Subgroups — #36563190

Order 22784.

734 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152000	C₂	1	1	yes	maximal, characteristic
2	#26022993	C₂	1	2	yes	maximal
3	#26021221	C₂	1	2	yes	maximal
4	#26020995	C₂	1	2	yes	maximal
5	#26020985	C₂	1	2	yes	maximal
6	#26020412	C₂	1	1	yes	maximal, characteristic
7	#26020149	C₂	1	2	yes	maximal
8	C₄×#15667946	C₂	1	2	yes	maximal
9	C₈₉×#1726	C₂	1	1	yes	maximal, characteristic
10	C₂×#15668004	C₂²	1	2	yes	—
11	#19152000	C₄	1	8	yes	complemented
12	C₂×#15667946	C₂²	1	2	yes	—
13	C₂×#15667946	C₂²	1	2	yes	—
14	C₂×#15667946	C₂²	1	2	yes	—
15	C₂×#15667944	C₂²	1	2	yes	—
16	#19151496	C₂²	1	4	yes	—
17	#19151495	—	2	4	no	—
18	#19151485	—	2	4	no	—
19	#19151468	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₈₉	C₂²	1	4	yes	—
21	#19151361	C₂²	1	4	yes	—
22	#19151361	C₂²	1	4	yes	—
23	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
29	#376×C₈₉	C₂²	1	1	yes	characteristic
30	#15668004	—	2	4	no	—
31	#15668004	C₂×C₄	1	8	yes	—
32	C₂×C₄⋊Dic₈₉	—	2	4	no	—
33	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
35	C₂×#13083553	—	2	4	no	—
36	C₂×#13083553	—	2	4	no	—
37	C₂×C₄×Dic₈₉	—	2	4	no	—
38	#15667946	—	2	4	no	—
39	#15667946	—	2	4	no	—
40	#15667946	C₂×C₄	1	8	yes	—
41	#15667944	C₂×C₄	1	8	yes	—
42	#15667919	—	2	4	no	—
43	#15667919	—	2	4	no	—
44	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
51	#97×C₈₉	—	2	2	no	—
52	#97×C₈₉	—	2	2	no	—
53	#97×C₈₉	Q₈	1	4	yes	—
54	C₄×C₈×C₈₉	—	2	2	no	—
55	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
56	C₄×C₈×C₈₉	Q₈	1	4	yes	—
57	#95×C₈₉	D₄	1	2	yes	—
58	#95×C₈₉	D₄	1	2	yes	—
59	C₂²×Dic₈₉	—	2	4	no	—
60	C₂²×Dic₈₉	—	2	4	no	—
61	C₄⋊Dic₈₉	—	2	4	no	—
62	C₄⋊Dic₈₉	—	2	4	no	—
63	C₄⋊Dic₈₉	—	2	4	no	—
64	C₄⋊Dic₈₉	—	8	4	no	—
65	C₄⋊Dic₈₉	C₂²×C₄	1	8	yes	—
66	#13083553	—	2	16	no	—
67	#13083553	—	8	4	no	—
68	C₄×Dic₈₉	—	8	4	no	—
69	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₈₉	—	2	1	no	—
75	C₂×C₈×C₈₉	—	2	1	no	—
76	C₂×C₈×C₈₉	—	2	1	no	—
77	C₂×C₈×C₈₉	—	2	1	no	—
78	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
81	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₈₉	#41	1	2	yes	—
85	C₂×C₈×C₈₉	#41	1	2	yes	—
86	C₄²×C₈₉	—	2	1	no	—
87	C₄²×C₈₉	—	2	1	no	—
88	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
89	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
90	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
91	C₄²×C₈₉	#41	1	2	yes	—
92	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	—	2	2	no	—
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	2	1	no	—
98	C₂×C₄×C₈₉	—	2	1	no	—
99	C₂×C₄×C₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	—	2	2	no	—
101	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
102	C₂×C₄×C₈₉	C₄⋊Q₈	1	4	yes	—
103	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
109	C₂×Dic₈₉	—	2	8	no	—
110	C₂×Dic₈₉	—	4	4	no	—
111	C₂×Dic₈₉	—	4	4	no	—
112	C₂×Dic₈₉	—	4	4	no	—
113	C₂×Dic₈₉	—	4	4	no	—
114	C₂×Dic₈₉	—	4	4	no	—
115	C₈×C₈₉	—	2	2	no	—
116	C₈×C₈₉	—	2	2	no	—
117	C₈×C₈₉	C₄×Q₈	1	4	yes	—
118	C₈×C₈₉	C₄×Q₈	1	4	yes	—
119	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
120	C₂²×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₂²×C₈₉	#532	1	1	yes	characteristic
122	C₂²×C₈₉	#576	1	2	yes	—
123	C₂²×C₈₉	#578	1	2	yes	—
124	C₄×C₈₉	—	2	4	no	—
125	C₄×C₈₉	—	2	4	no	—
126	C₄×C₈₉	—	4	1	no	—
127	C₄×C₈₉	—	4	1	no	—
128	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
129	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
130	C₄×C₈₉	#533	1	1	yes	characteristic
131	C₄×C₈₉	#533	1	1	yes	characteristic
132	Dic₈₉	—	4	8	no	—
133	Dic₈₉	—	8	4	no	—
134	#21723	—	89	1	no	maximal, Sylow
135	C₂×C₈₉	C₄×C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₈₉	#2951	1	2	yes	—
137	C₂×C₈₉	#3164	1	2	yes	—
138	C₂×C₈₉	#3176	1	1	yes	characteristic
139	C₂×C₈₉	#3177	1	1	yes	characteristic
140	C₂×#499	—	89	1	no	—
141	#2330	—	89	2	no	—
142	#1992	—	89	2	no	—
143	#1944	—	89	2	no	—
144	#1942	—	89	2	no	—
145	#1845	—	89	1	no	—
146	#1776	—	89	2	no	—
147	C₄×#107	—	89	2	no	—
148	#1726	D₈₉	1	1	yes	characteristic
149	C₈₉	#21723	1	1	yes	characteristic, complemented, Sylow
150	C₂×#125	—	89	2	no	—
151	#499	—	89	8	no	—
152	C₂×#107	—	89	2	no	—
153	C₂×#107	—	89	2	no	—
154	C₂×#107	—	89	2	no	—
155	C₂×#106	—	89	2	no	—
156	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
157	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
158	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
162	#389	—	89	4	no	—
163	#389	—	178	4	no	—
164	#387	—	178	4	no	—
165	#384	—	89	2	no	—
166	C₄×C₄⋊C₄	—	89	4	no	—
167	#376	C₂×D₈₉	1	1	yes	characteristic
168	#340	—	89	4	no	—
169	#340	—	89	4	no	—
170	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
174	#125	—	89	8	no	—
175	#125	—	178	4	no	—
176	C₂×C₄⋊C₄	—	89	4	no	—
177	C₂×C₄⋊C₄	—	89	4	no	—
178	C₂×C₄⋊C₄	—	178	4	no	—
179	C₂×C₄⋊C₄	—	178	4	no	—
180	C₂×C₄⋊C₄	—	178	4	no	—
181	C₂×C₄²	—	178	4	no	—
182	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
185	#107	—	89	8	no	—
186	#107	—	178	4	no	—
187	#107	—	178	4	no	—
188	#106	—	89	8	no	—
189	#97	—	2	2	no	—
190	#97	—	2	2	no	—
191	#97	Dic₁₇₈	1	4	yes	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	Dic₁₇₈	1	4	yes	—
194	C₄×C₈	C₄×D₈₉	1	4	yes	—
195	#95	—	178	4	no	—
196	#95	—	178	4	no	—
197	#95	D₃₅₆	1	2	yes	—
198	#95	D₃₅₆	1	2	yes	—
199	C₂²×C₄	—	178	4	no	—
200	C₂²×C₄	—	178	4	no	—
201	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
204	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
205	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
211	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
212	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
213	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
214	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
215	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
216	C₂×C₈	#13083574	1	2	yes	—
217	C₂×C₈	#13083574	1	2	yes	—
218	C₄⋊C₄	—	89	8	no	—
219	C₄⋊C₄	—	178	4	no	—
220	C₄⋊C₄	—	178	4	no	—
221	C₄⋊C₄	—	178	16	no	—
222	C₄⋊C₄	—	178	4	no	—
223	C₄⋊C₄	—	712	4	no	—
224	C₄⋊C₄	—	712	4	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	712	4	no	—
228	C₄²	C₂×C₄×D₈₉	1	2	yes	—
229	C₄²	C₂×Dic₁₇₈	1	2	yes	—
230	C₄²	C₂×Dic₁₇₈	1	2	yes	—
231	C₄²	#13083574	1	2	yes	—
232	C₂³	C₂²×D₃₅₆	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	356	4	no	—
242	C₂×C₄	—	356	4	no	—
243	C₂×C₄	—	356	4	no	—
244	C₂×C₄	—	356	4	no	—
245	C₂×C₄	—	356	4	no	—
246	C₂×C₄	—	178	8	no	—
247	C₂×C₄	C₄×D₃₅₆	1	4	yes	—
248	C₂×C₄	#15668012	1	4	yes	—
249	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
250	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
251	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
252	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
253	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
254	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₈	C₄×Dic₁₇₈	1	4	yes	—
258	C₈	C₄×Dic₁₇₈	1	4	yes	—
259	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
260	C₂²	C₂×#15668012	1	1	yes	characteristic
261	C₂²	#19152148	1	1	yes	characteristic
262	C₂²	#19152403	1	2	yes	—
263	C₂²	#19152415	1	2	yes	—
264	C₄	—	2	4	no	—
265	C₄	—	2	4	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	—	356	8	no	—
269	C₄	—	712	4	no	—
270	C₄	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
271	C₄	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
272	C₄	#19152155	1	1	yes	characteristic
273	C₄	#19152155	1	1	yes	characteristic
274	C₂	C₄×#15668012	1	1	yes	characteristic
275	C₂	#26029129	1	2	yes	—
276	C₂	#26030791	1	2	yes	—
277	C₂	#26030902	1	1	yes	characteristic
278	C₂	#26030911	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1726	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1726	D₈₉
hypocenter	C₈₉	#21723
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
11. #19152000	C₄	semidirect
149. C₈₉	#21723	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152000	2
#26022993	2
#26021221	2
#26020995	2
#26020985	2
#26020412	2
#26020149	2
C₄×#15667946	2
C₈₉×#1726	2
#21723	89