Subgroups — #36470866

Order 22784.

342 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151425	C₂	1	2	yes	maximal
2	C₂×#19151424	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
4	#26020415	C₂	1	2	yes	maximal
5	C₂×#366×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083549	C₂²	1	1	yes	characteristic
7	C₂×#15667946	C₂²	1	2	yes	—
8	C₂×#15667943	—	2	2	no	—
9	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
10	#19151463	—	2	2	no	—
11	#19151425	—	2	4	no	—
12	#19151424	—	2	2	no	—
13	C₁₆×Dic₈₉	C₄	1	4	yes	complemented
14	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#107×C₈₉	C₂²	1	2	yes	—
16	#366×C₈₉	—	2	2	no	—
17	C₂²×#10623494	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
20	C₂×#13083549	C₂×C₄	1	2	yes	—
21	C₂×#13083549	D₄	1	2	yes	—
22	C₂×#13083549	Q₈	1	2	yes	—
23	#15667946	—	2	2	no	—
24	#15667946	—	2	2	no	—
25	#15667943	—	4	4	no	—
26	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
27	C₈×Dic₈₉	Q₈	1	2	yes	—
28	#15667919	—	4	4	no	—
29	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
31	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
33	C₂×C₁₆×C₈₉	Q₈	1	2	yes	—
34	#107×C₈₉	—	2	2	no	—
35	#107×C₈₉	—	2	2	no	—
36	C₂²×Dic₈₉	—	4	2	no	—
37	C₂²×Dic₈₉	D₈	1	2	yes	—
38	C₂×#10623494	C₄⋊C₄	1	2	yes	—
39	C₂×#10623494	D₈	1	2	yes	—
40	C₂×#10623494	Q₁₆	1	2	yes	—
41	C₄⋊Dic₈₉	—	4	2	no	—
42	C₄⋊Dic₈₉	—	4	2	no	—
43	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
44	C₄×Dic₈₉	Q₁₆	1	2	yes	—
45	#13083549	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₈₉	—	4	2	no	—
47	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
54	C₄⋊C₄×C₈₉	—	4	2	no	—
55	C₄⋊C₄×C₈₉	—	4	2	no	—
56	C₁₆×C₈₉	C₄⋊C₄	1	4	yes	—
57	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₈₉	—	8	2	no	—
59	C₂×C₄×C₈₉	—	8	4	no	—
60	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×Dic₈₉	—	2	2	no	—
67	C₂×Dic₈₉	—	2	2	no	—
68	C₂×Dic₈₉	—	8	2	no	—
69	C₂×Dic₈₉	—	8	4	no	—
70	C₂×Dic₈₉	#107	1	4	yes	—
71	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
74	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
75	#10623494	#107	1	4	yes	—
76	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
77	C₂²×C₈₉	#493	1	1	yes	characteristic
78	C₂²×C₈₉	#500	1	1	yes	characteristic
79	C₂²×C₈₉	#508	1	2	yes	—
80	C₂²×C₈₉	#508	1	2	yes	—
81	C₄×C₈₉	—	16	2	no	—
82	C₄×C₈₉	#384	1	1	yes	characteristic
83	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
84	C₄×C₈₉	#494	1	1	yes	characteristic
85	C₄×C₈₉	#500	1	1	yes	characteristic
86	Dic₈₉	—	2	4	no	—
87	Dic₈₉	—	16	2	no	—
88	#12907	—	89	1	no	maximal, Sylow
89	C₂×C₈₉	#1845	1	1	yes	characteristic
90	C₂×C₈₉	#2158	1	2	yes	—
91	C₂×C₈₉	#2249	1	1	yes	characteristic
92	C₂×C₈₉	#2249	1	1	yes	characteristic
93	C₂×C₈₉	#2253	1	2	yes	—
94	C₂×#367	—	89	2	no	—
95	C₂×#366	—	89	1	no	—
96	C₂×#366	D₈₉	1	1	yes	characteristic
97	C₂×C₄×C₁₆	—	89	1	no	—
98	#1846	—	89	2	no	—
99	C₈₉	#12907	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₁₆	—	89	1	no	—
101	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
102	C₂×#107	—	89	2	no	—
103	C₂×#107	C₂×D₈₉	1	2	yes	—
104	C₂×#106	—	178	2	no	—
105	C₂×C₄×C₈	—	89	1	no	—
106	#382	—	178	2	no	—
107	#367	—	178	4	no	—
108	#366	—	2	2	no	—
109	#366	—	178	2	no	—
110	C₄×C₁₆	—	89	4	no	—
111	C₂²×C₈	—	89	1	no	—
112	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	178	2	no	—
115	C₂×C₄²	—	89	1	no	—
116	C₂×C₁₆	—	89	2	no	—
117	C₂×C₁₆	—	89	2	no	—
118	C₂×C₁₆	—	89	2	no	—
119	C₂×C₁₆	Dic₁₇₈	1	2	yes	—
120	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
121	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
122	#107	—	2	2	no	—
123	#107	—	2	2	no	—
124	#107	—	178	2	no	—
125	#107	—	178	2	no	—
126	#106	—	356	4	no	—
127	C₄×C₈	—	89	2	no	—
128	C₄×C₈	—	89	2	no	—
129	#95	—	356	4	no	—
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	—	89	2	no	—
132	C₂²×C₄	—	356	2	no	—
133	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
134	C₂×C₈	—	89	2	no	—
135	C₂×C₈	—	89	2	no	—
136	C₂×C₈	—	89	2	no	—
137	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
138	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
139	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
140	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
141	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
142	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	356	2	no	—
146	C₄⋊C₄	—	356	2	no	—
147	C₄²	—	89	2	no	—
148	C₄²	—	89	2	no	—
149	C₁₆	—	89	4	no	—
150	C₁₆	#13083553	1	4	yes	—
151	C₂³	D₈×D₈₉	1	1	yes	characteristic
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	8	4	no	—
154	C₂×C₄	—	89	4	no	—
155	C₂×C₄	—	178	2	no	—
156	C₂×C₄	—	178	2	no	—
157	C₂×C₄	—	712	2	no	—
158	C₂×C₄	—	712	4	no	—
159	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
160	C₂×C₄	#15668008	1	1	yes	characteristic
161	C₂×C₄	#15668011	1	1	yes	characteristic
162	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
163	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
164	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
165	C₈	—	89	4	no	—
166	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
167	C₈	C₂×#13083553	1	1	yes	characteristic
168	C₈	#15668010	1	1	yes	characteristic
169	C₈	#15668011	1	1	yes	characteristic
170	C₂²	#107×D₈₉	1	1	yes	characteristic
171	C₂²	#19151973	1	1	yes	characteristic
172	C₂²	#19152002	1	1	yes	characteristic
173	C₂²	#19152027	1	2	yes	—
174	C₂²	#19152031	1	2	yes	—
175	C₄	—	16	2	no	—
176	C₄	—	1424	2	no	—
177	C₄	—	178	4	no	—
178	C₄	#19151467	1	1	yes	characteristic
179	C₄	#107×D₈₉	1	1	yes	characteristic
180	C₄	#19151976	1	1	yes	characteristic
181	C₄	#19152002	1	1	yes	characteristic
182	C₂	#26020410	1	1	yes	characteristic
183	C₂	#26022033	1	2	yes	—
184	C₂	#26022510	1	1	yes	characteristic
185	C₂	#26022514	1	1	yes	characteristic
186	C₂	#26022529	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₈₉
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#366×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#366	D₈₉
hypocenter	C₈₉	#12907
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
13. C₁₆×Dic₈₉	C₄	semidirect
99. C₈₉	#12907	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151425	2
C₂×#19151424	2
C₂×C₁₆×Dic₈₉	2
#26020415	2
C₂×#366×C₈₉	2
#12907	89