Subgroups — #20245926

Order 6016.

656 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094000	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄²×D₄₇	C₂	1	1	yes	maximal, characteristic
4	#15689021	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689010	C₂	1	2	yes	maximal, complemented
6	#15688997	C₂	1	2	yes	maximal
7	#15688980	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#15688974	C₂	1	2	yes	maximal, complemented
10	#15688970	C₂	1	2	yes	maximal
11	C₄₇×#391	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
15	#13094000	C₄	1	8	yes	complemented
16	Q₈×Dic₄₇	—	2	4	no	—
17	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
20	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
21	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
22	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
23	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
24	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
25	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
26	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
29	C₄²×D₄₇	—	2	4	no	—
30	C₄₇⋊#95	C₂²	1	2	yes	—
31	C₄₇⋊#95	C₂²	1	2	yes	—
32	C₄₇⋊#95	C₂²	1	2	yes	—
33	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
36	#95×C₄₇	C₂²	1	2	yes	—
37	#95×C₄₇	C₂²	1	2	yes	—
38	C₂³×D₄₇	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₄₇	—	2	2	no	—
40	C₂×Q₈×C₄₇	—	2	2	no	—
41	C₂×Q₈×C₄₇	C₂×C₄	1	4	yes	complemented
42	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
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×Dic₄₇	C₂³	1	2	yes	—
48	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
49	C₂²×Dic₄₇	C₂³	1	2	yes	—
50	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
51	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₄₇	—	2	2	no	—
53	C₂×C₄×D₄₇	—	2	2	no	—
54	C₂×C₄×D₄₇	—	2	2	no	—
55	C₂×C₄×D₄₇	—	2	2	no	—
56	C₂×C₄×D₄₇	—	2	2	no	—
57	C₂×C₄×D₄₇	—	2	2	no	—
58	C₂×C₄×D₄₇	—	2	2	no	—
59	C₂×C₄×D₄₇	C₂×C₄	1	4	yes	—
60	C₂×C₄×D₄₇	Q₈	1	4	yes	complemented
61	C₄⋊C₄×C₄₇	—	2	2	no	—
62	C₄²×C₄₇	—	2	2	no	—
63	(C₂×D₄₇)⋊C₄	—	2	4	no	—
64	(C₂×D₄₇)⋊C₄	C₂×C₄	1	8	yes	—
65	C₄⋊Dic₄₇	—	2	2	no	—
66	C₄⋊Dic₄₇	—	2	4	no	—
67	C₄⋊Dic₄₇	C₂×C₄	1	4	yes	complemented
68	Dic₄₇⋊C₄	—	2	4	no	—
69	Dic₄₇⋊C₄	C₂×C₄	1	8	yes	complemented
70	C₄×Dic₄₇	—	2	2	no	—
71	C₄×Dic₄₇	—	2	2	no	—
72	C₄×Dic₄₇	—	2	4	no	—
73	C₄×Dic₄₇	—	2	4	no	—
74	C₄×Dic₄₇	D₄	1	4	yes	—
75	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂²×D₄₇	—	2	1	no	—
77	C₂²×D₄₇	—	2	1	no	—
78	C₂²×D₄₇	—	2	1	no	—
79	C₂²×D₄₇	—	2	1	no	—
80	C₂²×D₄₇	C₂²×C₄	1	2	yes	—
81	C₂²×D₄₇	C₂×Q₈	1	2	yes	—
82	C₂²×D₄₇	#41	1	2	yes	—
83	Q₈×C₄₇	—	2	8	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
94	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
97	C₂×C₄×C₄₇	#41	1	2	yes	—
98	C₂×C₄×C₄₇	#41	1	2	yes	—
99	C₂×Dic₄₇	—	2	1	no	—
100	C₂×Dic₄₇	—	2	2	no	—
101	C₂×Dic₄₇	—	2	2	no	—
102	C₂×Dic₄₇	—	2	1	no	—
103	C₂×Dic₄₇	—	2	1	no	—
104	C₂×Dic₄₇	—	2	1	no	—
105	C₂×Dic₄₇	—	2	2	no	—
106	C₂×Dic₄₇	—	2	2	no	—
107	C₂×Dic₄₇	—	2	2	no	—
108	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
109	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
110	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
111	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
112	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
113	C₂×Dic₄₇	#41	1	2	yes	—
114	C₄×D₄₇	—	2	8	no	—
115	C₄×D₄₇	—	2	8	no	—
116	C₄×D₄₇	—	2	8	no	—
117	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
118	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
120	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
122	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
123	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
124	C₂×D₄₇	—	2	2	no	—
125	C₂×D₄₇	—	2	2	no	—
126	C₂×D₄₇	—	2	2	no	—
127	C₂×D₄₇	—	2	2	no	—
128	C₂×D₄₇	—	2	2	no	—
129	C₂×D₄₇	—	2	2	no	—
130	C₂×D₄₇	C₄×Q₈	1	4	yes	complemented
131	C₄×C₄₇	—	2	4	no	—
132	C₄×C₄₇	—	2	2	no	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	4	2	no	—
135	C₄×C₄₇	C₄×D₄	1	4	yes	—
136	Dic₄₇	—	2	2	no	—
137	Dic₄₇	—	2	4	no	—
138	Dic₄₇	—	2	2	no	—
139	Dic₄₇	—	4	2	no	—
140	Dic₄₇	C₂²⋊Q₈	1	4	yes	complemented
141	C₄×C₂²⋊Q₈	—	47	1	no	maximal, Sylow
142	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
144	C₂×C₄₇	C₄×#41	1	1	yes	characteristic
145	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
146	C₂×C₄₇	#525	1	1	yes	characteristic
147	C₂×C₄₇	#529	1	1	yes	characteristic
148	C₂×C₄₇	#533	1	1	yes	characteristic
149	D₄₇	—	2	4	no	—
150	C₂×C₂²⋊Q₈	—	47	1	no	—
151	C₂×C₄×Q₈	—	47	1	no	—
152	C₂²×C₄²	—	47	1	no	—
153	#391	D₄₇	1	1	yes	characteristic, complemented
154	#389	—	47	1	no	—
155	#387	—	47	2	no	—
156	#385	—	47	2	no	—
157	#380	—	47	1	no	—
158	C₄×C₄⋊C₄	—	47	1	no	—
159	C₄×C₄⋊C₄	—	47	2	no	—
160	C₄×C₂²⋊C₄	—	47	2	no	—
161	C₄₇	C₄×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
163	C₂³×C₄	—	47	1	no	—
164	C₂³×C₄	—	47	1	no	—
165	C₂³×C₄	—	47	1	no	—
166	C₂²⋊Q₈	—	47	8	no	—
167	C₄×Q₈	—	94	4	no	—
168	C₂×C₄⋊C₄	—	47	2	no	—
169	C₂×C₄⋊C₄	—	47	1	no	—
170	C₂×C₄⋊C₄	—	47	1	no	—
171	C₂×C₄⋊C₄	—	47	2	no	—
172	C₂×C₄⋊C₄	—	47	2	no	—
173	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	—	47	2	no	—
175	C₂×C₂²⋊C₄	—	47	2	no	—
176	C₂×C₄²	—	47	1	no	—
177	C₂×C₄²	—	47	2	no	—
178	C₂×C₄²	—	47	2	no	—
179	C₂×C₄²	—	47	1	no	—
180	C₂×C₄²	—	47	1	no	—
181	C₂×C₄²	—	94	4	no	—
182	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
183	#95	—	47	2	no	—
184	#95	—	47	2	no	—
185	#95	—	47	2	no	—
186	#95	C₂×D₄₇	1	2	yes	—
187	#95	C₂×D₄₇	1	2	yes	—
188	C₂⁴	—	47	1	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	C₄×D₄₇	1	4	yes	complemented
192	C₂²×C₄	—	47	1	no	—
193	C₂²×C₄	—	47	1	no	—
194	C₂²×C₄	—	47	1	no	—
195	C₂²×C₄	—	47	2	no	—
196	C₂²×C₄	—	47	4	no	—
197	C₂²×C₄	—	47	4	no	—
198	C₂²×C₄	—	47	2	no	—
199	C₂²×C₄	—	94	2	no	—
200	C₂²×C₄	—	94	2	no	—
201	C₂²×C₄	—	94	2	no	—
202	C₂²×C₄	—	94	2	no	—
203	C₂²×C₄	—	94	2	no	—
204	C₂²×C₄	—	94	2	no	—
205	C₂²×C₄	—	94	2	no	—
206	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
207	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
208	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
210	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	47	8	no	—
213	C₄⋊C₄	—	47	4	no	—
214	C₄⋊C₄	—	94	4	no	—
215	C₄⋊C₄	—	94	2	no	—
216	C₄⋊C₄	—	94	4	no	—
217	C₂²⋊C₄	—	47	8	no	—
218	C₂²⋊C₄	—	94	4	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	47	4	no	—
221	C₄²	—	94	4	no	—
222	C₄²	—	94	2	no	—
223	C₄²	—	94	2	no	—
224	C₄²	—	94	4	no	—
225	C₂³	—	47	2	no	—
226	C₂³	—	47	2	no	—
227	C₂³	—	47	2	no	—
228	C₂³	—	94	1	no	—
229	C₂³	—	94	1	no	—
230	C₂³	—	94	1	no	—
231	C₂³	—	94	1	no	—
232	C₂³	C₂³×D₄₇	1	1	yes	characteristic
233	Q₈	—	2	8	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	47	2	no	—
244	C₂×C₄	—	47	2	no	—
245	C₂×C₄	—	47	2	no	—
246	C₂×C₄	—	47	4	no	—
247	C₂×C₄	—	47	2	no	—
248	C₂×C₄	—	47	2	no	—
249	C₂×C₄	—	94	8	no	—
250	C₂×C₄	—	94	8	no	—
251	C₂×C₄	—	94	1	no	—
252	C₂×C₄	—	94	2	no	—
253	C₂×C₄	—	94	1	no	—
254	C₂×C₄	—	94	1	no	—
255	C₂×C₄	—	94	2	no	—
256	C₂×C₄	—	94	1	no	—
257	C₂×C₄	—	94	8	no	—
258	C₂×C₄	—	94	2	no	—
259	C₂×C₄	—	94	2	no	—
260	C₂×C₄	—	94	2	no	—
261	C₂×C₄	C₂×C₄×D₄₇	1	4	yes	—
262	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
263	C₂×C₄	D₄×D₄₇	1	2	yes	—
264	C₂×C₄	#10625352	1	2	yes	—
265	C₂×C₄	Q₈×D₄₇	1	2	yes	—
266	C₂×C₄	#10625354	1	2	yes	—
267	C₂²	—	47	4	no	—
268	C₂²	—	94	2	no	—
269	C₂²	—	94	2	no	—
270	C₂²	—	94	2	no	—
271	C₂²	—	94	2	no	—
272	C₂²	—	94	2	no	—
273	C₂²	—	94	2	no	—
274	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
275	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
276	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
277	C₂²	#41×D₄₇	1	1	yes	characteristic
278	C₂²	#41×D₄₇	1	1	yes	characteristic
279	C₂²	C₂×#10625352	1	1	yes	characteristic
280	C₂²	C₂×#10625354	1	1	yes	characteristic
281	C₄	—	2	4	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	4	2	no	—
285	C₄	—	47	4	no	—
286	C₄	—	94	2	no	—
287	C₄	—	94	4	no	—
288	C₄	—	94	2	no	—
289	C₄	—	188	2	no	—
290	C₄	(C₂²⋊D₄₇)⋊C₄	1	4	yes	—
291	C₂	—	94	4	no	—
292	C₂	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
293	C₂	C₄×Q₈×D₄₇	1	1	yes	characteristic
294	C₂	C₄×#10625354	1	1	yes	characteristic
295	C₂	C₂²⋊Q₈×D₄₇	1	1	yes	characteristic
296	C₂	#15689636	1	1	yes	characteristic
297	C₂	#15689652	1	1	yes	characteristic
298	C₂	#15689678	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#391	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	#391	D₄₇
hypocenter	C₄₇	C₄×C₂²⋊Q₈
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×Dic₄₇	C₂	semidirect
4. #15689021	C₂	semidirect
5. #15689010	C₂	semidirect
8. C₄⋊C₄×Dic₄₇	C₂	semidirect
9. #15688974	C₂	semidirect
11. C₄₇×#391	C₂	semidirect
15. #13094000	C₄	semidirect
41. C₂×Q₈×C₄₇	C₂×C₄	semidirect
60. C₂×C₄×D₄₇	Q₈	semidirect
67. C₄⋊Dic₄₇	C₂×C₄	semidirect
69. Dic₄₇⋊C₄	C₂×C₄	semidirect
130. C₂×D₄₇	C₄×Q₈	semidirect
140. Dic₄₇	C₂²⋊Q₈	semidirect
153. #391	D₄₇	semidirect
161. C₄₇	C₄×C₂²⋊Q₈	semidirect
191. C₂×Q₈	C₄×D₄₇	semidirect