Subgroups — #36568319

Order 22784.

810 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028970	C₂	1	2	yes	maximal, complemented
2	#26021085	C₂	1	2	yes	maximal, complemented
3	#26020581	C₂	1	8	yes	maximal, complemented
4	#26020390	C₂	1	1	yes	maximal, characteristic, complemented
5	#26020003	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#3147	C₂	1	1	yes	maximal, characteristic
7	C₂×#15668019	C₂²	1	4	yes	—
8	#19151670	—	2	8	no	—
9	#19151622	C₂²	1	2	yes	—
10	C₂×#15667939	C₂²	1	8	yes	—
11	#19151565	C₂²	1	4	yes	—
12	C₂×#15667923	C₂²	1	1	yes	characteristic
13	#19151444	C₂²	1	1	yes	characteristic
14	#19151363	C₂²	1	4	yes	—
15	#19151352	C₂²	1	4	yes	—
16	C₂×#135×C₈₉	C₂²	1	4	yes	—
17	#532×C₈₉	C₂²	1	2	yes	—
18	#495×C₈₉	C₄	1	4	yes	complemented
19	#494×C₈₉	C₄	1	1	yes	characteristic, complemented
20	#493×C₈₉	C₄	1	1	yes	characteristic, complemented
21	#492×C₈₉	C₄	1	2	yes	complemented
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	#15668019	—	4	8	no	—
24	C₂×#13083558	—	2	4	no	—
25	C₂×#13083558	C₂³	1	4	yes	—
26	C₂²×#10623494	—	2	4	no	—
27	C₂²×#10623494	C₂³	1	2	yes	—
28	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
29	C₄⋊#10623494	—	2	4	no	—
30	#15667939	—	4	8	no	—
31	C₂²⋊#10623494	—	2	8	no	—
32	#15667923	D₄	1	4	yes	—
33	#15667919	—	2	2	no	—
34	C₂×#41×C₈₉	C₂³	1	4	yes	—
35	#135×C₈₉	—	4	8	no	—
36	C₂×Q₁₆×C₈₉	C₂×C₄	1	4	yes	—
37	C₂×SD₁₆×C₈₉	C₂×C₄	1	8	yes	—
38	C₂×D₈×C₈₉	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
40	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
41	#127×C₈₉	C₂×C₄	1	2	yes	—
42	#124×C₈₉	C₂×C₄	1	4	yes	—
43	#121×C₈₉	—	2	4	no	—
44	C₄×D₄×C₈₉	—	2	4	no	—
45	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
49	C₂²×Dic₈₉	—	2	2	no	—
50	#13083558	—	2	8	no	—
51	#13083558	—	4	8	no	—
52	C₂×#10623494	—	2	4	no	—
53	C₂×#10623494	—	2	4	no	—
54	C₂×#10623494	—	4	8	no	—
55	C₂×#10623494	C₂×D₄	1	4	yes	—
56	C₄×Dic₈₉	—	2	4	no	—
57	#41×C₈₉	—	2	8	no	—
58	#41×C₈₉	—	2	8	no	—
59	C₂×Q₈×C₈₉	C₂²×C₄	1	4	yes	—
60	C₂×D₄×C₈₉	—	2	4	no	—
61	C₂×D₄×C₈₉	—	2	4	no	—
62	C₂×D₄×C₈₉	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₈₉	—	2	4	no	—
64	C₂²×C₄×C₈₉	#41	1	2	yes	—
65	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
66	Q₁₆×C₈₉	—	4	4	no	—
67	SD₁₆×C₈₉	—	4	8	no	—
68	D₈×C₈₉	—	4	4	no	—
69	C₂×C₈×C₈₉	—	2	4	no	—
70	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
72	C₄⋊C₄×C₈₉	—	2	4	no	—
73	C₂²⋊C₄×C₈₉	—	2	8	no	—
74	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₈₉	C₂×D₄	1	2	yes	—
77	C₂³×C₈₉	—	2	4	no	—
78	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
79	Q₈×C₈₉	—	2	8	no	—
80	D₄×C₈₉	—	2	8	no	—
81	D₄×C₈₉	—	4	8	no	—
82	D₄×C₈₉	—	8	4	no	—
83	C₂×C₄×C₈₉	—	2	4	no	—
84	C₂×C₄×C₈₉	—	4	2	no	—
85	C₂×C₄×C₈₉	—	4	4	no	—
86	C₂×C₄×C₈₉	—	4	4	no	—
87	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
92	C₂×Dic₈₉	—	4	4	no	—
93	C₂×Dic₈₉	—	4	2	no	—
94	C₈×C₈₉	—	2	2	no	—
95	C₈×C₈₉	—	2	2	no	—
96	#10623494	—	2	4	no	—
97	#10623494	—	4	4	no	—
98	C₂²×C₈₉	—	2	2	no	—
99	C₂²×C₈₉	—	4	4	no	—
100	C₂²×C₈₉	—	4	8	no	—
101	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
102	C₂²×C₈₉	#540	1	2	yes	—
103	C₄×C₈₉	—	2	2	no	—
104	C₄×C₈₉	—	2	2	no	—
105	C₄×C₈₉	—	4	4	no	—
106	C₄×C₈₉	#518	1	2	yes	—
107	C₄×C₈₉	#535	1	2	yes	—
108	Dic₈₉	—	8	2	no	—
109	#22282	—	89	1	no	maximal, Sylow
110	C₂×C₈₉	—	2	2	no	—
111	C₂×C₈₉	—	8	4	no	—
112	C₂×C₈₉	#2328	1	1	yes	characteristic
113	C₂×C₈₉	#2957	1	2	yes	—
114	#3147	D₈₉	1	1	yes	characteristic
115	#2932	—	89	2	no	—
116	#1963	—	89	2	no	—
117	#1876	—	89	8	no	—
118	#1837	—	89	1	no	—
119	#1746	—	89	1	no	—
120	C₈₉	#22282	1	1	yes	characteristic, complemented, Sylow
121	C₂×#135	C₂×D₈₉	1	4	yes	—
122	C₂×#131	—	89	4	no	—
123	#532	C₂×D₈₉	1	2	yes	—
124	#495	Dic₈₉	1	4	yes	complemented
125	#494	Dic₈₉	1	1	yes	characteristic, complemented
126	#493	Dic₈₉	1	1	yes	characteristic, complemented
127	#492	Dic₈₉	1	2	yes	complemented
128	#435	—	178	8	no	—
129	#423	—	89	2	no	—
130	C₂×#104	—	89	8	no	—
131	#408	—	89	4	no	—
132	C₂×#97	—	89	1	no	—
133	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
134	#376	—	89	1	no	—
135	#341	—	89	4	no	—
136	#336	—	89	4	no	—
137	C₂×#41	C₂²×D₈₉	1	4	yes	—
138	#135	—	4	8	no	—
139	C₂×Q₁₆	C₂×Dic₈₉	1	4	yes	—
140	C₂×SD₁₆	C₂×Dic₈₉	1	8	yes	—
141	C₂×D₈	C₂×Dic₈₉	1	4	yes	—
142	#131	—	356	8	no	—
143	C₂×M₁₆	—	89	4	no	—
144	C₂×M₁₆	—	178	4	no	—
145	C₂²×C₈	—	89	2	no	—
146	C₂²×C₈	—	178	4	no	—
147	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
148	C₄⋊Q₈	C₂×Dic₈₉	1	2	yes	—
149	#127	C₂×Dic₈₉	1	2	yes	—
150	#124	C₂×Dic₈₉	1	4	yes	—
151	#121	—	2	4	no	—
152	C₄×D₄	—	2	4	no	—
153	C₂×C₄²	—	89	2	no	—
154	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
155	C₄⋊C₈	—	178	4	no	—
156	#104	—	356	8	no	—
157	C₂²⋊C₈	—	178	8	no	—
158	#97	—	89	4	no	—
159	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
160	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
161	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
162	#95	—	178	2	no	—
163	#41	—	2	8	no	—
164	#41	—	2	8	no	—
165	C₂×Q₈	C₂²×Dic₈₉	1	4	yes	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	C₂²×Dic₈₉	1	4	yes	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	178	2	no	—
171	C₂²×C₄	#13083573	1	2	yes	—
172	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
173	Q₁₆	—	4	4	no	—
174	SD₁₆	—	4	8	no	—
175	D₈	—	4	4	no	—
176	M₁₆	—	356	8	no	—
177	M₁₆	—	178	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	89	4	no	—
180	C₂×C₈	—	356	8	no	—
181	C₂×C₈	—	178	4	no	—
182	C₂×C₈	—	178	4	no	—
183	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
184	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	8	no	—
187	C₄²	—	178	4	no	—
188	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
189	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
190	C₄²	D₄×D₈₉	1	2	yes	—
191	C₂³	—	2	4	no	—
192	C₂³	C₂×#13083573	1	1	yes	characteristic
193	Q₈	—	2	8	no	—
194	D₄	—	2	8	no	—
195	D₄	—	4	8	no	—
196	D₄	—	8	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	356	4	no	—
202	C₂×C₄	—	356	2	no	—
203	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
204	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
205	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
206	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
207	C₂×C₄	#15668057	1	4	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	356	4	no	—
211	C₈	—	178	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	4	no	—
214	C₂²	—	4	8	no	—
215	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
216	C₂²	#19152202	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	4	no	—
220	C₄	—	712	2	no	—
221	C₄	#19152073	1	2	yes	—
222	C₄	#19152161	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	8	4	no	—
225	C₂	#26022978	1	1	yes	characteristic
226	C₂	#26029169	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting, prime radical	C₈₉×#3147	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#3147	D₈₉
hypocenter	C₈₉	#22282
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×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. #26028970	C₂	semidirect
2. #26021085	C₂	semidirect
3. #26020581	C₂	semidirect
4. #26020390	C₂	semidirect
5. #26020003	C₂	semidirect
18. #495×C₈₉	C₄	semidirect
19. #494×C₈₉	C₄	semidirect
20. #493×C₈₉	C₄	semidirect
21. #492×C₈₉	C₄	semidirect
120. C₈₉	#22282	semidirect
124. #495	Dic₈₉	semidirect
125. #494	Dic₈₉	semidirect
126. #493	Dic₈₉	semidirect
127. #492	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26028970	2
#26021085	2
#26020581	2
#26020390	2
#26020003	2
C₈₉×#3147	2
#22282	89