Subgroups — #20257193

Order 6016.

964 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₉₄	C₂	1	2	yes	maximal, complemented
2	#15689853	C₂	1	2	yes	maximal, complemented
3	#15689851	C₂	1	4	yes	maximal, complemented
4	#15689851	C₂	1	2	yes	maximal, complemented
5	#15689850	C₂	1	4	yes	maximal, complemented
6	#15689848	C₂	1	2	yes	maximal, complemented
7	#15689752	C₂	1	4	yes	maximal
8	#15689751	C₂	1	2	yes	maximal
9	#15689680	C₂	1	1	yes	maximal, characteristic
10	#15689675	C₂	1	4	yes	maximal
11	#15689600	C₂	1	2	yes	maximal
12	C₂×C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic
13	C₄₇×#533	C₂	1	1	yes	maximal, characteristic
14	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
15	#13094024	C₂²	1	4	yes	—
16	#13094024	C₂²	1	1	yes	characteristic
17	#13094024	C₂²	1	1	yes	characteristic
18	#13094023	C₂²	1	4	yes	—
19	#13094023	C₂²	1	8	yes	—
20	#13094023	C₂²	1	4	yes	—
21	#13094022	C₂²	1	8	yes	—
22	#13094022	C₂²	1	4	yes	—
23	#13094016	C₂²	1	4	yes	—
24	#13094016	C₂²	1	2	yes	—
25	#13094015	C₂²	1	4	yes	—
26	#13094014	C₂²	1	8	yes	—
27	#13094014	C₂²	1	8	yes	—
28	#13094014	C₂²	1	8	yes	—
29	#13094003	C₂²	1	2	yes	—
30	#13094003	C₂²	1	2	yes	—
31	#13094001	C₂²	1	8	yes	—
32	Q₈×Dic₄₇	C₂²	1	4	yes	—
33	C₄×Dic₉₄	C₂²	1	2	yes	—
34	C₄×Dic₉₄	C₂²	1	2	yes	—
35	C₄×Dic₉₄	C₂²	1	2	yes	—
36	C₄×Dic₉₄	C₂²	1	2	yes	—
37	C₄×Dic₉₄	C₂²	1	2	yes	—
38	C₄×Dic₉₄	C₂²	1	2	yes	—
39	C₄×Dic₉₄	C₂²	1	4	yes	—
40	C₄×Dic₉₄	C₂²	1	4	yes	—
41	#13093989	C₂²	1	4	yes	—
42	#13093989	C₂²	1	8	yes	—
43	#13093989	C₂²	1	4	yes	—
44	#13093989	C₂²	1	4	yes	—
45	#13093983	C₂²	1	2	yes	—
46	#13093980	C₂²	1	8	yes	—
47	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
48	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
49	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
50	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
51	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
52	#125×C₄₇	C₂²	1	2	yes	—
53	C₂²⋊Q₈×C₄₇	C₂²	1	2	yes	—
54	C₂²⋊Q₈×C₄₇	C₂²	1	2	yes	—
55	C₄×Q₈×C₄₇	C₂²	1	2	yes	—
56	C₄×Q₈×C₄₇	C₂²	1	2	yes	—
57	#117×C₄₇	C₂²	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
60	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
64	C₂²×Dic₄₇	C₂³	1	2	yes	—
65	C₂²×Dic₄₇	C₂³	1	2	yes	—
66	C₂×Dic₉₄	Q₈	1	8	yes	complemented
67	C₂×Dic₉₄	C₂³	1	2	yes	—
68	C₂×Dic₉₄	C₂³	1	8	yes	—
69	C₂×Dic₉₄	C₂³	1	2	yes	—
70	C₂×Dic₉₄	C₂³	1	2	yes	—
71	C₂×Dic₉₄	C₂³	1	2	yes	—
72	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
73	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
74	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
75	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
76	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
77	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
78	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
79	C₄²×C₄₇	C₂³	1	1	yes	characteristic
80	C₄²×C₄₇	C₂³	1	1	yes	characteristic
81	C₄²×C₄₇	C₂³	1	1	yes	characteristic
82	C₄²×C₄₇	C₂³	1	1	yes	characteristic
83	C₄²×C₄₇	C₂³	1	4	yes	—
84	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
85	C₄⋊Dic₄₇	C₂³	1	2	yes	—
86	C₄⋊Dic₄₇	C₂³	1	2	yes	—
87	C₄⋊Dic₄₇	C₂³	1	2	yes	—
88	C₄⋊Dic₄₇	C₂³	1	2	yes	—
89	C₄⋊Dic₄₇	C₂³	1	2	yes	—
90	C₄⋊Dic₄₇	C₂³	1	2	yes	—
91	C₄⋊Dic₄₇	C₂³	1	8	yes	—
92	Dic₄₇⋊C₄	C₂³	1	4	yes	—
93	Dic₄₇⋊C₄	C₂³	1	4	yes	—
94	Dic₄₇⋊C₄	C₂³	1	8	yes	—
95	Dic₄₇⋊C₄	C₂³	1	8	yes	—
96	Dic₄₇⋊C₄	C₂³	1	4	yes	—
97	Dic₄₇⋊C₄	C₂³	1	8	yes	—
98	Dic₄₇⋊C₄	C₂³	1	8	yes	—
99	Dic₄₇⋊C₄	C₂³	1	4	yes	—
100	C₄×Dic₄₇	C₂³	1	4	yes	—
101	C₄×Dic₄₇	C₂³	1	4	yes	—
102	C₄×Dic₄₇	C₂³	1	8	yes	—
103	C₄×Dic₄₇	C₂³	1	8	yes	—
104	C₄×Dic₄₇	C₂³	1	4	yes	—
105	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
106	Q₈×C₄₇	—	2	4	no	—
107	Q₈×C₄₇	—	4	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	4	1	no	—
110	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
111	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
113	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
114	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
119	C₂×Dic₄₇	—	4	2	no	—
120	C₂×Dic₄₇	C₂×Q₈	1	8	yes	—
121	C₂×Dic₄₇	C₂⁴	1	4	yes	—
122	C₂×Dic₄₇	C₂⁴	1	4	yes	—
123	C₂×Dic₄₇	C₂⁴	1	4	yes	—
124	C₂×Dic₄₇	C₂⁴	1	4	yes	—
125	Dic₉₄	—	2	4	no	—
126	Dic₉₄	—	4	2	no	—
127	Dic₉₄	—	4	8	no	—
128	Dic₉₄	C₂×Q₈	1	8	yes	—
129	Dic₉₄	C₂×Q₈	1	8	yes	—
130	C₂²×C₄₇	—	2	1	no	—
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	C₂²×Q₈	1	2	yes	—
133	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
134	C₄×C₄₇	—	2	1	no	—
135	C₄×C₄₇	—	2	4	no	—
136	C₄×C₄₇	—	2	1	no	—
137	C₄×C₄₇	—	2	4	no	—
138	C₄×C₄₇	—	2	1	no	—
139	C₄×C₄₇	C₂²×Q₈	1	2	yes	—
140	C₄×C₄₇	C₂²×Q₈	1	2	yes	—
141	C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
142	Dic₄₇	—	2	4	no	—
143	Dic₄₇	—	2	4	no	—
144	Dic₄₇	—	2	4	no	—
145	Dic₄₇	C₂²×Q₈	1	8	yes	—
146	#3502	—	47	1	no	maximal, Sylow
147	C₂×C₄₇	—	2	2	no	—
148	C₂×C₄₇	C₂³×Q₈	1	1	yes	characteristic
149	C₂×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
150	C₂×C₄₇	#585	1	1	yes	characteristic
151	Q₈²	—	47	2	no	—
152	Q₈²	—	47	2	no	—
153	#557	—	47	2	no	—
154	#557	—	47	4	no	—
155	#557	—	47	4	no	—
156	#557	—	47	2	no	—
157	#544	—	47	4	no	—
158	#544	—	47	2	no	—
159	#533	—	47	4	no	—
160	#533	—	47	1	no	—
161	#533	D₄₇	1	1	yes	characteristic
162	#519	—	47	2	no	—
163	C₂×C₄×Q₈	—	47	1	no	—
164	C₄₇	#3502	1	1	yes	characteristic, complemented, Sylow
165	C₂²×Q₈	—	47	1	no	—
166	C₄⋊Q₈	—	47	4	no	—
167	C₄⋊Q₈	—	47	8	no	—
168	C₄⋊Q₈	—	47	4	no	—
169	C₄⋊Q₈	—	47	8	no	—
170	C₄⋊Q₈	—	47	4	no	—
171	C₄⋊Q₈	—	47	1	no	—
172	C₄⋊Q₈	—	47	1	no	—
173	C₄⋊Q₈	—	47	4	no	—
174	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
175	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
176	#125	—	47	8	no	—
177	#125	—	47	8	no	—
178	#125	—	47	8	no	—
179	#125	—	47	4	no	—
180	#125	—	47	2	no	—
181	#125	—	47	4	no	—
182	#125	C₂×D₄₇	1	2	yes	—
183	C₂²⋊Q₈	—	47	2	no	—
184	C₂²⋊Q₈	—	47	2	no	—
185	C₂²⋊Q₈	—	47	8	no	—
186	C₂²⋊Q₈	C₂×D₄₇	1	2	yes	—
187	C₂²⋊Q₈	C₂×D₄₇	1	2	yes	—
188	C₄×Q₈	—	47	4	no	—
189	C₄×Q₈	—	47	4	no	—
190	C₄×Q₈	—	47	4	no	—
191	C₄×Q₈	—	47	8	no	—
192	C₄×Q₈	—	47	4	no	—
193	C₄×Q₈	—	47	2	no	—
194	C₄×Q₈	—	47	2	no	—
195	C₄×Q₈	—	47	2	no	—
196	C₄×Q₈	—	47	2	no	—
197	C₄×Q₈	—	47	4	no	—
198	C₄×Q₈	—	47	4	no	—
199	C₄×Q₈	—	47	2	no	—
200	C₄×Q₈	—	47	2	no	—
201	C₄×Q₈	C₂×D₄₇	1	2	yes	—
202	C₄×Q₈	C₂×D₄₇	1	2	yes	—
203	#117	—	47	2	no	—
204	#117	—	47	8	no	—
205	#117	C₂×D₄₇	1	2	yes	—
206	C₂×C₄⋊C₄	—	47	2	no	—
207	C₂×C₄⋊C₄	—	47	1	no	—
208	C₂×C₄²	—	47	2	no	—
209	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
210	C₂×Q₈	—	47	8	no	—
211	C₂×Q₈	—	47	2	no	—
212	C₂×Q₈	—	47	2	no	—
213	C₂×Q₈	—	47	2	no	—
214	C₂×Q₈	—	47	2	no	—
215	C₂×Q₈	—	47	8	no	—
216	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
217	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
218	C₂²×C₄	—	47	2	no	—
219	C₂²×C₄	—	47	2	no	—
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	—	47	4	no	—
225	C₄⋊C₄	—	47	8	no	—
226	C₄⋊C₄	—	47	8	no	—
227	C₄⋊C₄	—	47	4	no	—
228	C₄⋊C₄	—	47	8	no	—
229	C₄⋊C₄	—	47	4	no	—
230	C₄⋊C₄	—	47	8	no	—
231	C₄⋊C₄	—	47	2	no	—
232	C₄⋊C₄	—	47	2	no	—
233	C₄⋊C₄	—	47	2	no	—
234	C₄⋊C₄	—	47	2	no	—
235	C₄⋊C₄	—	47	2	no	—
236	C₄⋊C₄	—	47	2	no	—
237	C₄⋊C₄	—	47	8	no	—
238	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
239	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
240	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
241	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
242	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
243	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
244	C₂²⋊C₄	—	47	4	no	—
245	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
246	C₄²	—	47	4	no	—
247	C₄²	—	47	4	no	—
248	C₄²	—	47	8	no	—
249	C₄²	—	47	8	no	—
250	C₄²	—	47	4	no	—
251	C₄²	C₂²×D₄₇	1	1	yes	characteristic
252	C₄²	C₂²×D₄₇	1	1	yes	characteristic
253	C₄²	C₂²×D₄₇	1	1	yes	characteristic
254	C₄²	C₂²×D₄₇	1	1	yes	characteristic
255	C₄²	C₂²×D₄₇	1	4	yes	—
256	C₂³	C₂³×D₄₇	1	1	yes	characteristic
257	Q₈	—	2	4	no	—
258	Q₈	—	4	2	no	—
259	Q₈	—	47	8	no	—
260	Q₈	—	47	8	no	—
261	Q₈	—	94	4	no	—
262	Q₈	—	188	2	no	—
263	Q₈	—	188	8	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	47	8	no	—
267	C₂×C₄	—	47	4	no	—
268	C₂×C₄	—	47	4	no	—
269	C₂×C₄	—	47	4	no	—
270	C₂×C₄	—	47	4	no	—
271	C₂×C₄	—	188	2	no	—
272	C₂×C₄	Q₈×D₄₇	1	4	yes	—
273	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂³×D₄₇	1	4	yes	—
275	C₂×C₄	C₂³×D₄₇	1	4	yes	—
276	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
277	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
278	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
279	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	C₂×Q₈×D₄₇	1	2	yes	—
284	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	4	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	4	no	—
290	C₄	—	47	8	no	—
291	C₄	—	94	4	no	—
292	C₄	—	94	4	no	—
293	C₄	—	94	4	no	—
294	C₄	C₂×Q₈×D₄₇	1	2	yes	—
295	C₄	C₂×Q₈×D₄₇	1	2	yes	—
296	C₄	#13094071	1	2	yes	—
297	C₂	—	2	2	no	—
298	C₂	C₂²×Q₈×D₄₇	1	1	yes	characteristic
299	C₂	C₂×#13094071	1	1	yes	characteristic
300	C₂	#15689959	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	C₄₇×#533	C₂
hypercenter	#533	D₄₇
hypocenter	C₄₇	#3502
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₉₄	C₂	semidirect
2. #15689853	C₂	semidirect
3. #15689851	C₂	semidirect
4. #15689851	C₂	semidirect
5. #15689850	C₂	semidirect
6. #15689848	C₂	semidirect
66. C₂×Dic₉₄	Q₈	semidirect
164. C₄₇	#3502	semidirect