Subgroups — #36573373

Order 22784.

734 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26023209	C₂	1	2	yes	maximal
2	#26021022	C₂	1	4	yes	maximal
3	#26021011	C₂	1	2	yes	maximal
4	#26020383	C₂	1	2	yes	maximal
5	#26020380	C₂	1	1	yes	maximal, characteristic
6	C₄×#15667946	C₂	1	2	yes	maximal
7	C₄×#15667944	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#15667946	C₂²	1	4	yes	—
10	C₂×#15667944	C₂²	1	2	yes	—
11	#19151505	—	2	2	no	—
12	#19151484	—	2	4	no	—
13	#19151468	C₂²	1	2	yes	—
14	#19151464	C₂²	1	4	yes	—
15	C₄⋊C₄×Dic₈₉	—	2	2	no	—
16	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
17	C₄×C₄⋊Dic₈₉	C₂²	1	4	yes	—
18	#19151361	C₂²	1	2	yes	—
19	#19151361	C₂²	1	2	yes	—
20	#19151361	C₂²	1	1	yes	characteristic
21	#19151361	C₂²	1	4	yes	—
22	#19151361	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₈×Q₈×C₈₉	C₄	1	8	yes	complemented
25	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₈₉	—	2	2	no	—
30	C₂×C₄⋊Dic₈₉	—	2	2	no	—
31	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
32	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
33	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
34	C₂×C₄×Dic₈₉	—	2	2	no	—
35	C₂×C₄×Dic₈₉	—	2	4	no	—
36	#15667946	—	2	8	no	—
37	#15667944	—	2	4	no	—
38	#15667919	—	2	2	no	—
39	#15667919	—	2	4	no	—
40	#15667919	—	2	4	no	—
41	#15667919	—	2	4	no	—
42	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
46	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
47	C₄×Q₈×C₈₉	Q₈	1	4	yes	—
48	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
50	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
51	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
53	C₄⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
55	C₄×C₈×C₈₉	C₂×C₄	1	8	yes	—
56	C₂²×Dic₈₉	—	2	2	no	—
57	C₂²×Dic₈₉	—	2	2	no	—
58	C₂²×Dic₈₉	—	2	4	no	—
59	C₄⋊Dic₈₉	—	2	2	no	—
60	C₄⋊Dic₈₉	—	2	2	no	—
61	C₄⋊Dic₈₉	—	2	4	no	—
62	C₄⋊Dic₈₉	—	2	2	no	—
63	C₄⋊Dic₈₉	—	2	4	no	—
64	C₄⋊Dic₈₉	—	4	4	no	—
65	C₄⋊Dic₈₉	—	8	2	no	—
66	C₄⋊Dic₈₉	#41	1	4	yes	—
67	C₄×Dic₈₉	—	4	4	no	—
68	C₄×Dic₈₉	—	8	4	no	—
69	C₂×Q₈×C₈₉	—	2	4	no	—
70	C₂×Q₈×C₈₉	C₄⋊C₄	1	4	yes	—
71	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₈₉	—	2	2	no	—
77	C₂×C₈×C₈₉	—	2	4	no	—
78	C₂×C₈×C₈₉	—	2	2	no	—
79	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
82	C₄⋊C₄×C₈₉	C₄⋊C₄	1	8	yes	—
83	C₄⋊C₄×C₈₉	C₄⋊C₄	1	4	yes	—
84	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
85	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
86	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
87	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
88	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
89	Q₈×C₈₉	—	2	8	no	—
90	C₂×C₄×C₈₉	—	2	4	no	—
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
98	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₈₉	#135	1	4	yes	—
100	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
103	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
104	C₂×Dic₈₉	—	2	4	no	—
105	C₂×Dic₈₉	—	4	2	no	—
106	C₂×Dic₈₉	—	4	2	no	—
107	C₂×Dic₈₉	—	4	4	no	—
108	C₂×Dic₈₉	—	4	2	no	—
109	C₂×Dic₈₉	—	4	2	no	—
110	C₂×Dic₈₉	—	4	4	no	—
111	C₂×Dic₈₉	—	4	4	no	—
112	C₂×Dic₈₉	—	4	2	no	—
113	C₈×C₈₉	—	2	2	no	—
114	C₈×C₈₉	—	2	4	no	—
115	C₈×C₈₉	—	2	2	no	—
116	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₈₉	#548	1	2	yes	—
118	C₂²×C₈₉	C₂×#135	1	2	yes	—
119	C₂²×C₈₉	#576	1	1	yes	characteristic
120	C₂²×C₈₉	#578	1	1	yes	characteristic
121	C₄×C₈₉	—	2	4	no	—
122	C₄×C₈₉	—	2	4	no	—
123	C₄×C₈₉	—	2	2	no	—
124	C₄×C₈₉	#427	1	4	yes	—
125	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
126	C₄×C₈₉	#519	1	1	yes	characteristic
127	C₄×C₈₉	#557	1	2	yes	—
128	Dic₈₉	—	4	4	no	—
129	Dic₈₉	—	8	4	no	—
130	Dic₈₉	—	8	2	no	—
131	#22739	—	89	1	no	maximal, Sylow
132	C₂×C₈₉	#2351	1	1	yes	characteristic
133	C₂×C₈₉	C₂×#427	1	1	yes	characteristic
134	C₂×C₈₉	#2913	1	1	yes	characteristic
135	C₂×C₈₉	#3361	1	2	yes	—
136	C₂×C₈₉	#3364	1	2	yes	—
137	C₂×C₈×Q₈	D₈₉	1	1	yes	characteristic
138	#2355	—	89	2	no	—
139	#1949	—	89	4	no	—
140	#1947	—	89	2	no	—
141	#1835	—	89	2	no	—
142	#1834	—	89	1	no	—
143	C₄×#107	—	89	2	no	—
144	C₄×#106	—	89	1	no	—
145	C₈₉	#22739	1	1	yes	characteristic, complemented, Sylow
146	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
147	C₈×Q₈	Dic₈₉	1	8	yes	complemented
148	C₂×#107	—	89	4	no	—
149	C₂×#106	—	89	2	no	—
150	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
151	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
152	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
154	#391	—	178	2	no	—
155	#387	—	178	4	no	—
156	#384	—	89	2	no	—
157	#382	—	89	4	no	—
158	C₄×C₄⋊C₄	—	89	2	no	—
159	C₄×C₄⋊C₄	—	89	4	no	—
160	C₄×C₄⋊C₄	—	178	2	no	—
161	#340	—	89	2	no	—
162	#340	—	89	2	no	—
163	#340	—	89	1	no	—
164	#340	—	89	4	no	—
165	#340	—	89	1	no	—
166	C₂²×Q₈	D₃₅₆	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
170	C₄×Q₈	Dic₁₇₈	1	4	yes	—
171	C₄×Q₈	C₂×Dic₈₉	1	4	yes	—
172	C₂×C₄⋊C₄	—	89	2	no	—
173	C₂×C₄⋊C₄	—	89	2	no	—
174	C₂×C₄⋊C₄	—	89	4	no	—
175	C₂×C₄⋊C₄	—	178	2	no	—
176	C₂×C₄⋊C₄	—	178	2	no	—
177	C₂×C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
178	C₂×C₄⋊C₄	D₃₅₆	1	2	yes	—
179	C₂×C₄²	—	178	2	no	—
180	C₂×C₄²	—	178	4	no	—
181	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
182	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
183	#107	—	178	8	no	—
184	#106	—	178	4	no	—
185	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
186	C₄⋊C₈	C₂×Dic₈₉	1	8	yes	—
187	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
188	C₄×C₈	C₂×Dic₈₉	1	8	yes	—
189	#95	—	178	2	no	—
190	#95	—	178	4	no	—
191	#95	—	178	4	no	—
192	#95	—	178	4	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	C₄⋊Dic₈₉	1	4	yes	—
195	C₂²×C₄	—	178	2	no	—
196	C₂²×C₄	—	178	4	no	—
197	C₂²×C₄	—	178	2	no	—
198	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
199	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
200	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
202	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
207	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
208	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
209	C₄⋊C₄	—	89	4	no	—
210	C₄⋊C₄	—	356	4	no	—
211	C₄⋊C₄	—	178	2	no	—
212	C₄⋊C₄	—	178	4	no	—
213	C₄⋊C₄	—	178	2	no	—
214	C₄⋊C₄	—	178	4	no	—
215	C₄⋊C₄	—	178	2	no	—
216	C₄⋊C₄	—	712	2	no	—
217	C₄⋊C₄	C₄⋊Dic₈₉	1	8	yes	—
218	C₄⋊C₄	C₄⋊Dic₈₉	1	4	yes	—
219	C₄²	—	356	4	no	—
220	C₄²	—	712	4	no	—
221	C₄²	C₂²×Dic₈₉	1	4	yes	—
222	C₄²	C₂²×Dic₈₉	1	2	yes	—
223	C₄²	C₂×Dic₁₇₈	1	4	yes	—
224	C₄²	C₂×Dic₁₇₈	1	2	yes	—
225	C₂³	C₂²×D₃₅₆	1	1	yes	characteristic
226	Q₈	—	2	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	356	2	no	—
233	C₂×C₄	—	356	2	no	—
234	C₂×C₄	—	356	4	no	—
235	C₂×C₄	—	356	2	no	—
236	C₂×C₄	—	356	4	no	—
237	C₂×C₄	—	356	2	no	—
238	C₂×C₄	—	356	2	no	—
239	C₂×C₄	—	356	4	no	—
240	C₂×C₄	—	178	4	no	—
241	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
242	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
244	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
245	C₂×C₄	#15668034	1	4	yes	—
246	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
247	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
248	C₂×C₄	#41×D₈₉	1	2	yes	—
249	C₂×C₄	#15668060	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	2	4	no	—
252	C₈	—	2	2	no	—
253	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
254	C₂²	#19152260	1	2	yes	—
255	C₂²	C₂×#15668034	1	2	yes	—
256	C₂²	#19152403	1	1	yes	characteristic
257	C₂²	#19152415	1	1	yes	characteristic
258	C₄	—	2	4	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	356	4	no	—
262	C₄	—	712	4	no	—
263	C₄	—	712	2	no	—
264	C₄	#19151636	1	4	yes	—
265	C₄	#41×Dic₈₉	1	1	yes	characteristic
266	C₄	#19152076	1	1	yes	characteristic
267	C₄	#19152324	1	2	yes	—
268	C₂	#26023173	1	1	yes	characteristic
269	C₂	C₂×#19151636	1	1	yes	characteristic
270	C₂	#26028852	1	1	yes	characteristic
271	C₂	#26032914	1	2	yes	—
272	C₂	#26032939	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×C₈×Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×C₈×Q₈	D₈₉
hypocenter	C₈₉	#22739
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
24. C₈×Q₈×C₈₉	C₄	semidirect
145. C₈₉	#22739	semidirect
147. C₈×Q₈	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26023209	2
#26021022	2
#26021011	2
#26020383	2
#26020380	2
C₄×#15667946	2
C₄×#15667944	2
C₂×C₈×Q₈×C₈₉	2
#22739	89