Subgroups — #26369640

Order 12032.

718 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252240	C₂	1	2	yes	maximal, complemented
2	#20245264	C₂	1	4	yes	maximal, complemented
3	D₁₆×Dic₄₇	C₂	1	4	yes	maximal, complemented
4	#20245172	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245104	C₂	1	2	yes	maximal, complemented
6	#20244984	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
8	#15689598	—	2	2	no	—
9	C₂×#13093964	C₂²	1	1	yes	characteristic
10	#15689235	—	2	4	no	—
11	D₈×Dic₄₇	C₂²	1	8	yes	complemented
12	#15689161	C₂²	1	2	yes	—
13	#15689128	—	2	2	no	—
14	#15689078	C₂²	1	1	yes	characteristic
15	#15688951	C₂²	1	2	yes	complemented
16	#15688949	C₂²	1	2	yes	complemented
17	#15688924	C₂²	1	8	yes	complemented
18	#15688898	C₂²	1	2	yes	complemented
19	C₁₆×Dic₄₇	C₂²	1	2	yes	complemented
20	C₂²×D₈×C₄₇	C₂²	1	2	yes	—
21	C₂×D₁₆×C₄₇	C₄	1	8	yes	complemented
22	C₂×D₁₆×C₄₇	C₂²	1	2	yes	—
23	C₂×D₁₆×C₄₇	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#10625329	D₄	1	1	yes	characteristic
26	D₄×Dic₄₇	—	2	8	no	—
27	D₄×Dic₄₇	—	4	4	no	—
28	#13093983	—	2	2	no	—
29	#13093983	D₄	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₄₇	—	4	4	no	—
31	#13093964	—	2	2	no	—
32	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
33	#13093958	C₂³	1	4	yes	—
34	#13093956	—	2	2	no	—
35	#13093945	—	2	8	no	—
36	#13093934	D₄	1	2	yes	—
37	C₈×Dic₄₇	C₂³	1	2	yes	—
38	C₂²×D₄×C₄₇	—	2	2	no	—
39	C₂×D₈×C₄₇	—	2	2	no	—
40	C₂×D₈×C₄₇	—	2	4	no	—
41	C₂×D₈×C₄₇	C₂×C₄	1	8	yes	complemented
42	C₂×D₈×C₄₇	C₂³	1	4	yes	—
43	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
44	D₁₆×C₄₇	C₂×C₄	1	8	yes	complemented
45	D₁₆×C₄₇	C₂×C₄	1	8	yes	complemented
46	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
47	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
49	C₂⁴×C₄₇	—	4	2	no	—
50	C₂×D₄×C₄₇	—	2	8	no	—
51	C₂×D₄×C₄₇	—	2	4	no	—
52	C₂×D₄×C₄₇	—	4	2	no	—
53	C₂×D₄×C₄₇	—	4	4	no	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₄₇	—	4	8	no	—
56	D₈×C₄₇	—	2	8	no	—
57	D₈×C₄₇	—	2	4	no	—
58	D₈×C₄₇	C₂²×C₄	1	8	yes	—
59	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₄₇	#41	1	2	yes	—
62	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
63	C₂²⋊Dic₄₇	—	4	2	no	—
64	C₂²⋊Dic₄₇	—	4	8	no	—
65	C₂²⋊Dic₄₇	D₈	1	2	yes	—
66	C₄⋊Dic₄₇	—	2	4	no	—
67	C₄⋊Dic₄₇	D₈	1	2	yes	—
68	C₄×Dic₄₇	—	4	2	no	—
69	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
70	#10625329	—	2	2	no	—
71	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
72	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
73	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
74	C₄₇⋊C₁₆	—	2	2	no	—
75	C₂³×C₄₇	—	4	4	no	—
76	C₂³×C₄₇	—	4	8	no	—
77	C₂³×C₄₇	—	16	4	no	—
78	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
79	D₄×C₄₇	—	2	8	no	—
80	D₄×C₄₇	—	4	4	no	—
81	D₄×C₄₇	—	4	8	no	—
82	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₄₇	#135	1	2	yes	—
85	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
86	C₂×Dic₄₇	—	4	4	no	—
87	C₂×Dic₄₇	—	16	8	no	—
88	C₂×Dic₄₇	C₂×D₈	1	4	yes	—
89	C₈×C₄₇	C₄×D₄	1	2	yes	—
90	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
91	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
92	C₄₇⋊C₈	—	2	2	no	—
93	#33456	—	47	1	no	maximal, Sylow
94	C₂²×C₄₇	—	2	2	no	—
95	C₂²×C₄₇	—	4	8	no	—
96	C₂²×C₄₇	—	8	8	no	—
97	C₂²×C₄₇	—	16	8	no	—
98	C₂²×C₄₇	C₄×D₈	1	2	yes	—
99	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
100	C₄×C₄₇	C₄×D₈	1	2	yes	—
101	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
102	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
103	Dic₄₇	—	2	4	no	—
104	Dic₄₇	—	8	4	no	—
105	C₂²×D₁₆	D₄₇	1	1	yes	characteristic
106	#2948	—	47	2	no	—
107	#2177	—	47	4	no	—
108	C₄×D₁₆	—	47	4	no	—
109	#2158	—	47	1	no	—
110	#2140	—	47	2	no	—
111	#2108	—	47	1	no	—
112	C₂×C₄₇	—	2	2	no	—
113	C₂×C₄₇	—	8	8	no	—
114	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
115	C₂×C₄₇	#3415	1	2	yes	—
116	C₂²×D₈	C₂×D₄₇	1	2	yes	—
117	#518	—	94	2	no	—
118	C₂×D₁₆	Dic₄₇	1	8	yes	complemented
119	C₂×D₁₆	C₂×D₄₇	1	2	yes	—
120	C₂×D₁₆	C₂×D₄₇	1	2	yes	—
121	C₂×M₃₂	—	47	1	no	—
122	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
123	#442	—	94	4	no	—
124	C₄×D₈	—	47	8	no	—
125	#427	—	47	2	no	—
126	#418	—	94	2	no	—
127	#405	—	47	1	no	—
128	#367	—	47	2	no	—
129	#366	—	47	2	no	—
130	#357	—	47	8	no	—
131	#346	—	47	2	no	—
132	C₄×C₁₆	—	47	2	no	—
133	C₄₇	#33456	1	1	yes	characteristic, complemented, Sylow
134	C₂²×D₄	—	2	2	no	—
135	C₂×D₈	—	2	2	no	—
136	C₂×D₈	—	2	4	no	—
137	C₂×D₈	C₂×Dic₄₇	1	8	yes	complemented
138	C₂×D₈	C₂²×D₄₇	1	4	yes	—
139	C₂×M₁₆	—	47	1	no	—
140	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
141	C₄×D₄	—	94	8	no	—
142	C₄×D₄	—	188	4	no	—
143	#117	—	47	1	no	—
144	#117	—	94	2	no	—
145	C₂×C₂²⋊C₄	—	188	4	no	—
146	D₁₆	C₂×Dic₄₇	1	8	yes	complemented
147	D₁₆	C₂×Dic₄₇	1	8	yes	complemented
148	M₃₂	—	94	2	no	—
149	C₂×C₁₆	—	47	2	no	—
150	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	complemented
151	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
152	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
153	#107	—	47	4	no	—
154	#106	—	94	2	no	—
155	#102	—	94	8	no	—
156	#97	—	47	2	no	—
157	C₄×C₈	—	47	2	no	—
158	C₂⁴	—	4	2	no	—
159	C₂×D₄	—	2	8	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂²×C₄	—	188	8	no	—
164	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
165	D₈	—	2	8	no	—
166	D₈	—	2	4	no	—
167	D₈	C₂²×Dic₄₇	1	8	yes	—
168	M₁₆	—	94	2	no	—
169	C₂×C₈	—	47	2	no	—
170	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
171	C₂×C₈	#10625352	1	2	yes	—
172	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
173	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
174	C₄⋊C₄	—	47	2	no	—
175	C₄⋊C₄	—	94	4	no	—
176	C₂²⋊C₄	—	47	2	no	—
177	C₂²⋊C₄	—	188	8	no	—
178	C₂²⋊C₄	—	188	2	no	—
179	C₄²	—	47	2	no	—
180	C₄²	—	188	2	no	—
181	C₁₆	—	94	2	no	—
182	C₁₆	C₂²×Dic₄₇	1	2	yes	—
183	C₁₆	C₂²×Dic₄₇	1	2	yes	—
184	C₂³	—	4	4	no	—
185	C₂³	—	4	8	no	—
186	C₂³	—	16	4	no	—
187	C₂³	D₈×D₄₇	1	1	yes	characteristic
188	D₄	—	2	8	no	—
189	D₄	—	4	4	no	—
190	D₄	—	4	8	no	—
191	C₂×C₄	—	47	4	no	—
192	C₂×C₄	—	188	4	no	—
193	C₂×C₄	—	752	8	no	—
194	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
195	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
196	C₂×C₄	#13094043	1	2	yes	—
197	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
198	C₈	—	94	2	no	—
199	C₈	D₄×Dic₄₇	1	2	yes	—
200	C₈	C₂³×Dic₄₇	1	1	yes	characteristic
201	C₈	C₂×#10625352	1	1	yes	characteristic
202	C₂²	—	2	2	no	—
203	C₂²	—	4	8	no	—
204	C₂²	—	8	8	no	—
205	C₂²	—	16	8	no	—
206	C₂²	D₈×Dic₄₇	1	2	yes	—
207	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
208	C₄	—	94	4	no	—
209	C₄	—	376	4	no	—
210	C₄	D₈×Dic₄₇	1	2	yes	—
211	C₄	C₂×D₄×Dic₄₇	1	1	yes	characteristic
212	C₄	C₂×#13094043	1	1	yes	characteristic
213	C₂	—	2	2	no	—
214	C₂	—	8	8	no	—
215	C₂	C₂×D₈×Dic₄₇	1	1	yes	characteristic
216	C₂	#20256525	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting, prime radical	C₂²×D₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂²×D₁₆	D₄₇
hypocenter	C₄₇	#33456
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

Complements

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

N	H ≅ G/N	type
1. #20252240	C₂	semidirect
2. #20245264	C₂	semidirect
3. D₁₆×Dic₄₇	C₂	semidirect
4. #20245172	C₂	semidirect
5. #20245104	C₂	semidirect
6. #20244984	C₂	semidirect
11. D₈×Dic₄₇	C₂²	semidirect
15. #15688951	C₂²	semidirect
16. #15688949	C₂²	semidirect
17. #15688924	C₂²	semidirect
18. #15688898	C₂²	semidirect
19. C₁₆×Dic₄₇	C₂²	semidirect
21. C₂×D₁₆×C₄₇	C₄	semidirect
41. C₂×D₈×C₄₇	C₂×C₄	semidirect
44. D₁₆×C₄₇	C₂×C₄	semidirect
45. D₁₆×C₄₇	C₂×C₄	semidirect
46. C₂×C₁₆×C₄₇	C₂×C₄	semidirect
118. C₂×D₁₆	Dic₄₇	semidirect
133. C₄₇	#33456	semidirect
137. C₂×D₈	C₂×Dic₄₇	semidirect
146. D₁₆	C₂×Dic₄₇	semidirect
147. D₁₆	C₂×Dic₄₇	semidirect
150. C₂×C₁₆	C₂×Dic₄₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20252240	2
#20245264	2
D₁₆×Dic₄₇	2
#20245172	2
#20245104	2
#20244984	2
C₂²×D₁₆×C₄₇	2
#33456	47