Subgroups — #36563044

Order 22784.

1022 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030680	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151719	C₂	1	4	yes	maximal, complemented
3	#26028963	C₂	1	2	yes	maximal, complemented
4	#26021143	C₂	1	4	yes	maximal, complemented
5	#26021079	C₂	1	2	yes	maximal, complemented
6	C₄×#15667922	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1848	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668034	C₂²	1	4	yes	—
9	C₂×#15668020	C₂²	1	4	yes	—
10	#19152148	C₂²	1	2	yes	—
11	#19151982	C₄	1	4	yes	complemented
12	#19151978	C₄	1	1	yes	characteristic, complemented
13	#19151975	C₄	1	1	yes	characteristic, complemented
14	#19151971	C₄	1	2	yes	complemented
15	#19151719	—	2	16	no	—
16	#19151675	—	2	4	no	—
17	#19151663	—	2	4	no	—
18	C₂×#15667938	C₂²	1	8	yes	—
19	#19151566	C₂²	1	4	yes	—
20	C₂×#15667922	C₂²	1	2	yes	—
21	C₂×#15667922	C₂²	1	2	yes	—
22	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
23	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
24	C₂×#108×C₈₉	C₂²	1	4	yes	—
25	#423×C₈₉	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#13083574	C₂³	1	4	yes	—
28	#15668034	—	2	8	no	—
29	#15668034	—	2	8	no	—
30	C₂×Dic₃₅₆	C₂×C₄	1	4	yes	—
31	C₂×#13083563	C₂×C₄	1	8	yes	—
32	C₂×D₇₁₂	C₂×C₄	1	4	yes	—
33	#15668020	—	2	16	no	—
34	C₂×#13083557	—	2	4	no	—
35	C₂²×#10623494	C₂³	1	1	yes	characteristic
36	C₂²×#10623494	C₂³	1	1	yes	characteristic
37	C₂×C₈×D₈₉	—	2	4	no	—
38	#15668012	C₂×C₄	1	2	yes	—
39	#15668009	C₂×C₄	1	2	yes	—
40	#15668000	C₂×C₄	1	4	yes	—
41	#15667986	—	2	4	no	—
42	C₄×D₃₅₆	—	2	4	no	—
43	C₂×C₄×Dic₈₉	—	2	2	no	—
44	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
45	#15667938	—	2	16	no	—
46	#15667924	—	2	8	no	—
47	#15667922	—	2	4	no	—
48	#15667922	D₄	1	8	yes	complemented
49	C₄×#10623494	D₄	1	4	yes	—
50	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
51	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	#108×C₈₉	—	2	8	no	—
54	C₄⋊C₈×C₈₉	—	2	4	no	—
55	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
58	#13083574	—	2	8	no	—
59	#13083574	—	2	8	no	—
60	C₂×Dic₁₇₈	C₂²×C₄	1	4	yes	—
61	C₂×D₃₅₆	—	2	4	no	—
62	C₂×D₃₅₆	C₂²×C₄	1	4	yes	—
63	C₂×C₂²⋊D₈₉	—	2	4	no	—
64	C₂²×Dic₈₉	—	2	2	no	—
65	C₂×C₄×D₈₉	—	2	4	no	—
66	Dic₃₅₆	—	2	8	no	—
67	#13083563	—	2	16	no	—
68	D₇₁₂	—	2	8	no	—
69	#13083557	—	4	8	no	—
70	C₂×#10623494	—	2	2	no	—
71	C₂×#10623494	C₂×D₄	1	2	yes	—
72	C₂×#10623494	C₂×D₄	1	2	yes	—
73	C₂×#10623494	C₂×D₄	1	4	yes	—
74	C₈×D₈₉	—	4	8	no	—
75	C₄⋊Dic₈₉	—	2	4	no	—
76	#13083551	—	2	8	no	—
77	C₄×Dic₈₉	—	2	2	no	—
78	C₄×Dic₈₉	—	4	4	no	—
79	C₄×Dic₈₉	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₈₉	#41	1	2	yes	—
81	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
82	M₁₆×C₈₉	—	2	8	no	—
83	C₂×C₈×C₈₉	—	2	4	no	—
84	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
87	C₂×C₈×C₈₉	#41	1	4	yes	—
88	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₈₉	C₂×D₄	1	2	yes	—
91	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
92	C₂²×D₈₉	—	2	4	no	—
93	C₂×C₄×C₈₉	—	2	4	no	—
94	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
98	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
100	C₂²⋊D₈₉	—	4	8	no	—
101	C₂×Dic₈₉	—	2	4	no	—
102	C₂×Dic₈₉	—	2	4	no	—
103	C₂×Dic₈₉	—	4	4	no	—
104	D₃₅₆	—	2	8	no	—
105	D₃₅₆	—	4	8	no	—
106	C₄×D₈₉	—	4	4	no	—
107	C₄×D₈₉	—	4	4	no	—
108	Dic₁₇₈	—	2	8	no	—
109	C₈×C₈₉	—	4	4	no	—
110	C₈×C₈₉	C₄×D₄	1	4	yes	—
111	C₈×C₈₉	C₄×D₄	1	4	yes	—
112	#10623494	—	2	2	no	—
113	#10623494	#127	1	4	yes	—
114	C₂²×C₈₉	—	2	2	no	—
115	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
116	C₂²×C₈₉	#532	1	2	yes	—
117	C₂×D₈₉	—	4	4	no	—
118	C₂×D₈₉	—	4	8	no	—
119	C₄×C₈₉	—	2	2	no	—
120	C₄×C₈₉	#444	1	4	yes	complemented
121	C₄×C₈₉	C₂×C₄×D₄	1	2	yes	—
122	C₄×C₈₉	C₂×#127	1	1	yes	characteristic
123	C₄×C₈₉	#532	1	1	yes	characteristic
124	Dic₈₉	—	4	4	no	—
125	Dic₈₉	—	4	4	no	—
126	#21706	—	89	1	no	maximal, Sylow
127	C₂×C₈₉	—	2	2	no	—
128	C₂×C₈₉	C₄×#127	1	1	yes	characteristic
129	C₂×C₈₉	C₂×#444	1	2	yes	—
130	D₈₉	—	8	4	no	—
131	#3147	—	89	1	no	—
132	C₂×#444	—	89	4	no	—
133	#2931	—	89	2	no	—
134	#1975	—	89	4	no	—
135	#1962	—	89	2	no	—
136	#1848	D₈₉	1	1	yes	characteristic, complemented
137	C₄×#97	—	89	1	no	—
138	C₈₉	#21706	1	1	yes	characteristic, complemented, Sylow
139	C₂×#135	—	89	4	no	—
140	C₂×#131	—	89	4	no	—
141	#532	—	89	2	no	—
142	#495	—	89	4	no	—
143	#494	—	89	1	no	—
144	#493	—	89	1	no	—
145	#492	—	89	2	no	—
146	#444	—	178	16	no	—
147	#436	—	178	4	no	—
148	C₈×D₄	—	178	4	no	—
149	C₂×#108	C₂×D₈₉	1	4	yes	—
150	#423	C₂×D₈₉	1	2	yes	—
151	C₂×#104	—	89	8	no	—
152	#408	—	89	4	no	—
153	C₂×#97	—	89	2	no	—
154	C₂×#97	—	89	2	no	—
155	C₂×C₄×C₈	—	89	1	no	—
156	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
157	C₄³	—	89	1	no	—
158	C₂×#41	—	89	4	no	—
159	#135	—	178	8	no	—
160	#135	—	178	8	no	—
161	C₂×Q₁₆	—	89	4	no	—
162	C₂×SD₁₆	—	89	8	no	—
163	C₂×D₈	—	89	4	no	—
164	#131	—	178	16	no	—
165	C₂×M₁₆	—	178	4	no	—
166	C₂×M₁₆	C₂²×D₈₉	1	4	yes	—
167	C₂²×C₈	—	89	1	no	—
168	C₂²×C₈	—	89	1	no	—
169	C₂²×C₈	—	178	4	no	—
170	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
171	C₄⋊Q₈	—	89	2	no	—
172	#127	—	89	2	no	—
173	#124	—	89	4	no	—
174	#121	—	178	4	no	—
175	C₄×D₄	—	178	4	no	—
176	C₂×C₄²	—	89	2	no	—
177	C₂×C₄²	—	178	2	no	—
178	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
179	#108	—	2	8	no	—
180	C₄⋊C₈	—	2	4	no	—
181	#104	—	178	16	no	—
182	C₂²⋊C₈	—	178	8	no	—
183	#97	—	89	8	no	—
184	#97	—	178	4	no	—
185	C₄×C₈	—	89	4	no	—
186	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
187	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
188	C₄×C₈	C₄×D₈₉	1	2	yes	—
189	#41	—	178	8	no	—
190	#41	—	178	8	no	—
191	C₂×Q₈	—	89	4	no	—
192	C₂×D₄	—	89	4	no	—
193	C₂×D₄	—	178	4	no	—
194	C₂×D₄	—	178	4	no	—
195	C₂²×C₄	—	178	2	no	—
196	C₂²×C₄	—	178	4	no	—
197	C₂²×C₄	#13083575	1	2	yes	—
198	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
199	Q₁₆	—	178	8	no	—
200	SD₁₆	—	178	16	no	—
201	D₈	—	178	8	no	—
202	M₁₆	—	2	8	no	—
203	M₁₆	—	356	8	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	89	4	no	—
206	C₂×C₈	—	89	2	no	—
207	C₂×C₈	—	89	2	no	—
208	C₂×C₈	—	356	8	no	—
209	C₂×C₈	—	178	2	no	—
210	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
211	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
212	C₂×C₈	D₄×D₈₉	1	4	yes	—
213	C₂×C₈	#13083575	1	4	yes	—
214	C₄⋊C₄	—	178	4	no	—
215	C₂²⋊C₄	—	178	8	no	—
216	C₄²	—	89	4	no	—
217	C₄²	—	356	4	no	—
218	C₄²	—	178	2	no	—
219	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
220	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
221	C₄²	D₄×D₈₉	1	2	yes	—
222	C₂³	—	178	4	no	—
223	C₂³	C₂×#13083575	1	1	yes	characteristic
224	Q₈	—	178	8	no	—
225	D₄	—	356	8	no	—
226	D₄	—	356	8	no	—
227	D₄	—	178	8	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	356	4	no	—
230	C₂×C₄	—	356	4	no	—
231	C₂×C₄	—	356	4	no	—
232	C₂×C₄	—	178	4	no	—
233	C₂×C₄	—	178	4	no	—
234	C₂×C₄	#15667974	1	2	yes	—
235	C₂×C₄	#15667974	1	2	yes	—
236	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄×D₈₉	1	2	yes	—
238	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
239	C₂×C₄	C₂×#13083575	1	2	yes	—
240	C₈	—	4	4	no	—
241	C₈	—	89	4	no	—
242	C₈	—	178	2	no	—
243	C₈	#15667974	1	4	yes	—
244	C₈	#15667974	1	4	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	356	8	no	—
247	C₂²	—	356	4	no	—
248	C₂²	C₂×#15667974	1	1	yes	characteristic
249	C₂²	#19152149	1	2	yes	—
250	C₄	—	2	2	no	—
251	C₄	—	356	4	no	—
252	C₄	—	356	4	no	—
253	C₄	#19151719	1	4	yes	complemented
254	C₄	C₂×#15667974	1	2	yes	—
255	C₄	#127×D₈₉	1	1	yes	characteristic
256	C₄	#19152149	1	1	yes	characteristic
257	C₂	—	2	2	no	—
258	C₂	—	712	4	no	—
259	C₂	#26022814	1	1	yes	characteristic
260	C₂	C₂×#19151719	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1848	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#26030680	C₂
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1848	D₈₉
hypocenter	C₈₉	#21706
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
2. C₂×#19151719	C₂	semidirect
3. #26028963	C₂	semidirect
4. #26021143	C₂	semidirect
5. #26021079	C₂	semidirect
6. C₄×#15667922	C₂	semidirect
7. C₈₉×#1848	C₂	semidirect
11. #19151982	C₄	semidirect
12. #19151978	C₄	semidirect
13. #19151975	C₄	semidirect
14. #19151971	C₄	semidirect
48. #15667922	D₄	semidirect
120. C₄×C₈₉	#444	semidirect
136. #1848	D₈₉	semidirect
138. C₈₉	#21706	semidirect
253. C₄	#19151719	semidirect

Maximal subgroups

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

subgroup	index
#26030680	2
C₂×#19151719	2
#26028963	2
#26021143	2
#26021079	2
C₄×#15667922	2
C₈₉×#1848	2
#21706	89