Subgroups — #36707709

Order 22784.

690 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152030	C₂	1	1	yes	maximal, characteristic
2	#26029023	C₂	1	2	yes	maximal, complemented
3	#26022110	C₂	1	2	yes	maximal, complemented
4	#26022103	C₂	1	4	yes	maximal, complemented
5	#26022098	C₂	1	2	yes	maximal, complemented
6	#26021957	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#2158	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668034	C₂²	1	2	yes	—
10	#19152066	—	2	2	no	—
11	#19152030	C₄	1	8	yes	complemented
12	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
13	C₂×#15667957	C₂²	1	2	yes	—
14	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
15	C₂²×#13083549	C₂²	1	1	yes	characteristic
16	#19151692	C₂²	1	4	yes	—
17	#19151685	—	2	4	no	—
18	#19151680	C₂²	1	4	yes	—
19	#19151592	—	2	2	no	—
20	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
21	#19151401	C₂²	1	4	yes	—
22	#19151398	C₂²	1	4	yes	—
23	C₁₆×Dic₈₉	C₂²	1	2	yes	—
24	C₁₆×Dic₈₉	C₂²	1	2	yes	—
25	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	#427×C₈₉	C₂²	1	2	yes	—
27	#367×C₈₉	C₂²	1	2	yes	—
28	#366×C₈₉	C₂²	1	2	yes	—
29	C₂×#13083574	—	2	2	no	—
30	#15668034	—	2	4	no	—
31	#15668034	C₂×C₄	1	8	yes	complemented
32	C₂×Dic₃₅₆	C₂³	1	2	yes	—
33	C₂×#13083563	—	2	2	no	—
34	C₂×D₇₁₂	C₂³	1	2	yes	—
35	C₂²×#10623494	D₄	1	1	yes	characteristic
36	#15667977	—	2	4	no	—
37	#15667974	—	2	4	no	—
38	#15667972	—	4	4	no	—
39	#15667969	—	4	2	no	—
40	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
41	C₄²×D₈₉	—	4	2	no	—
42	Dic₇₁₂	C₂×C₄	1	4	yes	complemented
43	#15667957	C₂×C₄	1	8	yes	complemented
44	D₁₄₂₄	C₂×C₄	1	4	yes	complemented
45	C₂×#13083549	—	2	2	no	—
46	C₂×#13083549	C₂³	1	2	yes	—
47	#15667935	—	2	4	no	—
48	#15667932	—	2	4	no	—
49	C₈×Dic₈₉	D₄	1	2	yes	—
50	C₈×Dic₈₉	C₂³	1	2	yes	—
51	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
52	#117×C₈₉	—	2	2	no	—
53	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
54	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
55	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
56	#107×C₈₉	C₂³	1	4	yes	—
57	#106×C₈₉	—	2	2	no	—
58	#13083574	—	2	8	no	—
59	#13083574	—	4	4	no	—
60	C₂×Dic₁₇₈	—	2	2	no	—
61	C₂×D₃₅₆	—	2	2	no	—
62	C₂×C₂²⋊D₈₉	—	4	2	no	—
63	C₂²×Dic₈₉	D₈	1	2	yes	—
64	C₂×C₄×D₈₉	—	4	2	no	—
65	C₂×C₄×D₈₉	—	4	4	no	—
66	Dic₃₅₆	—	2	2	no	—
67	Dic₃₅₆	C₂²×C₄	1	4	yes	—
68	#13083563	—	2	8	no	—
69	D₇₁₂	—	2	2	no	—
70	D₇₁₂	C₂²×C₄	1	4	yes	—
71	C₂×#10623494	—	2	2	no	—
72	C₂×#10623494	C₂×D₄	1	2	yes	—
73	#13083553	—	4	4	no	—
74	#13083551	—	4	4	no	—
75	C₄×Dic₈₉	—	4	4	no	—
76	C₄×Dic₈₉	D₈	1	1	yes	characteristic
77	C₄×Dic₈₉	D₈	1	1	yes	characteristic
78	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
79	#13083549	—	2	2	no	—
80	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₈₉	#41	1	2	yes	—
84	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₈₉	—	2	4	no	—
86	C₂²⋊C₄×C₈₉	—	4	2	no	—
87	C₄²×C₈₉	—	4	2	no	—
88	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
89	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
90	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
91	C₂²×D₈₉	—	4	2	no	—
92	C₂×C₄×C₈₉	—	4	4	no	—
93	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₈₉	#135	1	2	yes	—
96	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
97	C₂²⋊D₈₉	—	4	8	no	—
98	C₂×Dic₈₉	—	2	4	no	—
99	C₂×Dic₈₉	—	4	2	no	—
100	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
101	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
102	D₃₅₆	—	2	4	no	—
103	D₃₅₆	—	4	2	no	—
104	C₄×D₈₉	—	4	8	no	—
105	C₄×D₈₉	—	8	8	no	—
106	Dic₁₇₈	—	2	4	no	—
107	Dic₁₇₈	—	4	2	no	—
108	C₈×C₈₉	C₄×D₄	1	2	yes	—
109	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
111	#10623494	—	2	2	no	—
112	C₂²×C₈₉	—	2	2	no	—
113	C₂²×C₈₉	C₄×D₈	1	2	yes	—
114	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
115	C₂×D₈₉	—	4	4	no	—
116	C₂×D₈₉	—	8	4	no	—
117	C₄×C₈₉	—	8	4	no	—
118	C₄×C₈₉	C₄×D₈	1	2	yes	—
119	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
120	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
121	Dic₈₉	—	2	2	no	—
122	Dic₈₉	—	4	4	no	—
123	Dic₈₉	#508	1	4	yes	complemented
124	C₄×#508	—	89	1	no	maximal, Sylow
125	C₂×C₈₉	—	2	2	no	—
126	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
127	C₂×C₈₉	C₂×#508	1	2	yes	—
128	D₈₉	—	8	4	no	—
129	C₂×#508	—	89	1	no	—
130	C₄×#135	—	89	2	no	—
131	C₄×Q₃₂	—	89	2	no	—
132	C₄×SD₃₂	—	89	4	no	—
133	C₄×D₁₆	—	89	2	no	—
134	#2158	D₈₉	1	1	yes	characteristic, complemented
135	#2138	—	89	2	no	—
136	C₂×C₄×C₁₆	—	89	1	no	—
137	C₈₉	C₄×#508	1	1	yes	characteristic, complemented, Sylow
138	C₂×#135	—	89	2	no	—
139	C₄×#41	—	178	2	no	—
140	#508	—	89	8	no	—
141	C₂×Q₃₂	—	89	1	no	—
142	C₂×SD₃₂	—	89	2	no	—
143	C₂×D₁₆	—	89	1	no	—
144	C₂²×C₁₆	—	89	1	no	—
145	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
146	C₄×Q₁₆	—	89	4	no	—
147	C₄×SD₁₆	—	178	4	no	—
148	C₄×D₈	—	89	4	no	—
149	#427	C₂×D₈₉	1	2	yes	—
150	#416	—	178	2	no	—
151	C₂×C₄×C₈	—	89	1	no	—
152	#367	C₂×D₈₉	1	2	yes	—
153	#366	C₂×D₈₉	1	2	yes	—
154	#358	—	89	4	no	—
155	#357	—	89	4	no	—
156	C₄×C₁₆	—	89	2	no	—
157	C₄×C₁₆	—	89	2	no	—
158	C₂×#41	—	178	2	no	—
159	#135	—	89	8	no	—
160	#135	—	178	4	no	—
161	C₂×Q₁₆	—	89	2	no	—
162	C₂×SD₁₆	—	178	2	no	—
163	C₂×D₈	—	89	2	no	—
164	C₂²×C₈	—	89	1	no	—
165	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
166	C₄×Q₈	—	178	4	no	—
167	C₄×D₄	—	356	4	no	—
168	C₄×D₄	—	178	4	no	—
169	#117	—	2	2	no	—
170	#117	—	356	2	no	—
171	C₂×C₄²	—	89	1	no	—
172	C₂×C₄²	—	356	2	no	—
173	Q₃₂	—	89	4	no	—
174	SD₃₂	—	89	8	no	—
175	D₁₆	—	89	4	no	—
176	C₂×C₁₆	—	89	2	no	—
177	C₂×C₁₆	—	178	2	no	—
178	C₂×C₁₆	C₄×D₈₉	1	4	yes	complemented
179	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
180	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
181	#107	C₂²×D₈₉	1	4	yes	—
182	#106	—	2	2	no	—
183	#103	—	178	4	no	—
184	#102	—	178	4	no	—
185	C₄×C₈	—	89	2	no	—
186	C₄×C₈	—	89	2	no	—
187	#41	—	356	4	no	—
188	#41	—	178	8	no	—
189	C₂×Q₈	—	178	2	no	—
190	C₂×D₄	—	356	2	no	—
191	C₂×D₄	—	178	2	no	—
192	C₂²×C₄	—	89	2	no	—
193	C₂²×C₄	—	356	4	no	—
194	C₂²×C₄	—	356	2	no	—
195	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
196	Q₁₆	—	89	4	no	—
197	Q₁₆	—	178	2	no	—
198	SD₁₆	—	178	8	no	—
199	D₈	—	89	4	no	—
200	D₈	—	178	2	no	—
201	C₂×C₈	—	89	2	no	—
202	C₂×C₈	—	178	2	no	—
203	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
204	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
205	C₂×C₈	#13083575	1	2	yes	—
206	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	356	4	no	—
209	C₂²⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	—	356	4	no	—
211	C₄²	—	4	2	no	—
212	C₄²	—	89	1	no	—
213	C₄²	—	89	1	no	—
214	C₄²	—	89	2	no	—
215	C₄²	—	356	4	no	—
216	C₁₆	—	178	2	no	—
217	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
218	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
219	C₂³	—	356	2	no	—
220	C₂³	D₈×D₈₉	1	1	yes	characteristic
221	Q₈	—	356	2	no	—
222	Q₈	—	178	4	no	—
223	D₄	—	356	8	no	—
224	D₄	—	356	2	no	—
225	D₄	—	178	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	89	2	no	—
228	C₂×C₄	—	89	2	no	—
229	C₂×C₄	—	356	2	no	—
230	C₂×C₄	—	356	8	no	—
231	C₂×C₄	—	178	4	no	—
232	C₂×C₄	—	712	8	no	—
233	C₂×C₄	#15667974	1	2	yes	—
234	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
235	C₂×C₄	#15668035	1	2	yes	—
236	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
237	C₈	—	178	2	no	—
238	C₈	#15667974	1	2	yes	—
239	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
240	C₈	C₂×#13083575	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	—	356	4	no	—
243	C₂²	—	712	4	no	—
244	C₂²	#19151680	1	2	yes	—
245	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
246	C₄	—	8	4	no	—
247	C₄	—	89	4	no	—
248	C₄	—	356	4	no	—
249	C₄	—	178	2	no	—
250	C₄	#19151680	1	2	yes	—
251	C₄	C₂×#15667974	1	1	yes	characteristic
252	C₄	C₂×#15668035	1	1	yes	characteristic
253	C₂	—	2	2	no	—
254	C₂	—	712	4	no	—
255	C₂	C₂×#19151680	1	1	yes	characteristic
256	C₂	#508×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2158	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#19152030	C₂
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2158	D₈₉
hypocenter	C₈₉	C₄×#508
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
2. #26029023	C₂	semidirect
3. #26022110	C₂	semidirect
4. #26022103	C₂	semidirect
5. #26022098	C₂	semidirect
6. #26021957	C₂	semidirect
7. C₂×C₁₆×Dic₈₉	C₂	semidirect
8. C₈₉×#2158	C₂	semidirect
11. #19152030	C₄	semidirect
31. #15668034	C₂×C₄	semidirect
42. Dic₇₁₂	C₂×C₄	semidirect
43. #15667957	C₂×C₄	semidirect
44. D₁₄₂₄	C₂×C₄	semidirect
53. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
123. Dic₈₉	#508	semidirect
134. #2158	D₈₉	semidirect
137. C₈₉	C₄×#508	semidirect
178. C₂×C₁₆	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152030	2
#26029023	2
#26022110	2
#26022103	2
#26022098	2
#26021957	2
C₂×C₁₆×Dic₈₉	2
C₈₉×#2158	2
C₄×#508	89