Subgroups — #36707848

Order 22784.

642 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029169	C₂	1	2	yes	maximal, complemented
2	#26022138	C₂	1	4	yes	maximal, complemented
3	#26022133	C₂	1	4	yes	maximal, complemented
4	#26022047	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151405	C₂	1	2	yes	maximal, complemented
6	#26021851	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#3414	C₂	1	1	yes	maximal, characteristic
8	#19152365	—	2	2	no	—
9	C₂²×#13083549	C₂²	1	1	yes	characteristic
10	#19151720	—	2	4	no	—
11	#19151715	C₂²	1	8	yes	complemented
12	C₂×#15667948	C₂²	1	2	yes	—
13	#19151615	—	2	2	no	—
14	#19151553	C₂²	1	1	yes	characteristic
15	#19151428	C₂²	1	4	yes	complemented
16	#19151405	C₂²	1	4	yes	complemented
17	#19151405	C₂²	1	4	yes	complemented
18	#19151373	C₂²	1	2	yes	complemented
19	C₁₆×Dic₈₉	C₂²	1	2	yes	complemented
20	C₂×#135×C₈₉	C₂²	1	2	yes	—
21	#510×C₈₉	C₄	1	4	yes	complemented
22	#509×C₈₉	C₄	1	4	yes	complemented
23	#508×C₈₉	C₂²	1	4	yes	—
24	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
25	#15668019	—	2	8	no	—
26	#15668019	—	4	4	no	—
27	C₂×#13083558	—	2	2	no	—
28	C₂×#13083558	—	4	4	no	—
29	C₂×#13083558	D₄	1	1	yes	characteristic
30	#15667971	D₄	1	1	yes	characteristic
31	C₂×#13083549	—	2	2	no	—
32	C₂×#13083549	C₂³	1	1	yes	characteristic
33	C₂×#13083549	C₂³	1	1	yes	characteristic
34	#15667948	—	2	2	no	—
35	#15667948	C₂³	1	4	yes	—
36	#15667939	—	2	8	no	—
37	#15667922	D₄	1	2	yes	—
38	C₈×Dic₈₉	C₂³	1	2	yes	—
39	C₂×#41×C₈₉	—	2	2	no	—
40	#135×C₈₉	—	2	2	no	—
41	#135×C₈₉	—	2	2	no	—
42	#135×C₈₉	C₂×C₄	1	4	yes	—
43	#135×C₈₉	C₂³	1	4	yes	—
44	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
45	C₂×SD₁₆×C₈₉	—	2	2	no	—
46	C₂×D₈×C₈₉	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	Q₃₂×C₈₉	C₂×C₄	1	4	yes	—
49	SD₃₂×C₈₉	C₂×C₄	1	8	yes	—
50	D₁₆×C₈₉	C₂×C₄	1	4	yes	—
51	M₃₂×C₈₉	C₂×C₄	1	4	yes	—
52	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
53	#13083558	—	2	4	no	—
54	#13083558	—	4	2	no	—
55	#13083558	—	4	8	no	—
56	#13083558	D₈	1	4	yes	—
57	C₂×#10623494	—	4	2	no	—
58	C₂×#10623494	—	4	8	no	—
59	C₂×#10623494	C₂×D₄	1	2	yes	—
60	C₄⋊Dic₈₉	—	2	1	no	—
61	C₂²⋊Dic₈₉	—	2	1	no	—
62	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
63	#13083549	—	2	1	no	—
64	#13083549	—	2	1	no	—
65	#41×C₈₉	—	2	4	no	—
66	#41×C₈₉	—	2	4	no	—
67	#41×C₈₉	—	4	2	no	—
68	#41×C₈₉	—	4	2	no	—
69	C₂×Q₈×C₈₉	—	2	2	no	—
70	C₂×D₄×C₈₉	—	2	2	no	—
71	C₂×D₄×C₈₉	—	4	2	no	—
72	C₂²×C₄×C₈₉	—	4	2	no	—
73	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
74	Q₁₆×C₈₉	—	2	2	no	—
75	Q₁₆×C₈₉	C₂²×C₄	1	4	yes	—
76	SD₁₆×C₈₉	—	2	4	no	—
77	SD₁₆×C₈₉	—	2	4	no	—
78	D₈×C₈₉	—	2	2	no	—
79	D₈×C₈₉	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₈₉	#41	1	2	yes	—
83	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
84	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
85	C₂³×C₈₉	—	4	2	no	—
86	C₂³×C₈₉	#135	1	1	yes	characteristic
87	Q₈×C₈₉	—	2	4	no	—
88	Q₈×C₈₉	—	4	2	no	—
89	D₄×C₈₉	—	2	4	no	—
90	D₄×C₈₉	—	4	4	no	—
91	D₄×C₈₉	—	4	4	no	—
92	D₄×C₈₉	—	4	2	no	—
93	C₂×C₄×C₈₉	—	4	2	no	—
94	C₂×C₄×C₈₉	—	4	4	no	—
95	C₂×C₄×C₈₉	—	4	4	no	—
96	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
100	C₂×Dic₈₉	—	2	2	no	—
101	C₈×C₈₉	C₄×D₄	1	2	yes	—
102	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
103	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
104	#10623494	—	4	4	no	—
105	#10623494	C₂×D₈	1	4	yes	—
106	C₂²×C₈₉	—	4	1	no	—
107	C₂²×C₈₉	—	4	4	no	—
108	C₂²×C₈₉	—	16	2	no	—
109	C₂²×C₈₉	C₄×D₈	1	2	yes	—
110	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
111	C₄×C₈₉	—	4	4	no	—
112	C₄×C₈₉	C₄×D₈	1	2	yes	—
113	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
114	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
115	Dic₈₉	—	4	2	no	—
116	#33464	—	89	1	no	maximal, Sylow
117	C₂×C₈₉	—	2	1	no	—
118	C₂×C₈₉	—	2	2	no	—
119	C₂×C₈₉	—	8	4	no	—
120	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
121	#3414	D₈₉	1	1	yes	characteristic
122	#2957	—	89	2	no	—
123	#2179	—	89	4	no	—
124	#2178	—	89	4	no	—
125	#2161	—	89	1	no	—
126	C₂×#359	—	89	2	no	—
127	#2108	—	89	1	no	—
128	C₈₉	#33464	1	1	yes	characteristic, complemented, Sylow
129	C₂×#135	C₂×D₈₉	1	2	yes	—
130	#568	—	178	2	no	—
131	#510	Dic₈₉	1	4	yes	complemented
132	#509	Dic₈₉	1	4	yes	complemented
133	#508	C₂×D₈₉	1	4	yes	—
134	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
135	C₂²×C₁₆	—	89	1	no	—
136	#444	—	178	4	no	—
137	#443	—	89	8	no	—
138	C₂×#108	—	89	2	no	—
139	#421	—	178	2	no	—
140	#405	—	89	1	no	—
141	#368	—	89	4	no	—
142	#359	—	89	4	no	—
143	#359	—	89	4	no	—
144	#346	—	89	2	no	—
145	C₄×C₁₆	—	89	2	no	—
146	C₂×#41	—	2	2	no	—
147	#135	—	2	2	no	—
148	#135	—	2	2	no	—
149	#135	C₂×Dic₈₉	1	4	yes	—
150	#135	C₂²×D₈₉	1	4	yes	—
151	C₂×Q₁₆	C₂×Dic₈₉	1	2	yes	—
152	C₂×SD₁₆	—	2	2	no	—
153	C₂×D₈	C₂×Dic₈₉	1	2	yes	—
154	#131	—	356	4	no	—
155	#131	—	178	8	no	—
156	C₂×M₁₆	—	89	1	no	—
157	C₂×M₁₆	—	356	4	no	—
158	C₂×M₁₆	—	178	2	no	—
159	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
160	#117	—	89	1	no	—
161	Q₃₂	C₂×Dic₈₉	1	4	yes	—
162	SD₃₂	C₂×Dic₈₉	1	8	yes	—
163	D₁₆	C₂×Dic₈₉	1	4	yes	—
164	M₃₂	C₂×Dic₈₉	1	4	yes	—
165	C₂×C₁₆	—	89	1	no	—
166	C₂×C₁₆	—	89	1	no	—
167	C₂×C₁₆	—	178	2	no	—
168	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
169	#108	—	89	4	no	—
170	#108	—	178	2	no	—
171	#104	—	178	8	no	—
172	#97	—	89	2	no	—
173	C₄×C₈	—	89	2	no	—
174	#41	—	2	4	no	—
175	#41	—	2	4	no	—
176	#41	—	4	2	no	—
177	#41	—	4	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
183	Q₁₆	—	2	2	no	—
184	Q₁₆	C₂²×Dic₈₉	1	4	yes	—
185	SD₁₆	—	2	4	no	—
186	SD₁₆	—	2	4	no	—
187	D₈	—	2	2	no	—
188	D₈	C₂²×Dic₈₉	1	4	yes	—
189	M₁₆	—	89	4	no	—
190	M₁₆	—	356	8	no	—
191	M₁₆	—	356	2	no	—
192	M₁₆	—	178	4	no	—
193	C₂×C₈	—	89	2	no	—
194	C₂×C₈	—	356	8	no	—
195	C₂×C₈	—	356	2	no	—
196	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
197	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
198	C₂×C₈	#13083573	1	2	yes	—
199	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
200	C₄⋊C₄	—	178	1	no	—
201	C₂²⋊C₄	—	178	1	no	—
202	C₄²	—	89	2	no	—
203	C₁₆	—	178	1	no	—
204	C₁₆	—	178	1	no	—
205	C₁₆	C₂²×Dic₈₉	1	4	yes	—
206	C₂³	—	4	2	no	—
207	C₂³	#15668031	1	1	yes	characteristic
208	Q₈	—	2	4	no	—
209	Q₈	—	4	2	no	—
210	D₄	—	2	4	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	4	no	—
213	D₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	178	2	no	—
218	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
219	C₂×C₄	D₈×D₈₉	1	2	yes	—
220	C₂×C₄	#15668031	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
222	C₈	—	89	4	no	—
223	C₈	—	356	4	no	—
224	C₈	D₄×Dic₈₉	1	2	yes	—
225	C₈	C₂³×Dic₈₉	1	1	yes	characteristic
226	C₈	C₂×#13083573	1	1	yes	characteristic
227	C₂²	—	4	1	no	—
228	C₂²	—	4	4	no	—
229	C₂²	—	16	2	no	—
230	C₂²	D₈×Dic₈₉	1	2	yes	—
231	C₂²	C₂×#15668031	1	1	yes	characteristic
232	C₄	—	4	4	no	—
233	C₄	—	356	2	no	—
234	C₄	D₈×Dic₈₉	1	2	yes	—
235	C₄	C₂×D₄×Dic₈₉	1	1	yes	characteristic
236	C₄	C₂×D₈×D₈₉	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	—	8	4	no	—
240	C₂	C₂×D₈×Dic₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting, prime radical	C₈₉×#3414	C₂
socle	C₂×C₈₉	C₂×C₄×D₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#3414	D₈₉
hypocenter	C₈₉	#33464
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #3414

Complements

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

N	H ≅ G/N	type
1. #26029169	C₂	semidirect
2. #26022138	C₂	semidirect
3. #26022133	C₂	semidirect
4. #26022047	C₂	semidirect
5. C₂×#19151405	C₂	semidirect
6. #26021851	C₂	semidirect
11. #19151715	C₂²	semidirect
15. #19151428	C₂²	semidirect
16. #19151405	C₂²	semidirect
17. #19151405	C₂²	semidirect
18. #19151373	C₂²	semidirect
19. C₁₆×Dic₈₉	C₂²	semidirect
21. #510×C₈₉	C₄	semidirect
22. #509×C₈₉	C₄	semidirect
128. C₈₉	#33464	semidirect
131. #510	Dic₈₉	semidirect
132. #509	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26029169	2
#26022138	2
#26022133	2
#26022047	2
C₂×#19151405	2
#26021851	2
C₈₉×#3414	2
#33464	89