Subgroups — #20247751

Order 6016.

632 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094024	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13094010	C₂	1	2	yes	maximal, complemented
3	C₂×#13093998	C₂	1	1	yes	maximal, characteristic
4	#15689039	C₂	1	2	yes	maximal, complemented
5	#15689036	C₂	1	2	yes	maximal, complemented
6	#15689025	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689025	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689015	C₂	1	2	yes	maximal, complemented
9	#15688983	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₉₄	C₂²	1	2	yes	—
13	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₄₇	C₂²	1	2	yes	—
15	#13094024	—	2	4	no	—
16	#13094010	—	2	8	no	—
17	#13093998	—	2	4	no	—
18	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
23	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
24	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
25	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
26	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
27	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
28	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
29	C₄₇⋊#95	C₂²	1	2	yes	—
30	C₄₇⋊#95	C₂²	1	2	yes	—
31	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
36	#95×C₄₇	C₂²	1	1	yes	characteristic
37	#95×C₄₇	C₂²	1	1	yes	characteristic
38	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
39	C₂³×D₄₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₄₇	—	2	2	no	—
41	C₂²×C₄×C₄₇	—	2	2	no	—
42	C₂²×C₄×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊D₄₇	—	2	4	no	—
50	C₂×C₂²⋊D₄₇	—	2	4	no	—
51	C₂×C₂²⋊D₄₇	—	4	2	no	—
52	C₂²×Dic₄₇	C₂³	1	2	yes	—
53	C₂²×Dic₄₇	C₂³	1	2	yes	—
54	C₂²×Dic₄₇	C₂³	1	2	yes	—
55	C₂×D₁₈₈	—	2	2	no	—
56	C₂×D₁₈₈	—	2	2	no	—
57	C₂×D₁₈₈	—	2	2	no	—
58	C₂×Dic₉₄	—	2	4	no	—
59	C₂×Dic₉₄	—	2	4	no	—
60	C₂×Dic₉₄	—	4	2	no	—
61	C₂²⋊C₄×C₄₇	—	2	2	no	—
62	C₂²⋊C₄×C₄₇	D₄	1	4	yes	complemented
63	C₄²×C₄₇	—	2	2	no	—
64	C₄²×C₄₇	—	2	2	no	—
65	C₂²⋊Dic₄₇	—	2	4	no	—
66	(C₂×D₄₇)⋊C₄	—	2	4	no	—
67	(C₂×D₄₇)⋊C₄	—	2	4	no	—
68	(C₂×D₄₇)⋊C₄	—	2	4	no	—
69	(C₂×D₄₇)⋊C₄	—	2	4	no	—
70	(C₂×D₄₇)⋊C₄	—	4	2	no	—
71	C₄⋊Dic₄₇	—	2	2	no	—
72	C₄⋊Dic₄₇	—	2	2	no	—
73	C₄⋊Dic₄₇	—	2	4	no	—
74	C₄⋊Dic₄₇	D₄	1	4	yes	—
75	C₄×Dic₄₇	—	2	4	no	—
76	C₂³×C₄₇	—	2	1	no	—
77	C₂³×C₄₇	—	2	1	no	—
78	C₂³×C₄₇	—	2	1	no	—
79	C₂³×C₄₇	—	2	1	no	—
80	C₂³×C₄₇	—	2	1	no	—
81	C₂³×C₄₇	—	2	1	no	—
82	C₂³×C₄₇	C₂×D₄	1	2	yes	—
83	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₂²×D₄₇	—	2	2	no	—
85	C₂²×D₄₇	—	2	2	no	—
86	C₂²×D₄₇	—	2	2	no	—
87	C₂²×D₄₇	—	2	2	no	—
88	C₂²×D₄₇	—	2	2	no	—
89	C₂²×D₄₇	—	2	2	no	—
90	C₂²×D₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	—	2	1	no	—
95	C₂×C₄×C₄₇	—	2	1	no	—
96	C₂×C₄×C₄₇	—	2	1	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	4	4	no	—
104	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₄₇	#41	1	2	yes	—
108	C₂×C₄×C₄₇	#41	1	2	yes	—
109	C₂×C₄×C₄₇	#41	1	2	yes	—
110	C₂²⋊D₄₇	—	4	8	no	—
111	C₂×Dic₄₇	—	2	2	no	—
112	C₂×Dic₄₇	—	2	2	no	—
113	C₂×Dic₄₇	—	2	2	no	—
114	C₂×Dic₄₇	—	2	2	no	—
115	C₂×Dic₄₇	—	2	2	no	—
116	C₂×Dic₄₇	—	2	2	no	—
117	C₂×Dic₄₇	—	2	2	no	—
118	C₂×Dic₄₇	—	2	2	no	—
119	C₂×Dic₄₇	C₂×D₄	1	4	yes	—
120	D₁₈₈	—	4	4	no	—
121	Dic₉₄	—	4	8	no	—
122	C₂²×C₄₇	—	2	2	no	—
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	4	1	no	—
126	C₂²×C₄₇	—	4	1	no	—
127	C₂²×C₄₇	—	4	1	no	—
128	C₂²×C₄₇	—	4	1	no	—
129	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
133	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
134	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
136	C₂×D₄₇	—	4	2	no	—
137	C₂×D₄₇	—	4	2	no	—
138	C₂×D₄₇	—	4	2	no	—
139	C₂×D₄₇	—	4	2	no	—
140	C₂×D₄₇	—	4	2	no	—
141	C₂×D₄₇	—	4	2	no	—
142	C₂×D₄₇	—	4	2	no	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	4	1	no	—
147	C₄×C₄₇	—	4	1	no	—
148	C₄×C₄₇	#124	1	4	yes	complemented
149	Dic₄₇	—	2	4	no	—
150	Dic₄₇	—	4	2	no	—
151	Dic₄₇	—	4	2	no	—
152	#2491	—	47	1	no	maximal, Sylow
153	C₂×C₄₇	—	4	2	no	—
154	C₂×C₄₇	C₂×#124	1	1	yes	characteristic
155	C₂×C₄₇	#532	1	1	yes	characteristic
156	C₂×C₄₇	#542	1	1	yes	characteristic
157	C₂×C₄₇	D₄²	1	1	yes	characteristic
158	C₂×C₄₇	#547	1	1	yes	characteristic
159	C₂×C₄₇	#548	1	1	yes	characteristic
160	C₂×C₄₇	#555	1	1	yes	characteristic
161	D₄₇	—	8	2	no	—
162	C₂×C₄⋊Q₈	—	47	1	no	—
163	C₂×#124	—	47	2	no	—
164	C₂×#121	—	47	1	no	—
165	#394	—	47	2	no	—
166	#393	—	47	2	no	—
167	#390	—	47	1	no	—
168	#390	—	47	1	no	—
169	#388	—	47	2	no	—
170	#380	—	47	1	no	—
171	C₄×C₄⋊C₄	—	47	1	no	—
172	C₄×C₂²⋊C₄	D₄₇	1	1	yes	characteristic, complemented
173	C₄₇	#2491	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₈	—	47	2	no	—
175	C₂²×D₄	—	47	2	no	—
176	C₂²×D₄	—	47	1	no	—
177	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
178	C₄⋊Q₈	—	94	4	no	—
179	#124	—	94	8	no	—
180	#121	—	94	4	no	—
181	C₂×C₄⋊C₄	—	47	1	no	—
182	C₂×C₄⋊C₄	—	47	1	no	—
183	C₂×C₄⋊C₄	—	47	2	no	—
184	C₂×C₄⋊C₄	—	47	1	no	—
185	C₂×C₂²⋊C₄	—	47	2	no	—
186	C₂×C₂²⋊C₄	—	47	2	no	—
187	C₂×C₂²⋊C₄	—	47	2	no	—
188	C₂×C₂²⋊C₄	—	47	2	no	—
189	C₂×C₂²⋊C₄	—	47	2	no	—
190	C₂×C₂²⋊C₄	—	47	2	no	—
191	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
193	C₂×C₄²	—	47	2	no	—
194	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
195	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
196	#95	—	47	2	no	—
197	#95	—	47	2	no	—
198	#95	C₂×D₄₇	1	1	yes	characteristic
199	#95	C₂×D₄₇	1	1	yes	characteristic
200	C₂⁴	—	47	2	no	—
201	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
202	C₂×Q₈	—	94	4	no	—
203	C₂×Q₈	—	94	4	no	—
204	C₂×Q₈	—	188	2	no	—
205	C₂×D₄	—	94	4	no	—
206	C₂×D₄	—	94	4	no	—
207	C₂×D₄	—	94	2	no	—
208	C₂×D₄	—	94	2	no	—
209	C₂×D₄	—	94	2	no	—
210	C₂×D₄	—	188	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	47	2	no	—
215	C₂²×C₄	—	47	2	no	—
216	C₂²×C₄	—	47	2	no	—
217	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
219	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
221	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
222	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
223	C₄⋊C₄	—	47	4	no	—
224	C₄⋊C₄	—	94	4	no	—
225	C₄⋊C₄	—	94	2	no	—
226	C₄⋊C₄	—	94	2	no	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	94	4	no	—
229	C₂²⋊C₄	—	94	4	no	—
230	C₂²⋊C₄	—	94	4	no	—
231	C₂²⋊C₄	—	94	4	no	—
232	C₂²⋊C₄	—	94	4	no	—
233	C₂²⋊C₄	—	188	2	no	—
234	C₂²⋊C₄	D₁₈₈	1	4	yes	complemented
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	94	4	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	94	2	no	—
245	C₂³	—	94	2	no	—
246	C₂³	—	94	2	no	—
247	C₂³	—	94	2	no	—
248	C₂³	—	94	2	no	—
249	C₂³	—	94	2	no	—
250	C₂³	—	94	2	no	—
251	C₂³	C₂×D₁₈₈	1	2	yes	—
252	C₂³	C₂³×D₄₇	1	1	yes	characteristic
253	Q₈	—	188	8	no	—
254	D₄	—	188	8	no	—
255	D₄	—	188	4	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	47	4	no	—
270	C₂×C₄	—	94	2	no	—
271	C₂×C₄	—	94	2	no	—
272	C₂×C₄	—	94	2	no	—
273	C₂×C₄	—	94	2	no	—
274	C₂×C₄	—	94	2	no	—
275	C₂×C₄	—	94	2	no	—
276	C₂×C₄	—	94	2	no	—
277	C₂×C₄	—	94	2	no	—
278	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
279	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
280	C₂×C₄	#10625350	1	2	yes	—
281	C₂×C₄	#10625350	1	2	yes	—
282	C₂×C₄	D₄×D₄₇	1	2	yes	—
283	C₂×C₄	#10625352	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	188	2	no	—
292	C₂²	—	188	2	no	—
293	C₂²	—	188	2	no	—
294	C₂²	—	188	2	no	—
295	C₂²	—	188	2	no	—
296	C₂²	—	188	2	no	—
297	C₂²	—	188	2	no	—
298	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
299	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
300	C₂²	C₂×#10625352	1	1	yes	characteristic
301	C₂²	C₂×#10625350	1	1	yes	characteristic
302	C₂²	C₂×#10625350	1	1	yes	characteristic
303	C₂²	#13094070	1	1	yes	characteristic
304	C₂²	#13094071	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	94	4	no	—
311	C₄	—	188	2	no	—
312	C₄	—	188	2	no	—
313	C₄	#13094010	1	4	yes	complemented
314	C₂	—	4	2	no	—
315	C₂	—	376	2	no	—
316	C₂	C₂×#13094010	1	1	yes	characteristic
317	C₂	#15689673	1	1	yes	characteristic
318	C₂	#15689744	1	1	yes	characteristic
319	C₂	D₄×D₁₈₈	1	1	yes	characteristic
320	C₂	#15689777	1	1	yes	characteristic
321	C₂	#15689782	1	1	yes	characteristic
322	C₂	#15689838	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄×C₂²⋊C₄×C₄₇	C₂
hypercenter	C₄×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	#2491
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#13094024	C₂	semidirect
2. C₂×#13094010	C₂	semidirect
4. #15689039	C₂	semidirect
5. #15689036	C₂	semidirect
6. #15689025	C₂	semidirect
7. #15689025	C₂	semidirect
8. #15689015	C₂	semidirect
9. #15688983	C₂	semidirect
10. C₄×C₄⋊Dic₄₇	C₂	semidirect
11. C₄×C₂²⋊C₄×C₄₇	C₂	semidirect
62. C₂²⋊C₄×C₄₇	D₄	semidirect
148. C₄×C₄₇	#124	semidirect
172. C₄×C₂²⋊C₄	D₄₇	semidirect
173. C₄₇	#2491	semidirect
234. C₂²⋊C₄	D₁₈₈	semidirect
313. C₄	#13094010	semidirect