Subgroups — #36707806

Order 22784.

646 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029120	C₂	1	2	yes	maximal
2	#26022123	C₂	1	4	yes	maximal, complemented
3	Q₃₂×Dic₈₉	C₂	1	4	yes	maximal, complemented
4	#26022036	C₂	1	1	yes	maximal, characteristic
5	#26021972	C₂	1	2	yes	maximal
6	#26021848	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#19152076	—	2	2	no	—
9	C₂×#15667952	C₂²	1	1	yes	characteristic
10	#19151705	—	2	4	no	—
11	Q₁₆×Dic₈₉	C₂²	1	8	yes	—
12	#19151636	C₂²	1	2	yes	—
13	#19151607	—	2	2	no	—
14	#19151553	C₂²	1	1	yes	characteristic
15	#19151426	C₂²	1	2	yes	—
16	#19151424	C₂²	1	2	yes	—
17	#19151402	C₂²	1	8	yes	—
18	#19151373	C₂²	1	2	yes	—
19	C₁₆×Dic₈₉	C₂²	1	2	yes	—
20	C₂²×Q₁₆×C₈₉	C₂²	1	2	yes	—
21	C₂×Q₃₂×C₈₉	C₄	1	8	yes	complemented
22	C₂×Q₃₂×C₈₉	C₂²	1	2	yes	—
23	C₂×Q₃₂×C₈₉	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#13083558	D₄	1	1	yes	characteristic
26	Q₈×Dic₈₉	—	2	8	no	—
27	Q₈×Dic₈₉	—	4	4	no	—
28	#15667971	—	2	2	no	—
29	#15667971	—	4	4	no	—
30	#15667971	D₄	1	1	yes	characteristic
31	#15667952	—	2	2	no	—
32	C₂×#13083549	C₂³	1	2	yes	—
33	#15667946	C₂³	1	4	yes	—
34	#15667944	—	2	2	no	—
35	#15667936	—	2	8	no	—
36	#15667922	D₄	1	2	yes	—
37	C₈×Dic₈₉	C₂³	1	2	yes	—
38	C₂²×Q₈×C₈₉	—	2	2	no	—
39	C₂×Q₁₆×C₈₉	—	2	2	no	—
40	C₂×Q₁₆×C₈₉	—	2	4	no	—
41	C₂×Q₁₆×C₈₉	C₂×C₄	1	8	yes	—
42	C₂×Q₁₆×C₈₉	C₂³	1	4	yes	—
43	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	Q₃₂×C₈₉	C₂×C₄	1	8	yes	—
45	Q₃₂×C₈₉	C₂×C₄	1	8	yes	—
46	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
47	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
49	#13083558	—	2	2	no	—
50	C₂×#10623494	C₂×D₄	1	2	yes	—
51	C₄⋊Dic₈₉	—	2	4	no	—
52	C₄⋊Dic₈₉	—	4	8	no	—
53	C₄⋊Dic₈₉	D₈	1	2	yes	—
54	C₂²⋊Dic₈₉	—	4	2	no	—
55	C₂²⋊Dic₈₉	D₈	1	2	yes	—
56	C₄×Dic₈₉	—	4	2	no	—
57	C₄×Dic₈₉	—	4	8	no	—
58	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
59	#13083549	—	2	2	no	—
60	C₂×Q₈×C₈₉	—	2	8	no	—
61	C₂×Q₈×C₈₉	—	2	4	no	—
62	C₂×Q₈×C₈₉	—	4	2	no	—
63	C₂×Q₈×C₈₉	—	4	4	no	—
64	C₂²×C₄×C₈₉	—	4	2	no	—
65	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
66	Q₁₆×C₈₉	—	2	8	no	—
67	Q₁₆×C₈₉	—	2	4	no	—
68	Q₁₆×C₈₉	C₂²×C₄	1	8	yes	—
69	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₈₉	#41	1	2	yes	—
72	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
73	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
74	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
75	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
76	Q₈×C₈₉	—	2	8	no	—
77	Q₈×C₈₉	—	4	4	no	—
78	Q₈×C₈₉	—	4	8	no	—
79	C₂×C₄×C₈₉	—	4	4	no	—
80	C₂×C₄×C₈₉	—	4	8	no	—
81	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₈₉	#135	1	2	yes	—
84	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
85	C₂×Dic₈₉	—	4	4	no	—
86	C₂×Dic₈₉	C₂×D₈	1	4	yes	—
87	C₈×C₈₉	C₄×D₄	1	2	yes	—
88	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
90	#10623494	—	2	2	no	—
91	C₂²×C₈₉	—	2	2	no	—
92	C₂²×C₈₉	C₄×D₈	1	2	yes	—
93	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₈₉	—	4	8	no	—
95	C₄×C₈₉	C₄×D₈	1	2	yes	—
96	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
97	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
98	Dic₈₉	—	2	4	no	—
99	Dic₈₉	—	8	4	no	—
100	#33458	—	89	1	no	maximal, Sylow
101	C₂×C₈₉	—	2	2	no	—
102	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
103	C₂×C₈₉	#3417	1	2	yes	—
104	C₂²×Q₃₂	D₈₉	1	1	yes	characteristic
105	#2950	—	89	2	no	—
106	#2176	—	89	4	no	—
107	C₄×Q₃₂	—	89	4	no	—
108	#2158	—	89	1	no	—
109	#2141	—	89	2	no	—
110	#2108	—	89	1	no	—
111	C₈₉	#33458	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₁₆	C₂×D₈₉	1	2	yes	—
113	#519	—	178	2	no	—
114	C₂×Q₃₂	Dic₈₉	1	8	yes	complemented
115	C₂×Q₃₂	C₂×D₈₉	1	2	yes	—
116	C₂×Q₃₂	C₂×D₈₉	1	2	yes	—
117	C₂×M₃₂	—	89	1	no	—
118	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
119	#441	—	178	4	no	—
120	C₄×Q₁₆	—	89	8	no	—
121	#427	—	89	2	no	—
122	#419	—	178	2	no	—
123	#405	—	89	1	no	—
124	#367	—	89	2	no	—
125	#366	—	89	2	no	—
126	#358	—	89	8	no	—
127	#346	—	89	2	no	—
128	C₄×C₁₆	—	89	2	no	—
129	C₂²×Q₈	—	2	2	no	—
130	C₂×Q₁₆	—	2	2	no	—
131	C₂×Q₁₆	—	2	4	no	—
132	C₂×Q₁₆	C₂×Dic₈₉	1	8	yes	—
133	C₂×Q₁₆	C₂²×D₈₉	1	4	yes	—
134	C₂×M₁₆	—	89	1	no	—
135	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
136	C₄×Q₈	—	356	4	no	—
137	C₄×Q₈	—	178	8	no	—
138	#117	—	89	1	no	—
139	#117	—	356	4	no	—
140	#117	—	178	2	no	—
141	Q₃₂	C₂×Dic₈₉	1	8	yes	—
142	Q₃₂	C₂×Dic₈₉	1	8	yes	—
143	M₃₂	—	178	2	no	—
144	C₂×C₁₆	—	89	2	no	—
145	C₂×C₁₆	C₂×Dic₈₉	1	4	yes	—
146	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
148	#107	—	89	4	no	—
149	#106	—	178	2	no	—
150	#103	—	178	8	no	—
151	#97	—	89	2	no	—
152	C₄×C₈	—	89	2	no	—
153	C₂×Q₈	—	2	8	no	—
154	C₂×Q₈	—	2	4	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
159	Q₁₆	—	2	8	no	—
160	Q₁₆	—	2	4	no	—
161	Q₁₆	C₂²×Dic₈₉	1	8	yes	—
162	M₁₆	—	178	2	no	—
163	C₂×C₈	—	89	2	no	—
164	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
165	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
166	C₂×C₈	#13083573	1	2	yes	—
167	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
168	C₄⋊C₄	—	89	2	no	—
169	C₄⋊C₄	—	356	8	no	—
170	C₄⋊C₄	—	178	4	no	—
171	C₂²⋊C₄	—	89	2	no	—
172	C₂²⋊C₄	—	356	2	no	—
173	C₄²	—	89	2	no	—
174	C₄²	—	356	8	no	—
175	C₄²	—	356	2	no	—
176	C₁₆	—	178	2	no	—
177	C₁₆	C₂²×Dic₈₉	1	2	yes	—
178	C₁₆	C₂²×Dic₈₉	1	2	yes	—
179	C₂³	D₈×D₈₉	1	1	yes	characteristic
180	Q₈	—	2	8	no	—
181	Q₈	—	4	8	no	—
182	Q₈	—	4	4	no	—
183	C₂×C₄	—	4	8	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	—	89	4	no	—
186	C₂×C₄	—	356	4	no	—
187	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
188	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
189	C₂×C₄	#15668031	1	2	yes	—
190	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
191	C₈	—	178	2	no	—
192	C₈	D₄×Dic₈₉	1	2	yes	—
193	C₈	C₂³×Dic₈₉	1	1	yes	characteristic
194	C₈	C₂×#13083573	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	D₈×Dic₈₉	1	2	yes	—
197	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
198	C₄	—	4	8	no	—
199	C₄	—	178	4	no	—
200	C₄	—	712	4	no	—
201	C₄	D₈×Dic₈₉	1	2	yes	—
202	C₄	C₂×D₄×Dic₈₉	1	1	yes	characteristic
203	C₄	C₂×#15668031	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×D₈×Dic₈₉	1	1	yes	characteristic
206	C₂	#26033401	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂²×Q₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂²×Q₃₂	D₈₉
hypocenter	C₈₉	#33458
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #26022123	C₂	semidirect
3. Q₃₂×Dic₈₉	C₂	semidirect
21. C₂×Q₃₂×C₈₉	C₄	semidirect
111. C₈₉	#33458	semidirect
114. C₂×Q₃₂	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26029120	2
#26022123	2
Q₃₂×Dic₈₉	2
#26022036	2
#26021972	2
#26021848	2
C₂²×Q₃₂×C₈₉	2
#33458	89