Subgroups — #26024849

Order 11392.

440 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19151527	C₂	1	2	yes	maximal, complemented
2	#19151524	C₂	1	2	yes	maximal, complemented
3	#19151500	C₂	1	1	yes	maximal, characteristic
4	#19151500	C₂	1	1	yes	maximal, characteristic
5	#19151500	C₂	1	1	yes	maximal, characteristic
6	#19151496	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151488	C₂	1	2	yes	maximal, complemented
8	#19151487	C₂	1	2	yes	maximal, complemented
9	#19151464	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₃₅₆	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	1	yes	characteristic
16	C₂×#13083553	C₂²	1	2	yes	—
17	C₂×#13083551	C₂²	1	2	yes	—
18	C₂×#13083551	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×C₄×Dic₈₉	C₂²	1	2	yes	—
24	#15667919	C₂²	1	2	yes	—
25	#15667919	C₂²	1	2	yes	—
26	#15667919	C₂²	1	2	yes	—
27	#15667919	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
33	#95×C₈₉	C₂²	1	1	yes	characteristic
34	#95×C₈₉	C₂²	1	1	yes	characteristic
35	C₂³×D₈₉	C₂³	1	2	yes	—
36	C₂×D₃₅₆	—	2	2	no	—
37	C₂×D₃₅₆	—	2	2	no	—
38	C₂×D₃₅₆	—	2	2	no	—
39	C₂²×Dic₈₉	C₂³	1	2	yes	—
40	C₂²×Dic₈₉	C₂³	1	2	yes	—
41	C₂²×Dic₈₉	C₂³	1	2	yes	—
42	C₄⋊Dic₈₉	—	2	2	no	—
43	C₄⋊Dic₈₉	—	2	2	no	—
44	C₄⋊Dic₈₉	—	2	2	no	—
45	#13083553	—	4	2	no	—
46	#13083551	—	2	4	no	—
47	#13083551	—	2	4	no	—
48	#13083551	—	4	2	no	—
49	#13083551	—	4	2	no	—
50	#13083551	—	4	2	no	—
51	#13083551	—	4	2	no	—
52	C₄×Dic₈₉	—	4	2	no	—
53	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
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	C₄²×C₈₉	—	2	2	no	—
63	C₄²×C₈₉	—	2	2	no	—
64	C₄²×C₈₉	—	2	2	no	—
65	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
66	C₂²×D₈₉	—	2	2	no	—
67	C₂²×D₈₉	—	2	2	no	—
68	C₂²×D₈₉	—	2	2	no	—
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	C₂×C₄×C₈₉	—	2	1	no	—
74	C₂×C₄×C₈₉	—	2	1	no	—
75	C₂×C₄×C₈₉	—	2	1	no	—
76	C₂×C₄×C₈₉	—	2	1	no	—
77	C₂×C₄×C₈₉	—	2	1	no	—
78	C₂×C₄×C₈₉	—	2	1	no	—
79	C₂×C₄×C₈₉	—	2	1	no	—
80	C₂×C₄×C₈₉	—	2	1	no	—
81	C₂×C₄×C₈₉	—	2	1	no	—
82	C₂×C₄×C₈₉	—	2	1	no	—
83	C₂×C₄×C₈₉	—	2	1	no	—
84	C₂×C₄×C₈₉	—	2	1	no	—
85	C₂×C₄×C₈₉	—	2	1	no	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	#41	1	2	yes	—
90	C₂×C₄×C₈₉	#41	1	2	yes	—
91	C₂×C₄×C₈₉	#41	1	2	yes	—
92	C₂×C₄×C₈₉	#41	1	2	yes	—
93	C₂×C₄×C₈₉	#41	1	2	yes	—
94	C₂×Dic₈₉	—	2	2	no	—
95	C₂×Dic₈₉	—	2	2	no	—
96	C₂×Dic₈₉	—	2	2	no	—
97	C₂×Dic₈₉	—	2	2	no	—
98	C₂×Dic₈₉	—	2	2	no	—
99	C₂×Dic₈₉	—	2	2	no	—
100	C₂×Dic₈₉	—	2	2	no	—
101	C₂×Dic₈₉	—	2	2	no	—
102	C₂×Dic₈₉	—	2	2	no	—
103	D₃₅₆	—	4	4	no	—
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	C₂×#41	1	1	yes	characteristic
109	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂×D₈₉	—	4	2	no	—
112	C₂×D₈₉	—	4	2	no	—
113	C₂×D₈₉	—	4	2	no	—
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	C₄×C₈₉	—	2	2	no	—
119	C₄×C₈₉	—	2	2	no	—
120	C₄×C₈₉	—	4	1	no	—
121	C₄×C₈₉	—	4	1	no	—
122	C₄×C₈₉	—	4	1	no	—
123	C₄×C₈₉	—	4	1	no	—
124	C₄×C₈₉	#126	1	4	yes	—
125	Dic₈₉	—	4	2	no	—
126	Dic₈₉	—	4	2	no	—
127	Dic₈₉	—	4	2	no	—
128	C₂×C₈₉	C₂×#126	1	1	yes	characteristic
129	C₂×C₈₉	#529	1	1	yes	characteristic
130	C₂×C₈₉	#540	1	1	yes	characteristic
131	C₂×C₈₉	#553	1	1	yes	characteristic
132	C₂×C₈₉	#555	1	1	yes	characteristic
133	C₂×C₈₉	#559	1	1	yes	characteristic
134	C₂×C₈₉	#559	1	1	yes	characteristic
135	D₈₉	—	8	2	no	—
136	#2513	—	89	1	no	maximal, Sylow
137	C₈₉	#2513	1	1	yes	characteristic, complemented, Sylow
138	#397	—	89	2	no	—
139	#397	—	89	2	no	—
140	#390	—	89	1	no	—
141	#390	—	89	1	no	—
142	#390	—	89	1	no	—
143	#389	—	89	1	no	—
144	#388	—	89	2	no	—
145	#388	—	89	2	no	—
146	#382	—	89	1	no	—
147	C₄×C₄⋊C₄	—	89	1	no	—
148	C₄×C₄⋊C₄	D₈₉	1	1	yes	characteristic, complemented
149	C₂²×D₄	—	89	1	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	1	no	—
154	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	—	89	2	no	—
157	C₂×C₂²⋊C₄	—	89	2	no	—
158	C₂×C₂²⋊C₄	—	89	2	no	—
159	C₂×C₂²⋊C₄	—	89	2	no	—
160	C₂×C₂²⋊C₄	—	89	2	no	—
161	C₂×C₂²⋊C₄	—	89	2	no	—
162	C₂×C₄²	—	89	2	no	—
163	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
164	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
165	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
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	1	yes	characteristic
171	#95	C₂×D₈₉	1	1	yes	characteristic
172	C₂⁴	—	89	2	no	—
173	C₂×D₄	—	178	2	no	—
174	C₂×D₄	—	178	2	no	—
175	C₂×D₄	—	178	2	no	—
176	C₂²×C₄	—	89	2	no	—
177	C₂²×C₄	—	89	2	no	—
178	C₂²×C₄	—	89	2	no	—
179	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
182	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
183	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
184	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	356	2	no	—
189	C₄⋊C₄	—	178	2	no	—
190	C₄⋊C₄	—	178	2	no	—
191	C₄⋊C₄	—	178	2	no	—
192	C₂²⋊C₄	—	356	2	no	—
193	C₂²⋊C₄	—	356	2	no	—
194	C₂²⋊C₄	—	356	2	no	—
195	C₂²⋊C₄	—	356	2	no	—
196	C₂²⋊C₄	—	178	4	no	—
197	C₂²⋊C₄	—	178	4	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	356	2	no	—
202	C₂³	—	178	2	no	—
203	C₂³	—	178	2	no	—
204	C₂³	—	178	2	no	—
205	C₂³	—	178	2	no	—
206	C₂³	—	178	2	no	—
207	C₂³	—	178	2	no	—
208	C₂³	—	178	2	no	—
209	C₂³	C₂³×D₈₉	1	1	yes	characteristic
210	D₄	—	356	4	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	178	2	no	—
228	C₂×C₄	—	178	2	no	—
229	C₂×C₄	—	178	2	no	—
230	C₂×C₄	—	178	2	no	—
231	C₂×C₄	—	178	2	no	—
232	C₂×C₄	—	178	2	no	—
233	C₂×C₄	—	178	2	no	—
234	C₂×C₄	—	178	2	no	—
235	C₂×C₄	—	178	2	no	—
236	C₂×C₄	#13083573	1	2	yes	—
237	C₂×C₄	#13083574	1	2	yes	—
238	C₂×C₄	#13083574	1	2	yes	—
239	C₂×C₄	#13083574	1	2	yes	—
240	C₂×C₄	#13083575	1	2	yes	—
241	C₂²	—	356	2	no	—
242	C₂²	—	356	2	no	—
243	C₂²	—	356	2	no	—
244	C₂²	—	356	2	no	—
245	C₂²	—	356	2	no	—
246	C₂²	—	356	2	no	—
247	C₂²	—	356	2	no	—
248	C₂²	C₂×#13083573	1	1	yes	characteristic
249	C₂²	C₂×#13083574	1	1	yes	characteristic
250	C₂²	C₂×#13083574	1	1	yes	characteristic
251	C₂²	C₂×#13083574	1	1	yes	characteristic
252	C₂²	C₂×#13083575	1	1	yes	characteristic
253	C₂²	#15668058	1	1	yes	characteristic
254	C₂²	#15668058	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	1	no	—
260	C₄	—	4	1	no	—
261	C₄	—	356	2	no	—
262	C₄	—	356	2	no	—
263	C₄	—	356	2	no	—
264	C₄	#15668007	1	4	yes	—
265	C₂	—	712	2	no	—
266	C₂	C₂×#15668007	1	1	yes	characteristic
267	C₂	#19152132	1	1	yes	characteristic
268	C₂	#19152203	1	1	yes	characteristic
269	C₂	#19152294	1	1	yes	characteristic
270	C₂	#19152313	1	1	yes	characteristic
271	C₂	#19152333	1	1	yes	characteristic
272	C₂	#19152333	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₄×C₄⋊C₄×C₈₉	C₂
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	C₄×C₄⋊C₄	D₈₉
hypocenter	C₈₉	#2513
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #19151527	C₂	semidirect
2. #19151524	C₂	semidirect
6. #19151496	C₂	semidirect
7. #19151488	C₂	semidirect
8. #19151487	C₂	semidirect
9. #19151464	C₂	semidirect
10. C₄×C₄⋊Dic₈₉	C₂	semidirect
11. C₄×C₄⋊C₄×C₈₉	C₂	semidirect
137. C₈₉	#2513	semidirect
148. C₄×C₄⋊C₄	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19151527	2
#19151524	2
#19151500	2
#19151496	2
#19151488	2
#19151487	2
#19151464	2
C₄×C₄⋊Dic₈₉	2
C₄×C₄⋊C₄×C₈₉	2
#2513	89