Subgroups — #36471795

Order 22784.

526 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151839	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151403	C₂	1	2	yes	maximal, complemented
3	#26020281	C₂	1	1	yes	maximal, characteristic
4	#26018344	C₂	1	2	yes	maximal, complemented
5	C₂³×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
7	C₂×#15667991	—	2	1	no	—
8	#19151839	—	2	2	no	—
9	#19151839	C₄	1	4	yes	complemented
10	C₂×#15667946	C₂²	1	2	yes	—
11	C₂×#15667946	C₂²	1	1	yes	characteristic
12	C₂×#15667937	—	2	1	no	—
13	#19151458	—	2	1	no	—
14	#19151403	—	2	4	no	—
15	#19151361	—	2	1	no	—
16	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₈₉	—	2	2	no	—
18	C₂²×C₁₆×C₈₉	—	2	4	no	—
19	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
20	C₂²×Dic₁₇₈	—	2	1	no	—
21	C₂×Dic₃₅₆	—	2	2	no	—
22	C₂×Dic₃₅₆	—	2	1	no	—
23	C₂×Dic₃₅₆	—	2	2	no	—
24	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
25	#15667991	—	2	4	no	—
26	#15667991	—	4	2	no	—
27	C₂×C₄⋊Dic₈₉	—	2	1	no	—
28	C₂×C₄⋊Dic₈₉	—	2	2	no	—
29	C₂×#13083553	—	4	1	no	—
30	C₂×C₂²⋊Dic₈₉	—	4	1	no	—
31	C₂×C₂²⋊Dic₈₉	—	4	1	no	—
32	#15667946	—	2	1	no	—
33	#15667946	—	2	2	no	—
34	#15667946	—	2	2	no	—
35	#15667946	C₂×C₄	1	2	yes	—
36	#15667937	—	2	4	no	—
37	#15667919	—	4	1	no	—
38	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	—	2	2	no	—
40	C₂²×C₈×C₈₉	—	2	2	no	—
41	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	D₄	1	2	yes	—
44	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂×C₁₆×C₈₉	—	2	8	no	—
46	C₂×C₁₆×C₈₉	—	2	2	no	—
47	C₂×C₁₆×C₈₉	—	2	2	no	—
48	C₂×C₁₆×C₈₉	D₄	1	4	yes	—
49	C₂×Dic₁₇₈	—	2	2	no	—
50	C₂×Dic₁₇₈	—	4	2	no	—
51	C₂×Dic₁₇₈	—	4	2	no	—
52	C₂×Dic₁₇₈	—	4	1	no	—
53	C₂²×Dic₈₉	—	4	1	no	—
54	C₂²×Dic₈₉	—	4	1	no	—
55	C₂²×Dic₈₉	—	4	2	no	—
56	Dic₃₅₆	—	2	4	no	—
57	Dic₃₅₆	—	4	1	no	—
58	Dic₃₅₆	—	4	1	no	—
59	C₄⋊Dic₈₉	—	2	2	no	—
60	C₄⋊Dic₈₉	—	4	1	no	—
61	C₄⋊Dic₈₉	—	4	2	no	—
62	C₄⋊Dic₈₉	—	4	2	no	—
63	#13083553	—	4	4	no	—
64	C₂²⋊Dic₈₉	—	4	4	no	—
65	C₂²⋊Dic₈₉	—	8	2	no	—
66	C₂⁴×C₈₉	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	—	2	2	no	—
68	C₂²×C₄×C₈₉	—	2	2	no	—
69	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₈₉	—	2	2	no	—
74	C₂×C₈×C₈₉	—	2	4	no	—
75	C₂×C₈×C₈₉	—	2	2	no	—
76	C₂×C₈×C₈₉	—	2	2	no	—
77	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₈₉	#41	1	2	yes	—
82	C₁₆×C₈₉	—	2	4	no	—
83	C₂³×C₈₉	—	2	2	no	—
84	C₂³×C₈₉	—	2	2	no	—
85	C₂³×C₈₉	#102	1	2	yes	—
86	C₂³×C₈₉	SD₃₂	1	2	yes	—
87	C₂³×C₈₉	Q₃₂	1	2	yes	—
88	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₈₉	—	2	2	no	—
90	C₂×C₄×C₈₉	—	2	2	no	—
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	2	4	no	—
93	C₂×C₄×C₈₉	#102	1	2	yes	—
94	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	#121	1	2	yes	—
96	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₈₉	#135	1	2	yes	—
98	C₂×Dic₈₉	—	4	2	no	—
99	C₂×Dic₈₉	—	4	2	no	—
100	C₂×Dic₈₉	—	8	2	no	—
101	C₂×Dic₈₉	—	8	2	no	—
102	C₂×Dic₈₉	—	8	2	no	—
103	C₂×Dic₈₉	—	8	2	no	—
104	C₂×Dic₈₉	—	8	2	no	—
105	Dic₁₇₈	—	4	2	no	—
106	Dic₁₇₈	—	4	2	no	—
107	Dic₁₇₈	—	8	2	no	—
108	C₈×C₈₉	—	2	2	no	—
109	C₈×C₈₉	C₄×D₄	1	2	yes	—
110	C₈×C₈₉	#120	1	1	yes	characteristic
111	C₈×C₈₉	#123	1	1	yes	characteristic
112	C₂²×C₈₉	—	2	2	no	—
113	C₂²×C₈₉	—	2	2	no	—
114	C₂²×C₈₉	—	2	4	no	—
115	C₂²×C₈₉	—	2	4	no	—
116	C₂²×C₈₉	#358	1	4	yes	complemented
117	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
118	C₂²×C₈₉	#466	1	2	yes	—
119	C₂²×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
120	C₂²×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
121	C₂²×C₈₉	#508	1	2	yes	—
122	C₄×C₈₉	—	2	2	no	—
123	C₄×C₈₉	#386	1	1	yes	characteristic
124	C₄×C₈₉	#416	1	1	yes	characteristic
125	C₄×C₈₉	#465	1	2	yes	—
126	Dic₈₉	—	8	2	no	—
127	Dic₈₉	—	8	2	no	—
128	Dic₈₉	—	16	2	no	—
129	#13060	—	89	1	no	maximal, Sylow
130	C₂×C₈₉	—	2	4	no	—
131	C₂×C₈₉	#1893	1	1	yes	characteristic
132	C₂×C₈₉	C₂×#358	1	1	yes	characteristic
133	C₂×C₈₉	#2138	1	1	yes	characteristic
134	C₂×C₈₉	#2216	1	2	yes	—
135	C₂×C₈₉	#2217	1	2	yes	—
136	C₂³×C₁₆	D₈₉	1	1	yes	characteristic
137	C₂×#467	—	89	1	no	—
138	C₂×#358	—	89	2	no	—
139	#1808	—	89	1	no	—
140	#1380	—	89	2	no	—
141	C₈₉	#13060	1	1	yes	characteristic, complemented, Sylow
142	C₂²×Q₁₆	—	89	1	no	—
143	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
144	C₂×C₂²⋊Q₈	—	178	1	no	—
145	C₂²×C₁₆	—	2	2	no	—
146	C₂²×C₁₆	—	2	4	no	—
147	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
148	#467	—	89	4	no	—
149	#467	—	178	2	no	—
150	C₂×#107	—	89	1	no	—
151	C₂×#107	—	89	2	no	—
152	C₂×#103	—	178	1	no	—
153	#380	—	178	1	no	—
154	#358	—	178	4	no	—
155	#340	—	178	1	no	—
156	C₂²×Q₈	—	178	1	no	—
157	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
158	C₂×Q₁₆	—	89	2	no	—
159	C₂×Q₁₆	—	178	1	no	—
160	C₂×Q₁₆	—	178	2	no	—
161	C₂×Q₁₆	—	178	2	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	C₄×D₈₉	1	2	yes	—
165	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
166	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
167	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
168	C₂²⋊Q₈	—	356	2	no	—
169	C₂²⋊Q₈	—	178	4	no	—
170	C₂×C₄⋊C₄	—	356	1	no	—
171	C₂×C₄⋊C₄	—	178	2	no	—
172	C₂×C₄⋊C₄	—	178	1	no	—
173	C₂×C₂²⋊C₄	—	356	1	no	—
174	C₂×C₂²⋊C₄	—	356	1	no	—
175	C₂×C₁₆	—	2	8	no	—
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	C₂²⋊D₈₉	1	4	yes	—
179	#107	—	89	2	no	—
180	#107	—	178	2	no	—
181	#107	—	178	2	no	—
182	#107	—	178	1	no	—
183	#103	—	178	4	no	—
184	#95	—	356	1	no	—
185	C₂⁴	D₇₁₂	1	1	yes	characteristic
186	C₂×Q₈	—	356	1	no	—
187	C₂×Q₈	—	356	2	no	—
188	C₂×Q₈	—	356	2	no	—
189	C₂×Q₈	—	178	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	356	1	no	—
193	C₂²×C₄	—	356	1	no	—
194	C₂²×C₄	—	356	2	no	—
195	C₂²×C₄	#13083551	1	2	yes	—
196	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
197	C₂²×C₄	#13083563	1	2	yes	—
198	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
199	Q₁₆	—	356	1	no	—
200	Q₁₆	—	356	1	no	—
201	Q₁₆	—	178	4	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	#13083551	1	2	yes	—
207	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
208	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
209	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
210	C₂×C₈	#13083574	1	2	yes	—
211	C₄⋊C₄	—	356	1	no	—
212	C₄⋊C₄	—	356	4	no	—
213	C₄⋊C₄	—	356	2	no	—
214	C₄⋊C₄	—	356	2	no	—
215	C₄⋊C₄	—	178	2	no	—
216	C₂²⋊C₄	—	356	4	no	—
217	C₂²⋊C₄	—	712	2	no	—
218	C₁₆	—	2	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	#15667934	1	2	yes	—
222	C₂³	#15667957	1	2	yes	—
223	C₂³	Dic₇₁₂	1	2	yes	—
224	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
225	Q₈	—	356	2	no	—
226	Q₈	—	356	2	no	—
227	Q₈	—	712	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	356	2	no	—
233	C₂×C₄	—	356	2	no	—
234	C₂×C₄	—	712	2	no	—
235	C₂×C₄	—	712	2	no	—
236	C₂×C₄	—	712	2	no	—
237	C₂×C₄	—	712	2	no	—
238	C₂×C₄	—	712	2	no	—
239	C₂×C₄	#15667934	1	2	yes	—
240	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
241	C₂×C₄	#15667986	1	2	yes	—
242	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
243	C₂×C₄	#15668034	1	2	yes	—
244	C₈	—	2	2	no	—
245	C₈	C₄×C₂²⋊D₈₉	1	2	yes	—
246	C₈	#15667981	1	1	yes	characteristic
247	C₈	#15667995	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	#19151403	1	4	yes	complemented
253	C₂²	C₂×#15667934	1	1	yes	characteristic
254	C₂²	#19151834	1	2	yes	—
255	C₂²	C₂×#15667957	1	1	yes	characteristic
256	C₂²	C₂×Dic₇₁₂	1	1	yes	characteristic
257	C₂²	#19152030	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	1424	2	no	—
260	C₄	—	712	2	no	—
261	C₄	—	712	2	no	—
262	C₄	#19151477	1	1	yes	characteristic
263	C₄	#19151595	1	1	yes	characteristic
264	C₄	#19151827	1	2	yes	—
265	C₂	—	2	4	no	—
266	C₂	#26020680	1	1	yes	characteristic
267	C₂	C₂×#19151403	1	1	yes	characteristic
268	C₂	#26021960	1	1	yes	characteristic
269	C₂	#26022343	1	2	yes	—
270	C₂	#26022348	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂³×C₁₆×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂⁴×C₈₉	D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂³×C₁₆	D₈₉
hypocenter	C₈₉	#13060
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#19151403	C₂	semidirect
4. #26018344	C₂	semidirect
9. #19151839	C₄	semidirect
116. C₂²×C₈₉	#358	semidirect
141. C₈₉	#13060	semidirect
252. C₂²	#19151403	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151839	2
C₂×#19151403	2
#26020281	2
#26018344	2
C₂³×C₁₆×C₈₉	2
#13060	89