Subgroups — #36472842

Order 22784.

410 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	C₂×#19152030	C₂	1	1	yes	maximal, characteristic
2	#26022036	C₂	1	1	yes	maximal, characteristic, complemented
3	#26021989	C₂	1	2	yes	maximal, complemented
4	#26021150	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668034	C₂²	1	2	yes	—
7	#19152030	—	2	1	no	—
8	#19152030	—	2	1	no	—
9	#19152030	—	2	1	no	—
10	#19152030	—	2	1	no	—
11	C₂×Dic₇₁₂	C₄	1	1	yes	characteristic, complemented
12	C₂×#15667957	C₄	1	2	yes	complemented
13	C₂×D₁₄₂₄	C₄	1	1	yes	characteristic, complemented
14	#19151636	C₂²	1	2	yes	—
15	#19151614	—	2	2	no	—
16	#19151571	—	2	2	no	—
17	#19151426	—	2	1	no	—
18	#19151424	—	2	1	no	—
19	#19151409	—	2	4	no	—
20	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
22	#405×C₈₉	C₂²	1	1	yes	characteristic
23	#346×C₈₉	—	2	1	no	—
24	C₄×C₁₆×C₈₉	—	2	1	no	—
25	C₂×#13083574	—	2	2	no	—
26	#15668034	—	2	2	no	—
27	#15668034	—	2	2	no	—
28	#15668034	—	2	2	no	—
29	#15668034	—	2	2	no	—
30	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
31	C₂×#13083563	—	2	2	no	—
32	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
33	#15667971	—	2	2	no	—
34	Dic₇₁₂	—	2	1	no	—
35	Dic₇₁₂	—	2	1	no	—
36	#15667957	—	2	2	no	—
37	#15667957	—	2	2	no	—
38	D₁₄₂₄	—	2	1	no	—
39	D₁₄₂₄	—	2	1	no	—
40	#15667946	—	2	2	no	—
41	#15667944	—	2	2	no	—
42	#15667938	—	4	4	no	—
43	#15667926	—	4	4	no	—
44	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	#117×C₈₉	D₄	1	1	yes	characteristic
47	M₃₂×C₈₉	—	2	1	no	—
48	M₃₂×C₈₉	D₄	1	2	yes	—
49	C₂×C₁₆×C₈₉	—	2	1	no	—
50	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
53	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
54	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
55	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
56	#97×C₈₉	—	2	1	no	—
57	C₄×C₈×C₈₉	—	2	1	no	—
58	#13083574	—	4	2	no	—
59	#13083574	—	4	2	no	—
60	#13083574	—	4	2	no	—
61	#13083574	—	4	2	no	—
62	C₂×Dic₁₇₈	—	2	2	no	—
63	C₂×D₃₅₆	—	2	2	no	—
64	C₂×C₂²⋊D₈₉	—	4	2	no	—
65	C₂×C₄×D₈₉	—	4	2	no	—
66	Dic₃₅₆	—	2	2	no	—
67	Dic₃₅₆	—	2	2	no	—
68	#13083563	—	4	2	no	—
69	#13083563	—	4	2	no	—
70	D₇₁₂	—	2	2	no	—
71	D₇₁₂	—	2	2	no	—
72	C₄⋊Dic₈₉	—	4	2	no	—
73	C₂²⋊Dic₈₉	—	4	2	no	—
74	C₄×Dic₈₉	—	4	2	no	—
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	M₁₆×C₈₉	—	2	1	no	—
77	M₁₆×C₈₉	D₈	1	2	yes	—
78	C₂×C₈×C₈₉	—	2	1	no	—
79	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
84	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
85	C₄⋊C₄×C₈₉	—	2	1	no	—
86	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
87	C₄²×C₈₉	—	2	1	no	—
88	C₁₆×C₈₉	—	2	2	no	—
89	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
94	C₂²×D₈₉	—	4	2	no	—
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
98	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
102	C₂²⋊D₈₉	—	8	2	no	—
103	C₂²⋊D₈₉	—	8	2	no	—
104	C₂×Dic₈₉	—	4	2	no	—
105	C₂×Dic₈₉	—	8	2	no	—
106	D₃₅₆	—	4	2	no	—
107	D₃₅₆	—	4	2	no	—
108	C₄×D₈₉	—	8	2	no	—
109	C₄×D₈₉	—	8	2	no	—
110	Dic₁₇₈	—	4	2	no	—
111	Dic₁₇₈	—	4	2	no	—
112	C₈×C₈₉	—	2	2	no	—
113	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
115	C₈×C₈₉	#121	1	1	yes	characteristic
116	C₈×C₈₉	#124	1	1	yes	characteristic
117	C₂²×C₈₉	—	4	1	no	—
118	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
119	C₂²×C₈₉	#466	1	1	yes	characteristic
120	C₂²×C₈₉	#493	1	1	yes	characteristic
121	C₂×D₈₉	—	8	2	no	—
122	C₂×D₈₉	—	16	2	no	—
123	C₄×C₈₉	—	4	2	no	—
124	C₄×C₈₉	#390	1	1	yes	characteristic
125	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
126	C₄×C₈₉	#466	1	1	yes	characteristic
127	C₄×C₈₉	#495	1	1	yes	characteristic
128	Dic₈₉	—	8	2	no	—
129	Dic₈₉	—	16	2	no	—
130	#13243	—	89	1	no	maximal, Sylow
131	C₂×C₈₉	—	2	1	no	—
132	C₂×C₈₉	—	2	1	no	—
133	C₂×C₈₉	—	2	1	no	—
134	C₂×C₈₉	#1970	1	1	yes	characteristic
135	D₈₉	—	16	2	no	—
136	C₂×#508	—	89	1	no	—
137	#2158	—	89	1	no	—
138	#2145	—	89	2	no	—
139	#2108	D₈₉	1	1	yes	characteristic, complemented
140	#1976	—	89	2	no	—
141	C₈₉	#13243	1	1	yes	characteristic, complemented, Sylow
142	C₂×#135	—	89	2	no	—
143	#508	—	178	1	no	—
144	#508	—	178	1	no	—
145	#508	—	178	1	no	—
146	#508	—	178	1	no	—
147	C₂×Q₃₂	—	89	1	no	—
148	C₂×SD₃₂	—	89	2	no	—
149	C₂×D₁₆	—	89	1	no	—
150	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
151	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
152	#427	—	89	2	no	—
153	#421	—	178	2	no	—
154	#410	—	178	2	no	—
155	#405	C₂×D₈₉	1	1	yes	characteristic
156	#367	—	178	1	no	—
157	#366	—	178	1	no	—
158	#360	—	178	4	no	—
159	#346	—	2	1	no	—
160	C₄×C₁₆	—	2	1	no	—
161	C₂×#41	—	178	2	no	—
162	#135	—	178	2	no	—
163	#135	—	178	2	no	—
164	#135	—	178	2	no	—
165	#135	—	178	2	no	—
166	C₂×Q₁₆	—	89	2	no	—
167	C₂×SD₁₆	—	178	2	no	—
168	C₂×D₈	—	89	2	no	—
169	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
171	#117	—	178	2	no	—
172	#117	D₃₅₆	1	1	yes	characteristic
173	Q₃₂	—	178	1	no	—
174	Q₃₂	—	178	1	no	—
175	SD₃₂	—	178	2	no	—
176	SD₃₂	—	178	2	no	—
177	D₁₆	—	178	1	no	—
178	D₁₆	—	178	1	no	—
179	M₃₂	—	2	1	no	—
180	M₃₂	D₃₅₆	1	2	yes	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
183	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
184	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
185	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
186	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
187	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
188	#107	—	178	2	no	—
189	#106	—	178	2	no	—
190	#104	—	356	4	no	—
191	#99	—	356	4	no	—
192	#97	—	2	1	no	—
193	C₄×C₈	—	2	1	no	—
194	#41	—	356	2	no	—
195	#41	—	356	2	no	—
196	#41	—	356	2	no	—
197	#41	—	356	2	no	—
198	C₂×Q₈	—	178	2	no	—
199	C₂×D₄	—	356	2	no	—
200	C₂×D₄	—	178	2	no	—
201	C₂²×C₄	—	356	2	no	—
202	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
203	Q₁₆	—	178	2	no	—
204	Q₁₆	—	178	2	no	—
205	SD₁₆	—	356	2	no	—
206	SD₁₆	—	356	2	no	—
207	D₈	—	178	2	no	—
208	D₈	—	178	2	no	—
209	M₁₆	—	2	1	no	—
210	M₁₆	D₇₁₂	1	2	yes	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
213	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
214	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
215	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
216	C₂×C₈	#13083574	1	1	yes	characteristic
217	C₂×C₈	#13083574	1	1	yes	characteristic
218	C₄⋊C₄	—	2	1	no	—
219	C₄⋊C₄	—	356	2	no	—
220	C₂²⋊C₄	—	356	2	no	—
221	C₂²⋊C₄	D₇₁₂	1	2	yes	—
222	C₄²	—	2	1	no	—
223	C₄²	—	356	2	no	—
224	C₁₆	—	2	2	no	—
225	C₁₆	#13083551	1	1	yes	characteristic
226	C₁₆	#13083551	1	1	yes	characteristic
227	C₁₆	#13083551	1	1	yes	characteristic
228	C₁₆	#13083551	1	1	yes	characteristic
229	C₂³	—	356	2	no	—
230	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
231	Q₈	—	356	2	no	—
232	Q₈	—	356	2	no	—
233	D₄	—	356	2	no	—
234	D₄	—	356	2	no	—
235	D₄	—	712	2	no	—
236	D₄	—	712	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	356	2	no	—
239	C₂×C₄	—	712	2	no	—
240	C₂×C₄	—	712	2	no	—
241	C₂×C₄	—	712	2	no	—
242	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
243	C₂×C₄	#15667986	1	1	yes	characteristic
244	C₂×C₄	#15668009	1	1	yes	characteristic
245	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
246	C₂×C₄	#15668034	1	1	yes	characteristic
247	C₂×C₄	#15668034	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	C₂×#13083551	1	1	yes	characteristic
250	C₈	C₄×D₃₅₆	1	1	yes	characteristic
251	C₈	#15667986	1	1	yes	characteristic
252	C₈	#15668000	1	1	yes	characteristic
253	C₂²	—	4	1	no	—
254	C₂²	—	1424	2	no	—
255	C₂²	—	712	2	no	—
256	C₂²	C₄×D₇₁₂	1	1	yes	characteristic
257	C₂²	#19151834	1	1	yes	characteristic
258	C₂²	#19151975	1	1	yes	characteristic
259	C₄	—	4	2	no	—
260	C₄	—	1424	2	no	—
261	C₄	—	712	2	no	—
262	C₄	#19151500	1	1	yes	characteristic
263	C₄	C₄×D₇₁₂	1	1	yes	characteristic
264	C₄	#19151834	1	1	yes	characteristic
265	C₄	#19151982	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	—	1424	2	no	—
270	C₂	#26021118	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19151834
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2108	C₂
socle	C₂×C₈₉	#1970
prime radical	C₂×#19152030	C₂
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2108	D₈₉
hypocenter	C₈₉	#13243
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #26022036	C₂	semidirect
3. #26021989	C₂	semidirect
4. #26021150	C₂	semidirect
5. C₈₉×#2108	C₂	semidirect
11. C₂×Dic₇₁₂	C₄	semidirect
12. C₂×#15667957	C₄	semidirect
13. C₂×D₁₄₂₄	C₄	semidirect
139. #2108	D₈₉	semidirect
141. C₈₉	#13243	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152030	2
#26022036	2
#26021989	2
#26021150	2
C₈₉×#2108	2
#13243	89