Subgroups — #26024039

Order 11392.

540 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668000	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#19151527	C₂	1	2	yes	maximal, complemented
4	#19151524	C₂	1	2	yes	maximal, complemented
5	#19151504	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151504	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151500	C₂	1	1	yes	maximal, characteristic
8	#19151496	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151469	C₂	1	2	yes	maximal, complemented
10	#19151460	C₂	1	2	yes	maximal, complemented
11	C₄×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
13	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₈₉	C₂²	1	2	yes	—
15	#15668000	—	2	4	no	—
16	C₄×D₃₅₆	—	2	4	no	—
17	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
19	C₂×#13083553	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	#15667919	C₂²	1	2	yes	—
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×C₄²×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
35	#95×C₈₉	C₂²	1	1	yes	characteristic
36	#95×C₈₉	C₂²	1	1	yes	characteristic
37	C₂³×D₈₉	C₂³	1	2	yes	—
38	C₂×Dic₁₇₈	—	2	2	no	—
39	C₂×Dic₁₇₈	—	2	2	no	—
40	C₂×Dic₁₇₈	—	2	2	no	—
41	C₂×D₃₅₆	—	2	2	no	—
42	C₂×D₃₅₆	—	2	2	no	—
43	C₂×D₃₅₆	D₄	1	4	yes	—
44	C₂²×Dic₈₉	C₂³	1	2	yes	—
45	C₂²×Dic₈₉	C₂³	1	2	yes	—
46	C₂²×Dic₈₉	C₂³	1	2	yes	—
47	C₂×C₄×D₈₉	—	2	4	no	—
48	C₂×C₄×D₈₉	—	2	4	no	—
49	C₂×C₄×D₈₉	—	4	2	no	—
50	C₄⋊Dic₈₉	—	2	2	no	—
51	C₄⋊Dic₈₉	—	2	2	no	—
52	#13083553	—	4	2	no	—
53	#13083551	—	2	4	no	—
54	#13083551	—	2	4	no	—
55	#13083551	—	2	4	no	—
56	#13083551	—	2	4	no	—
57	#13083551	—	4	2	no	—
58	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₈₉	—	2	2	no	—
66	C₄⋊C₄×C₈₉	—	2	2	no	—
67	C₄²×C₈₉	—	2	2	no	—
68	C₄²×C₈₉	—	2	2	no	—
69	C₄²×C₈₉	—	2	2	no	—
70	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
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	C₂²×D₈₉	—	2	2	no	—
77	C₂²×D₈₉	C₂×D₄	1	4	yes	—
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	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	1	no	—
91	C₂×C₄×C₈₉	—	2	1	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₈₉	#41	1	2	yes	—
96	C₂×C₄×C₈₉	#41	1	2	yes	—
97	C₂×C₄×C₈₉	#41	1	2	yes	—
98	C₂×C₄×C₈₉	#41	1	2	yes	—
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	C₂×Dic₈₉	—	2	2	no	—
104	C₂×Dic₈₉	—	2	2	no	—
105	C₂×Dic₈₉	—	2	2	no	—
106	C₂×Dic₈₉	—	2	2	no	—
107	C₂×Dic₈₉	—	2	2	no	—
108	D₃₅₆	—	2	8	no	—
109	C₄×D₈₉	—	4	8	no	—
110	Dic₁₇₈	—	4	4	no	—
111	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
112	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
113	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
114	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
115	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
116	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
118	C₂×D₈₉	—	2	4	no	—
119	C₂×D₈₉	—	2	4	no	—
120	C₂×D₈₉	—	2	4	no	—
121	C₂×D₈₉	—	4	2	no	—
122	C₂×D₈₉	—	4	2	no	—
123	C₂×D₈₉	—	4	2	no	—
124	C₂×D₈₉	—	4	2	no	—
125	C₄×C₈₉	—	2	2	no	—
126	C₄×C₈₉	—	2	2	no	—
127	C₄×C₈₉	—	4	1	no	—
128	C₄×C₈₉	—	4	1	no	—
129	C₄×C₈₉	—	4	1	no	—
130	C₄×C₈₉	—	4	1	no	—
131	C₄×C₈₉	#123	1	4	yes	—
132	Dic₈₉	—	4	2	no	—
133	Dic₈₉	—	4	2	no	—
134	Dic₈₉	—	4	2	no	—
135	C₂×C₈₉	C₂×#123	1	1	yes	characteristic
136	C₂×C₈₉	#529	1	1	yes	characteristic
137	C₂×C₈₉	#542	1	1	yes	characteristic
138	C₂×C₈₉	#546	1	1	yes	characteristic
139	C₂×C₈₉	#547	1	1	yes	characteristic
140	C₂×C₈₉	#556	1	1	yes	characteristic
141	C₂×C₈₉	#559	1	1	yes	characteristic
142	D₈₉	—	4	4	no	—
143	#2447	—	89	1	no	maximal, Sylow
144	C₈₉	#2447	1	1	yes	characteristic, complemented, Sylow
145	C₂×#124	—	89	1	no	—
146	C₂×C₄×D₄	—	89	1	no	—
147	#397	—	89	2	no	—
148	#397	—	89	2	no	—
149	#391	—	89	1	no	—
150	#391	—	89	1	no	—
151	#390	—	89	1	no	—
152	#389	—	89	1	no	—
153	#385	—	89	2	no	—
154	#381	—	89	2	no	—
155	C₄×C₄⋊C₄	D₈₉	1	1	yes	characteristic, complemented
156	C₂²×Q₈	—	89	1	no	—
157	C₂²×D₄	—	89	1	no	—
158	C₂³×C₄	—	89	2	no	—
159	#124	—	178	4	no	—
160	C₄×D₄	—	178	4	no	—
161	C₂×C₄⋊C₄	—	89	2	no	—
162	C₂×C₄⋊C₄	—	89	1	no	—
163	C₂×C₄⋊C₄	—	89	1	no	—
164	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
166	C₂×C₂²⋊C₄	—	89	2	no	—
167	C₂×C₂²⋊C₄	—	89	2	no	—
168	C₂×C₂²⋊C₄	—	89	2	no	—
169	C₂×C₂²⋊C₄	—	89	2	no	—
170	C₂×C₂²⋊C₄	—	89	2	no	—
171	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
172	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
173	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
174	#95	—	89	2	no	—
175	#95	—	89	2	no	—
176	#95	—	89	2	no	—
177	#95	—	89	2	no	—
178	#95	—	89	2	no	—
179	#95	C₂×D₈₉	1	1	yes	characteristic
180	#95	C₂×D₈₉	1	1	yes	characteristic
181	C₂⁴	—	89	2	no	—
182	C₂×Q₈	—	178	2	no	—
183	C₂×Q₈	—	178	2	no	—
184	C₂×Q₈	—	178	2	no	—
185	C₂×D₄	—	89	4	no	—
186	C₂×D₄	—	178	2	no	—
187	C₂×D₄	—	178	2	no	—
188	C₂²×C₄	—	89	2	no	—
189	C₂²×C₄	—	89	2	no	—
190	C₂²×C₄	—	89	2	no	—
191	C₂²×C₄	—	356	2	no	—
192	C₂²×C₄	—	178	4	no	—
193	C₂²×C₄	—	178	4	no	—
194	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
199	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	356	2	no	—
204	C₄⋊C₄	—	178	2	no	—
205	C₄⋊C₄	—	178	2	no	—
206	C₂²⋊C₄	—	356	2	no	—
207	C₂²⋊C₄	—	178	4	no	—
208	C₂²⋊C₄	—	178	4	no	—
209	C₂²⋊C₄	—	178	4	no	—
210	C₂²⋊C₄	—	178	4	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₂³	—	89	4	no	—
215	C₂³	—	178	2	no	—
216	C₂³	—	178	2	no	—
217	C₂³	—	178	2	no	—
218	C₂³	—	178	2	no	—
219	C₂³	—	178	2	no	—
220	C₂³	—	178	2	no	—
221	C₂³	C₂³×D₈₉	1	1	yes	characteristic
222	Q₈	—	356	4	no	—
223	D₄	—	178	8	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	356	8	no	—
241	C₂×C₄	—	178	2	no	—
242	C₂×C₄	—	178	2	no	—
243	C₂×C₄	—	178	2	no	—
244	C₂×C₄	—	178	2	no	—
245	C₂×C₄	—	178	2	no	—
246	C₂×C₄	—	178	2	no	—
247	C₂×C₄	—	178	2	no	—
248	C₂×C₄	—	178	2	no	—
249	C₂×C₄	—	178	2	no	—
250	C₂×C₄	D₄×D₈₉	1	2	yes	—
251	C₂×C₄	#13083574	1	2	yes	—
252	C₂×C₄	#13083574	1	2	yes	—
253	C₂×C₄	#13083574	1	2	yes	—
254	C₂×C₄	#13083575	1	2	yes	—
255	C₂²	—	356	2	no	—
256	C₂²	—	356	2	no	—
257	C₂²	—	356	2	no	—
258	C₂²	—	356	2	no	—
259	C₂²	—	178	4	no	—
260	C₂²	—	178	4	no	—
261	C₂²	—	178	4	no	—
262	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
263	C₂²	C₂×#13083574	1	1	yes	characteristic
264	C₂²	C₂×#13083574	1	1	yes	characteristic
265	C₂²	C₂×#13083574	1	1	yes	characteristic
266	C₂²	C₂×#13083575	1	1	yes	characteristic
267	C₂²	#15668058	1	1	yes	characteristic
268	C₂²	#15668059	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	356	2	no	—
276	C₄	—	356	2	no	—
277	C₄	—	356	2	no	—
278	C₄	#15667997	1	4	yes	—
279	C₂	—	356	4	no	—
280	C₂	C₂×#15667997	1	1	yes	characteristic
281	C₂	#19152132	1	1	yes	characteristic
282	C₂	#19152219	1	1	yes	characteristic
283	C₂	#19152244	1	1	yes	characteristic
284	C₂	#19152253	1	1	yes	characteristic
285	C₂	#19152321	1	1	yes	characteristic
286	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	#2447
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
3. #19151527	C₂	semidirect
4. #19151524	C₂	semidirect
5. #19151504	C₂	semidirect
6. #19151504	C₂	semidirect
8. #19151496	C₂	semidirect
9. #19151469	C₂	semidirect
10. #19151460	C₂	semidirect
11. C₄×C₄⋊C₄×C₈₉	C₂	semidirect
144. C₈₉	#2447	semidirect
155. 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
C₂×#15668000	2
C₂×C₄×D₃₅₆	2
#19151527	2
#19151524	2
#19151504	2
#19151500	2
#19151496	2
#19151469	2
#19151460	2
C₄×C₄⋊C₄×C₈₉	2
#2447	89