Subgroups — #36552521

Order 22784.

1126 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667946	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151553	C₂	1	1	yes	maximal, characteristic
3	#26022608	C₂	1	2	yes	maximal
4	#26020152	C₂	1	4	yes	maximal, complemented
5	#107×Dic₈₉	C₂	1	4	yes	maximal, complemented
6	#26019973	C₂	1	2	yes	maximal
7	C₂×#427×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×#13083558	C₂²	1	1	yes	characteristic
9	C₂×#15667971	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊Dic₈₉	C₂²	1	2	yes	—
11	C₂×#15667946	C₄	1	8	yes	complemented
12	C₂×#15667946	C₂²	1	2	yes	—
13	C₂×#15667946	C₂²	1	2	yes	—
14	#19151553	C₄	1	8	yes	complemented
15	C₂×#15667922	C₂²	1	2	yes	—
16	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
17	C₄⋊C₄×Dic₈₉	C₂²	1	8	yes	—
18	C₄×C₄⋊Dic₈₉	—	2	4	no	—
19	C₂²⋊C₄×Dic₈₉	—	2	4	no	—
20	#19151360	C₂²	1	8	yes	—
21	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#117×C₈₉	C₂²	1	2	yes	—
23	#427×C₈₉	C₄	1	8	yes	complemented
24	C₂×#107×C₈₉	C₂²	1	2	yes	—
25	C₂×#106×C₈₉	C₂²	1	2	yes	—
26	C₂³×Dic₈₉	—	2	2	no	—
27	C₂×#13083558	—	2	2	no	—
28	C₂×#13083558	—	2	2	no	—
29	C₂×#13083558	C₂×C₄	1	4	yes	—
30	C₂²×#10623494	C₂³	1	2	yes	—
31	#15667971	C₂×C₄	1	4	yes	—
32	#15667971	Q₈	1	4	yes	—
33	C₂×C₄⋊Dic₈₉	—	2	2	no	—
34	C₂×C₄⋊Dic₈₉	—	2	4	no	—
35	C₂×C₄⋊Dic₈₉	C₂×C₄	1	8	yes	—
36	C₂×C₄⋊Dic₈₉	D₄	1	2	yes	—
37	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
38	C₂×C₂²⋊Dic₈₉	D₄	1	2	yes	—
39	C₂×C₄×Dic₈₉	—	2	8	no	—
40	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
41	#15667946	C₂×C₄	1	8	yes	—
42	#15667946	C₂×C₄	1	8	yes	—
43	#15667922	C₂×C₄	1	8	yes	complemented
44	C₈×Dic₈₉	C₂×C₄	1	8	yes	complemented
45	#15667919	—	2	8	no	—
46	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
52	#117×C₈₉	—	2	4	no	—
53	#117×C₈₉	C₂×C₄	1	8	yes	—
54	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
55	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
56	C₂×C₄²×C₈₉	—	2	2	no	—
57	#107×C₈₉	C₂×C₄	1	8	yes	complemented
58	#106×C₈₉	C₂×C₄	1	8	yes	complemented
59	C₂²×Dic₈₉	—	2	4	no	—
60	C₂²×Dic₈₉	—	2	8	no	—
61	C₂²×Dic₈₉	—	8	4	no	—
62	C₂²×Dic₈₉	C₂×D₄	1	4	yes	—
63	#13083558	—	2	8	no	—
64	C₂×#10623494	—	2	2	no	—
65	C₂×#10623494	—	2	2	no	—
66	C₂×#10623494	C₂²×C₄	1	4	yes	—
67	C₄⋊Dic₈₉	—	2	8	no	—
68	C₄⋊Dic₈₉	—	2	4	no	—
69	C₄⋊Dic₈₉	C₄⋊C₄	1	8	yes	—
70	C₄⋊Dic₈₉	C₂²×C₄	1	8	yes	—
71	C₂²⋊Dic₈₉	C₄⋊C₄	1	8	yes	—
72	C₄×Dic₈₉	—	8	8	no	—
73	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
74	C₄×Dic₈₉	C₂×Q₈	1	4	yes	—
75	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	—	2	4	no	—
77	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₈₉	#41	1	2	yes	—
83	C₂²×C₄×C₈₉	#41	1	2	yes	—
84	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₄²	1	16	yes	complemented
86	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
92	C₄⋊C₄×C₈₉	—	2	4	no	—
93	C₄⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
94	C₂²⋊C₄×C₈₉	—	2	8	no	—
95	C₄²×C₈₉	—	2	8	no	—
96	C₂³×C₈₉	—	2	2	no	—
97	C₂³×C₈₉	—	2	2	no	—
98	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
99	C₂³×C₈₉	C₄×D₄	1	2	yes	—
100	C₂³×C₈₉	C₄×D₄	1	2	yes	—
101	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	—	2	8	no	—
103	C₂×C₄×C₈₉	—	4	4	no	—
104	C₂×C₄×C₈₉	—	4	4	no	—
105	C₂×C₄×C₈₉	C₂×C₄²	1	4	yes	—
106	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₈₉	#117	1	4	yes	—
108	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₈₉	C₄×Q₈	1	4	yes	—
111	C₂×C₄×C₈₉	C₄×Q₈	1	4	yes	—
112	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
118	C₂×Dic₈₉	—	2	4	no	—
119	C₂×Dic₈₉	—	2	4	no	—
120	C₂×Dic₈₉	—	2	8	no	—
121	C₂×Dic₈₉	—	4	8	no	—
122	C₂×Dic₈₉	—	8	8	no	—
123	C₂×Dic₈₉	C₂×C₄⋊C₄	1	8	yes	—
124	C₈×C₈₉	C₂×C₄²	1	4	yes	—
125	C₈×C₈₉	C₂×C₄²	1	4	yes	—
126	#10623494	—	2	4	no	—
127	C₂²×C₈₉	—	2	4	no	—
128	C₂²×C₈₉	—	2	2	no	—
129	C₂²×C₈₉	—	2	2	no	—
130	C₂²×C₈₉	—	2	4	no	—
131	C₂²×C₈₉	C₄×C₄⋊C₄	1	4	yes	—
132	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂²×C₈₉	#576	1	2	yes	—
136	C₂²×C₈₉	#578	1	2	yes	—
137	C₄×C₈₉	—	4	8	no	—
138	C₄×C₈₉	C₄×C₄⋊C₄	1	4	yes	—
139	C₄×C₈₉	C₂²×C₄²	1	1	yes	characteristic
140	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
141	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
142	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
143	Dic₈₉	—	2	8	no	—
144	Dic₈₉	—	4	8	no	—
145	#20455	—	89	1	no	maximal, Sylow
146	C₂×C₈₉	—	2	4	no	—
147	C₂×C₈₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₈₉	#2913	1	2	yes	—
149	C₂×C₈₉	#2948	1	1	yes	characteristic
150	C₂×C₈₉	#2950	1	1	yes	characteristic
151	C₂×C₈₉	#2951	1	2	yes	—
152	C₂×#427	D₈₉	1	1	yes	characteristic
153	C₂²×#107	—	89	1	no	—
154	C₂×#405	—	89	1	no	—
155	#2274	—	89	2	no	—
156	#1776	—	89	4	no	—
157	C₄×#107	—	89	4	no	—
158	#1736	—	89	2	no	—
159	C₈₉	#20455	1	1	yes	characteristic, complemented, Sylow
160	C₂²×M₁₆	—	89	1	no	—
161	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
162	C₂×#117	—	89	1	no	—
163	C₂×#117	C₂×D₈₉	1	2	yes	—
164	C₂²×C₄⋊C₄	—	89	2	no	—
165	#427	Dic₈₉	1	8	yes	complemented
166	C₂×#107	—	89	8	no	—
167	C₂×#107	—	89	2	no	—
168	C₂×#107	—	89	2	no	—
169	C₂×#107	C₂×D₈₉	1	2	yes	—
170	C₂×#106	C₂×D₈₉	1	2	yes	—
171	#405	—	89	8	no	—
172	C₂×#97	—	89	2	no	—
173	C₂×C₄×C₈	—	89	2	no	—
174	C₄×C₄⋊C₄	—	89	8	no	—
175	C₄×C₄⋊C₄	—	178	4	no	—
176	C₄×C₂²⋊C₄	—	178	4	no	—
177	#340	—	89	8	no	—
178	C₂³×C₄	—	178	2	no	—
179	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
180	C₂×M₁₆	—	89	4	no	—
181	C₂×M₁₆	—	178	2	no	—
182	C₂×M₁₆	—	178	2	no	—
183	C₂²×C₈	—	89	2	no	—
184	C₂²×C₈	C₄×D₈₉	1	4	yes	—
185	C₂²×C₈	C₄×D₈₉	1	4	yes	—
186	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
187	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
189	#117	—	2	4	no	—
190	#117	—	89	4	no	—
191	#117	—	89	4	no	—
192	#117	C₂×Dic₈₉	1	8	yes	—
193	C₂×C₄⋊C₄	—	89	2	no	—
194	C₂×C₄⋊C₄	—	89	8	no	—
195	C₂×C₄⋊C₄	—	89	4	no	—
196	C₂×C₄⋊C₄	—	178	2	no	—
197	C₂×C₄⋊C₄	—	178	4	no	—
198	C₂×C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₂²⋊C₄	—	89	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	89	2	no	—
203	C₂×C₄²	—	178	8	no	—
204	#107	—	89	8	no	—
205	#107	—	89	8	no	—
206	#107	C₂×Dic₈₉	1	8	yes	complemented
207	#106	C₂×Dic₈₉	1	8	yes	complemented
208	#97	—	89	8	no	—
209	C₄×C₈	—	89	8	no	—
210	#95	—	178	8	no	—
211	C₂⁴	D₄×D₈₉	1	1	yes	characteristic
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	89	4	no	—
214	C₂²×C₄	—	178	4	no	—
215	C₂²×C₄	—	178	8	no	—
216	C₂²×C₄	—	712	4	no	—
217	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
218	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
219	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
220	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
221	C₂²×C₄	Q₈×D₈₉	1	2	yes	—
222	C₂²×C₄	#13083573	1	2	yes	—
223	C₂²×C₄	#13083575	1	2	yes	—
224	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
225	M₁₆	—	178	8	no	—
226	C₂×C₈	—	89	4	no	—
227	C₂×C₈	—	178	2	no	—
228	C₂×C₈	—	178	2	no	—
229	C₂×C₈	C₄×Dic₈₉	1	16	yes	complemented
230	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
231	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
232	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
234	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
235	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	89	8	no	—
238	C₄⋊C₄	—	89	8	no	—
239	C₄⋊C₄	—	178	8	no	—
240	C₄⋊C₄	—	178	4	no	—
241	C₄⋊C₄	C₂²×Dic₈₉	1	8	yes	—
242	C₂²⋊C₄	—	2	8	no	—
243	C₂²⋊C₄	—	89	8	no	—
244	C₄²	—	2	8	no	—
245	C₄²	—	89	4	no	—
246	C₄²	—	89	4	no	—
247	C₄²	—	712	8	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	C₄⋊C₄×D₈₉	1	2	yes	—
251	C₂³	#15667974	1	2	yes	—
252	C₂³	D₄×Dic₈₉	1	2	yes	—
253	C₂³	C₂×D₄×D₈₉	1	1	yes	characteristic
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	89	8	no	—
258	C₂×C₄	—	356	8	no	—
259	C₂×C₄	—	178	8	no	—
260	C₂×C₄	—	178	4	no	—
261	C₂×C₄	—	178	4	no	—
262	C₂×C₄	—	712	8	no	—
263	C₂×C₄	C₂×C₄×Dic₈₉	1	4	yes	—
264	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
265	C₂×C₄	#15667970	1	4	yes	—
266	C₂×C₄	#15667974	1	2	yes	—
267	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
268	C₂×C₄	#15667977	1	4	yes	—
269	C₂×C₄	Q₈×Dic₈₉	1	4	yes	—
270	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
272	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
273	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
274	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
275	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
276	C₈	—	178	4	no	—
277	C₈	C₂×C₄×Dic₈₉	1	4	yes	—
278	C₈	C₂×C₄×Dic₈₉	1	4	yes	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	C₄⋊C₄×Dic₈₉	1	4	yes	—
284	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
285	C₂²	C₂×#15667974	1	1	yes	characteristic
286	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
287	C₂²	#19152400	1	2	yes	—
288	C₂²	#19152412	1	2	yes	—
289	C₄	—	4	8	no	—
290	C₄	—	356	8	no	—
291	C₄	—	178	8	no	—
292	C₄	C₄⋊C₄×Dic₈₉	1	4	yes	—
293	C₄	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
294	C₄	C₂×#15667970	1	1	yes	characteristic
295	C₄	C₂×#15667977	1	1	yes	characteristic
296	C₄	C₂×Q₈×Dic₈₉	1	1	yes	characteristic
297	C₂	—	2	4	no	—
298	C₂	C₂×C₄⋊C₄×Dic₈₉	1	1	yes	characteristic
299	C₂	#26028849	1	2	yes	—
300	C₂	#26029103	1	1	yes	characteristic
301	C₂	#26029119	1	1	yes	characteristic
302	C₂	#26029128	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₈₉
commutator	C₄×C₈₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×#427×C₈₉	C₂
socle	C₂³×C₈₉	C₂²×D₄
prime radical	C₂⁴×C₈₉	C₂×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×#427	D₈₉
hypocenter	C₈₉	#20455
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #26020152	C₂	semidirect
5. #107×Dic₈₉	C₂	semidirect
11. C₂×#15667946	C₄	semidirect
14. #19151553	C₄	semidirect
23. #427×C₈₉	C₄	semidirect
43. #15667922	C₂×C₄	semidirect
44. C₈×Dic₈₉	C₂×C₄	semidirect
57. #107×C₈₉	C₂×C₄	semidirect
58. #106×C₈₉	C₂×C₄	semidirect
85. C₂×C₈×C₈₉	C₄²	semidirect
159. C₈₉	#20455	semidirect
165. #427	Dic₈₉	semidirect
206. #107	C₂×Dic₈₉	semidirect
207. #106	C₂×Dic₈₉	semidirect
229. C₂×C₈	C₄×Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15667946	2
C₂×#19151553	2
#26022608	2
#26020152	2
#107×Dic₈₉	2
#26019973	2
C₂×#427×C₈₉	2
#20455	89