Subgroups — #36542125

Order 22784.

922 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030827	C₂	1	4	yes	maximal, complemented
2	#26030680	C₂	1	1	yes	maximal, characteristic, complemented
3	#26019868	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019866	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019859	C₂	1	1	yes	maximal, characteristic, complemented
6	#26019857	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019842	C₂	1	4	yes	maximal, complemented
8	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#3147	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668034	—	2	2	no	—
11	#19152148	C₂²	1	2	yes	—
12	#19152001	C₂²	1	4	yes	—
13	#19151982	—	2	2	no	—
14	#19151978	C₂²	1	1	yes	characteristic
15	#19151975	C₂²	1	1	yes	characteristic
16	#19151974	C₂²	1	4	yes	—
17	#19151971	C₂²	1	2	yes	—
18	#19151946	C₂²	1	4	yes	—
19	#19151944	C₂²	1	4	yes	—
20	#19151832	—	2	8	no	—
21	C₄×#13083559	—	2	8	no	—
22	C₂×C₄×#10623494	C₂²	1	2	yes	—
23	C₈×#10623494	C₂²	1	2	yes	—
24	C₈×#10623494	C₂²	1	1	yes	characteristic
25	C₈×#10623494	C₂²	1	1	yes	characteristic
26	C₂×#135×C₈₉	—	2	2	no	—
27	#532×C₈₉	C₂²	1	2	yes	—
28	#495×C₈₉	—	2	2	no	—
29	#494×C₈₉	C₂²	1	1	yes	characteristic
30	#493×C₈₉	C₂²	1	1	yes	characteristic
31	#492×C₈₉	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×#13083574	—	2	2	no	—
34	#15668034	—	4	8	no	—
35	C₂×Dic₃₅₆	—	2	2	no	—
36	C₂×#13083563	—	2	4	no	—
37	C₂×D₇₁₂	—	2	2	no	—
38	C₂×#13083559	—	2	8	no	—
39	C₂²×#10623494	D₄	1	4	yes	—
40	#15668012	C₂³	1	2	yes	—
41	#15668009	C₂³	1	2	yes	—
42	#15668000	—	2	2	no	—
43	#15667986	—	2	4	no	—
44	C₄×D₃₅₆	—	2	4	no	—
45	#15667945	—	2	8	no	—
46	#15667933	—	2	8	no	—
47	#15667932	—	2	8	no	—
48	C₄×#10623494	D₄	1	4	yes	—
49	C₄×#10623494	C₂³	1	2	yes	—
50	C₄×#10623494	C₂³	1	2	yes	—
51	C₂×#41×C₈₉	—	2	2	no	—
52	#135×C₈₉	—	4	8	no	—
53	C₂×Q₁₆×C₈₉	—	2	2	no	—
54	C₂×SD₁₆×C₈₉	—	2	4	no	—
55	C₂×D₈×C₈₉	—	2	2	no	—
56	C₂²×C₈×C₈₉	D₄	1	2	yes	—
57	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
58	#127×C₈₉	C₂³	1	2	yes	—
59	#124×C₈₉	—	2	2	no	—
60	#121×C₈₉	—	2	4	no	—
61	C₄×D₄×C₈₉	—	2	4	no	—
62	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₈₉	D₄	1	2	yes	—
64	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
66	#13083574	—	4	8	no	—
67	C₂×Dic₁₇₈	—	2	2	no	—
68	C₂×D₃₅₆	—	2	4	no	—
69	C₂×D₃₅₆	—	2	2	no	—
70	C₂×C₂²⋊D₈₉	—	4	2	no	—
71	C₂×C₄×D₈₉	—	4	2	no	—
72	Dic₃₅₆	—	4	4	no	—
73	#13083563	—	4	8	no	—
74	D₇₁₂	—	4	4	no	—
75	#13083559	—	4	16	no	—
76	C₂×#10623494	—	2	8	no	—
77	C₂×#10623494	C₂×D₄	1	4	yes	—
78	C₂×#10623494	C₂×D₄	1	4	yes	—
79	C₄⋊Dic₈₉	—	2	4	no	—
80	#13083551	—	4	4	no	—
81	#41×C₈₉	—	4	8	no	—
82	C₂×Q₈×C₈₉	—	2	2	no	—
83	C₂×D₄×C₈₉	—	2	4	no	—
84	C₂×D₄×C₈₉	—	2	2	no	—
85	C₂×D₄×C₈₉	—	4	2	no	—
86	C₂²×C₄×C₈₉	—	4	2	no	—
87	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
89	Q₁₆×C₈₉	—	4	4	no	—
90	SD₁₆×C₈₉	—	4	8	no	—
91	D₈×C₈₉	—	4	4	no	—
92	C₂×C₈×C₈₉	D₈	1	8	yes	complemented
93	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
95	C₄⋊C₄×C₈₉	—	2	4	no	—
96	C₂²⋊C₄×C₈₉	—	4	4	no	—
97	C₄²×C₈₉	C₂×D₄	1	2	yes	—
98	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
100	C₂³×C₈₉	—	4	2	no	—
101	C₂³×C₈₉	#127	1	1	yes	characteristic
102	C₂²×D₈₉	—	4	2	no	—
103	Q₈×C₈₉	—	4	4	no	—
104	D₄×C₈₉	—	4	4	no	—
105	D₄×C₈₉	—	4	8	no	—
106	D₄×C₈₉	—	8	4	no	—
107	C₂×C₄×C₈₉	—	2	4	no	—
108	C₂×C₄×C₈₉	—	4	2	no	—
109	C₂×C₄×C₈₉	—	8	4	no	—
110	C₂×C₄×C₈₉	#127	1	2	yes	—
111	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
113	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
114	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
115	C₂²⋊D₈₉	—	8	4	no	—
116	C₂×Dic₈₉	—	4	2	no	—
117	D₃₅₆	—	4	4	no	—
118	D₃₅₆	—	4	8	no	—
119	C₄×D₈₉	—	8	4	no	—
120	Dic₁₇₈	—	4	4	no	—
121	C₈×C₈₉	C₂×D₈	1	4	yes	—
122	C₈×C₈₉	C₂×D₈	1	4	yes	—
123	#10623494	—	2	4	no	—
124	#10623494	#135	1	8	yes	complemented
125	C₂²×C₈₉	—	2	2	no	—
126	C₂²×C₈₉	—	8	2	no	—
127	C₂²×C₈₉	—	8	4	no	—
128	C₂²×C₈₉	#493	1	2	yes	—
129	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
130	C₂×D₈₉	—	8	2	no	—
131	C₂×D₈₉	—	8	4	no	—
132	C₄×C₈₉	—	2	2	no	—
133	C₄×C₈₉	—	8	2	no	—
134	C₄×C₈₉	#493	1	2	yes	—
135	C₄×C₈₉	C₂²×D₈	1	2	yes	—
136	C₄×C₈₉	C₂×#135	1	4	yes	—
137	Dic₈₉	—	8	2	no	—
138	#19603	—	89	1	no	maximal, Sylow
139	C₂×C₈₉	—	2	2	no	—
140	C₂×C₈₉	—	16	2	no	—
141	C₂×C₈₉	C₂×#493	1	1	yes	characteristic
142	C₂×C₈₉	#3147	1	2	yes	—
143	D₈₉	—	16	2	no	—
144	#3169	—	89	4	no	—
145	#3147	—	89	1	no	—
146	#3147	D₈₉	1	1	yes	characteristic, complemented
147	#1713	—	89	1	no	—
148	#1712	—	89	1	no	—
149	#1709	—	89	1	no	—
150	#1709	—	89	1	no	—
151	#1706	—	89	4	no	—
152	C₂×C₈²	—	89	1	no	—
153	C₈₉	#19603	1	1	yes	characteristic, complemented, Sylow
154	C₂×#135	—	2	2	no	—
155	C₂×#135	—	178	2	no	—
156	#532	—	89	2	no	—
157	#532	C₂×D₈₉	1	2	yes	—
158	#500	—	89	4	no	—
159	#495	—	2	2	no	—
160	#495	—	178	2	no	—
161	#494	—	89	1	no	—
162	#494	C₂×D₈₉	1	1	yes	characteristic
163	#493	—	89	4	no	—
164	#493	—	89	1	no	—
165	#493	C₂×D₈₉	1	1	yes	characteristic
166	#492	—	89	2	no	—
167	#492	C₂×D₈₉	1	2	yes	—
168	#486	—	89	4	no	—
169	#486	—	89	4	no	—
170	#466	—	178	8	no	—
171	C₄×D₈	—	178	8	no	—
172	C₂×C₄×C₈	—	89	2	no	—
173	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
174	C₈²	—	89	2	no	—
175	C₈²	—	89	1	no	—
176	C₈²	—	89	1	no	—
177	C₂×#41	—	2	2	no	—
178	C₂×#41	—	178	2	no	—
179	#135	—	4	8	no	—
180	#135	—	356	8	no	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	178	2	no	—
183	C₂×SD₁₆	—	2	4	no	—
184	C₂×SD₁₆	—	178	4	no	—
185	C₂×D₈	—	2	2	no	—
186	C₂×D₈	—	178	8	no	—
187	C₂×D₈	—	178	2	no	—
188	C₂²×C₈	—	89	4	no	—
189	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
190	C₄⋊Q₈	—	89	2	no	—
191	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
192	#127	—	89	2	no	—
193	#127	C₂²×D₈₉	1	2	yes	—
194	#124	—	2	2	no	—
195	#124	—	178	2	no	—
196	#121	—	2	4	no	—
197	#121	—	178	4	no	—
198	C₄×D₄	—	2	4	no	—
199	C₄×D₄	—	178	4	no	—
200	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
201	#107	—	178	8	no	—
202	#102	—	178	8	no	—
203	#102	—	178	8	no	—
204	C₄×C₈	—	89	4	no	—
205	C₄×C₈	—	89	2	no	—
206	C₄×C₈	—	89	2	no	—
207	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
208	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
210	#41	—	4	8	no	—
211	#41	—	356	8	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	178	2	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	356	2	no	—
218	C₂×D₄	—	178	4	no	—
219	C₂×D₄	—	178	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	356	2	no	—
222	C₂²×C₄	D₄×D₈₉	1	2	yes	—
223	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
224	Q₁₆	—	4	4	no	—
225	Q₁₆	—	356	4	no	—
226	SD₁₆	—	4	8	no	—
227	SD₁₆	—	356	8	no	—
228	D₈	—	4	4	no	—
229	D₈	—	356	16	no	—
230	D₈	—	356	4	no	—
231	C₂×C₈	—	89	4	no	—
232	C₂×C₈	—	89	4	no	—
233	C₂×C₈	—	178	8	no	—
234	C₂×C₈	#13083559	1	8	yes	complemented
235	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
236	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	178	4	no	—
239	C₂²⋊C₄	—	4	4	no	—
240	C₂²⋊C₄	—	356	4	no	—
241	C₄²	D₄×D₈₉	1	2	yes	—
242	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
243	C₄²	C₂³×D₈₉	1	1	yes	characteristic
244	C₂³	—	4	2	no	—
245	C₂³	—	356	2	no	—
246	C₂³	#15668008	1	1	yes	characteristic
247	Q₈	—	4	4	no	—
248	Q₈	—	356	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	8	no	—
251	D₄	—	8	4	no	—
252	D₄	—	356	4	no	—
253	D₄	—	356	8	no	—
254	D₄	—	712	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	—	356	2	no	—
259	C₂×C₄	—	712	4	no	—
260	C₂×C₄	#15668008	1	2	yes	—
261	C₂×C₄	#15668008	1	1	yes	characteristic
262	C₂×C₄	C₂×#13083559	1	4	yes	—
263	C₂×C₄	C₂×D₄×D₈₉	1	2	yes	—
264	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
265	C₈	—	89	8	no	—
266	C₈	—	178	4	no	—
267	C₈	C₂×#13083559	1	4	yes	—
268	C₈	C₂×#13083559	1	4	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	8	4	no	—
272	C₂²	—	712	2	no	—
273	C₂²	—	712	4	no	—
274	C₂²	#19151974	1	2	yes	—
275	C₂²	C₂×#15668008	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	8	2	no	—
278	C₄	—	712	2	no	—
279	C₄	#19151974	1	2	yes	—
280	C₄	C₂²×#13083559	1	2	yes	—
281	C₄	#135×D₈₉	1	4	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	16	2	no	—
284	C₂	—	1424	2	no	—
285	C₂	C₂×#19151974	1	1	yes	characteristic
286	C₂	#26030676	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#15668008
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#3147	C₂
socle	C₂²×C₈₉	C₂×#127
hypercenter	#3147	D₈₉
hypocenter	C₈₉	#19603
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #26030827	C₂	semidirect
2. #26030680	C₂	semidirect
3. #26019868	C₂	semidirect
4. #26019866	C₂	semidirect
5. #26019859	C₂	semidirect
6. #26019857	C₂	semidirect
7. #26019842	C₂	semidirect
8. C₂×C₈×#10623494	C₂	semidirect
9. C₈₉×#3147	C₂	semidirect
92. C₂×C₈×C₈₉	D₈	semidirect
124. #10623494	#135	semidirect
146. #3147	D₈₉	semidirect
153. C₈₉	#19603	semidirect
234. C₂×C₈	#13083559	semidirect

Maximal subgroups

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

subgroup	index
#26030827	2
#26030680	2
#26019868	2
#26019866	2
#26019859	2
#26019857	2
#26019842	2
C₂×C₈×#10623494	2
C₈₉×#3147	2
#19603	89