Subgroups — #20248407

Order 6016.

468 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094023	C₂	1	1	yes	maximal, characteristic
2	#15689054	C₂	1	2	yes	maximal
3	#15689043	C₂	1	2	yes	maximal
4	#15689028	C₂	1	2	yes	maximal
5	#15689021	C₂	1	1	yes	maximal, characteristic
6	#15689019	C₂	1	1	yes	maximal, characteristic
7	#15689018	C₂	1	2	yes	maximal
8	#15688988	C₂	1	1	yes	maximal, characteristic
9	C₄⋊C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic
10	C₄×Dic₄₇⋊C₄	C₂	1	1	yes	maximal, characteristic
11	C₂×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₉₄	C₂²	1	2	yes	—
13	#13094023	—	2	4	no	—
14	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
19	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
23	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
24	C₄₇⋊#95	C₂²	1	1	yes	characteristic
25	C₄₇⋊#95	C₂²	1	1	yes	characteristic
26	C₄₇⋊#95	C₂²	1	2	yes	—
27	C₄₇⋊#95	C₂²	1	2	yes	—
28	C₄₇⋊#95	C₂²	1	2	yes	—
29	C₄₇⋊#95	C₂²	1	2	yes	—
30	C₄₇⋊#95	C₂²	1	2	yes	—
31	#125×C₄₇	—	2	4	no	—
32	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
34	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
35	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²×Dic₄₇	C₂³	1	2	yes	—
43	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
44	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
45	C₂²×Dic₄₇	C₂³	1	2	yes	—
46	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
47	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₉₄	—	2	4	no	—
49	C₂×Dic₉₄	—	2	4	no	—
50	C₂×Dic₉₄	—	4	2	no	—
51	C₄⋊C₄×C₄₇	—	2	4	no	—
52	C₄⋊C₄×C₄₇	—	2	4	no	—
53	C₄⋊C₄×C₄₇	—	2	2	no	—
54	C₄⋊C₄×C₄₇	D₄	1	4	yes	—
55	C₄²×C₄₇	—	2	2	no	—
56	C₄⋊Dic₄₇	—	2	2	no	—
57	C₄⋊Dic₄₇	Q₈	1	4	yes	complemented
58	Dic₄₇⋊C₄	—	2	4	no	—
59	Dic₄₇⋊C₄	—	2	2	no	—
60	Dic₄₇⋊C₄	—	2	4	no	—
61	Dic₄₇⋊C₄	—	2	2	no	—
62	Dic₄₇⋊C₄	—	4	2	no	—
63	C₄×Dic₄₇	—	2	2	no	—
64	C₄×Dic₄₇	—	4	2	no	—
65	C₄×Dic₄₇	Q₈	1	4	yes	complemented
66	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	1	no	—
70	C₂×C₄×C₄₇	—	2	1	no	—
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	—	2	1	no	—
73	C₂×C₄×C₄₇	—	2	1	no	—
74	C₂×C₄×C₄₇	—	2	2	no	—
75	C₂×C₄×C₄₇	—	2	1	no	—
76	C₂×C₄×C₄₇	—	2	1	no	—
77	C₂×C₄×C₄₇	—	2	2	no	—
78	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
79	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
81	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
82	C₂×Dic₄₇	—	2	1	no	—
83	C₂×Dic₄₇	—	2	1	no	—
84	C₂×Dic₄₇	—	2	1	no	—
85	C₂×Dic₄₇	—	2	1	no	—
86	C₂×Dic₄₇	—	2	2	no	—
87	C₂×Dic₄₇	—	2	1	no	—
88	C₂×Dic₄₇	—	2	2	no	—
89	C₂×Dic₄₇	—	2	1	no	—
90	C₂×Dic₄₇	—	2	1	no	—
91	C₂×Dic₄₇	—	2	1	no	—
92	C₂×Dic₄₇	—	2	2	no	—
93	C₂×Dic₄₇	—	2	2	no	—
94	C₂×Dic₄₇	—	2	2	no	—
95	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
96	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
97	C₂×Dic₄₇	C₂×Q₈	1	4	yes	—
98	C₂×Dic₄₇	#41	1	2	yes	—
99	C₂×Dic₄₇	#41	1	2	yes	—
100	Dic₉₄	—	4	8	no	—
101	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
102	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
103	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
104	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
105	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
106	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
107	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
108	C₄×C₄₇	—	2	4	no	—
109	C₄×C₄₇	—	4	1	no	—
110	C₄×C₄₇	—	4	1	no	—
111	C₄×C₄₇	—	4	2	no	—
112	C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
113	Dic₄₇	—	2	2	no	—
114	Dic₄₇	—	2	2	no	—
115	Dic₄₇	—	2	2	no	—
116	Dic₄₇	—	2	2	no	—
117	Dic₄₇	—	2	4	no	—
118	Dic₄₇	—	4	2	no	—
119	#2550	—	47	1	no	maximal, Sylow
120	C₂×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₂×C₄₇	#533	1	1	yes	characteristic
122	C₂×C₄₇	#544	1	1	yes	characteristic
123	C₂×C₄₇	#546	1	1	yes	characteristic
124	C₂×C₄₇	#548	1	1	yes	characteristic
125	C₂×C₄₇	#554	1	1	yes	characteristic
126	C₂×C₄₇	#557	1	1	yes	characteristic
127	C₂×C₄⋊Q₈	—	47	1	no	—
128	C₂×#125	D₄₇	1	1	yes	characteristic
129	#398	—	47	2	no	—
130	#395	—	47	2	no	—
131	#391	—	47	2	no	—
132	#389	—	47	2	no	—
133	#389	—	47	1	no	—
134	#389	—	47	1	no	—
135	#382	—	47	1	no	—
136	C₄×C₄⋊C₄	—	47	1	no	—
137	C₄×C₄⋊C₄	—	47	1	no	—
138	C₄₇	#2550	1	1	yes	characteristic, complemented, Sylow
139	C₂²×Q₈	—	47	2	no	—
140	C₄⋊Q₈	—	94	4	no	—
141	#125	—	2	4	no	—
142	C₂×C₄⋊C₄	—	47	1	no	—
143	C₂×C₄⋊C₄	—	47	2	no	—
144	C₂×C₄⋊C₄	—	47	1	no	—
145	C₂×C₄⋊C₄	—	47	1	no	—
146	C₂×C₄⋊C₄	—	47	1	no	—
147	C₂×C₄⋊C₄	—	47	2	no	—
148	C₂×C₄⋊C₄	—	47	2	no	—
149	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
150	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
151	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
152	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
153	C₂×C₄²	—	47	1	no	—
154	C₂×C₄²	—	47	1	no	—
155	C₂×C₄²	—	47	2	no	—
156	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
157	#95	—	47	1	no	—
158	#95	—	47	1	no	—
159	#95	—	47	2	no	—
160	#95	—	47	2	no	—
161	#95	—	47	2	no	—
162	#95	—	47	2	no	—
163	#95	—	47	2	no	—
164	C₂×Q₈	—	94	4	no	—
165	C₂×Q₈	—	94	4	no	—
166	C₂×Q₈	—	188	2	no	—
167	C₂²×C₄	—	47	1	no	—
168	C₂²×C₄	—	47	1	no	—
169	C₂²×C₄	—	47	2	no	—
170	C₂²×C₄	—	47	1	no	—
171	C₂²×C₄	—	47	1	no	—
172	C₂²×C₄	—	47	2	no	—
173	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
175	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
176	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
177	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	47	4	no	—
182	C₄⋊C₄	—	94	2	no	—
183	C₄⋊C₄	—	94	4	no	—
184	C₄⋊C₄	—	94	2	no	—
185	C₄⋊C₄	—	94	4	no	—
186	C₄⋊C₄	—	94	2	no	—
187	C₄⋊C₄	—	188	2	no	—
188	C₄⋊C₄	C₂²⋊D₄₇	1	4	yes	—
189	C₄²	—	2	2	no	—
190	C₄²	—	47	4	no	—
191	C₄²	—	94	2	no	—
192	C₄²	—	188	2	no	—
193	C₂³	C₂³×D₄₇	1	1	yes	characteristic
194	Q₈	—	188	8	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	47	2	no	—
207	C₂×C₄	—	47	2	no	—
208	C₂×C₄	—	47	4	no	—
209	C₂×C₄	—	47	2	no	—
210	C₂×C₄	—	47	2	no	—
211	C₂×C₄	—	94	1	no	—
212	C₂×C₄	—	94	1	no	—
213	C₂×C₄	—	94	2	no	—
214	C₂×C₄	—	94	1	no	—
215	C₂×C₄	—	94	1	no	—
216	C₂×C₄	—	94	1	no	—
217	C₂×C₄	—	94	2	no	—
218	C₂×C₄	—	94	1	no	—
219	C₂×C₄	—	94	1	no	—
220	C₂×C₄	—	94	1	no	—
221	C₂×C₄	—	94	2	no	—
222	C₂×C₄	—	94	2	no	—
223	C₂×C₄	—	94	2	no	—
224	C₂×C₄	Q₈×D₄₇	1	2	yes	—
225	C₂×C₄	Q₈×D₄₇	1	2	yes	—
226	C₂×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
227	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
228	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
229	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
230	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
231	C₂²	#41×D₄₇	1	1	yes	characteristic
232	C₂²	#41×D₄₇	1	1	yes	characteristic
233	C₂²	#13094070	1	1	yes	characteristic
234	C₂²	#13094071	1	1	yes	characteristic
235	C₄	—	2	4	no	—
236	C₄	—	4	1	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	2	no	—
239	C₄	—	94	2	no	—
240	C₄	—	94	2	no	—
241	C₄	—	94	2	no	—
242	C₄	—	94	4	no	—
243	C₄	—	94	2	no	—
244	C₄	—	188	2	no	—
245	C₄	#13094022	1	4	yes	—
246	C₂	C₂×#13094022	1	1	yes	characteristic
247	C₂	#15689679	1	1	yes	characteristic
248	C₂	#15689753	1	1	yes	characteristic
249	C₂	#15689761	1	1	yes	characteristic
250	C₂	#15689780	1	1	yes	characteristic
251	C₂	#15689829	1	1	yes	characteristic
252	C₂	#15689850	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×#125×C₄₇	C₂
hypercenter	C₂×#125	D₄₇
hypocenter	C₄₇	#2550
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
57. C₄⋊Dic₄₇	Q₈	semidirect
65. C₄×Dic₄₇	Q₈	semidirect
138. C₄₇	#2550	semidirect