Subgroups — #20256778

Order 6016.

1040 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689744	C₂	1	4	yes	maximal, complemented
2	#15689735	C₂	1	2	yes	maximal, complemented
3	#15689728	C₂	1	2	yes	maximal, complemented
4	#15689720	C₂	1	4	yes	maximal, complemented
5	#15689711	C₂	1	4	yes	maximal, complemented
6	#15689680	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689673	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689651	C₂	1	8	yes	maximal, complemented
9	#15689634	C₂	1	2	yes	maximal, complemented
10	C₄×#10625350	C₂	1	2	yes	maximal, complemented
11	C₂×#117×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625350	C₂²	1	2	yes	—
13	#13094024	C₂²	1	1	yes	characteristic, complemented
14	#13094024	C₂²	1	2	yes	complemented
15	#13094021	C₂²	1	1	yes	characteristic, complemented
16	#13094019	C₂²	1	8	yes	complemented
17	#13094017	C₂²	1	8	yes	—
18	#13094016	C₂²	1	2	yes	complemented
19	#13094014	C₂²	1	8	yes	complemented
20	#13094012	C₂²	1	2	yes	complemented
21	#13094010	C₂²	1	8	yes	—
22	#13094009	C₂²	1	8	yes	complemented
23	#13094007	C₂²	1	4	yes	—
24	#13094006	C₂²	1	8	yes	—
25	#13094003	C₂²	1	4	yes	—
26	#13094003	C₂²	1	4	yes	—
27	#13094002	C₂²	1	8	yes	complemented
28	#13093998	C₂²	1	4	yes	—
29	#13093998	C₂²	1	4	yes	—
30	#13093997	C₂²	1	8	yes	—
31	#13093993	C₂²	1	4	yes	—
32	C₄×Dic₉₄	C₂²	1	4	yes	complemented
33	C₄×Dic₉₄	C₂²	1	4	yes	complemented
34	C₄×D₁₈₈	C₂²	1	4	yes	complemented
35	C₄×D₁₈₈	C₂²	1	4	yes	complemented
36	C₄×C₂²⋊D₄₇	C₂²	1	8	yes	—
37	C₄×C₂²⋊D₄₇	C₂²	1	8	yes	—
38	#13093983	C₂²	1	2	yes	—
39	#13093981	C₂²	1	4	yes	complemented
40	C₄²×D₄₇	C₂²	1	4	yes	complemented
41	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
42	#117×C₄₇	C₂²	1	2	yes	—
43	#117×C₄₇	C₂²	1	2	yes	—
44	#117×C₄₇	C₂²	1	4	yes	—
45	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
46	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
49	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₄₇	C₂³	1	4	yes	—
58	C₂×C₂²⋊D₄₇	C₂³	1	8	yes	—
59	#10625350	—	2	8	no	—
60	C₂×D₁₈₈	C₂³	1	2	yes	—
61	C₂×D₁₈₈	C₂³	1	4	yes	—
62	C₂×C₄×D₄₇	C₂³	1	4	yes	—
63	C₂×C₄×D₄₇	C₂³	1	8	yes	—
64	C₂×Dic₉₄	C₂³	1	2	yes	—
65	C₂×Dic₉₄	C₂³	1	4	yes	—
66	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
67	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
68	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
69	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
70	C₄²×C₄₇	C₂³	1	2	yes	—
71	C₄²×C₄₇	C₂³	1	2	yes	—
72	C₄²×C₄₇	C₂³	1	2	yes	—
73	C₄²×C₄₇	C₂³	1	1	yes	characteristic
74	C₄²×C₄₇	C₂³	1	1	yes	characteristic
75	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
76	C₂²⋊Dic₄₇	C₂³	1	8	yes	—
77	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
78	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
79	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
80	C₄⋊Dic₄₇	C₂³	1	4	yes	—
81	C₄⋊Dic₄₇	C₂³	1	4	yes	—
82	C₄⋊Dic₄₇	C₂³	1	4	yes	—
83	Dic₄₇⋊C₄	C₂³	1	8	yes	—
84	Dic₄₇⋊C₄	C₂³	1	8	yes	—
85	Dic₄₇⋊C₄	C₂³	1	8	yes	—
86	C₄×Dic₄₇	C₂³	1	4	yes	—
87	C₄×Dic₄₇	C₂³	1	8	yes	—
88	C₂³×C₄₇	—	2	2	no	—
89	C₂³×C₄₇	—	4	2	no	—
90	C₂³×C₄₇	C₂⁴	1	2	yes	—
91	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
92	C₂²×D₄₇	C₂⁴	1	4	yes	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	4	2	no	—
97	C₂×C₄×C₄₇	—	4	2	no	—
98	C₂×C₄×C₄₇	#41	1	4	yes	—
99	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
100	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
101	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
102	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
103	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
105	C₂²⋊D₄₇	—	2	8	no	—
106	C₂²⋊D₄₇	—	4	8	no	—
107	C₂×Dic₄₇	C₂⁴	1	4	yes	—
108	C₂×Dic₄₇	C₂⁴	1	8	yes	—
109	D₁₈₈	—	2	4	no	—
110	D₁₈₈	—	4	4	no	—
111	C₄×D₄₇	—	2	8	no	—
112	C₄×D₄₇	—	4	8	no	—
113	Dic₉₄	—	2	4	no	—
114	Dic₉₄	—	4	4	no	—
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	2	4	no	—
118	C₂²×C₄₇	—	4	4	no	—
119	C₂²×C₄₇	C₂×#41	1	2	yes	—
120	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
121	C₂×D₄₇	—	2	4	no	—
122	C₂×D₄₇	—	2	8	no	—
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	—	2	2	no	—
125	C₄×C₄₇	—	2	4	no	—
126	C₄×C₄₇	—	2	2	no	—
127	C₄×C₄₇	C₂×#41	1	2	yes	—
128	C₄×C₄₇	C₂×#41	1	2	yes	—
129	Dic₄₇	—	2	4	no	—
130	Dic₄₇	—	2	8	no	—
131	#3455	—	47	1	no	maximal, Sylow
132	C₂×C₄₇	—	2	2	no	—
133	C₂×C₄₇	—	4	2	no	—
134	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
135	C₂×C₄₇	#585	1	2	yes	—
136	D₄₇	—	4	4	no	—
137	#542	—	47	4	no	—
138	#541	—	47	2	no	—
139	#540	—	47	2	no	—
140	#539	—	47	4	no	—
141	#538	—	47	4	no	—
142	#533	—	47	1	no	—
143	#532	—	47	1	no	—
144	#529	—	47	8	no	—
145	#525	—	47	2	no	—
146	C₄×#41	—	47	2	no	—
147	C₂×#117	D₄₇	1	1	yes	characteristic, complemented
148	C₄₇	#3455	1	1	yes	characteristic, complemented, Sylow
149	C₂×#41	—	47	2	no	—
150	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
151	C₄⋊Q₈	—	47	1	no	—
152	C₄⋊Q₈	—	47	2	no	—
153	#127	—	47	1	no	—
154	#126	—	47	8	no	—
155	#126	—	47	8	no	—
156	#125	—	47	8	no	—
157	#125	—	47	2	no	—
158	#124	—	47	8	no	—
159	#124	—	47	2	no	—
160	#123	—	47	4	no	—
161	#123	—	47	8	no	—
162	#123	—	47	8	no	—
163	C₂²⋊Q₈	—	47	4	no	—
164	C₂²⋊Q₈	—	47	4	no	—
165	C₂²⋊Q₈	—	47	8	no	—
166	#121	—	47	4	no	—
167	#121	—	47	4	no	—
168	#121	—	47	8	no	—
169	#120	—	47	4	no	—
170	C₄×Q₈	—	47	4	no	—
171	C₄×Q₈	—	47	4	no	—
172	C₄×D₄	—	47	8	no	—
173	C₄×D₄	—	47	8	no	—
174	C₄×D₄	—	47	4	no	—
175	C₄×D₄	—	47	4	no	—
176	#117	—	47	2	no	—
177	#117	—	47	4	no	—
178	#117	C₂×D₄₇	1	4	yes	—
179	#117	C₂×D₄₇	1	2	yes	—
180	#117	C₂×D₄₇	1	2	yes	—
181	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
182	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
183	C₂×C₄²	—	47	4	no	—
184	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
185	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
186	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
187	#41	—	94	8	no	—
188	C₂×Q₈	—	47	2	no	—
189	C₂×Q₈	—	47	4	no	—
190	C₂×D₄	—	47	4	no	—
191	C₂×D₄	—	47	8	no	—
192	C₂×D₄	—	47	2	no	—
193	C₂×D₄	—	47	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	47	4	no	—
196	C₂²×C₄	—	47	8	no	—
197	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
199	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
200	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
201	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
202	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
203	C₄⋊C₄	—	47	8	no	—
204	C₄⋊C₄	—	47	8	no	—
205	C₄⋊C₄	—	47	8	no	—
206	C₄⋊C₄	—	47	4	no	—
207	C₄⋊C₄	—	47	4	no	—
208	C₄⋊C₄	—	47	4	no	—
209	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
210	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
211	C₂²⋊C₄	—	47	4	no	—
212	C₂²⋊C₄	—	47	8	no	—
213	C₂²⋊C₄	—	47	8	no	—
214	C₂²⋊C₄	—	47	8	no	—
215	C₂²⋊C₄	—	47	8	no	—
216	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
217	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
218	C₄²	—	47	8	no	—
219	C₄²	—	47	4	no	—
220	C₄²	C₂²×D₄₇	1	2	yes	—
221	C₄²	C₂²×D₄₇	1	1	yes	characteristic
222	C₄²	C₂²×D₄₇	1	1	yes	characteristic
223	C₄²	C₂²×D₄₇	1	2	yes	—
224	C₄²	C₂²×D₄₇	1	2	yes	—
225	C₂³	—	2	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	47	4	no	—
228	C₂³	C₂³×D₄₇	1	2	yes	—
229	C₂³	C₂³×D₄₇	1	1	yes	characteristic
230	Q₈	—	94	4	no	—
231	Q₈	—	188	4	no	—
232	D₄	—	94	8	no	—
233	D₄	—	94	4	no	—
234	D₄	—	188	4	no	—
235	D₄	—	188	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	47	4	no	—
242	C₂×C₄	—	47	8	no	—
243	C₂×C₄	—	94	8	no	—
244	C₂×C₄	—	188	8	no	—
245	C₂×C₄	#10625350	1	4	yes	—
246	C₂×C₄	C₂³×D₄₇	1	4	yes	—
247	C₂×C₄	C₂³×D₄₇	1	2	yes	—
248	C₂×C₄	C₂³×D₄₇	1	2	yes	—
249	C₂×C₄	C₂³×D₄₇	1	2	yes	—
250	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
251	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	94	4	no	—
257	C₂²	—	94	8	no	—
258	C₂²	C₂×#10625350	1	2	yes	—
259	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	94	4	no	—
265	C₄	—	94	8	no	—
266	C₄	C₂×#10625350	1	2	yes	—
267	C₄	C₂×#10625350	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	4	2	no	—
270	C₂	—	188	4	no	—
271	C₂	C₂²×#10625350	1	1	yes	characteristic
272	C₂	#15689959	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Frattini	C₂²	C₂⁴×D₄₇
Fitting	C₂×#117×C₄₇	C₂
hypercenter	C₂×#117	D₄₇
hypocenter	C₄₇	#3455
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #15689744	C₂	semidirect
2. #15689735	C₂	semidirect
3. #15689728	C₂	semidirect
4. #15689720	C₂	semidirect
5. #15689711	C₂	semidirect
6. #15689680	C₂	semidirect
7. #15689673	C₂	semidirect
8. #15689651	C₂	semidirect
9. #15689634	C₂	semidirect
10. C₄×#10625350	C₂	semidirect
11. C₂×#117×C₄₇	C₂	semidirect
13. #13094024	C₂²	semidirect
14. #13094024	C₂²	semidirect
15. #13094021	C₂²	semidirect
16. #13094019	C₂²	semidirect
18. #13094016	C₂²	semidirect
19. #13094014	C₂²	semidirect
20. #13094012	C₂²	semidirect
22. #13094009	C₂²	semidirect
27. #13094002	C₂²	semidirect
32. C₄×Dic₉₄	C₂²	semidirect
33. C₄×Dic₉₄	C₂²	semidirect
34. C₄×D₁₈₈	C₂²	semidirect
35. C₄×D₁₈₈	C₂²	semidirect
39. #13093981	C₂²	semidirect
40. C₄²×D₄₇	C₂²	semidirect
147. C₂×#117	D₄₇	semidirect
148. C₄₇	#3455	semidirect