Subgroups — #36563042

Order 22784.

1046 conjugacy classes of subgroups.

The subgroup lattice has 268 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×#19151716	C₂	1	4	yes	maximal, complemented
3	#26028965	C₂	1	2	yes	maximal, complemented
4	#26021140	C₂	1	4	yes	maximal, complemented
5	#26021080	C₂	1	2	yes	maximal, complemented
6	#26020424	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668034	C₂²	1	4	yes	—
9	C₂×#15668018	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	#19151716	—	2	16	no	—
16	#19151677	—	2	4	no	—
17	C₈×D₃₅₆	—	2	4	no	—
18	C₂×#15667948	C₂²	1	4	yes	—
19	#19151622	C₂²	1	2	yes	—
20	C₂×#15667940	C₂²	1	8	yes	—
21	#19151564	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₄³×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	#15668018	—	2	16	no	—
34	C₂×#13083558	C₂³	1	4	yes	—
35	C₂×#13083557	—	2	4	no	—
36	C₂×C₈×D₈₉	—	2	4	no	—
37	#15668012	C₂×C₄	1	2	yes	—
38	#15668009	C₂×C₄	1	2	yes	—
39	#15668000	C₂×C₄	1	4	yes	—
40	#15667986	—	2	4	no	—
41	C₄×D₃₅₆	—	2	4	no	—
42	#15667948	—	2	8	no	—
43	C₄⋊#10623494	—	2	4	no	—
44	#15667940	—	2	16	no	—
45	#15667924	—	2	8	no	—
46	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
48	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	—	2	2	no	—
50	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
51	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₈₉	—	2	4	no	—
53	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₈₉	D₄	1	2	yes	—
57	C₄×C₈×C₈₉	D₄	1	2	yes	—
58	C₄×C₈×C₈₉	D₄	1	8	yes	complemented
59	#13083574	—	2	8	no	—
60	#13083574	—	2	8	no	—
61	C₂×Dic₁₇₈	C₂²×C₄	1	4	yes	—
62	C₂×D₃₅₆	—	2	4	no	—
63	C₂×D₃₅₆	C₂²×C₄	1	4	yes	—
64	C₂×C₂²⋊D₈₉	—	2	4	no	—
65	C₂×C₄×D₈₉	—	2	4	no	—
66	Dic₃₅₆	—	2	8	no	—
67	#13083563	—	2	16	no	—
68	D₇₁₂	—	2	8	no	—
69	#13083558	—	2	8	no	—
70	#13083557	—	4	8	no	—
71	C₂×#10623494	—	2	4	no	—
72	C₈×D₈₉	—	4	8	no	—
73	C₄⋊Dic₈₉	—	2	4	no	—
74	#13083551	—	2	8	no	—
75	C₂²×C₄×C₈₉	—	2	2	no	—
76	C₂²×C₄×C₈₉	#41	1	2	yes	—
77	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₈₉	—	2	2	no	—
79	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
83	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
85	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	#41	1	4	yes	—
87	C₄²×C₈₉	—	2	8	no	—
88	C₄²×C₈₉	—	2	2	no	—
89	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₈₉	C₂×D₄	1	4	yes	—
92	C₄²×C₈₉	C₂×D₄	1	2	yes	—
93	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
94	C₂²×D₈₉	—	2	4	no	—
95	C₂×C₄×C₈₉	—	2	8	no	—
96	C₂×C₄×C₈₉	—	2	4	no	—
97	C₂×C₄×C₈₉	—	2	4	no	—
98	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
102	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
104	C₂²⋊D₈₉	—	4	8	no	—
105	C₂×Dic₈₉	—	2	4	no	—
106	D₃₅₆	—	2	8	no	—
107	D₃₅₆	—	4	8	no	—
108	C₄×D₈₉	—	4	4	no	—
109	C₄×D₈₉	—	4	4	no	—
110	Dic₁₇₈	—	2	8	no	—
111	C₈×C₈₉	—	2	2	no	—
112	C₈×C₈₉	C₄×D₄	1	4	yes	—
113	C₈×C₈₉	C₄×D₄	1	4	yes	—
114	C₈×C₈₉	#127	1	2	yes	—
115	C₈×C₈₉	#127	1	2	yes	—
116	#10623494	—	4	4	no	—
117	C₂²×C₈₉	—	2	2	no	—
118	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂²×C₈₉	#532	1	2	yes	—
120	C₂×D₈₉	—	4	4	no	—
121	C₂×D₈₉	—	4	8	no	—
122	C₄×C₈₉	—	2	8	no	—
123	C₄×C₈₉	—	2	2	no	—
124	C₄×C₈₉	#443	1	4	yes	complemented
125	C₄×C₈₉	C₂×C₄×D₄	1	2	yes	—
126	C₄×C₈₉	C₂×#127	1	1	yes	characteristic
127	C₄×C₈₉	#532	1	1	yes	characteristic
128	Dic₈₉	—	4	4	no	—
129	#21705	—	89	1	no	maximal, Sylow
130	C₂×C₈₉	—	2	2	no	—
131	C₂×C₈₉	C₄×#127	1	1	yes	characteristic
132	C₂×C₈₉	C₂×#443	1	2	yes	—
133	D₈₉	—	8	4	no	—
134	#3147	—	89	1	no	—
135	C₂×#443	—	89	4	no	—
136	#2931	—	89	2	no	—
137	#1974	—	89	4	no	—
138	#1962	—	89	2	no	—
139	#1848	—	89	1	no	—
140	C₄²×C₈	D₈₉	1	1	yes	characteristic, complemented
141	C₈₉	#21705	1	1	yes	characteristic, complemented, Sylow
142	C₂×#135	—	89	4	no	—
143	C₂×#131	—	89	4	no	—
144	#532	—	89	2	no	—
145	#495	—	89	4	no	—
146	#494	—	89	1	no	—
147	#493	—	89	1	no	—
148	#492	—	89	2	no	—
149	#443	—	178	16	no	—
150	#436	—	178	4	no	—
151	C₈×D₄	—	178	4	no	—
152	C₂×#108	—	89	4	no	—
153	#423	—	89	2	no	—
154	C₂×#104	—	89	8	no	—
155	#408	—	89	4	no	—
156	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
157	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
159	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
160	C₄³	C₂×D₈₉	1	1	yes	characteristic
161	C₂×#41	—	89	4	no	—
162	#135	—	178	8	no	—
163	#135	—	178	8	no	—
164	C₂×Q₁₆	—	89	4	no	—
165	C₂×SD₁₆	—	89	8	no	—
166	C₂×D₈	—	89	4	no	—
167	#131	—	178	16	no	—
168	C₂×M₁₆	—	89	4	no	—
169	C₂×M₁₆	—	178	4	no	—
170	C₂²×C₈	—	178	4	no	—
171	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
173	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
174	C₄⋊Q₈	—	89	2	no	—
175	#127	—	89	2	no	—
176	#124	—	89	4	no	—
177	#121	—	178	4	no	—
178	C₄×D₄	—	178	4	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
181	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
182	#108	—	178	8	no	—
183	C₄⋊C₈	—	178	4	no	—
184	#104	—	178	16	no	—
185	C₂²⋊C₈	—	178	8	no	—
186	C₄×C₈	—	2	4	no	—
187	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
188	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
189	C₄×C₈	C₄×D₈₉	1	2	yes	—
190	C₄×C₈	D₃₅₆	1	2	yes	—
191	C₄×C₈	D₃₅₆	1	2	yes	—
192	C₄×C₈	D₃₅₆	1	8	yes	complemented
193	#41	—	178	8	no	—
194	#41	—	178	8	no	—
195	C₂×Q₈	—	89	4	no	—
196	C₂×D₄	—	89	4	no	—
197	C₂×D₄	—	178	4	no	—
198	C₂×D₄	—	178	4	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	178	4	no	—
201	C₂²×C₄	#13083574	1	2	yes	—
202	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
203	Q₁₆	—	178	8	no	—
204	SD₁₆	—	178	16	no	—
205	D₈	—	178	8	no	—
206	M₁₆	—	356	8	no	—
207	M₁₆	—	178	8	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	356	8	no	—
210	C₂×C₈	—	178	4	no	—
211	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
212	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
213	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
214	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
215	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
216	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
217	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
218	C₂×C₈	#13083574	1	4	yes	—
219	C₄⋊C₄	—	178	4	no	—
220	C₂²⋊C₄	—	178	8	no	—
221	C₄²	—	2	8	no	—
222	C₄²	—	2	2	no	—
223	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
224	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
225	C₄²	C₂×D₃₅₆	1	4	yes	—
226	C₄²	C₂×D₃₅₆	1	2	yes	—
227	C₂³	—	178	4	no	—
228	C₂³	C₂×#13083574	1	1	yes	characteristic
229	Q₈	—	178	8	no	—
230	D₄	—	356	8	no	—
231	D₄	—	356	8	no	—
232	D₄	—	178	8	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	356	4	no	—
237	C₂×C₄	—	356	4	no	—
238	C₂×C₄	—	178	4	no	—
239	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
240	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
241	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
242	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₃₅₆	1	2	yes	—
244	C₂×C₄	C₂×#13083574	1	2	yes	—
245	C₈	—	2	2	no	—
246	C₈	—	356	4	no	—
247	C₈	C₄×D₃₅₆	1	4	yes	—
248	C₈	C₄×D₃₅₆	1	4	yes	—
249	C₈	#15668009	1	2	yes	—
250	C₈	#15668009	1	2	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	356	8	no	—
253	C₂²	—	356	4	no	—
254	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
255	C₂²	#19152148	1	2	yes	—
256	C₄	—	2	8	no	—
257	C₄	—	2	2	no	—
258	C₄	—	356	4	no	—
259	C₄	#19151716	1	4	yes	complemented
260	C₄	C₂×C₄×D₃₅₆	1	2	yes	—
261	C₄	C₂×#15668009	1	1	yes	characteristic
262	C₄	#19152148	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	712	4	no	—
265	C₂	C₄×#15668009	1	1	yes	characteristic
266	C₂	C₂×#19151716	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄²×C₈×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#26030680	C₂
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄²×C₈	D₈₉
hypocenter	C₈₉	#21705
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#19151716	C₂	semidirect
3. #26028965	C₂	semidirect
4. #26021140	C₂	semidirect
5. #26021080	C₂	semidirect
6. #26020424	C₂	semidirect
7. C₄²×C₈×C₈₉	C₂	semidirect
11. #19151982	C₄	semidirect
12. #19151978	C₄	semidirect
13. #19151975	C₄	semidirect
14. #19151971	C₄	semidirect
58. C₄×C₈×C₈₉	D₄	semidirect
124. C₄×C₈₉	#443	semidirect
140. C₄²×C₈	D₈₉	semidirect
141. C₈₉	#21705	semidirect
192. C₄×C₈	D₃₅₆	semidirect
259. C₄	#19151716	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×#19151716	2
#26028965	2
#26021140	2
#26021080	2
#26020424	2
C₄²×C₈×C₈₉	2
#21705	89