Subgroups — #36470863

Order 22784.

374 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151426	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151425	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
4	#26020410	C₂	1	2	yes	maximal
5	C₂×#367×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083549	C₂²	1	1	yes	characteristic
7	C₂×#15667946	C₂²	1	2	yes	—
8	C₂×#15667945	—	2	2	no	—
9	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
10	#19151467	—	2	2	no	—
11	#19151426	—	2	2	no	—
12	#19151425	—	2	4	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	#367×C₈₉	—	2	2	no	—
17	C₂²×#10623494	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	C₂×#13083553	—	4	4	no	—
20	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
21	C₂×#13083549	C₂×C₄	1	2	yes	—
22	C₂×#13083549	D₄	1	2	yes	—
23	C₂×#13083549	Q₈	1	2	yes	—
24	#15667946	—	2	2	no	—
25	#15667946	—	2	2	no	—
26	#15667945	—	4	4	no	—
27	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
28	C₈×Dic₈₉	Q₈	1	2	yes	—
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	#13083553	—	8	8	no	—
44	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
45	C₄×Dic₈₉	Q₁₆	1	2	yes	—
46	#13083549	C₄⋊C₄	1	4	yes	complemented
47	C₂²×C₄×C₈₉	—	4	2	no	—
48	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂×D₄	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	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
55	C₄⋊C₄×C₈₉	—	4	2	no	—
56	C₄⋊C₄×C₈₉	—	4	2	no	—
57	C₁₆×C₈₉	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₈₉	—	8	2	no	—
60	C₂×C₄×C₈₉	—	8	4	no	—
61	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×Dic₈₉	—	8	2	no	—
68	C₂×Dic₈₉	—	8	4	no	—
69	C₂×Dic₈₉	#107	1	4	yes	—
70	C₂×Dic₈₉	SD₃₂	1	8	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	C₂×SD₃₂	1	4	yes	—
80	C₄×C₈₉	—	16	2	no	—
81	C₄×C₈₉	#384	1	1	yes	characteristic
82	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
83	C₄×C₈₉	#494	1	1	yes	characteristic
84	C₄×C₈₉	#500	1	1	yes	characteristic
85	Dic₈₉	—	16	2	no	—
86	Dic₈₉	#367	1	8	yes	complemented
87	#12906	—	89	1	no	maximal, Sylow
88	C₂×C₈₉	#1845	1	1	yes	characteristic
89	C₂×C₈₉	C₂×#367	1	2	yes	—
90	C₂×C₈₉	#2248	1	2	yes	—
91	C₂×C₈₉	#2254	1	2	yes	—
92	C₂×#367	—	89	1	no	—
93	C₂×#367	—	89	2	no	—
94	C₂×#367	D₈₉	1	1	yes	characteristic
95	C₂×C₄×C₁₆	—	89	1	no	—
96	#1845	—	89	2	no	—
97	C₈₉	#12906	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₁₆	—	89	1	no	—
99	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
100	C₂×#107	—	89	2	no	—
101	C₂×#107	—	178	2	no	—
102	C₂×#107	C₂×D₈₉	1	2	yes	—
103	C₂×C₄×C₈	—	89	1	no	—
104	#384	—	178	2	no	—
105	#367	—	2	2	no	—
106	#367	—	178	4	no	—
107	#367	—	178	2	no	—
108	C₄×C₁₆	—	89	4	no	—
109	C₂²×C₈	—	89	1	no	—
110	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	356	4	no	—
113	C₂×C₄⋊C₄	—	178	2	no	—
114	C₂×C₄²	—	89	1	no	—
115	C₂×C₁₆	—	89	2	no	—
116	C₂×C₁₆	—	89	2	no	—
117	C₂×C₁₆	—	89	2	no	—
118	C₂×C₁₆	Dic₁₇₈	1	2	yes	—
119	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
120	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
121	#107	—	2	2	no	—
122	#107	—	2	2	no	—
123	#107	—	356	4	no	—
124	#107	—	178	2	no	—
125	#107	—	178	2	no	—
126	C₄×C₈	—	89	2	no	—
127	C₄×C₈	—	89	2	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	—	89	2	no	—
130	C₂²×C₄	—	356	2	no	—
131	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
132	C₂×C₈	—	89	2	no	—
133	C₂×C₈	—	89	2	no	—
134	C₂×C₈	—	89	2	no	—
135	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
136	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
137	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
139	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
140	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	356	2	no	—
144	C₄⋊C₄	—	356	2	no	—
145	C₄⋊C₄	—	712	8	no	—
146	C₄²	—	89	2	no	—
147	C₄²	—	89	2	no	—
148	C₁₆	—	89	4	no	—
149	C₁₆	#13083553	1	4	yes	complemented
150	C₂³	D₈×D₈₉	1	1	yes	characteristic
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	—	89	4	no	—
154	C₂×C₄	—	89	8	no	—
155	C₂×C₄	—	712	2	no	—
156	C₂×C₄	—	712	4	no	—
157	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
158	C₂×C₄	#15668008	1	1	yes	characteristic
159	C₂×C₄	#15668011	1	1	yes	characteristic
160	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
161	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
162	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
163	C₈	—	89	4	no	—
164	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
165	C₈	C₂×#13083553	1	1	yes	characteristic
166	C₈	#15668010	1	1	yes	characteristic
167	C₈	#15668011	1	1	yes	characteristic
168	C₂²	#107×D₈₉	1	1	yes	characteristic
169	C₂²	#19151973	1	1	yes	characteristic
170	C₂²	#19152002	1	1	yes	characteristic
171	C₂²	SD₃₂×D₈₉	1	4	yes	—
172	C₄	—	16	2	no	—
173	C₄	—	89	8	no	—
174	C₄	—	1424	2	no	—
175	C₄	#19151467	1	1	yes	characteristic
176	C₄	#107×D₈₉	1	1	yes	characteristic
177	C₄	#19151976	1	1	yes	characteristic
178	C₄	#19152002	1	1	yes	characteristic
179	C₂	#26020410	1	1	yes	characteristic
180	C₂	#367×D₈₉	1	2	yes	—
181	C₂	#26022506	1	2	yes	—
182	C₂	#26022533	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₈₉
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#367×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#367	D₈₉
hypocenter	C₈₉	#12906
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
13. C₁₆×Dic₈₉	C₄	semidirect
46. #13083549	C₄⋊C₄	semidirect
57. C₁₆×C₈₉	C₄⋊C₄	semidirect
86. Dic₈₉	#367	semidirect
97. C₈₉	#12906	semidirect
149. C₁₆	#13083553	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151426	2
C₂×#19151425	2
C₂×C₁₆×Dic₈₉	2
#26020410	2
C₂×#367×C₈₉	2
#12906	89