Subgroups — #36469481

Order 22784.

482 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	C₂×#19151382	C₂	1	2	yes	maximal
2	C₄×#15667952	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151374	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018341	C₂	1	2	yes	maximal, complemented
5	C₄×#131×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667952	C₂²	1	2	yes	—
7	C₂²×#13083549	C₂²	1	2	yes	—
8	#19151382	C₄	1	8	yes	complemented
9	#19151374	—	2	1	no	—
10	#19151374	—	2	1	no	—
11	#19151374	—	2	1	no	—
12	C₄×#13083549	—	2	1	no	—
13	C₂×#131×C₈₉	C₄	1	1	yes	characteristic
14	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
15	C₄×#41×C₈₉	C₄	1	1	yes	characteristic
16	#405×C₈₉	—	2	1	no	—
17	#405×C₈₉	—	2	1	no	—
18	#405×C₈₉	C₄	1	1	yes	characteristic
19	#405×C₈₉	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₈₉	—	2	1	no	—
21	C₄×M₁₆×C₈₉	C₄	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₉	—	2	1	no	—
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	#15667952	—	2	2	no	—
25	#15667952	C₂×C₄	1	4	yes	complemented
26	C₂×#13083549	—	2	2	no	—
27	C₂×#13083549	—	2	2	no	—
28	C₂×#13083549	—	2	2	no	—
29	C₂×#13083549	C₂×C₄	1	4	yes	complemented
30	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
31	#131×C₈₉	—	2	2	no	—
32	#131×C₈₉	—	2	2	no	—
33	#131×C₈₉	C₈	1	4	yes	—
34	#131×C₈₉	C₂×C₄	1	4	yes	—
35	C₂×M₁₆×C₈₉	—	2	1	no	—
36	C₂×M₁₆×C₈₉	—	2	1	no	—
37	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	—	2	1	no	—
40	C₂²×C₈×C₈₉	—	2	1	no	—
41	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₈₉	C₈	1	2	yes	—
44	C₄×D₄×C₈₉	—	2	2	no	—
45	C₄×D₄×C₈₉	C₈	1	2	yes	—
46	#117×C₈₉	—	2	1	no	—
47	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	—	2	1	no	—
49	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
50	#97×C₈₉	—	2	1	no	—
51	#97×C₈₉	—	2	1	no	—
52	#97×C₈₉	—	2	1	no	—
53	C₄×C₈×C₈₉	—	2	1	no	—
54	C₄×C₈×C₈₉	—	2	1	no	—
55	C₄×C₈×C₈₉	—	2	1	no	—
56	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
57	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
58	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
59	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
60	#13083549	—	2	4	no	—
61	#13083549	—	2	4	no	—
62	#41×C₈₉	—	2	1	no	—
63	#41×C₈₉	—	2	1	no	—
64	#41×C₈₉	C₄²	1	2	yes	—
65	#41×C₈₉	C₂×C₈	1	2	yes	—
66	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
67	C₂×D₄×C₈₉	—	2	1	no	—
68	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	—	2	1	no	—
70	C₂²×C₄×C₈₉	—	2	2	no	—
71	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
72	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
73	M₁₆×C₈₉	—	2	1	no	—
74	M₁₆×C₈₉	—	2	1	no	—
75	M₁₆×C₈₉	—	2	4	no	—
76	M₁₆×C₈₉	—	2	4	no	—
77	M₁₆×C₈₉	C₄²	1	2	yes	—
78	M₁₆×C₈₉	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₈₉	—	2	1	no	—
80	C₂×C₈×C₈₉	—	2	1	no	—
81	C₂×C₈×C₈₉	—	2	4	no	—
82	C₂×C₈×C₈₉	—	2	4	no	—
83	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
86	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
87	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₈₉	#41	1	2	yes	—
91	C₄⋊C₄×C₈₉	—	2	2	no	—
92	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
93	C₂²⋊C₄×C₈₉	—	2	1	no	—
94	C₂²⋊C₄×C₈₉	—	2	2	no	—
95	C₄²×C₈₉	—	2	2	no	—
96	C₄²×C₈₉	—	2	1	no	—
97	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₈₉	C₂×C₈	1	2	yes	—
100	C₂³×C₈₉	—	2	1	no	—
101	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
102	Q₈×C₈₉	—	2	1	no	—
103	Q₈×C₈₉	C₄×C₈	1	2	yes	—
104	D₄×C₈₉	—	2	2	no	—
105	D₄×C₈₉	—	2	2	no	—
106	D₄×C₈₉	—	2	1	no	—
107	D₄×C₈₉	C₄×C₈	1	2	yes	—
108	C₂×C₄×C₈₉	—	2	1	no	—
109	C₂×C₄×C₈₉	—	2	2	no	—
110	C₂×C₄×C₈₉	—	2	2	no	—
111	C₂×C₄×C₈₉	—	2	2	no	—
112	C₂×C₄×C₈₉	—	4	2	no	—
113	C₂×C₄×C₈₉	—	4	4	no	—
114	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
115	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
116	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
122	C₈×C₈₉	—	2	2	no	—
123	C₈×C₈₉	—	2	2	no	—
124	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
125	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
126	C₈×C₈₉	C₄×D₄	1	2	yes	—
127	C₈×C₈₉	C₄×D₄	1	2	yes	—
128	C₂²×C₈₉	—	2	2	no	—
129	C₂²×C₈₉	—	2	2	no	—
130	C₂²×C₈₉	—	4	2	no	—
131	C₂²×C₈₉	C₄×M₁₆	1	1	yes	characteristic
132	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₂²×C₈₉	#431	1	1	yes	characteristic
134	C₄×C₈₉	—	2	2	no	—
135	C₄×C₈₉	—	2	2	no	—
136	C₄×C₈₉	—	2	2	no	—
137	C₄×C₈₉	—	4	2	no	—
138	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
140	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₄×C₈₉	#431	1	1	yes	characteristic
142	#12604	—	89	1	no	maximal, Sylow
143	C₂×C₈₉	—	2	1	no	—
144	C₂×C₈₉	—	2	1	no	—
145	C₂×C₈₉	—	4	2	no	—
146	C₂×C₈₉	C₄×C₂²⋊C₈	1	1	yes	characteristic
147	C₂×C₈₉	#2117	1	2	yes	—
148	C₄×#131	D₈₉	1	1	yes	characteristic
149	C₂×#350	—	89	2	no	—
150	C₄×M₃₂	—	89	1	no	—
151	C₂×#346	—	89	1	no	—
152	#1379	—	89	2	no	—
153	C₈₉	#12604	1	1	yes	characteristic, complemented, Sylow
154	C₂×#131	Dic₈₉	1	1	yes	characteristic
155	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
156	C₄×#41	Dic₈₉	1	1	yes	characteristic
157	C₂×M₃₂	—	89	2	no	—
158	C₂²×C₁₆	—	89	2	no	—
159	#405	—	2	1	no	—
160	#405	—	2	1	no	—
161	#405	Dic₈₉	1	1	yes	characteristic
162	#405	C₂×D₈₉	1	1	yes	characteristic
163	C₄×M₁₆	—	2	1	no	—
164	C₄×M₁₆	Dic₈₉	1	1	yes	characteristic
165	C₂×C₄×C₈	—	2	1	no	—
166	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
167	#350	—	89	8	no	—
168	#346	—	178	1	no	—
169	#346	—	178	1	no	—
170	#346	—	178	1	no	—
171	C₄×C₁₆	—	178	1	no	—
172	C₂×#41	C₂×Dic₈₉	1	1	yes	characteristic
173	#131	—	2	2	no	—
174	#131	—	2	2	no	—
175	#131	#10623494	1	4	yes	—
176	#131	C₄×D₈₉	1	4	yes	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	C₂×Dic₈₉	1	1	yes	characteristic
180	C₂×M₁₆	C₂×Dic₈₉	1	1	yes	characteristic
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
185	C₄×Q₈	#10623494	1	2	yes	—
186	C₄×D₄	—	2	2	no	—
187	C₄×D₄	#10623494	1	2	yes	—
188	#117	—	2	1	no	—
189	#117	C₂×Dic₈₉	1	1	yes	characteristic
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
192	M₃₂	—	89	4	no	—
193	M₃₂	—	178	2	no	—
194	C₂×C₁₆	—	89	4	no	—
195	C₂×C₁₆	—	178	2	no	—
196	C₂×C₁₆	—	178	2	no	—
197	C₂×C₁₆	—	178	2	no	—
198	#97	—	2	1	no	—
199	#97	—	2	1	no	—
200	#97	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
205	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
206	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
207	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
208	#41	—	2	1	no	—
209	#41	—	2	1	no	—
210	#41	C₄×Dic₈₉	1	2	yes	—
211	#41	C₂×#10623494	1	2	yes	—
212	C₂×Q₈	C₂×#10623494	1	1	yes	characteristic
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	C₂×#10623494	1	1	yes	characteristic
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	#13083558	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
219	M₁₆	—	2	1	no	—
220	M₁₆	—	2	1	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	C₄×Dic₈₉	1	2	yes	—
224	M₁₆	C₂×#10623494	1	2	yes	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
231	C₂×C₈	C₂×#10623494	1	2	yes	—
232	C₂×C₈	#13083558	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
234	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
235	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
236	C₂×C₈	#13083574	1	2	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	C₂×#10623494	1	2	yes	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	1	no	—
243	C₄²	C₂²⋊Dic₈₉	1	1	yes	characteristic
244	C₄²	C₂²⋊Dic₈₉	1	1	yes	characteristic
245	C₄²	C₂×#10623494	1	2	yes	—
246	C₁₆	—	178	4	no	—
247	C₁₆	—	178	4	no	—
248	C₂³	—	2	1	no	—
249	C₂³	C₂×#13083558	1	1	yes	characteristic
250	Q₈	—	2	1	no	—
251	Q₈	C₄×#10623494	1	2	yes	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	1	no	—
255	D₄	C₄×#10623494	1	2	yes	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	C₂²⋊#10623494	1	2	yes	—
263	C₂×C₄	C₂²⋊#10623494	1	2	yes	—
264	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
266	C₂×C₄	#15667971	1	1	yes	characteristic
267	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
268	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
269	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	2	2	no	—
272	C₈	C₂²⋊#10623494	1	2	yes	—
273	C₈	C₂²⋊#10623494	1	2	yes	—
274	C₈	C₄×C₂²⋊D₈₉	1	2	yes	—
275	C₈	C₄×C₂²⋊D₈₉	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	C₄×#13083558	1	1	yes	characteristic
280	C₂²	C₂×C₂²⋊#10623494	1	1	yes	characteristic
281	C₂²	#19151651	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	4	2	no	—
286	C₄	C₄×C₂²⋊Dic₈₉	1	1	yes	characteristic
287	C₄	C₂×C₄×#10623494	1	1	yes	characteristic
288	C₄	C₂×C₂²⋊#10623494	1	1	yes	characteristic
289	C₄	#19151651	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	4	2	no	—
293	C₂	C₄×C₂²⋊#10623494	1	1	yes	characteristic
294	C₂	#26021880	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₈₉
commutator	C₄×C₈₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₄×#131×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₈₉	C₂×C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄×#131	D₈₉
hypocenter	C₈₉	#12604
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. C₄×#15667952	C₂	semidirect
3. C₂×#19151374	C₂	semidirect
4. #26018341	C₂	semidirect
8. #19151382	C₄	semidirect
25. #15667952	C₂×C₄	semidirect
29. C₂×#13083549	C₂×C₄	semidirect
153. C₈₉	#12604	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151382	2
C₄×#15667952	2
C₂×#19151374	2
#26018341	2
C₄×#131×C₈₉	2
#12604	89