Subgroups — #36453589

Order 22784.

506 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030831	C₂	1	1	yes	maximal, characteristic
2	#26021171	C₂	1	2	yes	maximal, complemented
3	C₄×#15667946	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018748	C₂	1	2	yes	maximal, complemented
5	C₈×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152148	C₂²	1	2	yes	—
7	#19152003	C₄	1	1	yes	characteristic, complemented
8	#19151975	C₄	1	1	yes	characteristic, complemented
9	#19151945	C₄	1	2	yes	complemented
10	#19151834	—	2	1	no	—
11	#19151834	—	2	1	no	—
12	C₄×D₇₁₂	—	2	1	no	—
13	C₄×D₇₁₂	—	2	1	no	—
14	C₂×#15667946	—	2	1	no	—
15	C₂×#15667938	—	2	4	no	—
16	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
17	#19151361	—	2	1	no	—
18	#19151333	—	2	4	no	—
19	#431×C₈₉	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	#322×C₈₉	—	2	1	no	—
23	C₈²×C₈₉	—	2	1	no	—
24	C₂×#13083574	—	2	2	no	—
25	C₂×D₇₁₂	—	2	1	no	—
26	C₂×D₇₁₂	—	2	1	no	—
27	#15668012	C₂×C₄	1	2	yes	—
28	#15668009	C₂×C₄	1	2	yes	—
29	#15668000	—	2	2	no	—
30	#15667986	—	2	2	no	—
31	#15667986	—	2	2	no	—
32	C₄×D₃₅₆	—	2	2	no	—
33	C₄×D₃₅₆	—	2	2	no	—
34	C₂×C₄⋊Dic₈₉	—	2	2	no	—
35	C₂×C₄×Dic₈₉	—	2	2	no	—
36	#15667946	—	2	4	no	—
37	#15667946	—	2	1	no	—
38	#15667946	—	2	1	no	—
39	#15667938	—	4	8	no	—
40	#15667934	—	2	2	no	—
41	#15667934	—	2	2	no	—
42	#15667919	—	2	2	no	—
43	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
44	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₈₉	—	2	1	no	—
48	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
49	#97×C₈₉	—	2	1	no	—
50	C₄×C₈×C₈₉	—	2	1	no	—
51	C₄×C₈×C₈₉	—	2	1	no	—
52	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
55	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
56	#13083574	—	4	4	no	—
57	#13083574	—	4	4	no	—
58	C₂×Dic₁₇₈	—	2	2	no	—
59	C₂×D₃₅₆	—	2	2	no	—
60	C₂×D₃₅₆	—	2	2	no	—
61	C₂×D₃₅₆	—	2	2	no	—
62	C₂×C₂²⋊D₈₉	—	4	2	no	—
63	C₂²×Dic₈₉	—	4	2	no	—
64	C₂×C₄×D₈₉	—	4	2	no	—
65	D₇₁₂	—	4	2	no	—
66	D₇₁₂	—	4	2	no	—
67	C₄⋊Dic₈₉	—	2	4	no	—
68	C₄⋊Dic₈₉	—	2	2	no	—
69	C₄⋊Dic₈₉	—	2	2	no	—
70	C₄⋊Dic₈₉	—	4	2	no	—
71	#13083551	—	4	2	no	—
72	#13083551	—	4	2	no	—
73	C₄×Dic₈₉	—	4	4	no	—
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	M₁₆×C₈₉	—	2	2	no	—
78	M₁₆×C₈₉	D₈	1	4	yes	complemented
79	C₂×C₈×C₈₉	—	2	2	no	—
80	C₂×C₈×C₈₉	—	2	2	no	—
81	C₂×C₈×C₈₉	—	2	2	no	—
82	C₂×C₈×C₈₉	—	4	1	no	—
83	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₈₉	#41	1	1	yes	characteristic
90	C₄²×C₈₉	#41	1	1	yes	characteristic
91	C₂³×C₈₉	#127	1	1	yes	characteristic
92	C₂²×D₈₉	—	4	2	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	4	1	no	—
96	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₈₉	#135	1	2	yes	—
104	C₂²⋊D₈₉	—	8	2	no	—
105	C₂²⋊D₈₉	—	8	2	no	—
106	C₂×Dic₈₉	—	4	4	no	—
107	C₂×Dic₈₉	—	4	2	no	—
108	C₂×Dic₈₉	—	8	2	no	—
109	C₂×Dic₈₉	—	8	2	no	—
110	D₃₅₆	—	4	2	no	—
111	D₃₅₆	—	4	2	no	—
112	D₃₅₆	—	4	4	no	—
113	D₃₅₆	—	8	2	no	—
114	C₄×D₈₉	—	8	4	no	—
115	Dic₁₇₈	—	4	4	no	—
116	C₈×C₈₉	—	2	1	no	—
117	C₈×C₈₉	—	2	1	no	—
118	C₈×C₈₉	—	2	1	no	—
119	C₈×C₈₉	—	2	4	no	—
120	C₈×C₈₉	—	4	2	no	—
121	C₈×C₈₉	#104	1	2	yes	complemented
122	C₂²×C₈₉	—	2	1	no	—
123	C₂²×C₈₉	—	2	1	no	—
124	C₂²×C₈₉	#390	1	1	yes	characteristic
125	C₂²×C₈₉	#493	1	1	yes	characteristic
126	C₂²×C₈₉	#495	1	1	yes	characteristic
127	C₂×D₈₉	—	8	2	no	—
128	C₂×D₈₉	—	8	2	no	—
129	C₂×D₈₉	—	8	2	no	—
130	C₄×C₈₉	—	2	1	no	—
131	C₄×C₈₉	—	2	1	no	—
132	C₄×C₈₉	—	2	1	no	—
133	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
134	C₄×C₈₉	#421	1	1	yes	characteristic
135	C₄×C₈₉	C₄×D₈	1	2	yes	—
136	C₄×C₈₉	#466	1	2	yes	—
137	Dic₈₉	—	8	4	no	—
138	Dic₈₉	—	8	2	no	—
139	#9719	—	89	1	no	maximal, Sylow
140	C₂×C₈₉	—	2	1	no	—
141	C₂×C₈₉	—	2	1	no	—
142	C₂×C₈₉	#1970	1	1	yes	characteristic
143	C₂×C₈₉	#1976	1	1	yes	characteristic
144	C₂×C₈₉	#1980	1	1	yes	characteristic
145	D₈₉	—	16	2	no	—
146	#3169	—	89	1	no	—
147	#1980	—	89	2	no	—
148	C₄×#107	—	89	1	no	—
149	#1502	—	89	2	no	—
150	C₈×M₁₆	D₈₉	1	1	yes	characteristic, complemented
151	C₈₉	#9719	1	1	yes	characteristic, complemented, Sylow
152	#532	—	89	2	no	—
153	#500	—	89	1	no	—
154	#493	—	89	1	no	—
155	#486	—	89	2	no	—
156	#466	—	178	1	no	—
157	#466	—	178	1	no	—
158	C₄×D₈	—	178	1	no	—
159	C₄×D₈	—	178	1	no	—
160	#431	C₂×D₈₉	1	1	yes	characteristic
161	C₂×#107	—	178	1	no	—
162	C₂×#104	—	178	4	no	—
163	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
165	C₄×C₄⋊C₄	—	89	2	no	—
166	#340	—	178	1	no	—
167	#327	—	178	4	no	—
168	#322	—	2	1	no	—
169	C₈²	—	2	1	no	—
170	C₂×#41	—	178	2	no	—
171	C₂×D₈	—	178	1	no	—
172	C₂×D₈	—	178	1	no	—
173	C₂×M₁₆	D₃₅₆	1	2	yes	—
174	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
175	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
176	C₄⋊Q₈	—	89	2	no	—
177	#127	—	89	2	no	—
178	#124	—	178	2	no	—
179	#121	—	178	2	no	—
180	#121	—	178	2	no	—
181	C₄×D₄	—	178	2	no	—
182	C₄×D₄	—	178	2	no	—
183	C₂×C₄⋊C₄	—	178	2	no	—
184	C₂×C₄²	—	178	2	no	—
185	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
186	#107	—	178	4	no	—
187	#107	—	178	1	no	—
188	#107	—	178	1	no	—
189	C₄⋊C₈	—	2	1	no	—
190	#104	—	356	8	no	—
191	#102	—	178	2	no	—
192	#102	—	178	2	no	—
193	C₂²⋊C₈	D₃₅₆	1	2	yes	—
194	#97	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
198	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
199	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
200	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
201	#95	—	178	2	no	—
202	#41	—	356	4	no	—
203	#41	—	356	4	no	—
204	C₂×Q₈	—	178	2	no	—
205	C₂×D₄	—	356	2	no	—
206	C₂×D₄	—	178	2	no	—
207	C₂×D₄	—	178	2	no	—
208	C₂×D₄	—	178	2	no	—
209	C₂²×C₄	—	356	2	no	—
210	C₂²×C₄	—	356	2	no	—
211	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
214	D₈	—	356	2	no	—
215	D₈	—	356	2	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	D₇₁₂	1	4	yes	complemented
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	#13083551	1	1	yes	characteristic
223	C₂×C₈	#13083551	1	1	yes	characteristic
224	C₂×C₈	#13083551	1	1	yes	characteristic
225	C₂×C₈	#13083551	1	1	yes	characteristic
226	C₄⋊C₄	—	356	2	no	—
227	C₄⋊C₄	—	178	4	no	—
228	C₄⋊C₄	—	178	2	no	—
229	C₄⋊C₄	—	178	2	no	—
230	C₂²⋊C₄	—	356	2	no	—
231	C₂²⋊C₄	—	356	2	no	—
232	C₄²	—	356	4	no	—
233	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
234	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
235	C₄²	#13083574	1	1	yes	characteristic
236	C₄²	#13083574	1	1	yes	characteristic
237	C₂³	—	356	2	no	—
238	C₂³	#15668009	1	1	yes	characteristic
239	Q₈	—	356	4	no	—
240	D₄	—	356	4	no	—
241	D₄	—	356	2	no	—
242	D₄	—	356	2	no	—
243	D₄	—	712	2	no	—
244	D₄	—	712	2	no	—
245	D₄	—	712	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	356	4	no	—
250	C₂×C₄	—	356	2	no	—
251	C₂×C₄	—	712	2	no	—
252	C₂×C₄	—	712	2	no	—
253	C₂×C₄	—	712	4	no	—
254	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
255	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
256	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
257	C₂×C₄	#15667986	1	1	yes	characteristic
258	C₂×C₄	#15667986	1	1	yes	characteristic
259	C₂×C₄	#15668000	1	1	yes	characteristic
260	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
261	C₂×C₄	#15668034	1	2	yes	—
262	C₈	—	2	4	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	4	2	no	—
267	C₈	#15667938	1	2	yes	complemented
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	712	2	no	—
271	C₂²	—	712	2	no	—
272	C₂²	—	712	2	no	—
273	C₂²	#19151500	1	1	yes	characteristic
274	C₂²	#19151975	1	1	yes	characteristic
275	C₂²	#19151982	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	712	4	no	—
280	C₄	—	712	2	no	—
281	C₄	C₂×#15667938	1	1	yes	characteristic
282	C₄	#19151614	1	1	yes	characteristic
283	C₄	C₄×D₇₁₂	1	2	yes	—
284	C₄	#19151834	1	2	yes	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	1424	2	no	—
288	C₂	#26021118	1	1	yes	characteristic
289	C₂	#26021150	1	1	yes	characteristic
290	C₂	#26021171	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667986
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈×M₁₆×C₈₉	C₂
socle	C₂²×C₈₉	#390
prime radical	#26030831	C₂
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₈×M₁₆	D₈₉
hypocenter	C₈₉	#9719
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. #26021171	C₂	semidirect
3. C₄×#15667946	C₂	semidirect
4. #26018748	C₂	semidirect
5. C₈×M₁₆×C₈₉	C₂	semidirect
7. #19152003	C₄	semidirect
8. #19151975	C₄	semidirect
9. #19151945	C₄	semidirect
78. M₁₆×C₈₉	D₈	semidirect
121. C₈×C₈₉	#104	semidirect
150. C₈×M₁₆	D₈₉	semidirect
151. C₈₉	#9719	semidirect
217. M₁₆	D₇₁₂	semidirect
267. C₈	#15667938	semidirect

Maximal subgroups

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

subgroup	index
#26030831	2
#26021171	2
C₄×#15667946	2
#26018748	2
C₈×M₁₆×C₈₉	2
#9719	89