Subgroups — #36470624

Order 22784.

358 conjugacy classes of subgroups.

The subgroup lattice has 298 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	C₂×#19151427	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022034	C₂	1	1	yes	maximal, characteristic, complemented
4	#26020307	C₂	1	1	yes	maximal, characteristic
5	#26020303	C₂	1	1	yes	maximal, characteristic
6	#26018367	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668018	C₂²	1	1	yes	characteristic
9	#19152016	—	2	1	no	—
10	#19152016	—	2	1	no	—
11	#19152016	C₄	1	2	yes	complemented
12	#19152016	C₄	1	2	yes	complemented
13	C₂×#15667951	—	2	1	no	—
14	C₂²×#13083549	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₈₉	—	2	1	no	—
16	C₂×#15667948	C₂²	1	1	yes	characteristic
17	#19151636	C₂²	1	1	yes	characteristic
18	#19151616	—	2	1	no	—
19	#19151614	—	2	1	no	—
20	#19151427	—	2	1	no	—
21	#19151427	—	2	1	no	—
22	#19151425	—	2	1	no	—
23	#19151423	—	2	1	no	—
24	#19151369	—	2	1	no	—
25	#19151368	—	2	1	no	—
26	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
28	#427×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×#13083574	D₄	1	1	yes	characteristic
30	#15668018	—	2	1	no	—
31	#15668018	—	2	1	no	—
32	#15668018	C₂×C₄	1	2	yes	—
33	#15668018	Q₈	1	2	yes	—
34	C₂×#13083558	—	2	1	no	—
35	C₂×#13083558	D₄	1	1	yes	characteristic
36	C₂×#13083557	—	2	1	no	—
37	C₂×C₈×D₈₉	—	2	1	no	—
38	#15667971	—	2	1	no	—
39	#15667951	—	2	2	no	—
40	#15667951	—	2	2	no	—
41	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
42	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
43	C₂×#13083549	D₄	1	1	yes	characteristic
44	C₂×#13083549	D₄	1	1	yes	characteristic
45	C₂×#13083549	D₄	1	1	yes	characteristic
46	C₂×#13083549	D₄	1	1	yes	characteristic
47	C₁₆×D₈₉	—	2	2	no	—
48	C₁₆×D₈₉	—	2	2	no	—
49	#15667948	—	2	1	no	—
50	#15667948	—	2	1	no	—
51	#15667946	—	2	1	no	—
52	#15667944	—	2	1	no	—
53	#15667940	—	4	2	no	—
54	#15667938	—	4	2	no	—
55	C₂×M₁₆×C₈₉	—	2	1	no	—
56	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
57	#117×C₈₉	—	2	1	no	—
58	M₃₂×C₈₉	—	2	1	no	—
59	M₃₂×C₈₉	C₂×C₄	1	2	yes	—
60	C₂×C₁₆×C₈₉	—	2	1	no	—
61	#108×C₈₉	—	2	1	no	—
62	#108×C₈₉	—	2	1	no	—
63	#107×C₈₉	—	2	1	no	—
64	#106×C₈₉	—	2	1	no	—
65	#13083574	—	2	1	no	—
66	#13083574	—	2	1	no	—
67	#13083574	C₄⋊C₄	1	2	yes	—
68	#13083574	D₈	1	2	yes	—
69	C₂×Dic₁₇₈	Q₁₆	1	1	yes	characteristic
70	C₂×D₃₅₆	Q₁₆	1	1	yes	characteristic
71	C₂×C₂²⋊D₈₉	—	2	1	no	—
72	C₂×C₄×D₈₉	—	2	1	no	—
73	#13083558	—	2	1	no	—
74	#13083558	—	4	1	no	—
75	#13083558	—	4	1	no	—
76	#13083558	C₄⋊C₄	1	2	yes	—
77	#13083557	—	2	2	no	—
78	#13083557	—	2	2	no	—
79	C₂×#10623494	—	2	1	no	—
80	C₂×#10623494	—	4	1	no	—
81	C₈×D₈₉	—	2	2	no	—
82	C₈×D₈₉	—	2	2	no	—
83	C₄⋊Dic₈₉	—	4	1	no	—
84	C₂²⋊Dic₈₉	—	4	1	no	—
85	C₄×Dic₈₉	—	4	1	no	—
86	#13083549	C₂²⋊C₄	1	1	yes	characteristic
87	#13083549	C₂²⋊C₄	1	1	yes	characteristic
88	#13083549	C₂²⋊C₄	1	1	yes	characteristic
89	#13083549	C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
91	M₁₆×C₈₉	—	4	1	no	—
92	M₁₆×C₈₉	—	4	1	no	—
93	C₂×C₈×C₈₉	—	4	1	no	—
94	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
98	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
99	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
100	C₄⋊C₄×C₈₉	—	4	1	no	—
101	C₂²⋊C₄×C₈₉	—	4	1	no	—
102	C₄²×C₈₉	—	4	1	no	—
103	C₁₆×C₈₉	—	2	2	no	—
104	C₂³×C₈₉	#135	1	1	yes	characteristic
105	C₂²×D₈₉	—	2	1	no	—
106	C₂×C₄×C₈₉	—	8	1	no	—
107	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
113	C₂²⋊D₈₉	—	2	2	no	—
114	C₂²⋊D₈₉	—	2	2	no	—
115	C₂×Dic₈₉	—	2	1	no	—
116	C₂×Dic₈₉	—	8	1	no	—
117	D₃₅₆	—	2	1	no	—
118	D₃₅₆	#107	1	2	yes	—
119	C₄×D₈₉	—	2	2	no	—
120	C₄×D₈₉	—	2	2	no	—
121	Dic₁₇₈	—	2	1	no	—
122	Dic₁₇₈	#107	1	2	yes	—
123	C₈×C₈₉	—	8	1	no	—
124	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₈×C₈₉	#117	1	1	yes	characteristic
126	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
127	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
128	#10623494	—	2	2	no	—
129	#10623494	—	8	1	no	—
130	C₂²×C₈₉	—	4	1	no	—
131	C₂²×C₈₉	#427	1	1	yes	characteristic
132	C₂²×C₈₉	#466	1	1	yes	characteristic
133	C₂²×C₈₉	#467	1	1	yes	characteristic
134	C₂×D₈₉	—	2	2	no	—
135	C₂×D₈₉	—	4	2	no	—
136	C₄×C₈₉	—	16	1	no	—
137	C₄×C₈₉	#380	1	1	yes	characteristic
138	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
139	C₄×C₈₉	#466	1	1	yes	characteristic
140	C₄×C₈₉	#467	1	1	yes	characteristic
141	Dic₈₉	—	2	2	no	—
142	Dic₈₉	—	16	1	no	—
143	#12841	—	89	1	no	maximal, Sylow
144	C₂×C₈₉	—	2	1	no	—
145	C₂×C₈₉	—	2	1	no	—
146	C₂×C₈₉	—	2	1	no	—
147	C₂×C₈₉	#1808	1	1	yes	characteristic
148	D₈₉	—	4	2	no	—
149	C₂×#504	—	89	1	no	—
150	C₂×#368	—	89	1	no	—
151	#2158	—	89	1	no	—
152	#1814	—	89	1	no	—
153	#1814	—	89	1	no	—
154	#1389	—	89	1	no	—
155	#1389	D₈₉	1	1	yes	characteristic, complemented
156	C₈₉	#12841	1	1	yes	characteristic, complemented, Sylow
157	C₂×#131	—	89	1	no	—
158	#504	—	89	2	no	—
159	#504	—	89	2	no	—
160	#504	—	178	1	no	—
161	#504	—	178	1	no	—
162	C₂×M₃₂	—	178	1	no	—
163	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
164	C₂²×C₁₆	—	89	1	no	—
165	C₂²×C₁₆	—	178	1	no	—
166	C₂×#108	—	89	1	no	—
167	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
168	#427	—	89	1	no	—
169	#427	C₂×D₈₉	1	1	yes	characteristic
170	#421	—	178	1	no	—
171	#421	—	178	1	no	—
172	#368	—	178	1	no	—
173	#368	—	178	1	no	—
174	#367	—	178	1	no	—
175	#366	—	178	1	no	—
176	#344	—	178	1	no	—
177	#344	—	178	1	no	—
178	C₂×#41	—	89	1	no	—
179	#131	—	89	2	no	—
180	#131	—	89	2	no	—
181	#131	—	178	1	no	—
182	#131	—	178	1	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	89	1	no	—
185	C₂×M₁₆	—	178	1	no	—
186	C₂×M₁₆	—	178	1	no	—
187	C₂²×C₈	—	178	1	no	—
188	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
189	#117	—	2	1	no	—
190	#117	—	178	1	no	—
191	M₃₂	—	2	1	no	—
192	M₃₂	—	178	2	no	—
193	M₃₂	—	178	2	no	—
194	M₃₂	C₄×D₈₉	1	2	yes	—
195	C₂×C₁₆	—	2	1	no	—
196	C₂×C₁₆	—	89	1	no	—
197	C₂×C₁₆	—	89	1	no	—
198	C₂×C₁₆	—	89	1	no	—
199	C₂×C₁₆	—	89	1	no	—
200	C₂×C₁₆	—	89	1	no	—
201	C₂×C₁₆	—	89	1	no	—
202	C₂×C₁₆	—	178	2	no	—
203	C₂×C₁₆	—	178	2	no	—
204	#108	—	2	1	no	—
205	#108	—	2	1	no	—
206	#108	—	178	1	no	—
207	#108	—	178	1	no	—
208	#107	—	2	1	no	—
209	#107	—	178	1	no	—
210	#106	—	2	1	no	—
211	#106	—	178	1	no	—
212	#104	—	356	2	no	—
213	#104	—	356	2	no	—
214	#41	—	89	2	no	—
215	#41	—	89	2	no	—
216	#41	—	178	1	no	—
217	#41	—	178	1	no	—
218	C₂×Q₈	—	89	1	no	—
219	C₂×D₄	—	89	1	no	—
220	C₂×D₄	—	178	1	no	—
221	C₂²×C₄	—	178	1	no	—
222	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
223	M₁₆	—	4	1	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	89	2	no	—
226	M₁₆	—	356	1	no	—
227	M₁₆	—	356	1	no	—
228	M₁₆	—	178	1	no	—
229	M₁₆	—	178	2	no	—
230	M₁₆	—	178	2	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	356	1	no	—
233	C₂×C₈	—	178	1	no	—
234	C₂×C₈	—	178	2	no	—
235	C₂×C₈	—	178	2	no	—
236	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
237	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
238	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
239	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
240	C₂×C₈	#13083574	1	1	yes	characteristic
241	C₂×C₈	#13083574	1	1	yes	characteristic
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	356	1	no	—
244	C₂²⋊C₄	—	4	1	no	—
245	C₂²⋊C₄	—	356	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	356	1	no	—
248	C₁₆	—	2	2	no	—
249	C₁₆	—	89	1	no	—
250	C₁₆	—	89	1	no	—
251	C₁₆	—	89	1	no	—
252	C₁₆	—	89	1	no	—
253	C₂³	—	178	1	no	—
254	C₂³	#15668033	1	1	yes	characteristic
255	Q₈	—	89	2	no	—
256	Q₈	—	178	1	no	—
257	D₄	—	89	2	no	—
258	D₄	—	178	2	no	—
259	D₄	—	178	2	no	—
260	D₄	—	178	1	no	—
261	C₂×C₄	—	8	1	no	—
262	C₂×C₄	—	178	1	no	—
263	C₂×C₄	—	178	2	no	—
264	C₂×C₄	—	178	2	no	—
265	C₂×C₄	—	712	1	no	—
266	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
267	C₂×C₄	#15667985	1	1	yes	characteristic
268	C₂×C₄	#15667985	1	1	yes	characteristic
269	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
270	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
271	C₂×C₄	#15668033	1	1	yes	characteristic
272	C₈	—	8	1	no	—
273	C₈	—	178	2	no	—
274	C₈	—	712	1	no	—
275	C₈	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
276	C₈	#15667969	1	1	yes	characteristic
277	C₈	#15667990	1	1	yes	characteristic
278	C₈	#15667990	1	1	yes	characteristic
279	C₂²	—	4	1	no	—
280	C₂²	—	356	2	no	—
281	C₂²	—	178	2	no	—
282	C₂²	#19151633	1	1	yes	characteristic
283	C₂²	#19151833	1	1	yes	characteristic
284	C₂²	#19151838	1	1	yes	characteristic
285	C₄	—	16	1	no	—
286	C₄	—	1424	1	no	—
287	C₄	—	178	2	no	—
288	C₄	#19151457	1	1	yes	characteristic
289	C₄	#107×D₈₉	1	1	yes	characteristic
290	C₄	#19151833	1	1	yes	characteristic
291	C₄	#19151838	1	1	yes	characteristic
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	356	2	no	—
296	C₂	#26020280	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19151833
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1389	C₂
socle	C₂×C₈₉	#1808
prime radical	C₂×#13083574	D₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1389	D₈₉
hypocenter	C₈₉	#12841
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1389

Complements

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

N	H ≅ G/N	type
2. C₂×#19151427	C₂	semidirect
3. #26022034	C₂	semidirect
6. #26018367	C₂	semidirect
7. C₈₉×#1389	C₂	semidirect
11. #19152016	C₄	semidirect
12. #19152016	C₄	semidirect
155. #1389	D₈₉	semidirect
156. C₈₉	#12841	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
C₂×#19151427	2
#26022034	2
#26020307	2
#26020303	2
#26018367	2
C₈₉×#1389	2
#12841	89