Subgroups — #20257087

Order 6016.

1052 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689815	C₂	1	8	yes	maximal, complemented
2	#15689800	C₂	1	8	yes	maximal, complemented
3	#15689722	C₂	1	4	yes	maximal, complemented
4	#15689685	C₂	1	1	yes	maximal, characteristic
5	#15689659	C₂	1	4	yes	maximal, complemented
6	C₂×#13094008	C₂	1	2	yes	maximal, complemented
7	#15689596	C₂	1	2	yes	maximal, complemented
8	C₂×#13093980	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#543	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₂²⋊D₄₇	C₂²	1	2	yes	—
11	C₂³×Dic₄₇	C₂²	1	1	yes	characteristic
12	#13094019	C₂²	1	8	yes	complemented
13	#13094019	C₂²	1	8	yes	complemented
14	#13094015	C₂²	1	4	yes	complemented
15	#13094015	C₂²	1	8	yes	complemented
16	#13094013	C₂²	1	4	yes	complemented
17	#13094009	C₂²	1	8	yes	complemented
18	#13094008	C₂²	1	8	yes	—
19	#13094008	C₂²	1	4	yes	—
20	#13094008	C₂²	1	4	yes	complemented
21	#13094008	C₂²	1	4	yes	complemented
22	#13094000	C₂²	1	8	yes	complemented
23	#13093998	C₂²	1	4	yes	—
24	#13093994	C₂²	1	4	yes	—
25	Q₈×Dic₄₇	C₂²	1	4	yes	complemented
26	#13093986	C₂²	1	4	yes	complemented
27	C₄×C₂²⋊D₄₇	C₂²	1	8	yes	—
28	(C₂²⋊D₄₇)⋊C₄	C₂²	1	8	yes	complemented
29	#13093982	C₂²	1	8	yes	complemented
30	#13093980	C₂²	1	8	yes	complemented
31	#13093980	C₂²	1	4	yes	complemented
32	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	complemented
33	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	complemented
34	C₂×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
35	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	complemented
36	C₂²⋊C₄×D₄₇	C₂²	1	4	yes	complemented
37	C₂×C₄×Dic₄₇	C₂²	1	2	yes	complemented
38	C₂²⋊Q₈×C₄₇	C₂²	1	8	yes	complemented
39	C₂²⋊Q₈×C₄₇	C₂²	1	4	yes	complemented
40	C₂×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
42	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
43	C₂³×D₄₇	C₂³	1	2	yes	—
44	C₂×Q₈×C₄₇	C₂³	1	4	yes	—
45	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
47	C₂×C₂²⋊D₄₇	—	2	8	no	—
48	C₂×C₂²⋊D₄₇	C₂³	1	4	yes	—
49	C₂×C₂²⋊D₄₇	C₂³	1	4	yes	—
50	C₂²×Dic₄₇	—	2	4	no	—
51	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
52	C₂²×Dic₄₇	C₂³	1	4	yes	—
53	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
54	C₂²×Dic₄₇	C₂³	1	2	yes	—
55	C₂²×Dic₄₇	C₂³	1	2	yes	—
56	C₂×D₁₈₈	C₂³	1	4	yes	—
57	C₂×C₄×D₄₇	C₂³	1	8	yes	—
58	C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
59	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
60	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
61	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
62	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
63	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
64	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
65	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
66	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
67	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
68	C₄⋊Dic₄₇	C₂³	1	8	yes	—
69	C₄⋊Dic₄₇	C₂³	1	8	yes	—
70	C₄⋊Dic₄₇	C₂³	1	4	yes	—
71	Dic₄₇⋊C₄	C₂³	1	8	yes	—
72	Dic₄₇⋊C₄	C₂³	1	8	yes	—
73	C₄×Dic₄₇	C₂³	1	4	yes	—
74	C₄×Dic₄₇	C₂³	1	4	yes	—
75	C₄×Dic₄₇	C₂³	1	8	yes	—
76	C₂³×C₄₇	—	4	2	no	—
77	C₂³×C₄₇	#41	1	4	yes	—
78	C₂³×C₄₇	C₂⁴	1	2	yes	—
79	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂²×D₄₇	—	2	4	no	—
81	C₂²×D₄₇	—	4	4	no	—
82	C₂²×D₄₇	C₂⁴	1	4	yes	—
83	Q₈×C₄₇	—	4	4	no	—
84	C₂×C₄×C₄₇	—	4	4	no	—
85	C₂×C₄×C₄₇	—	4	2	no	—
86	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
87	C₂×C₄×C₄₇	C₂⁴	1	8	yes	—
88	C₂²⋊D₄₇	—	2	8	no	—
89	C₂²⋊D₄₇	—	2	8	no	—
90	C₂×Dic₄₇	—	2	8	no	—
91	C₂×Dic₄₇	—	2	2	no	—
92	C₂×Dic₄₇	—	2	2	no	—
93	C₂×Dic₄₇	—	2	4	no	—
94	C₂×Dic₄₇	—	4	2	no	—
95	C₂×Dic₄₇	C₂⁴	1	4	yes	—
96	C₂×Dic₄₇	C₂⁴	1	2	yes	—
97	C₂×Dic₄₇	C₂⁴	1	4	yes	—
98	C₂×Dic₄₇	C₂⁴	1	2	yes	—
99	D₁₈₈	—	4	4	no	—
100	C₄×D₄₇	—	4	8	no	—
101	C₂²×C₄₇	—	2	2	no	—
102	C₂²×C₄₇	—	2	4	no	—
103	C₂²×C₄₇	—	4	4	no	—
104	C₂²×C₄₇	C₂×#41	1	4	yes	—
105	C₂²×C₄₇	C₂×#41	1	2	yes	—
106	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
107	C₂×D₄₇	—	2	4	no	—
108	C₂×D₄₇	—	2	8	no	—
109	C₂×D₄₇	—	8	4	no	—
110	C₄×C₄₇	—	2	8	no	—
111	C₄×C₄₇	—	2	4	no	—
112	Dic₄₇	—	2	2	no	—
113	Dic₄₇	—	2	2	no	—
114	Dic₄₇	—	2	4	no	—
115	Dic₄₇	—	2	4	no	—
116	#3491	—	47	1	no	maximal, Sylow
117	C₂×C₄₇	—	2	4	no	—
118	C₂×C₄₇	—	2	2	no	—
119	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
120	C₂×C₄₇	#585	1	2	yes	—
121	D₄₇	—	4	4	no	—
122	#552	—	47	8	no	—
123	#551	—	47	8	no	—
124	#543	D₄₇	1	1	yes	characteristic, complemented
125	#539	—	47	4	no	—
126	#534	—	47	1	no	—
127	#529	—	47	4	no	—
128	C₂×#123	—	47	2	no	—
129	#518	—	47	2	no	—
130	C₂×#117	—	47	1	no	—
131	C₄₇	#3491	1	1	yes	characteristic, complemented, Sylow
132	C₂²×D₄	—	47	2	no	—
133	C₂³×C₄	—	47	1	no	—
134	#126	—	47	8	no	—
135	#126	—	47	8	no	—
136	#125	—	47	4	no	—
137	#125	—	47	8	no	—
138	#124	—	47	4	no	—
139	#123	—	47	8	no	—
140	#123	—	47	4	no	—
141	#123	—	47	4	no	—
142	#123	—	47	8	no	—
143	#123	—	47	4	no	—
144	C₂²⋊Q₈	—	47	8	no	—
145	C₂²⋊Q₈	C₂×D₄₇	1	4	yes	complemented
146	C₂²⋊Q₈	C₂×D₄₇	1	8	yes	complemented
147	#121	—	47	4	no	—
148	#120	—	47	4	no	—
149	C₄×Q₈	—	47	4	no	—
150	C₄×D₄	—	47	8	no	—
151	C₄×D₄	—	47	4	no	—
152	C₄×D₄	—	47	8	no	—
153	#117	—	47	8	no	—
154	#117	—	47	8	no	—
155	#117	—	47	4	no	—
156	C₂×C₄⋊C₄	—	47	2	no	—
157	C₂×C₄⋊C₄	—	47	4	no	—
158	C₂×C₂²⋊C₄	—	47	1	no	—
159	C₂×C₂²⋊C₄	—	47	4	no	—
160	C₂×C₂²⋊C₄	—	47	4	no	—
161	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
162	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
163	C₂×C₄²	—	47	2	no	—
164	C₂⁴	—	47	2	no	—
165	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
166	C₂×Q₈	C₂²×D₄₇	1	4	yes	—
167	C₂×D₄	—	47	4	no	—
168	C₂×D₄	—	47	4	no	—
169	C₂×D₄	—	47	4	no	—
170	C₂×D₄	—	94	8	no	—
171	C₂²×C₄	—	47	1	no	—
172	C₂²×C₄	—	47	4	no	—
173	C₂²×C₄	—	47	2	no	—
174	C₂²×C₄	—	47	1	no	—
175	C₂²×C₄	—	47	2	no	—
176	C₂²×C₄	—	47	8	no	—
177	C₂²×C₄	—	94	4	no	—
178	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
179	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
180	C₄⋊C₄	—	47	8	no	—
181	C₄⋊C₄	—	47	8	no	—
182	C₄⋊C₄	—	47	8	no	—
183	C₄⋊C₄	—	47	8	no	—
184	C₄⋊C₄	—	47	4	no	—
185	C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
186	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
187	C₂²⋊C₄	—	47	4	no	—
188	C₂²⋊C₄	—	47	8	no	—
189	C₂²⋊C₄	—	47	8	no	—
190	C₂²⋊C₄	—	47	8	no	—
191	C₂²⋊C₄	—	47	8	no	—
192	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
193	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
194	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
195	C₄²	—	47	4	no	—
196	C₄²	—	47	4	no	—
197	C₄²	—	47	8	no	—
198	C₂³	—	4	2	no	—
199	C₂³	—	47	4	no	—
200	C₂³	—	94	4	no	—
201	C₂³	—	188	4	no	—
202	C₂³	#10625354	1	4	yes	—
203	C₂³	C₂³×D₄₇	1	2	yes	—
204	C₂³	C₂³×D₄₇	1	1	yes	characteristic
205	Q₈	—	4	4	no	—
206	D₄	—	94	8	no	—
207	D₄	—	94	8	no	—
208	D₄	—	188	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	47	2	no	—
212	C₂×C₄	—	47	4	no	—
213	C₂×C₄	—	47	4	no	—
214	C₂×C₄	—	47	2	no	—
215	C₂×C₄	—	94	8	no	—
216	C₂×C₄	—	94	2	no	—
217	C₂×C₄	—	94	2	no	—
218	C₂×C₄	—	94	4	no	—
219	C₂×C₄	—	188	8	no	—
220	C₂×C₄	—	188	2	no	—
221	C₂×C₄	C₂³×D₄₇	1	4	yes	—
222	C₂×C₄	C₂³×D₄₇	1	8	yes	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	4	no	—
225	C₂²	—	4	4	no	—
226	C₂²	—	94	4	no	—
227	C₂²	—	94	8	no	—
228	C₂²	—	376	4	no	—
229	C₂²	C₂×#10625354	1	4	yes	—
230	C₂²	C₂×#10625354	1	2	yes	—
231	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
232	C₄	—	2	8	no	—
233	C₄	—	2	4	no	—
234	C₄	—	94	2	no	—
235	C₄	—	94	2	no	—
236	C₄	—	94	4	no	—
237	C₄	—	94	4	no	—
238	C₂	—	2	4	no	—
239	C₂	—	2	2	no	—
240	C₂	—	188	4	no	—
241	C₂	C₂²×#10625354	1	1	yes	characteristic
242	C₂	#15689958	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	C₄₇×#543	C₂
hypercenter	#543	D₄₇
hypocenter	C₄₇	#3491
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #543

Complements

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

N	H ≅ G/N	type
1. #15689815	C₂	semidirect
2. #15689800	C₂	semidirect
3. #15689722	C₂	semidirect
5. #15689659	C₂	semidirect
6. C₂×#13094008	C₂	semidirect
7. #15689596	C₂	semidirect
8. C₂×#13093980	C₂	semidirect
9. C₄₇×#543	C₂	semidirect
12. #13094019	C₂²	semidirect
13. #13094019	C₂²	semidirect
14. #13094015	C₂²	semidirect
15. #13094015	C₂²	semidirect
16. #13094013	C₂²	semidirect
17. #13094009	C₂²	semidirect
20. #13094008	C₂²	semidirect
21. #13094008	C₂²	semidirect
22. #13094000	C₂²	semidirect
25. Q₈×Dic₄₇	C₂²	semidirect
26. #13093986	C₂²	semidirect
28. (C₂²⋊D₄₇)⋊C₄	C₂²	semidirect
29. #13093982	C₂²	semidirect
30. #13093980	C₂²	semidirect
31. #13093980	C₂²	semidirect
32. C₂×C₄⋊Dic₄₇	C₂²	semidirect
33. C₂×Dic₄₇⋊C₄	C₂²	semidirect
35. C₂×(C₂×D₄₇)⋊C₄	C₂²	semidirect
36. C₂²⋊C₄×D₄₇	C₂²	semidirect
37. C₂×C₄×Dic₄₇	C₂²	semidirect
38. C₂²⋊Q₈×C₄₇	C₂²	semidirect
39. C₂²⋊Q₈×C₄₇	C₂²	semidirect
124. #543	D₄₇	semidirect
131. C₄₇	#3491	semidirect
145. C₂²⋊Q₈	C₂×D₄₇	semidirect
146. C₂²⋊Q₈	C₂×D₄₇	semidirect