Subgroups — #36563208

Order 22784.

750 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152003	C₂	1	1	yes	maximal, characteristic
2	#26023000	C₂	1	2	yes	maximal
3	#26021225	C₂	1	2	yes	maximal
4	#26020990	C₂	1	2	yes	maximal
5	#26020985	C₂	1	2	yes	maximal
6	#26020416	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×#15668012	C₂²	1	2	yes	—
11	#19152003	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	#19151502	—	2	4	no	—
17	#19151496	C₂²	1	4	yes	—
18	#19151481	—	2	4	no	—
19	#19151464	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	C₂²×Dic₁₇₈	—	2	2	no	—
31	#15668012	—	2	4	no	—
32	#15668012	C₂×C₄	1	8	yes	—
33	C₂×C₄⋊Dic₈₉	—	2	2	no	—
34	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
35	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
36	C₂×#13083553	—	2	4	no	—
37	C₂×C₄×Dic₈₉	—	2	4	no	—
38	#15667946	—	2	4	no	—
39	#15667946	C₂×C₄	1	8	yes	—
40	#15667946	C₂×C₄	1	8	yes	—
41	#15667944	—	2	4	no	—
42	#15667919	—	2	4	no	—
43	#15667919	—	2	2	no	—
44	#15667919	—	2	2	no	—
45	#15667919	—	2	4	no	—
46	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	#97×C₈₉	—	2	2	no	—
54	#97×C₈₉	—	2	2	no	—
55	#97×C₈₉	Q₈	1	4	yes	complemented
56	C₄×C₈×C₈₉	—	2	2	no	—
57	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₈₉	Q₈	1	4	yes	complemented
59	#95×C₈₉	D₄	1	2	yes	—
60	#95×C₈₉	D₄	1	2	yes	—
61	C₂×Dic₁₇₈	—	2	8	no	—
62	C₂×Dic₁₇₈	—	8	4	no	—
63	C₂²×Dic₈₉	—	2	4	no	—
64	C₂²×Dic₈₉	—	2	4	no	—
65	C₄⋊Dic₈₉	—	2	8	no	—
66	C₄⋊Dic₈₉	—	2	4	no	—
67	C₄⋊Dic₈₉	—	2	4	no	—
68	C₄⋊Dic₈₉	—	2	4	no	—
69	C₄⋊Dic₈₉	C₂²×C₄	1	8	yes	—
70	#13083553	—	8	4	no	—
71	C₄×Dic₈₉	—	8	4	no	—
72	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₈₉	—	2	1	no	—
78	C₂×C₈×C₈₉	—	2	1	no	—
79	C₂×C₈×C₈₉	—	2	1	no	—
80	C₂×C₈×C₈₉	—	2	1	no	—
81	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₈₉	#41	1	2	yes	—
88	C₂×C₈×C₈₉	#41	1	2	yes	—
89	C₄²×C₈₉	—	2	1	no	—
90	C₄²×C₈₉	—	2	1	no	—
91	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
92	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
93	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
94	C₄²×C₈₉	#41	1	2	yes	—
95	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	—	2	2	no	—
99	C₂×C₄×C₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	—	2	2	no	—
101	C₂×C₄×C₈₉	—	2	2	no	—
102	C₂×C₄×C₈₉	—	2	2	no	—
103	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₈₉	C₄⋊Q₈	1	4	yes	—
106	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
112	C₂×Dic₈₉	—	2	8	no	—
113	C₂×Dic₈₉	—	4	4	no	—
114	C₂×Dic₈₉	—	4	4	no	—
115	C₂×Dic₈₉	—	4	4	no	—
116	C₂×Dic₈₉	—	4	4	no	—
117	C₂×Dic₈₉	—	4	4	no	—
118	Dic₁₇₈	—	8	8	no	—
119	C₈×C₈₉	—	2	2	no	—
120	C₈×C₈₉	—	2	2	no	—
121	C₈×C₈₉	C₄×Q₈	1	4	yes	—
122	C₈×C₈₉	C₄×Q₈	1	4	yes	—
123	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂²×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
125	C₂²×C₈₉	#532	1	1	yes	characteristic
126	C₂²×C₈₉	#576	1	2	yes	—
127	C₂²×C₈₉	#578	1	2	yes	—
128	C₄×C₈₉	—	2	4	no	—
129	C₄×C₈₉	—	2	2	no	—
130	C₄×C₈₉	—	2	2	no	—
131	C₄×C₈₉	—	4	2	no	—
132	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
133	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
134	C₄×C₈₉	#533	1	1	yes	characteristic
135	C₄×C₈₉	#533	1	1	yes	characteristic
136	Dic₈₉	—	4	8	no	—
137	Dic₈₉	—	8	4	no	—
138	#21725	—	89	1	no	maximal, Sylow
139	C₂×C₈₉	C₄×C₄⋊Q₈	1	1	yes	characteristic
140	C₂×C₈₉	#2948	1	1	yes	characteristic
141	C₂×C₈₉	#2950	1	1	yes	characteristic
142	C₂×C₈₉	#3164	1	2	yes	—
143	C₂×C₈₉	#3175	1	2	yes	—
144	C₂×#500	—	89	1	no	—
145	#2331	—	89	2	no	—
146	#1993	—	89	2	no	—
147	#1943	—	89	2	no	—
148	#1942	—	89	2	no	—
149	#1846	—	89	1	no	—
150	#1776	—	89	2	no	—
151	C₄×#107	—	89	2	no	—
152	#1726	D₈₉	1	1	yes	characteristic
153	C₈₉	#21725	1	1	yes	characteristic, complemented, Sylow
154	C₂×C₄⋊Q₈	—	89	2	no	—
155	#500	—	89	8	no	—
156	C₂×#107	—	89	2	no	—
157	C₂×#107	—	89	2	no	—
158	C₂×#107	—	89	2	no	—
159	C₂×#106	—	89	2	no	—
160	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
161	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
162	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
166	#391	—	178	4	no	—
167	#389	—	89	4	no	—
168	#387	—	178	4	no	—
169	#382	—	89	2	no	—
170	C₄×C₄⋊C₄	—	89	4	no	—
171	#376	C₂×D₈₉	1	1	yes	characteristic
172	#340	—	89	4	no	—
173	#340	—	89	4	no	—
174	C₂²×Q₈	—	178	2	no	—
175	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
179	C₄⋊Q₈	—	89	8	no	—
180	C₄⋊Q₈	—	178	4	no	—
181	C₂×C₄⋊C₄	—	89	4	no	—
182	C₂×C₄⋊C₄	—	89	4	no	—
183	C₂×C₄⋊C₄	—	178	2	no	—
184	C₂×C₄⋊C₄	—	178	4	no	—
185	C₂×C₄²	—	178	4	no	—
186	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
187	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
188	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
189	#107	—	89	8	no	—
190	#107	—	89	8	no	—
191	#107	—	178	4	no	—
192	#106	—	178	4	no	—
193	#97	—	2	2	no	—
194	#97	—	2	2	no	—
195	#97	Dic₁₇₈	1	4	yes	complemented
196	C₄×C₈	—	2	2	no	—
197	C₄×C₈	Dic₁₇₈	1	4	yes	complemented
198	C₄×C₈	C₄×D₈₉	1	4	yes	—
199	#95	—	178	4	no	—
200	#95	—	178	2	no	—
201	#95	—	178	2	no	—
202	#95	—	178	4	no	—
203	#95	D₃₅₆	1	2	yes	—
204	#95	D₃₅₆	1	2	yes	—
205	C₂×Q₈	—	178	8	no	—
206	C₂×Q₈	—	712	4	no	—
207	C₂²×C₄	—	178	4	no	—
208	C₂²×C₄	—	178	4	no	—
209	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
210	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
213	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
219	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
220	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
221	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
222	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
223	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
224	C₂×C₈	#13083574	1	2	yes	—
225	C₂×C₈	#13083574	1	2	yes	—
226	C₄⋊C₄	—	89	8	no	—
227	C₄⋊C₄	—	178	4	no	—
228	C₄⋊C₄	—	178	4	no	—
229	C₄⋊C₄	—	178	8	no	—
230	C₄⋊C₄	—	178	4	no	—
231	C₄⋊C₄	—	712	4	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	712	4	no	—
235	C₄²	C₂×C₄×D₈₉	1	2	yes	—
236	C₄²	C₂×Dic₁₇₈	1	2	yes	—
237	C₄²	C₂×Dic₁₇₈	1	2	yes	—
238	C₄²	#13083574	1	2	yes	—
239	C₂³	C₂²×D₃₅₆	1	1	yes	characteristic
240	Q₈	—	712	8	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	356	4	no	—
249	C₂×C₄	—	356	4	no	—
250	C₂×C₄	—	356	4	no	—
251	C₂×C₄	—	356	4	no	—
252	C₂×C₄	—	356	4	no	—
253	C₂×C₄	—	178	8	no	—
254	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
255	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
256	C₂×C₄	#15668012	1	4	yes	—
257	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
258	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
259	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
260	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
261	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
262	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	2	2	no	—
265	C₈	C₄×Dic₁₇₈	1	4	yes	—
266	C₈	C₄×Dic₁₇₈	1	4	yes	—
267	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
268	C₂²	C₂×#15668012	1	1	yes	characteristic
269	C₂²	#19152148	1	1	yes	characteristic
270	C₂²	#19152403	1	2	yes	—
271	C₂²	#19152415	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	4	no	—
275	C₄	—	4	2	no	—
276	C₄	—	356	8	no	—
277	C₄	—	712	4	no	—
278	C₄	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
279	C₄	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
280	C₄	#19152155	1	1	yes	characteristic
281	C₄	#19152155	1	1	yes	characteristic
282	C₂	C₄×#15668012	1	1	yes	characteristic
283	C₂	#26029105	1	1	yes	characteristic
284	C₂	#26029121	1	1	yes	characteristic
285	C₂	#26030791	1	2	yes	—
286	C₂	#26030891	1	2	yes	—

	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	#21725
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. #19152003	C₄	semidirect
55. #97×C₈₉	Q₈	semidirect
58. C₄×C₈×C₈₉	Q₈	semidirect
153. C₈₉	#21725	semidirect
195. #97	Dic₁₇₈	semidirect
197. C₄×C₈	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152003	2
#26023000	2
#26021225	2
#26020990	2
#26020985	2
#26020416	2
#26020149	2
C₄×#15667946	2
C₈₉×#1726	2
#21725	89