Subgroups — #36542086

Order 22784.

814 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030807	C₂	1	4	yes	maximal, complemented
2	#26030766	C₂	1	1	yes	maximal, characteristic, complemented
3	#26019851	C₂	1	2	yes	maximal, complemented
4	#26019846	C₂	1	2	yes	maximal, complemented
5	#26019841	C₂	1	2	yes	maximal, complemented
6	#26019838	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#3147	C₂	1	1	yes	maximal, characteristic
9	#19152155	C₂²	1	2	yes	—
10	#19152003	C₂²	1	2	yes	—
11	#19152000	—	2	2	no	—
12	#19151996	C₂²	1	2	yes	—
13	#19151994	C₂²	1	4	yes	—
14	#19151970	C₂²	1	4	yes	—
15	#19151950	C₂²	1	4	yes	—
16	#19151944	C₂²	1	4	yes	—
17	#19151825	—	2	8	no	—
18	C₄×#13083562	—	2	8	no	—
19	#19151636	—	2	2	no	—
20	C₂×C₄×#10623494	C₂²	1	2	yes	—
21	C₈×#10623494	C₂²	1	2	yes	—
22	C₈×#10623494	C₂²	1	1	yes	characteristic
23	C₈×#10623494	C₂²	1	1	yes	characteristic
24	C₂×#135×C₈₉	—	2	2	no	—
25	#532×C₈₉	C₂²	1	2	yes	—
26	#495×C₈₉	—	2	2	no	—
27	#494×C₈₉	C₂²	1	1	yes	characteristic
28	#493×C₈₉	C₂²	1	1	yes	characteristic
29	#492×C₈₉	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#13083562	—	2	8	no	—
32	C₂²×#10623494	D₄	1	4	yes	—
33	#15668012	C₂³	1	4	yes	—
34	#15668004	—	2	2	no	—
35	#15667991	—	2	4	no	—
36	C₄×Dic₁₇₈	—	2	4	no	—
37	#15667971	—	2	2	no	—
38	#15667946	—	2	4	no	—
39	#15667944	—	2	4	no	—
40	#15667943	—	2	8	no	—
41	#15667935	—	2	8	no	—
42	#15667933	—	2	8	no	—
43	C₄×#10623494	D₄	1	4	yes	—
44	C₄×#10623494	C₂³	1	2	yes	—
45	C₄×#10623494	C₂³	1	2	yes	—
46	C₂×#41×C₈₉	—	2	2	no	—
47	#135×C₈₉	—	4	8	no	—
48	C₂×Q₁₆×C₈₉	—	2	2	no	—
49	C₂×SD₁₆×C₈₉	—	2	4	no	—
50	C₂×D₈×C₈₉	—	2	2	no	—
51	C₂²×C₈×C₈₉	D₄	1	2	yes	—
52	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
53	#127×C₈₉	C₂³	1	2	yes	—
54	#124×C₈₉	—	2	2	no	—
55	#121×C₈₉	—	2	4	no	—
56	C₄×D₄×C₈₉	—	2	4	no	—
57	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₈₉	D₄	1	2	yes	—
59	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₇₈	—	2	4	no	—
62	#13083562	—	4	16	no	—
63	C₂×#10623494	—	2	8	no	—
64	C₂×#10623494	C₂×D₄	1	4	yes	—
65	C₂×#10623494	C₂×D₄	1	4	yes	—
66	C₄⋊Dic₈₉	—	2	4	no	—
67	C₄⋊Dic₈₉	—	2	4	no	—
68	#13083553	—	4	4	no	—
69	C₂²⋊Dic₈₉	—	4	2	no	—
70	C₄×Dic₈₉	—	4	2	no	—
71	#41×C₈₉	—	4	8	no	—
72	C₂×Q₈×C₈₉	—	2	2	no	—
73	C₂×D₄×C₈₉	—	2	4	no	—
74	C₂×D₄×C₈₉	—	2	2	no	—
75	C₂×D₄×C₈₉	—	4	2	no	—
76	C₂²×C₄×C₈₉	—	4	2	no	—
77	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
79	Q₁₆×C₈₉	—	4	4	no	—
80	SD₁₆×C₈₉	—	4	8	no	—
81	D₈×C₈₉	—	4	4	no	—
82	C₂×C₈×C₈₉	SD₁₆	1	8	yes	complemented
83	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
85	C₄⋊C₄×C₈₉	—	2	4	no	—
86	C₂²⋊C₄×C₈₉	—	4	4	no	—
87	C₄²×C₈₉	C₂×D₄	1	2	yes	—
88	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
90	C₂³×C₈₉	—	4	2	no	—
91	C₂³×C₈₉	#127	1	1	yes	characteristic
92	Q₈×C₈₉	—	4	4	no	—
93	D₄×C₈₉	—	4	4	no	—
94	D₄×C₈₉	—	4	8	no	—
95	D₄×C₈₉	—	8	4	no	—
96	C₂×C₄×C₈₉	—	2	4	no	—
97	C₂×C₄×C₈₉	—	4	2	no	—
98	C₂×C₄×C₈₉	—	8	4	no	—
99	C₂×C₄×C₈₉	#127	1	2	yes	—
100	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
102	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
103	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
104	C₂×Dic₈₉	—	4	4	no	—
105	Dic₁₇₈	—	4	8	no	—
106	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
107	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
108	#10623494	—	2	4	no	—
109	#10623494	#135	1	8	yes	complemented
110	C₂²×C₈₉	—	2	2	no	—
111	C₂²×C₈₉	—	8	2	no	—
112	C₂²×C₈₉	—	8	4	no	—
113	C₂²×C₈₉	#492	1	2	yes	—
114	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
115	C₄×C₈₉	—	2	2	no	—
116	C₄×C₈₉	—	8	2	no	—
117	C₄×C₈₉	#492	1	2	yes	—
118	C₄×C₈₉	C₂²×SD₁₆	1	2	yes	—
119	C₄×C₈₉	C₂×#135	1	4	yes	—
120	Dic₈₉	—	8	4	no	—
121	#19596	—	89	1	no	maximal, Sylow
122	C₂×C₈₉	—	2	2	no	—
123	C₂×C₈₉	—	16	2	no	—
124	C₂×C₈₉	C₂×#492	1	1	yes	characteristic
125	C₂×C₈₉	#3147	1	2	yes	—
126	#3167	—	89	4	no	—
127	#3160	—	89	1	no	—
128	#3147	D₈₉	1	1	yes	characteristic
129	#1708	—	89	2	no	—
130	#1707	—	89	2	no	—
131	#1706	—	89	2	no	—
132	#1705	—	89	2	no	—
133	C₂×C₈²	—	89	1	no	—
134	C₈₉	#19596	1	1	yes	characteristic, complemented, Sylow
135	C₂×#135	—	2	2	no	—
136	#533	—	89	2	no	—
137	#532	C₂×D₈₉	1	2	yes	—
138	#500	—	89	2	no	—
139	#499	—	178	2	no	—
140	#498	—	89	4	no	—
141	#498	—	89	2	no	—
142	#495	—	2	2	no	—
143	#494	C₂×D₈₉	1	1	yes	characteristic
144	#493	C₂×D₈₉	1	1	yes	characteristic
145	#492	—	89	4	no	—
146	#492	C₂×D₈₉	1	2	yes	—
147	#487	—	89	4	no	—
148	#486	—	89	4	no	—
149	#465	—	178	8	no	—
150	C₄×SD₁₆	—	178	8	no	—
151	#427	—	178	2	no	—
152	C₂×C₄×C₈	—	89	2	no	—
153	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
154	C₈²	—	89	1	no	—
155	C₈²	—	89	2	no	—
156	C₈²	—	89	1	no	—
157	C₂×#41	—	2	2	no	—
158	#135	—	4	8	no	—
159	C₂×Q₁₆	—	2	2	no	—
160	C₂×SD₁₆	—	2	4	no	—
161	C₂×SD₁₆	—	178	8	no	—
162	C₂×D₈	—	2	2	no	—
163	C₂²×C₈	—	89	4	no	—
164	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
165	C₄⋊Q₈	—	89	4	no	—
166	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
167	#127	C₂²×D₈₉	1	2	yes	—
168	#125	—	178	2	no	—
169	#124	—	2	2	no	—
170	C₂²⋊Q₈	—	178	4	no	—
171	#121	—	2	4	no	—
172	C₄×Q₈	—	178	4	no	—
173	C₄×D₄	—	2	4	no	—
174	#117	—	178	2	no	—
175	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
176	#107	—	178	4	no	—
177	#106	—	178	8	no	—
178	#106	—	178	4	no	—
179	#103	—	178	8	no	—
180	#102	—	178	8	no	—
181	C₄×C₈	—	89	4	no	—
182	C₄×C₈	—	89	2	no	—
183	C₄×C₈	—	89	2	no	—
184	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
185	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
186	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
187	#41	—	4	8	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	178	4	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	D₄×D₈₉	1	2	yes	—
195	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
196	Q₁₆	—	4	4	no	—
197	SD₁₆	—	4	8	no	—
198	SD₁₆	—	356	16	no	—
199	D₈	—	4	4	no	—
200	C₂×C₈	—	89	4	no	—
201	C₂×C₈	—	89	4	no	—
202	C₂×C₈	—	178	8	no	—
203	C₂×C₈	#13083562	1	8	yes	complemented
204	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
205	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	356	4	no	—
208	C₄⋊C₄	—	178	4	no	—
209	C₄⋊C₄	—	178	4	no	—
210	C₂²⋊C₄	—	4	4	no	—
211	C₂²⋊C₄	—	356	2	no	—
212	C₄²	—	356	2	no	—
213	C₄²	D₄×D₈₉	1	2	yes	—
214	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
215	C₄²	C₂³×D₈₉	1	1	yes	characteristic
216	C₂³	—	4	2	no	—
217	C₂³	#15668008	1	1	yes	characteristic
218	Q₈	—	4	4	no	—
219	Q₈	—	356	8	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	8	no	—
222	D₄	—	8	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	8	4	no	—
226	C₂×C₄	—	356	4	no	—
227	C₂×C₄	#15668008	1	2	yes	—
228	C₂×C₄	#15668008	1	1	yes	characteristic
229	C₂×C₄	C₂×#13083562	1	4	yes	—
230	C₂×C₄	C₂×D₄×D₈₉	1	2	yes	—
231	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
232	C₈	—	89	8	no	—
233	C₈	—	178	4	no	—
234	C₈	C₂×#13083562	1	4	yes	—
235	C₈	C₂×#13083562	1	4	yes	—
236	C₂²	—	2	2	no	—
237	C₂²	—	8	2	no	—
238	C₂²	—	8	4	no	—
239	C₂²	#19151970	1	2	yes	—
240	C₂²	C₂×#15668008	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	712	4	no	—
244	C₄	#19151970	1	2	yes	—
245	C₄	C₂²×#13083562	1	2	yes	—
246	C₄	#135×D₈₉	1	4	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	16	2	no	—
249	C₂	C₂×#19151970	1	1	yes	characteristic
250	C₂	#26030676	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#15668008
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting, prime radical	C₈₉×#3147	C₂
socle	C₂²×C₈₉	C₂×#127
hypercenter	#3147	D₈₉
hypocenter	C₈₉	#19596
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. #26030807	C₂	semidirect
2. #26030766	C₂	semidirect
3. #26019851	C₂	semidirect
4. #26019846	C₂	semidirect
5. #26019841	C₂	semidirect
6. #26019838	C₂	semidirect
7. C₂×C₈×#10623494	C₂	semidirect
82. C₂×C₈×C₈₉	SD₁₆	semidirect
109. #10623494	#135	semidirect
134. C₈₉	#19596	semidirect
203. C₂×C₈	#13083562	semidirect

Maximal subgroups

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

subgroup	index
#26030807	2
#26030766	2
#26019851	2
#26019846	2
#26019841	2
#26019838	2
C₂×C₈×#10623494	2
C₈₉×#3147	2
#19596	89