₂²	1	2	yes	—
43	C₄×D₁₈₈	C₂²	1	2	yes	—
44	C₄×D₁₈₈	C₂²	1	2	yes	—
45	D₄×Dic₄₇	C₂²	1	4	yes	—
46	#13093986	C₂²	1	4	yes	—
47	#13093983	C₂²	1	2	yes	—
48	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
49	C₄⋊C₄×D₄₇	C₂²	1	4	yes	—
50	C₂×#41×C₄₇	C₂²	1	2	yes	—
51	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
52	#127×C₄₇	C₂²	1	1	yes	characteristic
53	#124×C₄₇	C₂²	1	2	yes	—
54	#121×C₄₇	C₂²	1	2	yes	—
55	#121×C₄₇	C₂²	1	2	yes	—
56	C₄×D₄×C₄₇	C₂²	1	2	yes	—
57	C₄×D₄×C₄₇	C₂²	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
59	#41×C₄₇	—	4	4	no	—
60	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
61	C₂×D₄×C₄₇	C₂³	1	2	yes	—
62	C₂×D₄×C₄₇	C₂³	1	2	yes	—
63	C₂×D₄×C₄₇	C₂³	1	4	yes	—
64	C₂×D₄×C₄₇	C₂³	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
69	C₂×C₂²⋊D₄₇	C₂³	1	4	yes	—
70	C₂×C₂²⋊D₄₇	C₂³	1	8	yes	—
71	C₂×C₂²⋊D₄₇	C₂³	1	8	yes	—
72	C₂²×Dic₄₇	C₂³	1	8	yes	—
73	#10625352	—	4	8	no	—
74	#10625350	—	4	8	no	—
75	#10625350	—	4	4	no	—
76	C₂×D₁₈₈	C₂³	1	2	yes	—
77	C₂×D₁₈₈	C₂³	1	2	yes	—
78	C₂×D₁₈₈	C₂³	1	2	yes	—
79	C₂×C₄×D₄₇	C₂³	1	4	yes	—
80	C₂×C₄×D₄₇	C₂³	1	8	yes	—
81	C₂×Dic₉₄	C₂³	1	2	yes	—
82	C₂×Dic₉₄	C₂³	1	2	yes	—
83	C₂×Dic₉₄	C₂³	1	2	yes	—
84	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
85	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
86	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
87	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
88	C₄²×C₄₇	C₂³	1	1	yes	characteristic
89	C₄²×C₄₇	C₂³	1	1	yes	characteristic
90	C₄²×C₄₇	C₂³	1	1	yes	characteristic
91	C₄²×C₄₇	C₂³	1	1	yes	characteristic
92	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
93	C₂²⋊Dic₄₇	C₂³	1	8	yes	—
94	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
95	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
96	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
97	C₄⋊Dic₄₇	C₂³	1	2	yes	—
98	C₄⋊Dic₄₇	C₂³	1	2	yes	—
99	C₄⋊Dic₄₇	C₂³	1	2	yes	—
100	C₄⋊Dic₄₇	C₂³	1	8	yes	—
101	C₄⋊Dic₄₇	C₂³	1	2	yes	—
102	C₄⋊Dic₄₇	C₂³	1	2	yes	—
103	C₄⋊Dic₄₇	C₂³	1	2	yes	—
104	Dic₄₇⋊C₄	C₂³	1	8	yes	—
105	Dic₄₇⋊C₄	C₂³	1	4	yes	—
106	Dic₄₇⋊C₄	C₂³	1	4	yes	—
107	C₄×Dic₄₇	C₂³	1	4	yes	—
108	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
109	C₂³×C₄₇	C₂⁴	1	4	yes	—
110	C₂²×D₄₇	C₂⁴	1	4	yes	—
111	Q₈×C₄₇	—	4	2	no	—
112	D₄×C₄₇	—	2	4	no	—
113	D₄×C₄₇	—	2	4	no	—
114	D₄×C₄₇	—	4	4	no	—
115	D₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	8	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
123	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₄₇	—	4	8	no	—
128	C₂²⋊D₄₇	—	4	4	no	—
129	C₂²⋊D₄₇	—	4	8	no	—
130	C₂×Dic₄₇	—	4	8	no	—
131	C₂×Dic₄₇	C₂⁴	1	4	yes	—
132	C₂×Dic₄₇	C₂⁴	1	4	yes	—
133	C₂×Dic₄₇	C₂⁴	1	4	yes	—
134	D₁₈₈	—	2	4	no	—
135	D₁₈₈	—	2	4	no	—
136	D₁₈₈	—	4	2	no	—
137	C₄×D₄₇	—	2	8	no	—
138	C₄×D₄₇	—	4	8	no	—
139	Dic₉₄	—	2	4	no	—
140	Dic₉₄	—	2	4	no	—
141	Dic₉₄	—	4	2	no	—
142	C₂²×C₄₇	—	2	4	no	—
143	C₂²×C₄₇	—	2	1	no	—
144	C₂²×C₄₇	—	2	4	no	—
145	C₂²×C₄₇	—	2	1	no	—
146	C₂²×C₄₇	—	2	1	no	—
147	C₂²×C₄₇	—	2	4	no	—
148	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
149	C₂×D₄₇	—	2	4	no	—
150	C₂×D₄₇	—	2	4	no	—
151	C₂×D₄₇	—	2	4	no	—
152	C₄×C₄₇	—	2	1	no	—
153	C₄×C₄₇	—	2	4	no	—
154	C₄×C₄₇	—	2	1	no	—
155	C₄×C₄₇	—	2	1	no	—
156	C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
157	C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
158	C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
159	Dic₄₇	—	2	4	no	—
160	Dic₄₇	—	2	4	no	—
161	Dic₄₇	—	2	4	no	—
162	#3548	—	47	1	no	maximal, Sylow
163	C₂×C₄₇	—	4	4	no	—
164	C₂×C₄₇	—	4	1	no	—
165	C₂×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
167	C₂×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
168	D₄₇	—	4	4	no	—
169	#562	—	47	4	no	—
170	#554	—	47	2	no	—
171	#554	—	47	2	no	—
172	#550	—	47	2	no	—
173	#550	—	47	2	no	—
174	#547	—	47	2	no	—
175	#547	—	47	2	no	—
176	#540	—	47	4	no	—
177	#537	—	47	4	no	—
178	#537	—	47	4	no	—
179	#533	—	47	1	no	—
180	#532	—	47	1	no	—
181	#532	D₄₇	1	1	yes	characteristic, complemented
182	C₄₇	#3548	1	1	yes	characteristic, complemented, Sylow
183	C₂×#41	—	47	4	no	—
184	C₂×#41	—	47	2	no	—
185	C₂×#41	—	47	4	no	—
186	C₂×#41	C₂×D₄₇	1	2	yes	—
187	C₄⋊Q₈	—	47	1	no	—
188	C₄⋊Q₈	—	47	1	no	—
189	C₄⋊Q₈	—	47	1	no	—
190	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
191	#127	—	47	4	no	—
192	#127	—	47	1	no	—
193	#127	C₂×D₄₇	1	1	yes	characteristic
194	#125	—	47	4	no	—
195	#125	—	47	2	no	—
196	#124	—	47	2	no	—
197	#124	C₂×D₄₇	1	2	yes	—
198	#123	—	47	8	no	—
199	#123	—	47	8	no	—
200	C₂²⋊Q₈	—	47	8	no	—
201	C₂²⋊Q₈	—	47	2	no	—
202	C₂²⋊Q₈	—	47	2	no	—
203	C₂²⋊Q₈	—	47	4	no	—
204	C₂²⋊Q₈	—	47	4	no	—
205	C₂²⋊Q₈	—	47	8	no	—
206	#121	—	47	8	no	—
207	#121	—	47	8	no	—
208	#121	—	47	4	no	—
209	#121	—	47	8	no	—
210	#121	—	47	2	no	—
211	#121	—	47	2	no	—
212	#121	—	47	4	no	—
213	#121	—	47	8	no	—
214	#121	C₂×D₄₇	1	2	yes	—
215	#121	C₂×D₄₇	1	2	yes	—
216	C₄×Q₈	—	47	2	no	—
217	C₄×Q₈	—	47	2	no	—
218	C₄×D₄	—	47	4	no	—
219	C₄×D₄	—	47	4	no	—
220	C₄×D₄	—	47	2	no	—
221	C₄×D₄	—	47	2	no	—
222	C₄×D₄	C₂×D₄₇	1	2	yes	—
223	C₄×D₄	C₂×D₄₇	1	2	yes	—
224	#117	—	47	2	no	—
225	C₂×C₄⋊C₄	—	47	4	no	—
226	C₂×C₄⋊C₄	—	47	4	no	—
227	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
228	#41	—	4	4	no	—
229	#41	—	188	4	no	—
230	#41	—	188	8	no	—
231	#41	—	188	8	no	—
232	C₂×Q₈	—	47	2	no	—
233	C₂×Q₈	—	47	2	no	—
234	C₂×Q₈	—	47	2	no	—
235	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
236	C₂×D₄	—	47	4	no	—
237	C₂×D₄	—	47	8	no	—
238	C₂×D₄	—	47	8	no	—
239	C₂×D₄	—	47	2	no	—
240	C₂×D₄	—	47	2	no	—
241	C₂×D₄	—	47	2	no	—
242	C₂×D₄	C₂²×D₄₇	1	2	yes	—
243	C₂×D₄	C₂²×D₄₇	1	2	yes	—
244	C₂×D₄	C₂²×D₄₇	1	4	yes	—
245	C₂×D₄	C₂²×D₄₇	1	2	yes	—
246	C₂²×C₄	—	47	8	no	—
247	C₂²×C₄	—	47	4	no	—
248	C₂²×C₄	—	47	8	no	—
249	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
251	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
252	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
253	C₄⋊C₄	—	47	8	no	—
254	C₄⋊C₄	—	47	4	no	—
255	C₄⋊C₄	—	47	4	no	—
256	C₄⋊C₄	—	47	2	no	—
257	C₄⋊C₄	—	47	2	no	—
258	C₄⋊C₄	—	47	2	no	—
259	C₄⋊C₄	—	47	8	no	—
260	C₄⋊C₄	—	47	2	no	—
261	C₄⋊C₄	—	47	2	no	—
262	C₄⋊C₄	—	47	2	no	—
263	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
264	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
265	C₂²⋊C₄	—	47	4	no	—
266	C₂²⋊C₄	—	47	8	no	—
267	C₂²⋊C₄	—	47	8	no	—
268	C₂²⋊C₄	—	47	4	no	—
269	C₂²⋊C₄	—	47	4	no	—
270	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
271	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
272	C₄²	—	47	4	no	—
273	C₄²	C₂²×D₄₇	1	1	yes	characteristic
274	C₄²	C₂²×D₄₇	1	1	yes	characteristic
275	C₄²	C₂²×D₄₇	1	1	yes	characteristic
276	C₄²	C₂²×D₄₇	1	1	yes	characteristic
277	C₂³	—	47	4	no	—
278	C₂³	C₂³×D₄₇	1	1	yes	characteristic
279	C₂³	C₂³×D₄₇	1	4	yes	—
280	Q₈	—	4	2	no	—
281	Q₈	—	94	4	no	—
282	Q₈	—	94	4	no	—
283	Q₈	—	188	2	no	—
284	D₄	—	2	4	no	—
285	D₄	—	2	4	no	—
286	D₄	—	4	4	no	—
287	D₄	—	4	2	no	—
288	D₄	—	94	4	no	—
289	D₄	—	94	4	no	—
290	D₄	—	188	2	no	—
291	D₄	—	188	8	no	—
292	D₄	—	188	4	no	—
293	D₄	—	188	8	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	47	4	no	—
299	C₂×C₄	—	47	4	no	—
300	C₂×C₄	—	47	4	no	—
301	C₂×C₄	—	94	8	no	—
302	C₂×C₄	—	188	8	no	—
303	C₂×C₄	—	188	8	no	—
304	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
305	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₄₇	1	4	yes	—
308	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
309	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
311	C₂²	—	2	1	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	94	4	no	—
318	C₂²	—	94	4	no	—
319	C₂²	—	94	4	no	—
320	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	94	4	no	—
326	C₄	—	94	4	no	—
327	C₄	—	94	4	no	—
328	C₄	#13094069	1	2	yes	—
329	C₄	#13094070	1	2	yes	—
330	C₄	#13094071	1	2	yes	—
331	C₂	—	4	1	no	—
332	C₂	—	4	4	no	—
333	C₂	—	188	4	no	—
334	C₂	C₂×#13094069	1	1	yes	characteristic
335	C₂	C₂×#13094070	1	1	yes	characteristic
336	C₂	C₂×#13094071	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	C₄₇×#532	C₂
hypercenter	#532	D₄₇
hypocenter	C₄₇	#3548
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #532

Complements

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

N	H ≅ G/N	type
1. #15689867	C₂	semidirect
2. #15689834	C₂	semidirect
3. #15689830	C₂	semidirect
4. #15689794	C₂	semidirect
5. #15689793	C₂	semidirect
6. #15689778	C₂	semidirect
7. #15689777	C₂	semidirect
8. #15689726	C₂	semidirect
9. #15689702	C₂	semidirect
10. #15689701	C₂	semidirect
11. #15689680	C₂	semidirect
12. #15689673	C₂	semidirect
13. C₄₇×#532	C₂	semidirect
181. #532	D₄₇	semidirect
182. C₄₇	#3548	semidirect