Subgroups — #36441571

Order 22784.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#13083549	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151361	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1808	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₈₉	—	2	1	no	—
5	C₂²×#13083549	—	2	1	no	—
6	C₂²⋊#13083549	—	2	1	no	—
7	C₂²⋊#13083549	C₄	1	2	yes	complemented
8	#19151361	—	2	1	no	—
9	#19151361	C₄	1	2	yes	complemented
10	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×#107×C₈₉	C₄	1	2	yes	complemented
12	#380×C₈₉	—	2	1	no	—
13	#340×C₈₉	—	2	1	no	—
14	C₂³×Dic₈₉	—	4	1	no	—
15	C₂×C₄⋊Dic₈₉	—	2	4	no	—
16	C₂×C₄⋊Dic₈₉	—	2	2	no	—
17	C₂×C₄⋊Dic₈₉	—	4	2	no	—
18	C₂×C₄⋊Dic₈₉	—	4	1	no	—
19	C₂×#13083549	—	2	2	no	—
20	C₂×#13083549	—	4	2	no	—
21	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₈₉	—	4	2	no	—
23	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₂×C₂²⋊C₄×C₈₉	—	4	1	no	—
31	#107×C₈₉	—	4	2	no	—
32	#95×C₈₉	—	4	1	no	—
33	C₂²×Dic₈₉	—	4	4	no	—
34	C₂²×Dic₈₉	—	4	2	no	—
35	C₂²×Dic₈₉	—	8	2	no	—
36	C₂²×Dic₈₉	—	8	1	no	—
37	C₂²×Dic₈₉	—	8	1	no	—
38	C₄⋊Dic₈₉	—	2	4	no	—
39	C₄⋊Dic₈₉	—	4	4	no	—
40	C₄⋊Dic₈₉	—	4	2	no	—
41	#13083549	—	4	2	no	—
42	C₂⁴×C₈₉	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	—	2	2	no	—
44	C₂²×C₄×C₈₉	—	2	2	no	—
45	C₂²×C₄×C₈₉	—	4	2	no	—
46	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
52	C₂×C₈×C₈₉	—	2	2	no	—
53	C₂×C₈×C₈₉	—	2	2	no	—
54	C₂×C₈×C₈₉	—	4	1	no	—
55	C₂×C₈×C₈₉	—	4	1	no	—
56	C₂×C₈×C₈₉	—	4	1	no	—
57	C₂×C₈×C₈₉	—	4	1	no	—
58	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
59	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
62	C₄⋊C₄×C₈₉	—	4	2	no	—
63	C₄⋊C₄×C₈₉	—	4	2	no	—
64	C₂²⋊C₄×C₈₉	—	8	1	no	—
65	C₂²⋊C₄×C₈₉	—	8	1	no	—
66	C₂³×C₈₉	—	2	2	no	—
67	C₂³×C₈₉	—	2	2	no	—
68	C₂³×C₈₉	#102	1	1	yes	characteristic
69	C₂³×C₈₉	#102	1	1	yes	characteristic
70	C₂³×C₈₉	#107	1	1	yes	characteristic
71	C₂³×C₈₉	D₁₆	1	1	yes	characteristic
72	C₂³×C₈₉	SD₃₂	1	1	yes	characteristic
73	C₂³×C₈₉	SD₃₂	1	1	yes	characteristic
74	C₂³×C₈₉	Q₃₂	1	1	yes	characteristic
75	C₂×C₄×C₈₉	—	2	2	no	—
76	C₂×C₄×C₈₉	—	2	2	no	—
77	C₂×C₄×C₈₉	—	2	2	no	—
78	C₂×C₄×C₈₉	—	2	1	no	—
79	C₂×C₄×C₈₉	—	2	1	no	—
80	C₂×C₄×C₈₉	—	2	1	no	—
81	C₂×C₄×C₈₉	—	2	1	no	—
82	C₂×C₄×C₈₉	—	8	2	no	—
83	C₂×C₄×C₈₉	—	8	2	no	—
84	C₂×C₄×C₈₉	—	8	2	no	—
85	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
86	C₂×C₄×C₈₉	#99	1	2	yes	—
87	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
88	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
89	C₂×C₄×C₈₉	#104	1	2	yes	—
90	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
91	C₂×Dic₈₉	—	4	4	no	—
92	C₂×Dic₈₉	—	8	4	no	—
93	C₂×Dic₈₉	—	8	4	no	—
94	C₂×Dic₈₉	—	8	2	no	—
95	C₂×Dic₈₉	—	8	2	no	—
96	C₈×C₈₉	—	2	1	no	—
97	C₈×C₈₉	—	2	1	no	—
98	C₈×C₈₉	—	2	1	no	—
99	C₈×C₈₉	—	2	1	no	—
100	C₂²×C₈₉	—	2	1	no	—
101	C₂²×C₈₉	—	2	1	no	—
102	C₂²×C₈₉	—	2	2	no	—
103	C₂²×C₈₉	—	2	2	no	—
104	C₂²×C₈₉	—	2	2	no	—
105	C₂²×C₈₉	—	2	2	no	—
106	C₂²×C₈₉	—	2	1	no	—
107	C₂²×C₈₉	—	2	1	no	—
108	C₂²×C₈₉	#327	1	2	yes	—
109	C₂²×C₈₉	#340	1	1	yes	characteristic
110	C₂²×C₈₉	#357	1	1	yes	characteristic
111	C₂²×C₈₉	#357	1	1	yes	characteristic
112	C₂²×C₈₉	#358	1	1	yes	characteristic
113	C₂²×C₈₉	#358	1	1	yes	characteristic
114	C₂²×C₈₉	#360	1	2	yes	—
115	C₂²×C₈₉	#366	1	1	yes	characteristic
116	C₂²×C₈₉	#367	1	1	yes	characteristic
117	C₄×C₈₉	—	2	1	no	—
118	C₄×C₈₉	—	2	1	no	—
119	C₄×C₈₉	—	16	2	no	—
120	C₄×C₈₉	#328	1	2	yes	—
121	C₄×C₈₉	#339	1	1	yes	characteristic
122	C₄×C₈₉	#342	1	1	yes	characteristic
123	Dic₈₉	—	8	4	no	—
124	#7258	—	89	1	no	maximal, Sylow
125	C₂×C₈₉	—	2	1	no	—
126	C₂×C₈₉	—	2	1	no	—
127	C₂×C₈₉	—	2	1	no	—
128	C₂×C₈₉	—	2	1	no	—
129	C₂×C₈₉	#1294	1	1	yes	characteristic
130	C₂×C₈₉	#1347	1	2	yes	—
131	C₂×C₈₉	#1348	1	2	yes	—
132	C₂×C₈₉	#1380	1	1	yes	characteristic
133	C₂×C₈₉	#1386	1	1	yes	characteristic
134	C₂×C₂²⋊C₁₆	—	89	1	no	—
135	#1808	D₈₉	1	1	yes	characteristic
136	C₂×#340	—	89	1	no	—
137	C₈₉	#7258	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
139	C₂²×C₄⋊C₄	—	178	1	no	—
140	C₂²×C₁₆	—	178	1	no	—
141	C₂×#107	Dic₈₉	1	2	yes	complemented
142	#380	—	2	1	no	—
143	C₂²⋊C₁₆	—	89	2	no	—
144	C₂²⋊C₁₆	—	178	1	no	—
145	#340	—	2	1	no	—
146	#340	—	89	2	no	—
147	#340	—	178	1	no	—
148	C₂³×C₄	—	356	1	no	—
149	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
150	C₂²×C₈	—	4	2	no	—
151	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
152	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
153	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
154	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
155	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
156	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	356	2	no	—
159	C₂×C₄⋊C₄	—	356	1	no	—
160	C₂×C₄⋊C₄	—	178	4	no	—
161	C₂×C₄⋊C₄	—	178	2	no	—
162	C₂×C₂²⋊C₄	—	4	1	no	—
163	C₂×C₁₆	—	356	2	no	—
164	C₂×C₁₆	—	178	2	no	—
165	#107	—	4	2	no	—
166	#95	—	4	1	no	—
167	C₂⁴	#13083559	1	1	yes	characteristic
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	—	356	4	no	—
172	C₂²×C₄	—	356	2	no	—
173	C₂²×C₄	—	712	2	no	—
174	C₂²×C₄	—	712	1	no	—
175	C₂²×C₄	—	712	1	no	—
176	C₂²×C₄	#13083551	1	1	yes	characteristic
177	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
178	C₂²×C₄	#13083553	1	1	yes	characteristic
179	C₂²×C₄	#13083561	1	1	yes	characteristic
180	C₂²×C₄	#13083562	1	1	yes	characteristic
181	C₂²×C₄	#13083564	1	1	yes	characteristic
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	1	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
189	C₂×C₈	#13083551	1	1	yes	characteristic
190	C₂×C₈	#13083553	1	1	yes	characteristic
191	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	356	2	no	—
195	C₄⋊C₄	—	356	4	no	—
196	C₄⋊C₄	—	178	4	no	—
197	C₂²⋊C₄	—	8	1	no	—
198	C₂²⋊C₄	—	8	1	no	—
199	C₁₆	—	356	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	#15667932	1	1	yes	characteristic
203	C₂³	#15667933	1	1	yes	characteristic
204	C₂³	#15667945	1	1	yes	characteristic
205	C₂³	#15667953	1	1	yes	characteristic
206	C₂³	#15667955	1	1	yes	characteristic
207	C₂³	#15667956	1	1	yes	characteristic
208	C₂³	#15667958	1	1	yes	characteristic
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	—	356	4	no	—
220	C₂×C₄	—	712	2	no	—
221	C₂×C₄	—	712	4	no	—
222	C₂×C₄	—	712	4	no	—
223	C₂×C₄	—	712	2	no	—
224	C₂×C₄	#15667919	1	1	yes	characteristic
225	C₂×C₄	#15667927	1	2	yes	—
226	C₂×C₄	#15667935	1	1	yes	characteristic
227	C₂×C₄	#15667936	1	1	yes	characteristic
228	C₂×C₄	#15667939	1	2	yes	—
229	C₂×C₄	#15667943	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	#19151332	1	2	yes	—
243	C₂²	#19151360	1	1	yes	characteristic
244	C₂²	#19151398	1	1	yes	characteristic
245	C₂²	#19151399	1	1	yes	characteristic
246	C₂²	#19151401	1	1	yes	characteristic
247	C₂²	#19151402	1	1	yes	characteristic
248	C₂²	#19151408	1	2	yes	—
249	C₂²	#19151423	1	1	yes	characteristic
250	C₂²	#19151425	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	16	2	no	—
254	C₄	—	712	4	no	—
255	C₄	#19151335	1	2	yes	—
256	C₄	#19151357	1	1	yes	characteristic
257	C₄	#19151365	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	#26018130	1	1	yes	characteristic
263	C₂	#26018261	1	2	yes	—
264	C₂	#26018264	1	2	yes	—
265	C₂	#26018343	1	1	yes	characteristic
266	C₂	#26018357	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667933
commutator	C₂×C₈×C₈₉	C₄²
Frattini	C₂³×C₈	C₂×D₈₉
Fitting	C₈₉×#1808	C₂
socle	C₂³×C₈₉	#102
prime radical	C₂⁴×C₈₉	D₈
cosocle	C₂³×C₈×C₈₉	C₂²
hypercenter	#1808	D₈₉
hypocenter	C₈₉	#7258
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1808

Complements

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

N	H ≅ G/N	type
7. C₂²⋊#13083549	C₄	semidirect
9. #19151361	C₄	semidirect
11. C₂×#107×C₈₉	C₄	semidirect
137. C₈₉	#7258	semidirect
141. C₂×#107	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#13083549	2
C₂×#19151361	2
C₈₉×#1808	2
#7258	89