Subgroups — #36520178

Order 22784.

712 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028885	C₂	1	2	yes	maximal
2	#26018555	C₂	1	4	yes	maximal, complemented
3	#26018552	C₂	1	8	yes	maximal, complemented
4	C₈₉×#2920	C₂	1	1	yes	maximal, characteristic
5	#19151655	C₄	1	8	yes	—
6	#19151651	C₂²	1	8	yes	—
7	#19151622	C₄	1	4	yes	—
8	#19151562	C₄	1	4	yes	—
9	C₄×#13083558	C₂²	1	4	yes	—
10	#19151320	C₂²	1	8	yes	complemented
11	#19151319	C₂²	1	4	yes	complemented
12	C₈×#10623494	C₂²	1	4	yes	complemented
13	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
14	#432×C₈₉	C₄	1	4	yes	—
15	#431×C₈₉	C₂²	1	4	yes	—
16	#423×C₈₉	C₄	1	2	yes	—
17	#407×C₈₉	C₄	1	2	yes	—
18	C₄×M₁₆×C₈₉	C₂²	1	2	yes	—
19	C₂×#13083558	C₂×C₄	1	8	yes	—
20	C₄⋊#10623494	C₂×C₄	1	16	yes	—
21	C₂²⋊#10623494	C₂×C₄	1	16	yes	—
22	#15667923	C₂×C₄	1	8	yes	—
23	C₄×#10623494	C₂³	1	8	yes	—
24	C₂³×C₄×C₈₉	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
33	C₄⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
34	C₂²⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
35	#97×C₈₉	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₈₉	C₂³	1	4	yes	—
37	#13083558	—	4	8	no	—
38	C₂×#10623494	C₂²×C₄	1	16	yes	—
39	C₂⁴×C₈₉	C₄²	1	1	yes	characteristic
40	C₂²×C₄×C₈₉	—	4	2	no	—
41	C₂²×C₄×C₈₉	—	4	4	no	—
42	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
44	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
45	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
46	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	M₁₆×C₈₉	—	4	4	no	—
51	C₂×C₈×C₈₉	C₂²×C₄	1	8	yes	—
52	C₄²×C₈₉	C₄²	1	4	yes	—
53	C₄²×C₈₉	C₄²	1	2	yes	—
54	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
55	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
56	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
58	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
59	C₂³×C₈₉	—	4	2	no	—
60	C₂³×C₈₉	—	4	1	no	—
61	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
62	C₂³×C₈₉	C₂×C₄²	1	2	yes	—
63	C₂×C₄×C₈₉	—	2	2	no	—
64	C₂×C₄×C₈₉	—	2	8	no	—
65	C₂×C₄×C₈₉	—	2	4	no	—
66	C₂×C₄×C₈₉	—	2	4	no	—
67	C₂×C₄×C₈₉	—	4	2	no	—
68	C₂×C₄×C₈₉	—	4	4	no	—
69	C₂×C₄×C₈₉	—	4	2	no	—
70	C₂×C₄×C₈₉	—	4	1	no	—
71	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
73	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
74	C₂×C₄×C₈₉	C₂×C₄²	1	4	yes	—
75	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
76	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
77	C₈×C₈₉	—	2	8	no	—
78	#10623494	—	2	16	no	—
79	C₂²×C₈₉	—	2	4	no	—
80	C₂²×C₈₉	—	2	2	no	—
81	C₂²×C₈₉	—	2	1	no	—
82	C₂²×C₈₉	—	2	2	no	—
83	C₂²×C₈₉	—	4	4	no	—
84	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
85	C₄×C₈₉	—	2	4	no	—
86	C₄×C₈₉	—	2	2	no	—
87	C₄×C₈₉	—	2	1	no	—
88	C₄×C₈₉	—	2	2	no	—
89	C₄×C₈₉	#568	1	4	yes	—
90	C₄×C₈₉	#568	1	2	yes	—
91	#17339	—	89	1	no	maximal, Sylow
92	C₂×C₈₉	—	4	2	no	—
93	C₂×C₈₉	—	4	1	no	—
94	C₂×C₈₉	#2873	1	2	yes	—
95	C₂×C₈₉	#2873	1	1	yes	characteristic
96	#2920	—	89	2	no	—
97	#2920	D₈₉	1	1	yes	characteristic
98	#1455	—	89	4	no	—
99	#1454	—	89	8	no	—
100	C₈₉	#17339	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
102	#432	—	89	8	no	—
103	#432	Dic₈₉	1	4	yes	—
104	#431	—	89	8	no	—
105	#431	C₂×D₈₉	1	4	yes	—
106	#423	—	89	4	no	—
107	#423	Dic₈₉	1	2	yes	—
108	#407	—	89	4	no	—
109	#407	Dic₈₉	1	2	yes	—
110	C₄×M₁₆	—	89	4	no	—
111	C₄×M₁₆	C₂×D₈₉	1	2	yes	—
112	#322	—	89	4	no	—
113	#322	—	89	8	no	—
114	C₈²	—	89	4	no	—
115	C₂³×C₄	C₄×D₈₉	1	2	yes	—
116	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
117	C₂×M₁₆	—	89	8	no	—
118	C₂×M₁₆	C₂×Dic₈₉	1	4	yes	—
119	C₂×C₄²	C₄×D₈₉	1	2	yes	—
120	C₂×C₄²	C₄×D₈₉	1	4	yes	—
121	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
122	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
123	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
124	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
125	C₄⋊C₈	—	89	16	no	—
126	C₄⋊C₈	C₂×Dic₈₉	1	8	yes	—
127	C₂²⋊C₈	—	89	16	no	—
128	C₂²⋊C₈	C₂×Dic₈₉	1	8	yes	—
129	#97	—	89	8	no	—
130	#97	C₂×Dic₈₉	1	4	yes	—
131	C₄×C₈	—	89	8	no	—
132	C₄×C₈	C₂²×D₈₉	1	4	yes	—
133	C₂⁴	C₄×Dic₈₉	1	1	yes	characteristic
134	C₂²×C₄	—	4	4	no	—
135	C₂²×C₄	—	4	2	no	—
136	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
137	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
138	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
139	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
140	C₂²×C₄	C₂×C₄×D₈₉	1	4	yes	—
141	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
142	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
143	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
144	M₁₆	—	4	4	no	—
145	M₁₆	—	356	8	no	—
146	C₂×C₈	—	89	16	no	—
147	C₂×C₈	C₂²×Dic₈₉	1	8	yes	—
148	C₄²	C₄×Dic₈₉	1	4	yes	—
149	C₄²	C₄×Dic₈₉	1	2	yes	—
150	C₄²	C₂×C₄×D₈₉	1	2	yes	—
151	C₄²	C₂×C₄×D₈₉	1	4	yes	—
152	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
153	C₄²	C₂²×Dic₈₉	1	2	yes	—
154	C₄²	C₂³×D₈₉	1	1	yes	characteristic
155	C₂³	—	4	2	no	—
156	C₂³	—	4	1	no	—
157	C₂³	C₂×C₄×Dic₈₉	1	2	yes	—
158	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
169	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
170	C₂×C₄	C₂×C₄×Dic₈₉	1	4	yes	—
171	C₂×C₄	C₂²×C₄×D₈₉	1	2	yes	—
172	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
173	C₈	—	2	8	no	—
174	C₈	—	178	16	no	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	4	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	4	4	no	—
180	C₂²	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	—	2	4	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	2	no	—
185	C₄	#19152364	1	4	yes	—
186	C₄	#19152365	1	2	yes	—
187	C₂	—	4	1	no	—
188	C₂	—	4	2	no	—
189	C₂	#26028670	1	2	yes	—
190	C₂	#26028671	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂²×C₄²
Fitting	C₈₉×#2920	C₂
prime radical	C₂⁴×C₈₉	C₄²
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#2920	D₈₉
hypocenter	C₈₉	#17339
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
2. #26018555	C₂	semidirect
3. #26018552	C₂	semidirect
10. #19151320	C₂²	semidirect
11. #19151319	C₂²	semidirect
12. C₈×#10623494	C₂²	semidirect
100. C₈₉	#17339	semidirect

Maximal subgroups

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

subgroup	index
#26028885	2
#26018555	2
#26018552	2
C₈₉×#2920	2
#17339	89