Subgroups — #20254714

Order 6016.

398 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689673	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689420	C₂	1	2	yes	maximal, complemented
3	#15689418	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689413	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689357	C₂	1	2	yes	maximal, complemented
6	#15689354	C₂	1	2	yes	maximal, complemented
7	#15689335	C₂	1	2	yes	maximal, complemented
8	#15689323	C₂	1	1	yes	maximal, characteristic, complemented
9	#15689318	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#529	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625350	—	2	1	no	—
13	C₂×#10625336	C₂²	1	2	yes	—
14	C₂×#10625334	C₂²	1	2	yes	—
15	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
16	#13094024	C₂²	1	1	yes	characteristic
17	#13094021	C₂²	1	1	yes	characteristic
18	#13094012	—	2	1	no	—
19	#13093998	—	2	1	no	—
20	#13093998	C₂²	1	2	yes	—
21	C₄×D₁₈₈	—	2	1	no	—
22	C₄×D₁₈₈	C₂²	1	2	yes	—
23	#13093957	C₂²	1	2	yes	—
24	#13093955	C₂²	1	2	yes	—
25	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
28	#13093948	C₂²	1	2	yes	—
29	#13093945	C₂²	1	2	yes	—
30	#13093944	C₂²	1	2	yes	—
31	#13093944	C₂²	1	2	yes	—
32	#126×C₄₇	—	2	1	no	—
33	#125×C₄₇	C₂²	1	1	yes	characteristic
34	#124×C₄₇	C₂²	1	1	yes	characteristic
35	#123×C₄₇	—	2	1	no	—
36	C₂²⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
37	#121×C₄₇	C₂²	1	1	yes	characteristic
38	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₄₇	—	2	1	no	—
40	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
41	#117×C₄₇	—	2	1	no	—
42	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
43	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₄₇	—	2	1	no	—
45	C₂×D₄×C₄₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	—	2	1	no	—
47	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₄₇	—	2	2	no	—
51	#10625350	—	4	4	no	—
52	C₂×D₁₈₈	—	2	1	no	—
53	C₂×D₁₈₈	—	2	1	no	—
54	C₂×D₁₈₈	C₂³	1	2	yes	—
55	C₂×C₄×D₄₇	—	2	2	no	—
56	C₂×Dic₉₄	—	2	1	no	—
57	C₄⋊C₄×C₄₇	—	2	1	no	—
58	C₄⋊C₄×C₄₇	—	2	1	no	—
59	C₄⋊C₄×C₄₇	—	2	1	no	—
60	C₄⋊C₄×C₄₇	—	2	1	no	—
61	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
63	C₂²⋊C₄×C₄₇	—	2	1	no	—
64	C₂²⋊C₄×C₄₇	—	2	1	no	—
65	C₂²⋊C₄×C₄₇	—	2	1	no	—
66	C₂²⋊C₄×C₄₇	—	2	1	no	—
67	C₂²⋊C₄×C₄₇	—	2	1	no	—
68	C₄²×C₄₇	—	2	1	no	—
69	C₄²×C₄₇	D₄	1	1	yes	characteristic
70	C₄²×C₄₇	D₄	1	1	yes	characteristic
71	C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	C₄²×C₄₇	C₂³	1	1	yes	characteristic
73	#10625336	—	4	2	no	—
74	#10625334	—	4	2	no	—
75	(C₂×D₄₇)⋊C₄	—	2	2	no	—
76	(C₂×D₄₇)⋊C₄	—	2	2	no	—
77	C₄⋊Dic₄₇	—	2	1	no	—
78	C₄⋊Dic₄₇	C₂³	1	2	yes	—
79	C₂×C₄₇⋊C₈	—	4	2	no	—
80	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
81	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
82	C₂³×C₄₇	—	2	1	no	—
83	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₄₇	—	2	2	no	—
85	Q₈×C₄₇	—	2	1	no	—
86	Q₈×C₄₇	—	2	1	no	—
87	D₄×C₄₇	—	2	1	no	—
88	D₄×C₄₇	—	2	1	no	—
89	D₄×C₄₇	—	4	2	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	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	—	4	2	no	—
98	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₄₇	—	4	4	no	—
105	C₂×Dic₄₇	—	2	2	no	—
106	D₁₈₈	—	2	2	no	—
107	D₁₈₈	—	2	2	no	—
108	D₁₈₈	—	4	2	no	—
109	D₁₈₈	—	4	2	no	—
110	C₄×D₄₇	—	4	4	no	—
111	Dic₉₄	—	4	2	no	—
112	C₄₇⋊C₈	—	2	2	no	—
113	C₄₇⋊C₈	—	2	2	no	—
114	C₂²×C₄₇	—	2	1	no	—
115	C₂²×C₄₇	—	2	1	no	—
116	C₂²×C₄₇	—	4	1	no	—
117	C₂²×C₄₇	—	4	1	no	—
118	C₂²×C₄₇	—	4	1	no	—
119	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
120	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₄₇	—	4	2	no	—
122	C₂×D₄₇	—	4	2	no	—
123	C₂×D₄₇	—	4	2	no	—
124	C₄×C₄₇	—	2	1	no	—
125	C₄×C₄₇	—	2	1	no	—
126	C₄×C₄₇	—	2	1	no	—
127	C₄×C₄₇	—	2	1	no	—
128	C₄×C₄₇	—	4	1	no	—
129	C₄×C₄₇	—	4	1	no	—
130	C₄×C₄₇	—	4	1	no	—
131	C₄×C₄₇	#135	1	2	yes	—
132	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
133	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
134	Dic₄₇	—	4	2	no	—
135	#3247	—	47	1	no	maximal, Sylow
136	C₂×C₄₇	—	2	2	no	—
137	C₂×C₄₇	—	8	1	no	—
138	C₂×C₄₇	#537	1	1	yes	characteristic
139	C₂×C₄₇	C₂×#135	1	1	yes	characteristic
140	C₂×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
141	D₄₇	—	8	2	no	—
142	#532	—	47	1	no	—
143	#529	D₄₇	1	1	yes	characteristic, complemented
144	#478	—	47	2	no	—
145	#477	—	47	1	no	—
146	#476	—	47	1	no	—
147	#466	—	47	2	no	—
148	#465	—	47	2	no	—
149	#463	—	47	2	no	—
150	#460	—	47	1	no	—
151	#459	—	47	1	no	—
152	C₂×C₄⋊C₈	—	47	1	no	—
153	C₄₇	#3247	1	1	yes	characteristic, complemented, Sylow
154	C₂×#41	—	94	1	no	—
155	C₂×SD₁₆	—	47	2	no	—
156	C₂×D₈	—	47	2	no	—
157	C₂²×C₈	—	47	2	no	—
158	C₄⋊Q₈	—	47	1	no	—
159	#127	—	47	1	no	—
160	#126	—	2	1	no	—
161	#125	C₂×D₄₇	1	1	yes	characteristic
162	#124	—	94	1	no	—
163	#124	C₂×D₄₇	1	1	yes	characteristic
164	#123	—	2	1	no	—
165	C₂²⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
166	#121	—	47	2	no	—
167	#121	—	94	1	no	—
168	#121	C₂×D₄₇	1	1	yes	characteristic
169	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	—	47	2	no	—
172	C₄×D₄	—	94	1	no	—
173	C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
174	#117	—	2	1	no	—
175	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
176	#107	—	47	2	no	—
177	#106	—	47	2	no	—
178	C₄⋊C₈	—	47	1	no	—
179	C₄⋊C₈	—	47	1	no	—
180	C₄⋊C₈	—	47	2	no	—
181	#103	—	47	2	no	—
182	#102	—	47	2	no	—
183	#102	—	47	2	no	—
184	#102	—	47	2	no	—
185	#41	—	188	4	no	—
186	C₂×Q₈	—	94	1	no	—
187	C₂×Q₈	C₂²×D₄₇	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	47	2	no	—
190	C₂×D₄	—	94	2	no	—
191	C₂×D₄	—	94	1	no	—
192	C₂×D₄	—	94	1	no	—
193	C₂×D₄	C₂²×D₄₇	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	94	2	no	—
196	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
199	SD₁₆	—	188	2	no	—
200	D₈	—	188	2	no	—
201	C₂×C₈	—	47	2	no	—
202	C₂×C₈	—	47	2	no	—
203	C₂×C₈	—	188	2	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	47	2	no	—
209	C₄⋊C₄	—	94	1	no	—
210	C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
211	C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	94	2	no	—
218	C₂²⋊C₄	—	94	2	no	—
219	C₄²	—	2	1	no	—
220	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
221	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
222	C₄²	C₂²×D₄₇	1	1	yes	characteristic
223	C₄²	C₂²×D₄₇	1	1	yes	characteristic
224	C₂³	—	2	1	no	—
225	C₂³	—	94	2	no	—
226	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
227	Q₈	—	2	1	no	—
228	Q₈	—	2	1	no	—
229	Q₈	—	188	2	no	—
230	D₄	—	2	1	no	—
231	D₄	—	2	1	no	—
232	D₄	—	4	2	no	—
233	D₄	—	94	2	no	—
234	D₄	—	94	2	no	—
235	D₄	—	188	2	no	—
236	D₄	—	188	4	no	—
237	D₄	—	188	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	94	2	no	—
247	C₂×C₄	—	188	4	no	—
248	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
253	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
254	C₈	—	94	2	no	—
255	C₈	—	94	2	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	188	2	no	—
262	C₂²	—	188	2	no	—
263	C₂²	—	188	2	no	—
264	C₂²	#13094049	1	2	yes	—
265	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	188	2	no	—
274	C₄	#13094044	1	2	yes	—
275	C₄	#13094070	1	1	yes	characteristic
276	C₄	#13094071	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	8	1	no	—
279	C₂	—	376	2	no	—
280	C₂	#15689701	1	1	yes	characteristic
281	C₂	C₂×#13094044	1	1	yes	characteristic
282	C₂	C₂×#13094049	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#529	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#529	D₄₇
hypocenter	C₄₇	#3247
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. #15689673	C₂	semidirect
2. #15689420	C₂	semidirect
3. #15689418	C₂	semidirect
4. #15689413	C₂	semidirect
5. #15689357	C₂	semidirect
6. #15689354	C₂	semidirect
7. #15689335	C₂	semidirect
8. #15689323	C₂	semidirect
9. #15689318	C₂	semidirect
10. C₂×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
11. C₄₇×#529	C₂	semidirect
143. #529	D₄₇	semidirect
153. C₄₇	#3247	semidirect