Subgroups — #20252392

Order 6016.

370 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689657	C₂	1	1	yes	maximal, characteristic
2	#15689253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689202	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689199	C₂	1	2	yes	maximal, complemented
6	#15689192	C₂	1	2	yes	maximal, complemented
7	C₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689185	C₂	1	2	yes	maximal, complemented
9	#15689176	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689080	C₂	1	2	yes	maximal, complemented
11	C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625324	C₂²	1	2	yes	—
13	C₂×C₈×D₄₇	C₂²	1	2	yes	—
14	#13094018	C₄	1	2	yes	—
15	#13094016	C₄	1	1	yes	characteristic
16	#13094012	C₄	1	1	yes	characteristic
17	#13094007	C₄	1	2	yes	—
18	#13094003	C₄	1	1	yes	characteristic
19	#13093998	C₄	1	1	yes	characteristic
20	C₄×Dic₉₄	C₂²	1	1	yes	characteristic
21	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₄₇	C₂²	1	2	yes	—
23	#13093981	C₂²	1	2	yes	—
24	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
25	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
26	Dic₄₇⋊C₈	C₂²	1	2	yes	—
27	Dic₄₇⋊C₈	C₂²	1	2	yes	—
28	C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
29	(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	—
30	(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	—
31	#13093934	C₂²	1	2	yes	—
32	C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
33	C₈×Dic₄₇	C₂²	1	2	yes	—
34	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
37	#97×C₄₇	C₂²	1	1	yes	characteristic
38	#97×C₄₇	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
45	C₂×D₁₈₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₄₇	C₂³	1	2	yes	—
47	C₂×Dic₉₄	C₂×C₄	1	1	yes	characteristic
48	M₁₆×C₄₇	—	2	2	no	—
49	M₁₆×C₄₇	—	2	2	no	—
50	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
59	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
60	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
61	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
62	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
63	C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
64	C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
65	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
66	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
67	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
68	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
69	C₄×Dic₄₇	C₂³	1	2	yes	—
70	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
71	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
72	#10625324	—	2	4	no	—
73	C₈×D₄₇	—	2	4	no	—
74	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×D₄₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₄₇	—	2	2	no	—
77	C₂×C₄×C₄₇	—	2	1	no	—
78	C₂×C₄×C₄₇	—	2	1	no	—
79	C₂×C₄×C₄₇	—	4	1	no	—
80	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₂²⋊D₄₇	—	4	2	no	—
87	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
88	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
89	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
90	D₁₈₈	—	2	2	no	—
91	C₄×D₄₇	—	2	2	no	—
92	C₄×D₄₇	—	2	2	no	—
93	Dic₉₄	—	2	2	no	—
94	C₈×C₄₇	—	2	1	no	—
95	C₈×C₄₇	—	2	1	no	—
96	C₈×C₄₇	#41	1	2	yes	—
97	C₈×C₄₇	#41	1	2	yes	—
98	C₄₇⋊C₈	—	2	2	no	—
99	C₄₇⋊C₈	—	2	2	no	—
100	C₂²×C₄₇	—	2	1	no	—
101	C₂²×C₄₇	—	2	1	no	—
102	C₂²×C₄₇	—	2	1	no	—
103	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×D₄₇	—	2	2	no	—
105	C₂×D₄₇	—	2	2	no	—
106	C₂×D₄₇	—	2	2	no	—
107	C₄×C₄₇	—	2	1	no	—
108	C₄×C₄₇	—	2	1	no	—
109	C₄×C₄₇	—	2	1	no	—
110	C₄×C₄₇	—	2	1	no	—
111	C₄×C₄₇	#131	1	2	yes	—
112	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
113	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
114	Dic₄₇	—	2	2	no	—
115	Dic₄₇	—	2	2	no	—
116	Dic₄₇	—	2	2	no	—
117	#2969	—	47	1	no	maximal, Sylow
118	C₂×C₄₇	—	4	1	no	—
119	C₂×C₄₇	C₄×#41	1	1	yes	characteristic
120	C₂×C₄₇	C₂×#131	1	1	yes	characteristic
121	C₂×C₄₇	#568	1	1	yes	characteristic
122	D₄₇	—	4	2	no	—
123	#529	—	47	1	no	—
124	#446	—	47	1	no	—
125	C₈×Q₈	—	47	1	no	—
126	#436	—	47	2	no	—
127	#436	—	47	1	no	—
128	#435	—	47	2	no	—
129	C₈×D₄	—	47	1	no	—
130	#433	—	47	2	no	—
131	#431	—	47	1	no	—
132	#405	—	47	2	no	—
133	C₄×M₁₆	D₄₇	1	1	yes	characteristic, complemented
134	C₄₇	#2969	1	1	yes	characteristic, complemented, Sylow
135	C₂×M₁₆	—	47	2	no	—
136	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
137	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
138	C₂²×C₈	—	47	2	no	—
139	#126	—	47	2	no	—
140	#125	—	47	1	no	—
141	#124	—	47	1	no	—
142	#123	—	47	2	no	—
143	C₂²⋊Q₈	—	47	1	no	—
144	#121	—	47	1	no	—
145	C₄×Q₈	—	47	1	no	—
146	C₄×D₄	—	47	2	no	—
147	C₄×D₄	—	47	1	no	—
148	#117	—	47	2	no	—
149	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
150	C₄⋊C₈	—	47	2	no	—
151	C₄⋊C₈	—	47	2	no	—
152	C₄⋊C₈	—	47	1	no	—
153	C₄⋊C₈	—	47	1	no	—
154	C₂²⋊C₈	—	47	2	no	—
155	C₂²⋊C₈	—	47	2	no	—
156	C₂²⋊C₈	—	47	2	no	—
157	#97	—	47	2	no	—
158	#97	C₂×D₄₇	1	1	yes	characteristic
159	#97	C₂×D₄₇	1	1	yes	characteristic
160	C₄×C₈	—	47	2	no	—
161	C₄×C₈	—	47	2	no	—
162	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
163	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
164	C₂×Q₈	—	47	1	no	—
165	C₂×D₄	—	47	2	no	—
166	C₂×D₄	—	47	1	no	—
167	C₂²×C₄	—	47	2	no	—
168	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
169	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
171	M₁₆	—	2	2	no	—
172	M₁₆	—	2	2	no	—
173	M₁₆	—	94	4	no	—
174	C₂×C₈	—	47	2	no	—
175	C₂×C₈	—	47	2	no	—
176	C₂×C₈	—	94	4	no	—
177	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
179	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
180	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
181	C₄⋊C₄	—	47	2	no	—
182	C₄⋊C₄	—	47	2	no	—
183	C₄⋊C₄	—	47	2	no	—
184	C₄⋊C₄	—	47	2	no	—
185	C₄⋊C₄	—	47	1	no	—
186	C₄⋊C₄	—	47	1	no	—
187	C₂²⋊C₄	—	47	2	no	—
188	C₂²⋊C₄	—	47	2	no	—
189	C₂²⋊C₄	—	47	2	no	—
190	C₂²⋊C₄	—	47	2	no	—
191	C₂²⋊C₄	—	47	2	no	—
192	C₄²	—	47	2	no	—
193	C₄²	C₄×D₄₇	1	1	yes	characteristic
194	C₄²	C₄×D₄₇	1	1	yes	characteristic
195	C₄²	C₂²×D₄₇	1	1	yes	characteristic
196	C₄²	C₂²×D₄₇	1	1	yes	characteristic
197	C₂³	—	47	2	no	—
198	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
199	Q₈	—	94	2	no	—
200	D₄	—	94	2	no	—
201	D₄	—	188	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	47	2	no	—
207	C₂×C₄	—	47	2	no	—
208	C₂×C₄	—	47	2	no	—
209	C₂×C₄	—	94	2	no	—
210	C₂×C₄	—	94	2	no	—
211	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	94	2	no	—
220	C₈	—	94	2	no	—
221	C₈	#10625350	1	2	yes	—
222	C₈	#10625350	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	94	2	no	—
227	C₂²	—	94	2	no	—
228	C₂²	—	94	2	no	—
229	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	94	2	no	—
235	C₄	—	94	2	no	—
236	C₄	—	94	2	no	—
237	C₄	#13094032	1	2	yes	—
238	C₄	C₂×#10625350	1	1	yes	characteristic
239	C₄	C₂×#10625350	1	1	yes	characteristic
240	C₂	—	4	1	no	—
241	C₂	—	188	2	no	—
242	C₂	C₄×#10625350	1	1	yes	characteristic
243	C₂	C₂×#13094032	1	1	yes	characteristic
244	C₂	#15689889	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₄×M₁₆×C₄₇	C₂
hypercenter	C₄×M₁₆	D₄₇
hypocenter	C₄₇	#2969
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #15689253	C₂	semidirect
3. C₈×Dic₉₄	C₂	semidirect
4. #15689202	C₂	semidirect
5. #15689199	C₂	semidirect
6. #15689192	C₂	semidirect
7. C₈×D₁₈₈	C₂	semidirect
8. #15689185	C₂	semidirect
9. #15689176	C₂	semidirect
10. #15689080	C₂	semidirect
11. C₄×M₁₆×C₄₇	C₂	semidirect
133. C₄×M₁₆	D₄₇	semidirect
134. C₄₇	#2969	semidirect