Subgroups — #20257209

Order 6016.

988 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689851	C₂	1	4	yes	maximal, complemented
2	#15689833	C₂	1	4	yes	maximal, complemented
3	#15689827	C₂	1	8	yes	maximal, complemented
4	#15689751	C₂	1	2	yes	maximal, complemented
5	#15689750	C₂	1	4	yes	maximal, complemented
6	#15689679	C₂	1	2	yes	maximal, complemented
7	C₂×#13094016	C₂	1	1	yes	maximal, characteristic, complemented
8	#125×D₄₇	C₂	1	2	yes	maximal, complemented
9	#15689603	C₂	1	2	yes	maximal
10	C₄×#10625350	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#533	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625350	C₂²	1	1	yes	characteristic
13	#13094024	C₂²	1	4	yes	—
14	#13094023	C₂²	1	4	yes	—
15	#13094022	C₂²	1	8	yes	—
16	#13094016	C₂²	1	2	yes	—
17	#13094016	C₂²	1	4	yes	—
18	#13094016	C₂²	1	4	yes	—
19	#13094016	C₂²	1	2	yes	—
20	#13094015	C₂²	1	4	yes	—
21	#13094014	C₂²	1	8	yes	—
22	#13094014	C₂²	1	8	yes	—
23	#13094002	C₂²	1	8	yes	—
24	#13094002	C₂²	1	8	yes	—
25	#13094001	C₂²	1	8	yes	—
26	#13094000	C₂²	1	8	yes	—
27	C₄×Dic₉₄	C₂²	1	2	yes	—
28	C₄×Dic₉₄	C₂²	1	4	yes	—
29	#13093989	C₂²	1	8	yes	—
30	C₄×D₁₈₈	C₂²	1	2	yes	—
31	C₄×D₁₈₈	C₂²	1	4	yes	—
32	C₄×C₂²⋊D₄₇	C₂²	1	4	yes	—
33	(C₂²⋊D₄₇)⋊C₄	C₂²	1	8	yes	—
34	#13093983	C₂²	1	1	yes	characteristic
35	#13093982	C₂²	1	4	yes	—
36	#13093981	C₂²	1	2	yes	—
37	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
38	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
39	C₄⋊C₄×D₄₇	C₂²	1	4	yes	—
40	C₄⋊C₄×D₄₇	C₂²	1	8	yes	—
41	C₄²×D₄₇	C₂²	1	2	yes	—
42	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
43	#125×C₄₇	C₂²	1	1	yes	characteristic
44	#125×C₄₇	C₂²	1	1	yes	characteristic
45	C₂²⋊Q₈×C₄₇	C₂²	1	4	yes	—
46	C₄×Q₈×C₄₇	C₂²	1	4	yes	—
47	#117×C₄₇	C₂²	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
49	C₂×Q₈×C₄₇	C₂³	1	4	yes	—
50	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₄₇	C₂³	1	2	yes	—
53	C₂²×Dic₄₇	C₂³	1	4	yes	—
54	#10625350	Q₈	1	8	yes	complemented
55	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₄₇	C₂³	1	4	yes	—
57	C₂×C₄×D₄₇	C₂³	1	8	yes	—
58	C₂×C₄×D₄₇	C₂³	1	2	yes	—
59	C₂×Dic₉₄	C₂³	1	8	yes	—
60	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
62	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
63	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
64	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
65	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
66	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
67	C₄²×C₄₇	C₂³	1	2	yes	—
68	C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	C₄²×C₄₇	C₂³	1	1	yes	characteristic
70	C₄²×C₄₇	C₂³	1	4	yes	—
71	C₂²⋊Dic₄₇	C₂³	1	2	yes	—
72	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
73	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
74	C₄⋊Dic₄₇	C₂³	1	4	yes	—
75	C₄⋊Dic₄₇	C₂³	1	8	yes	—
76	C₄⋊Dic₄₇	C₂³	1	4	yes	—
77	C₄⋊Dic₄₇	C₂³	1	2	yes	—
78	Dic₄₇⋊C₄	C₂³	1	8	yes	—
79	Dic₄₇⋊C₄	C₂³	1	8	yes	—
80	Dic₄₇⋊C₄	C₂³	1	8	yes	—
81	Dic₄₇⋊C₄	C₂³	1	8	yes	—
82	Dic₄₇⋊C₄	C₂³	1	4	yes	—
83	Dic₄₇⋊C₄	C₂³	1	8	yes	—
84	C₄×Dic₄₇	C₂³	1	2	yes	—
85	C₄×Dic₄₇	C₂³	1	4	yes	—
86	C₄×Dic₄₇	C₂³	1	8	yes	—
87	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
88	C₂²×D₄₇	C₂⁴	1	2	yes	—
89	Q₈×C₄₇	—	4	4	no	—
90	C₂×C₄×C₄₇	—	4	2	no	—
91	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
92	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
93	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
94	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
95	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
96	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₄₇	C₂×Q₈	1	8	yes	—
99	C₂×Dic₄₇	—	4	4	no	—
100	C₂×Dic₄₇	C₂⁴	1	4	yes	—
101	C₂×Dic₄₇	C₂⁴	1	8	yes	—
102	C₂×Dic₄₇	C₂⁴	1	2	yes	—
103	D₁₈₈	C₂×Q₈	1	4	yes	—
104	C₄×D₄₇	—	4	4	no	—
105	C₄×D₄₇	—	4	8	no	—
106	C₄×D₄₇	C₂×Q₈	1	8	yes	—
107	Dic₉₄	—	4	8	no	—
108	Dic₉₄	C₂×Q₈	1	4	yes	—
109	C₂²×C₄₇	—	2	2	no	—
110	C₂²×C₄₇	C₂²×Q₈	1	2	yes	—
111	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
112	C₂×D₄₇	—	2	4	no	—
113	C₂×D₄₇	C₂²×Q₈	1	4	yes	—
114	C₄×C₄₇	—	2	4	no	—
115	C₄×C₄₇	—	2	2	no	—
116	C₄×C₄₇	—	2	4	no	—
117	C₄×C₄₇	—	2	2	no	—
118	C₄×C₄₇	C₂²×Q₈	1	2	yes	—
119	C₄×C₄₇	C₂²×Q₈	1	2	yes	—
120	Dic₄₇	—	2	4	no	—
121	Dic₄₇	—	2	8	no	—
122	Dic₄₇	C₂²×Q₈	1	4	yes	—
123	#3504	—	47	1	no	maximal, Sylow
124	C₂×C₄₇	—	2	2	no	—
125	C₂×C₄₇	C₂³×Q₈	1	1	yes	characteristic
126	C₂×C₄₇	#585	1	2	yes	—
127	D₄₇	—	2	4	no	—
128	#557	—	47	4	no	—
129	#554	—	47	8	no	—
130	#554	—	47	4	no	—
131	#544	—	47	4	no	—
132	#544	—	47	2	no	—
133	#533	—	47	2	no	—
134	#533	D₄₇	1	1	yes	characteristic, complemented
135	C₂×#125	—	47	2	no	—
136	C₂×#125	—	47	1	no	—
137	#520	—	47	2	no	—
138	C₄×#41	—	47	1	no	—
139	C₄₇	#3504	1	1	yes	characteristic, complemented, Sylow
140	C₂×#41	—	47	1	no	—
141	C₄⋊Q₈	—	47	8	no	—
142	C₄⋊Q₈	—	47	4	no	—
143	C₄⋊Q₈	—	47	4	no	—
144	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
145	#125	—	47	4	no	—
146	#125	—	47	8	no	—
147	#125	—	47	8	no	—
148	#125	—	47	2	no	—
149	#125	—	47	2	no	—
150	#125	—	47	4	no	—
151	#125	—	47	4	no	—
152	#125	C₂×D₄₇	1	1	yes	characteristic
153	#125	C₂×D₄₇	1	1	yes	characteristic
154	C₂²⋊Q₈	—	47	8	no	—
155	C₂²⋊Q₈	—	47	8	no	—
156	C₂²⋊Q₈	—	47	8	no	—
157	C₂²⋊Q₈	—	47	8	no	—
158	C₂²⋊Q₈	C₂×D₄₇	1	4	yes	—
159	C₄×Q₈	—	47	8	no	—
160	C₄×Q₈	—	47	2	no	—
161	C₄×Q₈	—	47	4	no	—
162	C₄×Q₈	C₂×D₄₇	1	4	yes	—
163	C₄×D₄	—	47	4	no	—
164	C₄×D₄	—	47	8	no	—
165	C₄×D₄	—	47	2	no	—
166	C₄×D₄	—	47	4	no	—
167	#117	—	47	1	no	—
168	#117	—	47	4	no	—
169	#117	—	47	2	no	—
170	#117	C₂×D₄₇	1	2	yes	—
171	C₂×C₄⋊C₄	—	47	2	no	—
172	C₂×C₄⋊C₄	—	47	4	no	—
173	C₂×C₄⋊C₄	—	47	4	no	—
174	C₂×C₄⋊C₄	—	47	8	no	—
175	C₂×C₄²	—	47	2	no	—
176	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
177	#41	—	47	8	no	—
178	C₂×Q₈	—	47	1	no	—
179	C₂×Q₈	—	47	8	no	—
180	C₂×Q₈	C₂²×D₄₇	1	4	yes	—
181	C₂×D₄	—	47	2	no	—
182	C₂×D₄	—	47	1	no	—
183	C₂²×C₄	—	47	2	no	—
184	C₂²×C₄	—	47	4	no	—
185	C₂²×C₄	—	47	4	no	—
186	C₂²×C₄	—	47	8	no	—
187	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
188	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
189	C₄⋊C₄	—	47	8	no	—
190	C₄⋊C₄	—	47	8	no	—
191	C₄⋊C₄	—	47	8	no	—
192	C₄⋊C₄	—	47	8	no	—
193	C₄⋊C₄	—	47	4	no	—
194	C₄⋊C₄	—	47	8	no	—
195	C₄⋊C₄	—	47	4	no	—
196	C₄⋊C₄	—	47	4	no	—
197	C₄⋊C₄	—	47	8	no	—
198	C₄⋊C₄	—	47	2	no	—
199	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
200	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
201	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
202	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
203	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
204	C₂²⋊C₄	—	47	2	no	—
205	C₂²⋊C₄	—	47	4	no	—
206	C₂²⋊C₄	—	47	8	no	—
207	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
208	C₄²	—	47	2	no	—
209	C₄²	—	47	4	no	—
210	C₄²	—	47	8	no	—
211	C₄²	C₂²×D₄₇	1	1	yes	characteristic
212	C₄²	C₂²×D₄₇	1	1	yes	characteristic
213	C₄²	C₂²×D₄₇	1	2	yes	—
214	C₄²	C₂²×D₄₇	1	4	yes	—
215	C₂³	—	47	2	no	—
216	C₂³	C₂³×D₄₇	1	1	yes	characteristic
217	Q₈	—	4	4	no	—
218	Q₈	—	47	4	no	—
219	Q₈	—	188	8	no	—
220	D₄	—	47	8	no	—
221	D₄	—	47	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	47	2	no	—
224	C₂×C₄	—	47	8	no	—
225	C₂×C₄	—	47	4	no	—
226	C₂×C₄	—	47	8	no	—
227	C₂×C₄	—	188	4	no	—
228	C₂×C₄	—	188	8	no	—
229	C₂×C₄	—	188	4	no	—
230	C₂×C₄	Q₈×D₄₇	1	4	yes	—
231	C₂×C₄	C₂³×D₄₇	1	2	yes	—
232	C₂×C₄	C₂³×D₄₇	1	4	yes	—
233	C₂×C₄	C₂³×D₄₇	1	2	yes	—
234	C₂×C₄	C₂³×D₄₇	1	4	yes	—
235	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
236	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
237	C₂²	—	2	2	no	—
238	C₂²	—	47	4	no	—
239	C₂²	—	94	4	no	—
240	C₂²	C₂×Q₈×D₄₇	1	2	yes	—
241	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	4	no	—
245	C₄	—	2	4	no	—
246	C₄	—	47	4	no	—
247	C₄	—	94	8	no	—
248	C₄	—	94	4	no	—
249	C₄	C₂×Q₈×D₄₇	1	2	yes	—
250	C₄	C₂×Q₈×D₄₇	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	94	4	no	—
253	C₂	C₂²×Q₈×D₄₇	1	1	yes	characteristic
254	C₂	#15689959	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	C₄₇×#533	C₂
hypercenter	#533	D₄₇
hypocenter	C₄₇	#3504
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. #15689851	C₂	semidirect
2. #15689833	C₂	semidirect
3. #15689827	C₂	semidirect
4. #15689751	C₂	semidirect
5. #15689750	C₂	semidirect
6. #15689679	C₂	semidirect
7. C₂×#13094016	C₂	semidirect
8. #125×D₄₇	C₂	semidirect
11. C₄₇×#533	C₂	semidirect
54. #10625350	Q₈	semidirect
134. #533	D₄₇	semidirect
139. C₄₇	#3504	semidirect