Subgroups — #36445455

Order 22784.

444 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028965	C₂	1	1	yes	maximal, characteristic
2	#26018912	C₂	1	1	yes	maximal, characteristic
3	#26018843	C₂	1	1	yes	maximal, characteristic
4	C₈×#13083558	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018143	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018142	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668018	C₄	1	2	yes	—
9	#19152148	C₂²	1	1	yes	characteristic
10	#19151677	—	2	1	no	—
11	#19151677	—	2	1	no	—
12	C₈×D₃₅₆	—	2	1	no	—
13	C₈×D₃₅₆	—	2	1	no	—
14	#19151651	C₂²	1	1	yes	characteristic
15	#19151622	C₂²	1	1	yes	characteristic
16	#19151564	C₄	1	2	yes	—
17	C₄×#13083558	C₂²	1	1	yes	characteristic
18	#19151346	C₄	1	2	yes	—
19	#19151341	C₄	1	2	yes	—
20	#19151337	C₄	1	4	yes	—
21	#19151320	—	2	1	no	—
22	C₈×#10623494	—	2	1	no	—
23	#431×C₈₉	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	#322×C₈₉	—	2	1	no	—
27	#322×C₈₉	—	2	1	no	—
28	C₂×#13083574	C₂×C₄	1	2	yes	—
29	#15668018	—	4	4	no	—
30	C₂×#13083558	—	2	1	no	—
31	C₂×#13083558	C₂×C₄	1	2	yes	—
32	C₂×#13083557	—	2	2	no	—
33	C₂×C₈×D₈₉	—	2	2	no	—
34	#15668012	C₂×C₄	1	1	yes	characteristic
35	#15668009	C₂×C₄	1	1	yes	characteristic
36	#15668000	C₂×C₄	1	2	yes	—
37	#15667986	—	2	1	no	—
38	#15667986	—	2	1	no	—
39	C₄×D₃₅₆	—	2	1	no	—
40	C₄×D₃₅₆	—	2	1	no	—
41	C₄⋊#10623494	—	2	1	no	—
42	C₄⋊#10623494	—	2	1	no	—
43	C₄⋊#10623494	C₂×C₄	1	2	yes	—
44	C₂²⋊#10623494	—	2	1	no	—
45	#15667924	—	2	2	no	—
46	#15667924	—	2	2	no	—
47	#15667923	C₂×C₄	1	2	yes	—
48	C₄×#10623494	—	2	1	no	—
49	C₄×#10623494	—	2	1	no	—
50	C₂×M₁₆×C₈₉	—	2	1	no	—
51	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
55	C₂²⋊C₈×C₈₉	—	2	1	no	—
56	#97×C₈₉	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₈₉	—	2	1	no	—
58	C₄×C₈×C₈₉	—	2	1	no	—
59	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
60	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
61	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
62	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
63	#13083574	—	4	2	no	—
64	#13083574	—	4	2	no	—
65	C₂×Dic₁₇₈	C₄²	1	2	yes	—
66	C₂×D₃₅₆	—	2	1	no	—
67	C₂×D₃₅₆	—	2	1	no	—
68	C₂×D₃₅₆	C₄²	1	2	yes	—
69	C₂×C₂²⋊D₈₉	—	2	2	no	—
70	C₂×C₄×D₈₉	—	2	2	no	—
71	#13083558	—	2	4	no	—
72	#13083558	—	4	2	no	—
73	#13083557	—	4	4	no	—
74	C₂×#10623494	—	2	2	no	—
75	C₂×#10623494	—	2	2	no	—
76	C₂×#10623494	—	2	2	no	—
77	C₈×D₈₉	—	4	4	no	—
78	C₄⋊Dic₈₉	—	2	1	no	—
79	C₄⋊Dic₈₉	—	2	1	no	—
80	#13083551	—	2	2	no	—
81	#13083551	—	2	2	no	—
82	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
85	M₁₆×C₈₉	—	4	2	no	—
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	—	2	2	no	—
88	C₂×C₈×C₈₉	—	2	2	no	—
89	C₂×C₈×C₈₉	—	4	1	no	—
90	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂³×C₈₉	#117	1	1	yes	characteristic
99	C₂²×D₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	—	2	1	no	—
101	C₂×C₄×C₈₉	—	2	1	no	—
102	C₂×C₄×C₈₉	—	2	2	no	—
103	C₂×C₄×C₈₉	—	2	2	no	—
104	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
110	C₂²⋊D₈₉	—	4	4	no	—
111	C₂×Dic₈₉	—	2	2	no	—
112	D₃₅₆	—	4	2	no	—
113	D₃₅₆	—	4	2	no	—
114	D₃₅₆	—	4	2	no	—
115	C₄×D₈₉	—	4	2	no	—
116	C₄×D₈₉	—	4	2	no	—
117	Dic₁₇₈	—	4	2	no	—
118	C₈×C₈₉	—	2	1	no	—
119	C₈×C₈₉	—	2	1	no	—
120	C₈×C₈₉	—	2	1	no	—
121	C₈×C₈₉	—	4	2	no	—
122	C₈×C₈₉	—	4	2	no	—
123	C₈×C₈₉	#100	1	2	yes	—
124	#10623494	—	2	4	no	—
125	#10623494	—	4	2	no	—
126	#10623494	—	4	2	no	—
127	C₂²×C₈₉	—	2	1	no	—
128	C₂²×C₈₉	—	2	1	no	—
129	C₂²×C₈₉	—	2	1	no	—
130	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×D₈₉	—	4	2	no	—
132	C₂×D₈₉	—	4	2	no	—
133	C₂×D₈₉	—	4	2	no	—
134	C₄×C₈₉	—	2	1	no	—
135	C₄×C₈₉	—	2	1	no	—
136	C₄×C₈₉	—	2	1	no	—
137	C₄×C₈₉	C₂×#100	1	1	yes	characteristic
138	C₄×C₈₉	#413	1	1	yes	characteristic
139	C₄×C₈₉	#443	1	2	yes	—
140	C₄×C₈₉	#444	1	2	yes	—
141	Dic₈₉	—	4	2	no	—
142	#8245	—	89	1	no	maximal, Sylow
143	C₂×C₈₉	—	4	1	no	—
144	C₂×C₈₉	C₄×#100	1	1	yes	characteristic
145	C₂×C₈₉	#1764	1	1	yes	characteristic
146	C₂×C₈₉	#1765	1	1	yes	characteristic
147	D₈₉	—	8	2	no	—
148	#2931	—	89	1	no	—
149	#1543	—	89	1	no	—
150	#1525	—	89	1	no	—
151	#1451	D₈₉	1	1	yes	characteristic, complemented
152	C₈×M₁₆	—	89	1	no	—
153	#1299	—	89	1	no	—
154	#1299	—	89	1	no	—
155	C₈₉	#8245	1	1	yes	characteristic, complemented, Sylow
156	C₂×#131	—	89	2	no	—
157	#532	—	89	1	no	—
158	#436	—	178	1	no	—
159	#436	—	178	1	no	—
160	C₈×D₄	—	178	1	no	—
161	C₈×D₄	—	178	1	no	—
162	#431	—	89	1	no	—
163	#431	C₂×D₈₉	1	1	yes	characteristic
164	#423	—	89	1	no	—
165	#408	—	89	2	no	—
166	C₄×M₁₆	—	89	1	no	—
167	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
169	#333	—	89	2	no	—
170	#331	—	89	2	no	—
171	#329	—	89	4	no	—
172	#322	—	2	1	no	—
173	#322	—	2	1	no	—
174	#322	—	178	1	no	—
175	C₈²	—	178	1	no	—
176	C₂×#41	—	89	2	no	—
177	#131	—	356	4	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	89	2	no	—
180	C₂×M₁₆	—	178	1	no	—
181	C₂×M₁₆	—	178	2	no	—
182	C₂²×C₈	—	178	2	no	—
183	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
184	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
185	C₄⋊Q₈	—	89	1	no	—
186	#127	—	89	1	no	—
187	#124	—	89	2	no	—
188	#121	—	178	1	no	—
189	#121	—	178	1	no	—
190	C₄×D₄	—	178	1	no	—
191	C₄×D₄	—	178	1	no	—
192	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
193	C₄⋊C₈	—	89	2	no	—
194	C₄⋊C₈	—	178	1	no	—
195	C₄⋊C₈	—	178	1	no	—
196	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
197	C₂²⋊C₈	—	2	1	no	—
198	C₂²⋊C₈	—	178	1	no	—
199	C₂²⋊C₈	—	178	2	no	—
200	C₂²⋊C₈	—	178	2	no	—
201	#97	—	89	2	no	—
202	#97	C₄×D₈₉	1	2	yes	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	178	1	no	—
206	C₄×C₈	—	178	1	no	—
207	C₄×C₈	D₃₅₆	1	1	yes	characteristic
208	C₄×C₈	D₃₅₆	1	1	yes	characteristic
209	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
210	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
211	#41	—	356	2	no	—
212	#41	—	356	2	no	—
213	C₂×Q₈	—	89	2	no	—
214	C₂×D₄	—	89	2	no	—
215	C₂×D₄	—	178	2	no	—
216	C₂×D₄	—	178	1	no	—
217	C₂×D₄	—	178	1	no	—
218	C₂²×C₄	—	178	2	no	—
219	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
220	C₂²×C₄	#13083574	1	1	yes	characteristic
221	C₂²×C₄	#13083574	1	1	yes	characteristic
222	M₁₆	—	4	2	no	—
223	M₁₆	—	356	4	no	—
224	M₁₆	—	356	2	no	—
225	M₁₆	—	178	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	356	4	no	—
231	C₂×C₈	—	178	2	no	—
232	C₂×C₈	—	178	2	no	—
233	C₂×C₈	—	178	2	no	—
234	C₂×C₈	#13083551	1	1	yes	characteristic
235	C₂×C₈	#13083551	1	1	yes	characteristic
236	C₂×C₈	#13083551	1	1	yes	characteristic
237	C₂×C₈	#13083551	1	1	yes	characteristic
238	C₄⋊C₄	—	178	1	no	—
239	C₄⋊C₄	—	178	1	no	—
240	C₂²⋊C₄	—	178	2	no	—
241	C₂²⋊C₄	—	178	2	no	—
242	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
243	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
244	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
245	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
246	C₂³	—	178	2	no	—
247	C₂³	#15667969	1	1	yes	characteristic
248	Q₈	—	356	2	no	—
249	D₄	—	356	2	no	—
250	D₄	—	356	4	no	—
251	D₄	—	356	2	no	—
252	D₄	—	356	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	356	2	no	—
258	C₂×C₄	—	356	2	no	—
259	C₂×C₄	—	178	2	no	—
260	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
261	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
262	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
263	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
264	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
265	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	4	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	356	2	no	—
272	C₈	—	356	2	no	—
273	C₈	—	178	4	no	—
274	C₈	#15667928	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	356	2	no	—
279	C₂²	—	356	2	no	—
280	C₂²	—	356	2	no	—
281	C₂²	C₄×#13083551	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	356	2	no	—
286	C₄	C₂×#15667928	1	1	yes	characteristic
287	C₄	#19151580	1	1	yes	characteristic
288	C₄	#19151713	1	2	yes	—
289	C₄	#19151721	1	2	yes	—
290	C₂	—	4	1	no	—
291	C₂	—	712	2	no	—
292	C₂	C₄×#15667928	1	1	yes	characteristic
293	C₂	#26020091	1	1	yes	characteristic
294	C₂	#26020096	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#13083551
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1451	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	#19152148	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1451	D₈₉
hypocenter	C₈₉	#8245
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1451

Complements

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

N	H ≅ G/N	type
4. C₈×#13083558	C₂	semidirect
5. #26018143	C₂	semidirect
6. #26018142	C₂	semidirect
7. C₈₉×#1451	C₂	semidirect
151. #1451	D₈₉	semidirect
155. C₈₉	#8245	semidirect

Maximal subgroups

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

subgroup	index
#26028965	2
#26018912	2
#26018843	2
C₈×#13083558	2
#26018143	2
#26018142	2
C₈₉×#1451	2
#8245	89