Subgroups — #36472189

Order 22784.

324 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151383	C₂	1	1	yes	maximal, characteristic
2	#26021865	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151373	C₂	1	1	yes	maximal, characteristic, complemented
4	#26020091	C₂	1	1	yes	maximal, characteristic
5	#26018341	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018333	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668018	C₂²	1	1	yes	characteristic
9	C₂×#15667952	C₂²	1	1	yes	characteristic
10	C₂²×#13083549	C₂²	1	1	yes	characteristic
11	C₂×#15667938	C₄	1	1	yes	characteristic
12	#19151592	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
14	#19151383	—	2	2	no	—
15	#19151377	—	2	2	no	—
16	#19151373	—	2	2	no	—
17	#19151362	C₄	1	1	yes	characteristic
18	#19151357	C₄	1	1	yes	characteristic
19	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	#405×C₈₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆×C₈₉	—	2	2	no	—
23	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
24	#15668018	—	2	1	no	—
25	#15668018	—	2	1	no	—
26	#15668018	—	2	1	no	—
27	#15668018	—	2	1	no	—
28	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
29	C₂×#13083557	—	2	1	no	—
30	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₈₉	—	2	1	no	—
32	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
33	#15667952	—	2	1	no	—
34	#15667952	—	2	1	no	—
35	C₂×#13083549	—	2	1	no	—
36	C₂×#13083549	—	2	1	no	—
37	C₂×#13083549	—	2	1	no	—
38	C₂×#13083549	D₄	1	2	yes	—
39	#15667938	—	2	2	no	—
40	#15667935	C₈	1	2	yes	—
41	#15667932	C₈	1	2	yes	—
42	C₈×Dic₈₉	—	2	1	no	—
43	C₈×Dic₈₉	—	2	1	no	—
44	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
47	M₃₂×C₈₉	—	2	1	no	—
48	M₃₂×C₈₉	—	2	1	no	—
49	C₂×C₁₆×C₈₉	—	2	1	no	—
50	C₂×C₁₆×C₈₉	—	2	1	no	—
51	C₂×C₁₆×C₈₉	—	2	1	no	—
52	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
53	#97×C₈₉	—	2	1	no	—
54	C₄×C₈×C₈₉	—	2	1	no	—
55	#13083574	—	2	1	no	—
56	#13083574	—	2	1	no	—
57	#13083574	—	2	1	no	—
58	#13083574	—	2	1	no	—
59	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
60	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
61	C₂×C₂²⋊D₈₉	—	2	1	no	—
62	C₂²×Dic₈₉	—	2	1	no	—
63	C₂×C₄×D₈₉	—	2	1	no	—
64	#13083558	—	2	1	no	—
65	#13083558	—	2	1	no	—
66	#13083557	—	4	1	no	—
67	#13083557	—	4	1	no	—
68	C₂×#10623494	—	2	1	no	—
69	C₂×#10623494	—	2	1	no	—
70	C₂×#10623494	—	2	1	no	—
71	C₂×#10623494	C₂×C₈	1	2	yes	—
72	C₈×D₈₉	—	4	1	no	—
73	C₈×D₈₉	—	4	1	no	—
74	C₄×Dic₈₉	—	2	1	no	—
75	C₄×Dic₈₉	M₁₆	1	2	yes	—
76	#13083549	—	2	2	no	—
77	#13083549	—	4	1	no	—
78	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	M₁₆×C₈₉	—	2	1	no	—
80	M₁₆×C₈₉	—	2	1	no	—
81	C₂×C₈×C₈₉	—	2	1	no	—
82	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
85	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
86	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
87	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
88	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
89	C₂²⋊C₄×C₈₉	—	2	1	no	—
90	C₄²×C₈₉	—	2	1	no	—
91	C₁₆×C₈₉	—	2	2	no	—
92	C₁₆×C₈₉	—	4	1	no	—
93	C₂³×C₈₉	#131	1	1	yes	characteristic
94	C₂²×D₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
102	C₂²⋊D₈₉	—	4	1	no	—
103	C₂²⋊D₈₉	—	4	1	no	—
104	C₂×Dic₈₉	—	2	1	no	—
105	C₂×Dic₈₉	—	2	2	no	—
106	C₂×Dic₈₉	—	4	2	no	—
107	D₃₅₆	—	2	1	no	—
108	D₃₅₆	—	2	1	no	—
109	C₄×D₈₉	—	4	1	no	—
110	C₄×D₈₉	—	4	1	no	—
111	Dic₁₇₈	—	2	1	no	—
112	Dic₁₇₈	—	2	1	no	—
113	C₈×C₈₉	—	4	1	no	—
114	C₈×C₈₉	#121	1	1	yes	characteristic
115	C₈×C₈₉	#123	1	1	yes	characteristic
116	C₈×C₈₉	#124	1	1	yes	characteristic
117	C₈×C₈₉	#126	1	1	yes	characteristic
118	#10623494	—	2	2	no	—
119	#10623494	—	4	1	no	—
120	C₂²×C₈₉	—	2	2	no	—
121	C₂²×C₈₉	#433	1	1	yes	characteristic
122	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
123	C₂²×C₈₉	#435	1	1	yes	characteristic
124	C₂×D₈₉	—	4	2	no	—
125	C₂×D₈₉	—	4	1	no	—
126	C₄×C₈₉	—	4	2	no	—
127	C₄×C₈₉	#388	1	1	yes	characteristic
128	C₄×C₈₉	#431	1	1	yes	characteristic
129	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
130	C₄×C₈₉	#436	1	1	yes	characteristic
131	Dic₈₉	—	4	2	no	—
132	Dic₈₉	—	4	1	no	—
133	#13125	—	89	1	no	maximal, Sylow
134	C₂×C₈₉	—	2	1	no	—
135	C₂×C₈₉	—	2	1	no	—
136	C₂×C₈₉	#1923	1	1	yes	characteristic
137	C₂×C₈₉	#2171	1	2	yes	—
138	D₈₉	—	8	1	no	—
139	#2151	D₈₉	1	1	yes	characteristic, complemented
140	C₂×#350	—	89	1	no	—
141	#2113	—	89	1	no	—
142	C₂×#346	—	89	1	no	—
143	#1764	—	89	1	no	—
144	#1379	—	89	1	no	—
145	#1376	—	89	1	no	—
146	C₈₉	#13125	1	1	yes	characteristic, complemented, Sylow
147	C₂×#131	—	89	1	no	—
148	C₂×M₃₂	—	89	1	no	—
149	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
150	C₂²×C₁₆	—	89	1	no	—
151	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
152	C₂×#104	—	89	1	no	—
153	#416	—	89	1	no	—
154	#405	C₂×D₈₉	1	1	yes	characteristic
155	C₂×C₄×C₈	—	89	1	no	—
156	C₄⋊C₁₆	—	2	2	no	—
157	#350	—	178	2	no	—
158	C₂²⋊C₁₆	—	178	2	no	—
159	#346	—	178	2	no	—
160	#341	—	89	1	no	—
161	#339	—	89	1	no	—
162	C₂×#41	—	89	1	no	—
163	#131	—	178	1	no	—
164	#131	—	178	1	no	—
165	#131	—	178	1	no	—
166	#131	—	178	1	no	—
167	C₂×M₁₆	—	89	1	no	—
168	C₂×M₁₆	—	178	1	no	—
169	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
170	C₂²×C₈	—	89	1	no	—
171	C₂²×C₈	—	178	1	no	—
172	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
173	#117	C₄×D₈₉	1	1	yes	characteristic
174	C₂×C₄²	—	89	1	no	—
175	M₃₂	—	2	1	no	—
176	M₃₂	—	2	1	no	—
177	M₃₂	—	178	1	no	—
178	M₃₂	—	178	1	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	89	2	no	—
183	C₂×C₁₆	—	178	1	no	—
184	C₂×C₁₆	—	178	1	no	—
185	C₂×C₁₆	—	178	1	no	—
186	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
187	#104	—	178	2	no	—
188	#103	—	89	2	no	—
189	#102	—	89	2	no	—
190	#97	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	178	1	no	—
193	C₄×C₈	—	178	1	no	—
194	#41	—	178	1	no	—
195	#41	—	178	1	no	—
196	#41	—	178	1	no	—
197	#41	—	178	1	no	—
198	C₂×Q₈	—	89	1	no	—
199	C₂×D₄	—	89	1	no	—
200	C₂×D₄	—	178	1	no	—
201	C₂²×C₄	—	178	1	no	—
202	C₂²×C₄	—	178	1	no	—
203	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
204	M₁₆	—	2	1	no	—
205	M₁₆	—	2	1	no	—
206	M₁₆	—	356	1	no	—
207	M₁₆	—	356	1	no	—
208	M₁₆	—	178	1	no	—
209	M₁₆	—	178	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	89	2	no	—
212	C₂×C₈	—	356	1	no	—
213	C₂×C₈	—	356	1	no	—
214	C₂×C₈	—	178	1	no	—
215	C₂×C₈	—	178	1	no	—
216	C₂×C₈	—	178	1	no	—
217	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
218	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
219	C₂×C₈	#13083573	1	1	yes	characteristic
220	C₂×C₈	#13083574	1	1	yes	characteristic
221	C₂×C₈	#13083575	1	1	yes	characteristic
222	C₂×C₈	#13083575	1	1	yes	characteristic
223	C₄⋊C₄	C₈×D₈₉	1	2	yes	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	89	2	no	—
227	C₄²	—	178	1	no	—
228	C₁₆	—	2	2	no	—
229	C₁₆	—	4	1	no	—
230	C₁₆	—	356	1	no	—
231	C₁₆	—	178	2	no	—
232	C₂³	—	178	1	no	—
233	C₂³	#15668018	1	1	yes	characteristic
234	Q₈	—	178	1	no	—
235	Q₈	—	178	1	no	—
236	D₄	—	356	1	no	—
237	D₄	—	356	1	no	—
238	D₄	—	178	1	no	—
239	D₄	—	178	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	356	2	no	—
242	C₂×C₄	—	356	1	no	—
243	C₂×C₄	—	356	1	no	—
244	C₂×C₄	—	178	1	no	—
245	C₂×C₄	—	178	2	no	—
246	C₂×C₄	#15667970	1	1	yes	characteristic
247	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
248	C₂×C₄	#15667974	1	1	yes	characteristic
249	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
250	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
251	C₂×C₄	#15668020	1	1	yes	characteristic
252	C₈	—	4	1	no	—
253	C₈	—	356	1	no	—
254	C₈	—	178	2	no	—
255	C₈	#15667984	1	1	yes	characteristic
256	C₈	#15667997	1	1	yes	characteristic
257	C₈	#15668001	1	1	yes	characteristic
258	C₈	#15668007	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	356	2	no	—
261	C₂²	—	356	1	no	—
262	C₂²	#19151658	1	1	yes	characteristic
263	C₂²	C₈×C₂²⋊D₈₉	1	1	yes	characteristic
264	C₂²	#19151668	1	1	yes	characteristic
265	C₄	—	4	2	no	—
266	C₄	—	356	2	no	—
267	C₄	—	356	1	no	—
268	C₄	#19151488	1	1	yes	characteristic
269	C₄	#19151650	1	1	yes	characteristic
270	C₄	#19151663	1	1	yes	characteristic
271	C₄	#19151674	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	712	1	no	—
275	C₂	#26020857	1	1	yes	characteristic
276	C₂	#26022092	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2151	C₂
socle	C₂²×C₈₉	C₈×D₄
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2151	D₈₉
hypocenter	C₈₉	#13125
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ #2151

Complements

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

N	H ≅ G/N	type
3. C₂×#19151373	C₂	semidirect
5. #26018341	C₂	semidirect
6. #26018333	C₂	semidirect
7. C₈₉×#2151	C₂	semidirect
139. #2151	D₈₉	semidirect
146. C₈₉	#13125	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151383	2
#26021865	2
C₂×#19151373	2
#26020091	2
#26018341	2
#26018333	2
C₈₉×#2151	2
#13125	89