Subgroups — #36469549

Order 22784.

382 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152016	C₂	1	1	yes	maximal, characteristic
2	#26021872	C₂	1	1	yes	maximal, characteristic
3	#26021856	C₂	1	1	yes	maximal, characteristic, complemented
4	#26020053	C₂	1	1	yes	maximal, characteristic
5	#26018342	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018341	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2110	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668018	C₂²	1	1	yes	characteristic
9	#19152016	—	2	2	no	—
10	#19152016	—	2	2	no	—
11	C₂×#15667952	C₂²	1	1	yes	characteristic
12	C₂×#15667951	C₄	1	2	yes	complemented
13	C₂²×#13083549	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₈₉	C₄	1	2	yes	complemented
15	#19151580	C₄	1	1	yes	characteristic
16	#19151571	C₄	1	1	yes	characteristic
17	#19151553	C₂²	1	1	yes	characteristic
18	#19151379	C₄	1	4	yes	complemented
19	#19151375	—	2	2	no	—
20	#19151370	C₄	1	1	yes	characteristic
21	#19151368	C₄	1	1	yes	characteristic
22	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
24	#405×C₈₉	C₂²	1	1	yes	characteristic
25	#347×C₈₉	—	2	2	no	—
26	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
27	#15668018	—	2	1	no	—
28	#15668018	—	2	1	no	—
29	#15668018	—	2	1	no	—
30	#15668018	—	2	1	no	—
31	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
32	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
33	C₂×#13083557	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
35	#15667971	C₂×C₄	1	1	yes	characteristic
36	#15667952	—	2	1	no	—
37	#15667952	C₂×C₄	1	2	yes	—
38	#15667951	—	2	4	no	—
39	C₂×#13083549	—	2	1	no	—
40	C₂×#13083549	—	2	1	no	—
41	C₂×#13083549	—	2	1	no	—
42	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
43	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
44	C₁₆×D₈₉	—	2	4	no	—
45	#15667930	—	2	1	no	—
46	#15667928	—	2	1	no	—
47	#15667926	—	2	2	no	—
48	#15667922	—	2	1	no	—
49	C₈×Dic₈₉	—	2	1	no	—
50	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
52	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₈₉	—	2	1	no	—
54	M₃₂×C₈₉	C₂×C₄	1	2	yes	—
55	M₃₂×C₈₉	D₄	1	2	yes	—
56	M₃₂×C₈₉	D₄	1	2	yes	—
57	C₂×C₁₆×C₈₉	—	2	1	no	—
58	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
59	#97×C₈₉	—	2	1	no	—
60	C₄×C₈×C₈₉	—	2	1	no	—
61	#13083574	—	2	1	no	—
62	#13083574	—	2	1	no	—
63	#13083574	—	2	1	no	—
64	#13083574	—	2	1	no	—
65	C₂×Dic₁₇₈	M₁₆	1	1	yes	characteristic
66	C₂×D₃₅₆	M₁₆	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₉	M₁₆	1	2	yes	—
68	C₂×C₄×D₈₉	C₄²	1	2	yes	—
69	#13083558	—	2	1	no	—
70	#13083558	—	2	1	no	—
71	#13083558	—	2	1	no	—
72	#13083558	—	2	1	no	—
73	#13083557	—	2	2	no	—
74	#13083557	—	2	2	no	—
75	C₂×#10623494	—	2	1	no	—
76	C₂×#10623494	C₄²	1	2	yes	—
77	C₈×D₈₉	—	2	2	no	—
78	C₈×D₈₉	—	2	2	no	—
79	C₄⋊Dic₈₉	—	2	1	no	—
80	C₂²⋊Dic₈₉	—	2	1	no	—
81	C₄×Dic₈₉	—	2	1	no	—
82	#13083549	—	2	1	no	—
83	#13083549	—	2	1	no	—
84	#13083549	—	2	2	no	—
85	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	M₁₆×C₈₉	—	2	1	no	—
87	M₁₆×C₈₉	—	2	1	no	—
88	C₂×C₈×C₈₉	—	2	1	no	—
89	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
94	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
95	C₄⋊C₄×C₈₉	—	2	1	no	—
96	C₂²⋊C₄×C₈₉	—	2	1	no	—
97	C₄²×C₈₉	—	2	1	no	—
98	C₁₆×C₈₉	—	2	2	no	—
99	C₁₆×C₈₉	C₂²⋊C₄	1	4	yes	—
100	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×D₈₉	#97	1	2	yes	—
102	C₂×C₄×C₈₉	—	4	1	no	—
103	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
109	C₂²⋊D₈₉	—	2	2	no	—
110	C₂²⋊D₈₉	—	2	2	no	—
111	C₂×Dic₈₉	—	4	1	no	—
112	C₂×Dic₈₉	#97	1	2	yes	—
113	D₃₅₆	—	2	1	no	—
114	D₃₅₆	—	2	1	no	—
115	C₄×D₈₉	—	2	2	no	—
116	C₄×D₈₉	—	2	2	no	—
117	Dic₁₇₈	—	2	1	no	—
118	Dic₁₇₈	—	2	1	no	—
119	C₈×C₈₉	—	4	1	no	—
120	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₈×C₈₉	#117	1	1	yes	characteristic
122	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
123	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
124	#10623494	—	2	2	no	—
125	#10623494	—	4	1	no	—
126	C₂²×C₈₉	—	4	1	no	—
127	C₂²×C₈₉	C₂×#97	1	1	yes	characteristic
128	C₂²×C₈₉	#435	1	1	yes	characteristic
129	C₂²×C₈₉	#435	1	1	yes	characteristic
130	C₂×D₈₉	—	2	2	no	—
131	C₂×D₈₉	—	4	2	no	—
132	C₄×C₈₉	—	8	1	no	—
133	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₈₉	#405	1	1	yes	characteristic
135	C₄×C₈₉	#435	1	1	yes	characteristic
136	C₄×C₈₉	#435	1	1	yes	characteristic
137	Dic₈₉	—	2	2	no	—
138	Dic₈₉	—	8	1	no	—
139	#12617	—	89	1	no	maximal, Sylow
140	C₂×C₈₉	—	2	1	no	—
141	C₂×C₈₉	—	2	1	no	—
142	C₂×C₈₉	—	2	1	no	—
143	C₂×C₈₉	#1752	1	1	yes	characteristic
144	D₈₉	—	4	2	no	—
145	C₂×#504	—	89	1	no	—
146	#2115	—	89	1	no	—
147	#2110	—	89	1	no	—
148	#2110	D₈₉	1	1	yes	characteristic, complemented
149	#1757	—	89	1	no	—
150	#1379	—	89	1	no	—
151	#1379	—	89	1	no	—
152	C₈₉	#12617	1	1	yes	characteristic, complemented, Sylow
153	C₂×#131	—	89	1	no	—
154	#504	—	178	2	no	—
155	#504	—	178	2	no	—
156	C₂×M₃₂	—	89	2	no	—
157	C₂×M₃₂	—	89	1	no	—
158	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
159	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
160	C₂²×C₁₆	—	89	1	no	—
161	C₂²×C₁₆	—	89	2	no	—
162	#413	—	89	1	no	—
163	#410	—	89	1	no	—
164	#405	—	89	1	no	—
165	#405	C₂×D₈₉	1	1	yes	characteristic
166	#349	—	89	4	no	—
167	#347	—	2	2	no	—
168	#347	—	178	2	no	—
169	#344	—	89	1	no	—
170	#344	—	89	1	no	—
171	C₂×#41	—	89	1	no	—
172	#131	—	178	1	no	—
173	#131	—	178	1	no	—
174	#131	—	178	1	no	—
175	#131	—	178	1	no	—
176	C₂×M₁₆	—	89	1	no	—
177	C₂×M₁₆	—	89	2	no	—
178	C₂×M₁₆	—	89	1	no	—
179	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
180	C₂²×C₈	—	89	2	no	—
181	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
182	#117	—	89	1	no	—
183	#117	C₄×D₈₉	1	1	yes	characteristic
184	M₃₂	—	2	1	no	—
185	M₃₂	—	89	2	no	—
186	M₃₂	—	178	1	no	—
187	M₃₂	—	178	4	no	—
188	M₃₂	C₄×D₈₉	1	2	yes	—
189	M₃₂	D₃₅₆	1	2	yes	—
190	M₃₂	C₂²⋊D₈₉	1	2	yes	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	89	1	no	—
193	C₂×C₁₆	—	89	1	no	—
194	C₂×C₁₆	—	178	1	no	—
195	C₂×C₁₆	—	178	1	no	—
196	C₂×C₁₆	—	178	1	no	—
197	C₂×C₁₆	—	178	4	no	—
198	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
199	#101	—	178	1	no	—
200	#100	—	178	1	no	—
201	#99	—	178	2	no	—
202	#97	—	2	1	no	—
203	#97	—	178	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	178	1	no	—
206	#41	—	178	1	no	—
207	#41	—	178	1	no	—
208	#41	—	178	1	no	—
209	#41	—	178	1	no	—
210	C₂×Q₈	—	89	1	no	—
211	C₂×D₄	—	89	2	no	—
212	C₂×D₄	—	89	1	no	—
213	C₂²×C₄	—	89	2	no	—
214	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
215	M₁₆	—	2	1	no	—
216	M₁₆	—	2	1	no	—
217	M₁₆	—	178	1	no	—
218	M₁₆	—	178	1	no	—
219	M₁₆	—	178	1	no	—
220	M₁₆	—	178	1	no	—
221	M₁₆	—	178	2	no	—
222	M₁₆	—	178	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	89	2	no	—
225	C₂×C₈	—	178	1	no	—
226	C₂×C₈	—	178	2	no	—
227	C₂×C₈	—	178	2	no	—
228	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
230	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
231	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
232	C₂×C₈	#13083574	1	1	yes	characteristic
233	C₂×C₈	#13083574	1	1	yes	characteristic
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	178	1	no	—
236	C₂²⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	178	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	178	1	no	—
240	C₁₆	—	2	2	no	—
241	C₁₆	—	178	2	no	—
242	C₁₆	—	178	1	no	—
243	C₁₆	—	178	1	no	—
244	C₁₆	#13083551	1	4	yes	—
245	C₂³	—	89	2	no	—
246	C₂³	M₁₆×D₈₉	1	1	yes	characteristic
247	Q₈	—	178	1	no	—
248	Q₈	—	178	1	no	—
249	D₄	—	178	2	no	—
250	D₄	—	178	2	no	—
251	D₄	—	178	1	no	—
252	D₄	—	178	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	89	2	no	—
255	C₂×C₄	—	356	1	no	—
256	C₂×C₄	—	178	2	no	—
257	C₂×C₄	—	178	2	no	—
258	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
259	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
260	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
261	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
262	C₂×C₄	#15668020	1	1	yes	characteristic
263	C₂×C₄	#15668020	1	1	yes	characteristic
264	C₈	—	4	1	no	—
265	C₈	—	356	1	no	—
266	C₈	—	178	2	no	—
267	C₈	C₂×#13083551	1	1	yes	characteristic
268	C₈	#15667969	1	1	yes	characteristic
269	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
270	C₈	C₄×D₃₅₆	1	1	yes	characteristic
271	C₂²	—	4	1	no	—
272	C₂²	—	356	2	no	—
273	C₂²	—	178	2	no	—
274	C₂²	#97×D₈₉	1	1	yes	characteristic
275	C₂²	#19151667	1	1	yes	characteristic
276	C₂²	#19151671	1	1	yes	characteristic
277	C₄	—	8	1	no	—
278	C₄	—	178	2	no	—
279	C₄	—	712	1	no	—
280	C₄	C₄×#13083551	1	1	yes	characteristic
281	C₄	#19151555	1	1	yes	characteristic
282	C₄	#19151667	1	1	yes	characteristic
283	C₄	#19151671	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	356	2	no	—
288	C₂	#26020029	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#15667969
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2110	C₂
socle	C₂×C₈₉	#1752
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2110	D₈₉
hypocenter	C₈₉	#12617
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ #2110

Complements

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

N	H ≅ G/N	type
3. #26021856	C₂	semidirect
5. #26018342	C₂	semidirect
6. #26018341	C₂	semidirect
7. C₈₉×#2110	C₂	semidirect
12. C₂×#15667951	C₄	semidirect
14. C₂×C₁₆×D₈₉	C₄	semidirect
18. #19151379	C₄	semidirect
148. #2110	D₈₉	semidirect
152. C₈₉	#12617	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152016	2
#26021872	2
#26021856	2
#26020053	2
#26018342	2
#26018341	2
C₈₉×#2110	2
#12617	89