Subgroups — #26024841

Order 11392.

496 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668007	C₂	1	2	yes	maximal, complemented
2	#19151527	C₂	1	2	yes	maximal, complemented
3	#19151520	C₂	1	2	yes	maximal, complemented
4	#19151500	C₂	1	1	yes	maximal, characteristic
5	#19151499	C₂	1	1	yes	maximal, characteristic
6	#19151499	C₂	1	1	yes	maximal, characteristic
7	#19151487	C₂	1	2	yes	maximal, complemented
8	#19151467	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊C₄×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#389×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
13	#15668007	—	2	8	no	—
14	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
17	C₂×#13083553	C₂²	1	2	yes	—
18	C₂×#13083553	C₂²	1	2	yes	—
19	C₂×#13083551	C₂²	1	2	yes	—
20	C₂×#13083551	C₂²	1	2	yes	—
21	C₂×#13083551	C₂²	1	2	yes	—
22	C₂×#13083551	C₂²	1	2	yes	—
23	C₂×#13083551	C₂²	1	2	yes	—
24	C₂×#13083551	C₂²	1	2	yes	—
25	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
28	#15667919	C₂²	1	1	yes	characteristic
29	#15667919	C₂²	1	1	yes	characteristic
30	#15667919	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
34	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
35	#95×C₈₉	C₂²	1	2	yes	—
36	C₂³×D₈₉	C₂³	1	2	yes	—
37	C₂×D₃₅₆	—	2	2	no	—
38	C₂×D₃₅₆	—	2	2	no	—
39	C₂×D₃₅₆	—	2	2	no	—
40	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
41	C₂²×Dic₈₉	C₂³	1	2	yes	—
42	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
43	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
44	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
45	C₄⋊Dic₈₉	—	2	2	no	—
46	C₄⋊Dic₈₉	—	2	2	no	—
47	C₄⋊Dic₈₉	D₄	1	4	yes	complemented
48	#13083553	—	2	4	no	—
49	#13083553	—	4	2	no	—
50	#13083551	—	2	4	no	—
51	#13083551	—	2	4	no	—
52	#13083551	—	2	4	no	—
53	#13083551	—	2	4	no	—
54	#13083551	—	4	2	no	—
55	#13083551	—	4	2	no	—
56	C₄×Dic₈₉	—	2	2	no	—
57	C₄×Dic₈₉	—	2	2	no	—
58	C₄×Dic₈₉	—	2	4	no	—
59	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
61	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
62	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₈₉	—	2	4	no	—
65	C₄⋊C₄×C₈₉	—	2	2	no	—
66	C₄⋊C₄×C₈₉	—	2	2	no	—
67	C₄²×C₈₉	—	2	2	no	—
68	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
69	C₂²×D₈₉	—	2	2	no	—
70	C₂²×D₈₉	—	2	2	no	—
71	C₂²×D₈₉	—	2	2	no	—
72	C₂²×D₈₉	—	2	2	no	—
73	C₂²×D₈₉	—	2	2	no	—
74	C₂²×D₈₉	—	2	2	no	—
75	C₂²×D₈₉	—	2	2	no	—
76	C₂×C₄×C₈₉	—	2	1	no	—
77	C₂×C₄×C₈₉	—	2	1	no	—
78	C₂×C₄×C₈₉	—	2	2	no	—
79	C₂×C₄×C₈₉	—	2	2	no	—
80	C₂×C₄×C₈₉	—	2	1	no	—
81	C₂×C₄×C₈₉	—	2	2	no	—
82	C₂×C₄×C₈₉	—	2	2	no	—
83	C₂×C₄×C₈₉	—	2	2	no	—
84	C₂×C₄×C₈₉	—	2	1	no	—
85	C₂×C₄×C₈₉	—	2	2	no	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₈₉	#41	1	2	yes	—
89	C₂×C₄×C₈₉	#41	1	2	yes	—
90	C₂×C₄×C₈₉	#41	1	2	yes	—
91	C₂×Dic₈₉	—	2	1	no	—
92	C₂×Dic₈₉	—	2	1	no	—
93	C₂×Dic₈₉	—	2	1	no	—
94	C₂×Dic₈₉	—	2	1	no	—
95	C₂×Dic₈₉	—	2	1	no	—
96	C₂×Dic₈₉	—	2	1	no	—
97	C₂×Dic₈₉	—	2	2	no	—
98	C₂×Dic₈₉	—	2	1	no	—
99	C₂×Dic₈₉	—	2	1	no	—
100	C₂×Dic₈₉	—	2	2	no	—
101	C₂×Dic₈₉	—	2	1	no	—
102	C₂×Dic₈₉	—	2	2	no	—
103	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
104	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
105	C₂×Dic₈₉	#41	1	2	yes	—
106	D₃₅₆	—	4	4	no	—
107	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
108	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
109	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
110	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
111	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
112	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
114	C₂×D₈₉	—	4	2	no	—
115	C₂×D₈₉	—	4	2	no	—
116	C₂×D₈₉	—	4	2	no	—
117	C₂×D₈₉	—	4	2	no	—
118	C₂×D₈₉	—	4	2	no	—
119	C₂×D₈₉	—	4	2	no	—
120	C₂×D₈₉	—	4	2	no	—
121	C₄×C₈₉	—	2	2	no	—
122	C₄×C₈₉	—	4	2	no	—
123	C₄×C₈₉	—	4	1	no	—
124	C₄×C₈₉	—	4	2	no	—
125	C₄×C₈₉	#126	1	4	yes	complemented
126	Dic₈₉	—	2	2	no	—
127	Dic₈₉	—	2	2	no	—
128	Dic₈₉	—	2	2	no	—
129	Dic₈₉	—	4	2	no	—
130	Dic₈₉	—	4	1	no	—
131	C₂×C₈₉	C₂×#126	1	1	yes	characteristic
132	C₂×C₈₉	#532	1	1	yes	characteristic
133	C₂×C₈₉	#542	1	1	yes	characteristic
134	C₂×C₈₉	#547	1	1	yes	characteristic
135	C₂×C₈₉	#550	1	1	yes	characteristic
136	C₂×C₈₉	#553	1	1	yes	characteristic
137	C₂×C₈₉	#556	1	1	yes	characteristic
138	D₈₉	—	8	2	no	—
139	#2512	—	89	1	no	maximal, Sylow
140	C₈₉	#2512	1	1	yes	characteristic, complemented, Sylow
141	C₂×#126	—	89	2	no	—
142	#397	—	89	2	no	—
143	#396	—	89	2	no	—
144	#390	—	89	1	no	—
145	#390	—	89	1	no	—
146	#390	—	89	1	no	—
147	#389	D₈₉	1	1	yes	characteristic, complemented
148	#388	—	89	2	no	—
149	#384	—	89	1	no	—
150	C₄×C₄⋊C₄	—	89	1	no	—
151	C₄×C₄⋊C₄	—	89	1	no	—
152	C₂²×D₄	—	89	1	no	—
153	#126	—	178	8	no	—
154	C₂×C₄⋊C₄	—	89	1	no	—
155	C₂×C₄⋊C₄	—	89	2	no	—
156	C₂×C₄⋊C₄	—	89	2	no	—
157	C₂×C₄⋊C₄	—	89	1	no	—
158	C₂×C₄⋊C₄	—	89	1	no	—
159	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
160	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
161	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
162	C₂×C₂²⋊C₄	—	89	2	no	—
163	C₂×C₂²⋊C₄	—	89	2	no	—
164	C₂×C₂²⋊C₄	—	89	2	no	—
165	C₂×C₂²⋊C₄	—	89	2	no	—
166	C₂×C₂²⋊C₄	—	89	2	no	—
167	C₂×C₂²⋊C₄	—	89	2	no	—
168	C₂×C₄²	—	89	2	no	—
169	C₂×C₄²	—	89	1	no	—
170	C₂×C₄²	—	89	1	no	—
171	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
172	#95	—	89	1	no	—
173	#95	—	89	2	no	—
174	#95	—	89	1	no	—
175	#95	C₂×D₈₉	1	2	yes	—
176	C₂⁴	—	89	2	no	—
177	C₂×D₄	—	178	2	no	—
178	C₂×D₄	—	178	2	no	—
179	C₂×D₄	—	178	2	no	—
180	C₂²×C₄	—	89	1	no	—
181	C₂²×C₄	—	89	1	no	—
182	C₂²×C₄	—	89	1	no	—
183	C₂²×C₄	—	89	1	no	—
184	C₂²×C₄	—	89	2	no	—
185	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
187	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
188	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	89	4	no	—
194	C₄⋊C₄	—	356	2	no	—
195	C₄⋊C₄	—	178	4	no	—
196	C₄⋊C₄	—	178	2	no	—
197	C₄⋊C₄	—	178	2	no	—
198	C₂²⋊C₄	—	356	2	no	—
199	C₂²⋊C₄	—	356	2	no	—
200	C₂²⋊C₄	—	178	4	no	—
201	C₂²⋊C₄	—	178	4	no	—
202	C₂²⋊C₄	—	178	4	no	—
203	C₂²⋊C₄	—	178	4	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	178	4	no	—
206	C₄²	—	178	2	no	—
207	C₄²	—	178	2	no	—
208	C₂³	—	178	2	no	—
209	C₂³	—	178	2	no	—
210	C₂³	—	178	2	no	—
211	C₂³	—	178	2	no	—
212	C₂³	—	178	2	no	—
213	C₂³	—	178	2	no	—
214	C₂³	—	178	2	no	—
215	C₂³	C₂³×D₈₉	1	1	yes	characteristic
216	D₄	—	356	4	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	89	2	no	—
229	C₂×C₄	—	89	2	no	—
230	C₂×C₄	—	89	2	no	—
231	C₂×C₄	—	178	2	no	—
232	C₂×C₄	—	178	1	no	—
233	C₂×C₄	—	178	2	no	—
234	C₂×C₄	—	178	1	no	—
235	C₂×C₄	—	178	1	no	—
236	C₂×C₄	—	178	2	no	—
237	C₂×C₄	—	178	1	no	—
238	C₂×C₄	—	178	1	no	—
239	C₂×C₄	—	178	1	no	—
240	C₂×C₄	—	178	1	no	—
241	C₂×C₄	—	178	1	no	—
242	C₂×C₄	—	178	1	no	—
243	C₂×C₄	D₄×D₈₉	1	2	yes	—
244	C₂×C₄	#13083573	1	2	yes	—
245	C₂×C₄	#13083574	1	2	yes	—
246	C₂×C₄	#13083575	1	2	yes	—
247	C₂²	—	356	2	no	—
248	C₂²	—	356	2	no	—
249	C₂²	—	356	2	no	—
250	C₂²	—	356	2	no	—
251	C₂²	—	356	2	no	—
252	C₂²	—	356	2	no	—
253	C₂²	—	356	2	no	—
254	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
255	C₂²	#41×D₈₉	1	1	yes	characteristic
256	C₂²	C₂×#13083573	1	1	yes	characteristic
257	C₂²	C₂×#13083574	1	1	yes	characteristic
258	C₂²	C₂×#13083575	1	1	yes	characteristic
259	C₂²	#15668058	1	1	yes	characteristic
260	C₂²	#15668060	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	2	no	—
265	C₄	—	356	2	no	—
266	C₄	—	356	1	no	—
267	C₄	—	178	2	no	—
268	C₄	—	178	2	no	—
269	C₄	—	178	2	no	—
270	C₄	#15668007	1	4	yes	complemented
271	C₂	—	712	2	no	—
272	C₂	C₂×#15668007	1	1	yes	characteristic
273	C₂	#19152143	1	1	yes	characteristic
274	C₂	#19152221	1	1	yes	characteristic
275	C₂	#19152251	1	1	yes	characteristic
276	C₂	#19152269	1	1	yes	characteristic
277	C₂	#19152301	1	1	yes	characteristic
278	C₂	#19152320	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#389×C₈₉	C₂
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	#389	D₈₉
hypocenter	C₈₉	#2512
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
1. C₂×#15668007	C₂	semidirect
2. #19151527	C₂	semidirect
3. #19151520	C₂	semidirect
7. #19151487	C₂	semidirect
8. #19151467	C₂	semidirect
9. C₄⋊C₄×Dic₈₉	C₂	semidirect
10. C₄×C₄⋊Dic₈₉	C₂	semidirect
11. #389×C₈₉	C₂	semidirect
47. C₄⋊Dic₈₉	D₄	semidirect
125. C₄×C₈₉	#126	semidirect
140. C₈₉	#2512	semidirect
147. #389	D₈₉	semidirect
270. C₄	#15668007	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668007	2
#19151527	2
#19151520	2
#19151500	2
#19151499	2
#19151487	2
#19151467	2
C₄⋊C₄×Dic₈₉	2
C₄×C₄⋊Dic₈₉	2
#389×C₈₉	2
#2512	89