Subgroups — #26024686

Order 11392.

428 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19151536	C₂	1	2	yes	maximal, complemented
2	#19151520	C₂	1	2	yes	maximal
3	#19151497	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151487	C₂	1	2	yes	maximal
5	#19151481	C₂	1	2	yes	maximal, complemented
6	#19151467	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151455	C₂	1	1	yes	maximal, characteristic
8	#19151455	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13083551	C₂	1	1	yes	maximal, characteristic
11	#389×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
16	C₂×#13083553	C₂²	1	2	yes	—
17	C₂×#13083553	C₂²	1	2	yes	—
18	C₂×#13083551	C₂²	1	1	yes	characteristic
19	C₂×#13083551	C₂²	1	2	yes	—
20	C₂×#13083551	C₂²	1	2	yes	—
21	C₂×#13083551	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
24	#15667919	C₂²	1	1	yes	characteristic
25	#15667919	C₂²	1	1	yes	characteristic
26	#15667919	C₂²	1	2	yes	—
27	#15667919	C₂²	1	2	yes	—
28	#15667919	C₂²	1	2	yes	—
29	#15667919	C₂²	1	2	yes	—
30	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
33	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
34	#95×C₈₉	C₂²	1	2	yes	—
35	C₂³×D₈₉	C₂³	1	1	yes	characteristic
36	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
37	C₂²×Dic₈₉	C₂³	1	2	yes	—
38	C₂²×Dic₈₉	C₂³	1	2	yes	—
39	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
40	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
41	C₂×C₄×D₈₉	—	2	2	no	—
42	C₂×C₄×D₈₉	—	2	2	no	—
43	C₂×C₄×D₈₉	—	2	2	no	—
44	C₄⋊Dic₈₉	—	2	2	no	—
45	C₄⋊Dic₈₉	—	2	2	no	—
46	C₄⋊Dic₈₉	—	2	4	no	—
47	#13083553	—	4	2	no	—
48	#13083553	—	4	2	no	—
49	#13083551	—	2	2	no	—
50	#13083551	—	4	2	no	—
51	#13083551	—	4	2	no	—
52	#13083551	Q₈	1	4	yes	—
53	C₄×Dic₈₉	—	2	2	no	—
54	C₄×Dic₈₉	—	4	2	no	—
55	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
56	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
57	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₈₉	—	2	2	no	—
61	C₄⋊C₄×C₈₉	—	2	2	no	—
62	C₄⋊C₄×C₈₉	—	4	2	no	—
63	C₄²×C₈₉	—	2	2	no	—
64	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
65	C₂²×D₈₉	—	2	1	no	—
66	C₂²×D₈₉	—	2	1	no	—
67	C₂²×D₈₉	—	2	1	no	—
68	C₂²×D₈₉	—	2	1	no	—
69	C₂²×D₈₉	—	2	1	no	—
70	C₂²×D₈₉	—	2	1	no	—
71	C₂²×D₈₉	C₂×Q₈	1	2	yes	—
72	C₂×C₄×C₈₉	—	2	1	no	—
73	C₂×C₄×C₈₉	—	2	1	no	—
74	C₂×C₄×C₈₉	—	2	1	no	—
75	C₂×C₄×C₈₉	—	2	2	no	—
76	C₂×C₄×C₈₉	—	2	2	no	—
77	C₂×C₄×C₈₉	—	2	1	no	—
78	C₂×C₄×C₈₉	—	2	2	no	—
79	C₂×C₄×C₈₉	—	2	1	no	—
80	C₂×C₄×C₈₉	—	2	2	no	—
81	C₂×C₄×C₈₉	—	2	2	no	—
82	C₂×C₄×C₈₉	—	2	2	no	—
83	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
84	C₂×C₄×C₈₉	#41	1	2	yes	—
85	C₂×C₄×C₈₉	#41	1	2	yes	—
86	C₂×C₄×C₈₉	#41	1	2	yes	—
87	C₂×Dic₈₉	—	2	1	no	—
88	C₂×Dic₈₉	—	2	1	no	—
89	C₂×Dic₈₉	—	2	1	no	—
90	C₂×Dic₈₉	—	2	1	no	—
91	C₂×Dic₈₉	—	2	2	no	—
92	C₂×Dic₈₉	—	2	2	no	—
93	C₂×Dic₈₉	—	2	2	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	2	no	—
99	C₂×Dic₈₉	—	2	2	no	—
100	C₂×Dic₈₉	C₂×Q₈	1	2	yes	—
101	C₂×Dic₈₉	#41	1	2	yes	—
102	C₄×D₈₉	—	4	4	no	—
103	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
104	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
105	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
106	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
107	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
108	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
109	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂×D₈₉	—	2	2	no	—
111	C₂×D₈₉	—	2	2	no	—
112	C₂×D₈₉	—	2	2	no	—
113	C₂×D₈₉	—	4	1	no	—
114	C₂×D₈₉	—	4	1	no	—
115	C₂×D₈₉	—	4	1	no	—
116	C₂×D₈₉	—	4	1	no	—
117	C₄×C₈₉	—	2	2	no	—
118	C₄×C₈₉	—	4	2	no	—
119	C₄×C₈₉	—	4	1	no	—
120	C₄×C₈₉	—	4	2	no	—
121	C₄×C₈₉	#126	1	4	yes	—
122	Dic₈₉	—	2	2	no	—
123	Dic₈₉	—	2	2	no	—
124	Dic₈₉	—	4	1	no	—
125	Dic₈₉	—	4	2	no	—
126	Dic₈₉	—	4	2	no	—
127	C₂×C₈₉	C₂×#126	1	1	yes	characteristic
128	C₂×C₈₉	#533	1	1	yes	characteristic
129	C₂×C₈₉	#540	1	1	yes	characteristic
130	C₂×C₈₉	#553	1	1	yes	characteristic
131	C₂×C₈₉	#554	1	1	yes	characteristic
132	C₂×C₈₉	#554	1	1	yes	characteristic
133	C₂×C₈₉	#555	1	1	yes	characteristic
134	D₈₉	—	4	2	no	—
135	#2497	—	89	1	no	maximal, Sylow
136	C₈₉	#2497	1	1	yes	characteristic, complemented, Sylow
137	#400	—	89	2	no	—
138	#396	—	89	2	no	—
139	#389	—	89	1	no	—
140	#389	D₈₉	1	1	yes	characteristic, complemented
141	#388	—	89	2	no	—
142	#387	—	89	2	no	—
143	#384	—	89	1	no	—
144	#380	—	89	1	no	—
145	#380	—	89	1	no	—
146	C₄×C₄⋊C₄	—	89	1	no	—
147	C₄×C₂²⋊C₄	—	89	1	no	—
148	C₂³×C₄	—	89	1	no	—
149	C₂×C₄⋊C₄	—	89	2	no	—
150	C₂×C₄⋊C₄	—	89	2	no	—
151	C₂×C₄⋊C₄	—	89	1	no	—
152	C₂×C₄⋊C₄	—	89	1	no	—
153	C₂×C₄⋊C₄	—	89	2	no	—
154	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
157	C₂×C₂²⋊C₄	—	89	1	no	—
158	C₂×C₂²⋊C₄	—	89	2	no	—
159	C₂×C₂²⋊C₄	—	89	2	no	—
160	C₂×C₂²⋊C₄	—	89	1	no	—
161	C₂×C₄²	—	89	2	no	—
162	C₂×C₄²	—	89	1	no	—
163	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
164	#95	—	89	1	no	—
165	#95	—	89	1	no	—
166	#95	—	89	2	no	—
167	#95	—	89	2	no	—
168	#95	—	89	2	no	—
169	#95	—	89	2	no	—
170	#95	C₂×D₈₉	1	2	yes	—
171	C₂⁴	—	89	1	no	—
172	C₂²×C₄	—	89	1	no	—
173	C₂²×C₄	—	89	1	no	—
174	C₂²×C₄	—	89	1	no	—
175	C₂²×C₄	—	89	2	no	—
176	C₂²×C₄	—	89	2	no	—
177	C₂²×C₄	—	178	2	no	—
178	C₂²×C₄	—	178	2	no	—
179	C₂²×C₄	—	178	2	no	—
180	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
182	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
183	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
184	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	356	2	no	—
189	C₄⋊C₄	—	356	2	no	—
190	C₄⋊C₄	—	178	4	no	—
191	C₄⋊C₄	—	178	2	no	—
192	C₄⋊C₄	—	178	2	no	—
193	C₂²⋊C₄	—	89	4	no	—
194	C₂²⋊C₄	—	356	2	no	—
195	C₂²⋊C₄	—	356	2	no	—
196	C₂²⋊C₄	—	178	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	356	2	no	—
199	C₄²	—	178	2	no	—
200	C₂³	—	89	2	no	—
201	C₂³	—	178	1	no	—
202	C₂³	—	178	1	no	—
203	C₂³	—	178	1	no	—
204	C₂³	—	178	1	no	—
205	C₂³	—	178	1	no	—
206	C₂³	—	178	1	no	—
207	C₂³	C₂³×D₈₉	1	1	yes	characteristic
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	89	2	no	—
220	C₂×C₄	—	89	2	no	—
221	C₂×C₄	—	356	4	no	—
222	C₂×C₄	—	178	2	no	—
223	C₂×C₄	—	178	1	no	—
224	C₂×C₄	—	178	2	no	—
225	C₂×C₄	—	178	1	no	—
226	C₂×C₄	—	178	2	no	—
227	C₂×C₄	—	178	1	no	—
228	C₂×C₄	—	178	2	no	—
229	C₂×C₄	—	178	2	no	—
230	C₂×C₄	—	178	2	no	—
231	C₂×C₄	—	178	1	no	—
232	C₂×C₄	—	178	1	no	—
233	C₂×C₄	—	178	1	no	—
234	C₂×C₄	—	178	1	no	—
235	C₂×C₄	Q₈×D₈₉	1	2	yes	—
236	C₂×C₄	#13083573	1	2	yes	—
237	C₂×C₄	#13083574	1	2	yes	—
238	C₂×C₄	#13083575	1	2	yes	—
239	C₂²	—	356	1	no	—
240	C₂²	—	356	1	no	—
241	C₂²	—	356	1	no	—
242	C₂²	—	356	1	no	—
243	C₂²	—	178	2	no	—
244	C₂²	—	178	2	no	—
245	C₂²	—	178	2	no	—
246	C₂²	C₂×Q₈×D₈₉	1	1	yes	characteristic
247	C₂²	#41×D₈₉	1	1	yes	characteristic
248	C₂²	C₂×#13083573	1	1	yes	characteristic
249	C₂²	C₂×#13083574	1	1	yes	characteristic
250	C₂²	C₂×#13083575	1	1	yes	characteristic
251	C₂²	#15668057	1	1	yes	characteristic
252	C₂²	#15668058	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	2	no	—
257	C₄	—	356	2	no	—
258	C₄	—	356	2	no	—
259	C₄	—	356	1	no	—
260	C₄	—	178	2	no	—
261	C₄	—	178	2	no	—
262	C₄	#15668007	1	4	yes	—
263	C₂	—	356	2	no	—
264	C₂	C₂×#15668007	1	1	yes	characteristic
265	C₂	#19152152	1	1	yes	characteristic
266	C₂	#19152199	1	1	yes	characteristic
267	C₂	#19152301	1	1	yes	characteristic
268	C₂	#19152306	1	1	yes	characteristic
269	C₂	#19152309	1	1	yes	characteristic
270	C₂	#19152312	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#389×C₈₉	C₂
prime radical	C₂³×D₈₉	C₂³
hypercenter	#389	D₈₉
hypocenter	C₈₉	#2497
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. #19151536	C₂	semidirect
3. #19151497	C₂	semidirect
5. #19151481	C₂	semidirect
6. #19151467	C₂	semidirect
9. C₄×C₄⋊Dic₈₉	C₂	semidirect
11. #389×C₈₉	C₂	semidirect
136. C₈₉	#2497	semidirect
140. #389	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19151536	2
#19151520	2
#19151497	2
#19151487	2
#19151481	2
#19151467	2
#19151455	2
C₄×C₄⋊Dic₈₉	2
C₄×#13083551	2
#389×C₈₉	2
#2497	89