Subgroups — #36472642

Order 22784.

614 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151975	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151400	C₂	1	4	yes	maximal, complemented
3	#26020412	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₇₁₂	—	2	1	no	—
6	C₂²×D₇₁₂	C₂²	1	2	yes	—
7	C₂×#15668009	—	2	1	no	—
8	#19151975	—	2	2	no	—
9	#19151975	C₄	1	4	yes	complemented
10	C₂×#15667946	—	2	1	no	—
11	C₂×#15667946	C₂²	1	2	yes	—
12	#19151468	—	2	1	no	—
13	#19151400	—	2	8	no	—
14	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₈₉	—	2	2	no	—
17	C₂²×D₃₅₆	—	2	2	no	—
18	C₂²×D₃₅₆	—	4	2	no	—
19	C₂×D₇₁₂	—	2	4	no	—
20	C₂×D₇₁₂	—	2	4	no	—
21	C₂×D₇₁₂	—	2	2	no	—
22	C₂×D₇₁₂	—	2	4	no	—
23	C₂×D₇₁₂	—	4	2	no	—
24	C₂×D₇₁₂	—	4	2	no	—
25	C₂×D₇₁₂	C₂×C₄	1	4	yes	—
26	#15668009	—	2	4	no	—
27	#15668009	—	4	2	no	—
28	C₂×C₄⋊Dic₈₉	—	2	2	no	—
29	C₂×C₄⋊Dic₈₉	—	4	2	no	—
30	#15667946	—	2	4	no	—
31	#15667946	—	4	2	no	—
32	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₈₉	—	2	2	no	—
36	C₂×C₁₆×C₈₉	—	2	2	no	—
37	C₂×C₁₆×C₈₉	D₄	1	4	yes	complemented
38	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₈₉	D₄	1	2	yes	—
40	C₂³×D₈₉	—	4	2	no	—
41	C₂×D₃₅₆	—	2	4	no	—
42	C₂×D₃₅₆	—	4	2	no	—
43	C₂×D₃₅₆	—	4	2	no	—
44	C₂×D₃₅₆	—	4	2	no	—
45	C₂×D₃₅₆	—	4	2	no	—
46	C₂×D₃₅₆	—	4	8	no	—
47	C₂×D₃₅₆	—	4	2	no	—
48	C₂×D₃₅₆	—	8	4	no	—
49	C₂×D₃₅₆	—	8	4	no	—
50	C₂²×Dic₈₉	—	4	2	no	—
51	D₇₁₂	—	2	8	no	—
52	D₇₁₂	—	4	8	no	—
53	D₇₁₂	—	4	4	no	—
54	C₄⋊Dic₈₉	—	4	2	no	—
55	C₄⋊Dic₈₉	—	4	2	no	—
56	C₄⋊Dic₈₉	—	8	4	no	—
57	C₂²×C₄×C₈₉	D₈	1	2	yes	—
58	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₈₉	D₈	1	2	yes	—
61	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
62	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₈₉	#41	1	2	yes	—
66	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₈₉	SD₁₆	1	2	yes	—
68	C₁₆×C₈₉	—	2	4	no	—
69	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
70	C₂²×D₈₉	—	4	4	no	—
71	C₂²×D₈₉	—	8	2	no	—
72	C₂²×D₈₉	—	8	2	no	—
73	C₂²×D₈₉	—	8	4	no	—
74	C₂²×D₈₉	—	8	4	no	—
75	C₂×C₄×C₈₉	#102	1	4	yes	—
76	C₂×C₄×C₈₉	D₁₆	1	4	yes	—
77	C₂×C₄×C₈₉	SD₃₂	1	4	yes	—
78	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
80	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×Dic₈₉	—	8	2	no	—
85	C₂×Dic₈₉	—	8	4	no	—
86	D₃₅₆	—	4	4	no	—
87	D₃₅₆	—	4	4	no	—
88	D₃₅₆	—	8	16	no	—
89	D₃₅₆	—	8	2	no	—
90	D₃₅₆	—	8	2	no	—
91	C₈×C₈₉	#102	1	4	yes	—
92	C₈×C₈₉	C₄×D₄	1	2	yes	—
93	C₈×C₈₉	#121	1	2	yes	—
94	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
95	C₂²×C₈₉	#486	1	1	yes	characteristic
96	C₂²×C₈₉	#493	1	1	yes	characteristic
97	C₂²×C₈₉	C₂×D₁₆	1	2	yes	—
98	C₂²×C₈₉	C₂×SD₃₂	1	2	yes	—
99	C₂×D₈₉	—	8	8	no	—
100	C₂×D₈₉	—	8	4	no	—
101	C₂×D₈₉	—	16	4	no	—
102	C₂×D₈₉	—	16	2	no	—
103	C₂×D₈₉	—	16	2	no	—
104	C₄×C₈₉	#357	1	8	yes	complemented
105	C₄×C₈₉	#390	1	1	yes	characteristic
106	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
107	C₄×C₈₉	#486	1	1	yes	characteristic
108	C₄×C₈₉	#492	1	1	yes	characteristic
109	Dic₈₉	—	16	2	no	—
110	#13202	—	89	1	no	maximal, Sylow
111	C₂×C₈₉	#1957	1	1	yes	characteristic
112	C₂×C₈₉	C₂×#357	1	2	yes	—
113	C₂×C₈₉	#2240	1	2	yes	—
114	C₂×C₈₉	#2246	1	1	yes	characteristic
115	C₂×C₈₉	#2248	1	1	yes	characteristic
116	D₈₉	—	16	4	no	—
117	C₂×#493	—	89	1	no	—
118	C₂×#357	—	89	4	no	—
119	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
120	#1845	—	89	1	no	—
121	C₈₉	#13202	1	1	yes	characteristic, complemented, Sylow
122	C₂²×D₈	—	89	2	no	—
123	C₂²×D₈	—	178	1	no	—
124	C₂×#127	—	178	1	no	—
125	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
126	#493	—	89	4	no	—
127	#493	—	178	2	no	—
128	C₂×#107	—	89	2	no	—
129	C₂×#107	—	178	1	no	—
130	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
131	#384	—	178	1	no	—
132	#357	—	178	8	no	—
133	C₄×C₁₆	—	2	2	no	—
134	C₂²×D₄	—	356	2	no	—
135	C₂²×D₄	—	178	2	no	—
136	C₂×D₈	—	89	4	no	—
137	C₂×D₈	—	356	2	no	—
138	C₂×D₈	—	356	2	no	—
139	C₂×D₈	—	178	4	no	—
140	C₂×D₈	—	178	2	no	—
141	C₂×D₈	—	178	4	no	—
142	C₂×D₈	—	178	4	no	—
143	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
144	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
145	#127	—	356	2	no	—
146	#127	—	178	4	no	—
147	C₂×C₄⋊C₄	—	356	2	no	—
148	C₂×C₄⋊C₄	—	178	2	no	—
149	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	D₃₅₆	1	4	yes	complemented
153	#107	—	356	2	no	—
154	#107	—	178	4	no	—
155	C₄×C₈	C₄×D₈₉	1	2	yes	—
156	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
157	C₂⁴	—	356	2	no	—
158	C₂×D₄	—	356	2	no	—
159	C₂×D₄	—	356	2	no	—
160	C₂×D₄	—	356	2	no	—
161	C₂×D₄	—	356	2	no	—
162	C₂×D₄	—	356	2	no	—
163	C₂×D₄	—	356	8	no	—
164	C₂×D₄	—	178	4	no	—
165	C₂×D₄	—	712	4	no	—
166	C₂×D₄	—	712	4	no	—
167	C₂²×C₄	—	356	2	no	—
168	C₂²×C₄	D₇₁₂	1	2	yes	—
169	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
170	D₈	—	356	4	no	—
171	D₈	—	356	8	no	—
172	D₈	—	178	8	no	—
173	C₂×C₈	#13083551	1	2	yes	—
174	C₂×C₈	D₇₁₂	1	2	yes	—
175	C₂×C₈	#13083563	1	2	yes	—
176	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
177	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
178	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
179	C₂×C₈	#13083574	1	2	yes	—
180	C₄⋊C₄	—	356	2	no	—
181	C₄⋊C₄	—	356	2	no	—
182	C₄⋊C₄	—	712	4	no	—
183	C₄²	#13083551	1	2	yes	—
184	C₄²	#13083563	1	2	yes	—
185	C₁₆	—	2	4	no	—
186	C₂³	—	356	4	no	—
187	C₂³	—	712	2	no	—
188	C₂³	—	712	2	no	—
189	C₂³	—	712	4	no	—
190	C₂³	—	712	4	no	—
191	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
192	D₄	—	356	4	no	—
193	D₄	—	356	4	no	—
194	D₄	—	712	16	no	—
195	D₄	—	712	2	no	—
196	D₄	—	712	2	no	—
197	C₂×C₄	—	712	2	no	—
198	C₂×C₄	—	712	4	no	—
199	C₂×C₄	#15667934	1	4	yes	—
200	C₂×C₄	D₁₄₂₄	1	4	yes	—
201	C₂×C₄	#15667957	1	4	yes	—
202	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
203	C₂×C₄	#15668000	1	1	yes	characteristic
204	C₂×C₄	#15668009	1	1	yes	characteristic
205	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
206	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
207	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
208	C₈	#15667934	1	4	yes	—
209	C₈	C₄×D₃₅₆	1	2	yes	—
210	C₈	#15667986	1	2	yes	—
211	C₂²	—	1424	4	no	—
212	C₂²	—	1424	2	no	—
213	C₂²	—	1424	2	no	—
214	C₂²	—	712	8	no	—
215	C₂²	—	712	4	no	—
216	C₂²	C₂×#15667934	1	1	yes	characteristic
217	C₂²	#19151945	1	1	yes	characteristic
218	C₂²	#19151975	1	1	yes	characteristic
219	C₂²	C₂×D₁₄₂₄	1	2	yes	—
220	C₂²	C₂×#15667957	1	2	yes	—
221	C₄	—	1424	2	no	—
222	C₄	#19151400	1	8	yes	complemented
223	C₄	#19151500	1	1	yes	characteristic
224	C₄	C₂×#15667934	1	1	yes	characteristic
225	C₄	#19151945	1	1	yes	characteristic
226	C₄	#19151971	1	1	yes	characteristic
227	C₂	—	1424	4	no	—
228	C₂	#26021058	1	1	yes	characteristic
229	C₂	C₂×#19151400	1	2	yes	—
230	C₂	#26022476	1	2	yes	—
231	C₂	#26022502	1	1	yes	characteristic
232	C₂	#26022508	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂×#19151975	C₂
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	#13202
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#19151400	C₂	semidirect
3. #26020412	C₂	semidirect
4. C₂×C₄×C₁₆×C₈₉	C₂	semidirect
9. #19151975	C₄	semidirect
37. C₂×C₁₆×C₈₉	D₄	semidirect
104. C₄×C₈₉	#357	semidirect
119. C₂×C₄×C₁₆	D₈₉	semidirect
121. C₈₉	#13202	semidirect
152. C₂×C₁₆	D₃₅₆	semidirect
222. C₄	#19151400	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151975	2
C₂×#19151400	2
#26020412	2
C₂×C₄×C₁₆×C₈₉	2
#13202	89