Subgroups — #36708094

Order 22784.

444 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022161	C₂	1	8	yes	maximal, complemented
2	#26022059	C₂	1	4	yes	maximal
3	#26022055	C₂	1	2	yes	maximal
4	C₈₉×#2960	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#13083549	C₂²	1	4	yes	—
6	C₄⋊#13083549	C₂²	1	8	yes	—
7	C₂²⋊#13083549	C₂²	1	4	yes	—
8	#19151374	C₂²	1	4	yes	—
9	#19151374	C₂²	1	4	yes	—
10	C₄×#13083549	C₂²	1	4	yes	—
11	#533×C₈₉	C₄	1	1	yes	characteristic
12	#446×C₈₉	C₄	1	4	yes	—
13	C₈×Q₈×C₈₉	C₂²	1	4	yes	—
14	#424×C₈₉	C₄	1	2	yes	—
15	#423×C₈₉	C₄	1	1	yes	characteristic
16	#405×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#13083549	C₂³	1	8	yes	—
19	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
22	C₄⋊Q₈×C₈₉	C₈	1	2	yes	—
23	#125×C₈₉	C₈	1	2	yes	—
24	C₂²⋊Q₈×C₈₉	C₈	1	4	yes	—
25	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
26	#117×C₈₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
29	C₄⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
30	#97×C₈₉	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₈₉	C₂³	1	4	yes	—
33	C₄×C₈×C₈₉	C₂³	1	2	yes	—
34	#13083549	—	2	8	no	—
35	C₂×Q₈×C₈₉	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
37	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
38	M₁₆×C₈₉	—	4	2	no	—
39	C₂×C₈×C₈₉	—	2	2	no	—
40	C₂×C₈×C₈₉	—	4	1	no	—
41	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
43	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
44	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
45	C₄⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
46	C₄⋊C₄×C₈₉	C₂×C₈	1	8	yes	—
47	C₄⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
48	C₂²⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
49	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
51	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
52	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
53	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
54	Q₈×C₈₉	—	4	4	no	—
55	C₂×C₄×C₈₉	—	2	1	no	—
56	C₂×C₄×C₈₉	—	2	1	no	—
57	C₂×C₄×C₈₉	—	4	2	no	—
58	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
60	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
61	C₂×C₄×C₈₉	C₂²×C₈	1	8	yes	—
62	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
63	C₈×C₈₉	—	2	1	no	—
64	C₈×C₈₉	—	2	2	no	—
65	C₈×C₈₉	—	2	4	no	—
66	C₈×C₈₉	2₋¹⁺⁴	1	2	yes	—
67	C₂²×C₈₉	—	2	2	no	—
68	C₂²×C₈₉	—	2	1	no	—
69	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
70	C₄×C₈₉	—	2	2	no	—
71	C₄×C₈₉	—	2	2	no	—
72	C₄×C₈₉	—	2	8	no	—
73	C₄×C₈₉	—	2	1	no	—
74	C₄×C₈₉	#519	1	1	yes	characteristic
75	C₄×C₈₉	#568	1	1	yes	characteristic
76	#33501	—	89	1	no	maximal, Sylow
77	C₂×C₈₉	—	4	1	no	—
78	C₂×C₈₉	#2982	1	1	yes	characteristic
79	C₂×C₈₉	#3407	1	2	yes	—
80	#2960	D₈₉	1	1	yes	characteristic
81	#2183	—	89	8	no	—
82	#2164	—	89	4	no	—
83	#2163	—	89	2	no	—
84	C₈₉	#33501	1	1	yes	characteristic, complemented, Sylow
85	#533	Dic₈₉	1	1	yes	characteristic
86	#446	Dic₈₉	1	4	yes	—
87	C₈×Q₈	C₂×D₈₉	1	4	yes	—
88	#424	Dic₈₉	1	2	yes	—
89	#423	Dic₈₉	1	1	yes	characteristic
90	#405	C₂×D₈₉	1	2	yes	—
91	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
92	C₄⋊C₁₆	—	89	4	no	—
93	C₄⋊C₁₆	—	89	8	no	—
94	C₂²⋊C₁₆	—	89	4	no	—
95	#346	—	89	4	no	—
96	#346	—	89	4	no	—
97	C₄×C₁₆	—	89	4	no	—
98	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
99	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
100	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
101	C₄⋊Q₈	#10623494	1	2	yes	—
102	#125	#10623494	1	2	yes	—
103	C₂²⋊Q₈	#10623494	1	4	yes	—
104	C₄×Q₈	C₂×Dic₈₉	1	4	yes	—
105	#117	C₂×Dic₈₉	1	2	yes	—
106	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
107	C₂×C₁₆	—	89	8	no	—
108	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
109	C₄⋊C₈	C₂×Dic₈₉	1	8	yes	—
110	#97	C₂×Dic₈₉	1	4	yes	—
111	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
112	C₄×C₈	C₂²×D₈₉	1	4	yes	—
113	C₄×C₈	C₂²×D₈₉	1	2	yes	—
114	C₂×Q₈	C₂×#10623494	1	4	yes	—
115	C₂²×C₄	C₂×#10623494	1	2	yes	—
116	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
117	M₁₆	—	4	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
121	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
122	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
123	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
124	C₄⋊C₄	C₂×#10623494	1	4	yes	—
125	C₄⋊C₄	C₂×#10623494	1	8	yes	—
126	C₄⋊C₄	C₂×#10623494	1	4	yes	—
127	C₂²⋊C₄	C₂×#10623494	1	4	yes	—
128	C₄²	C₂×#10623494	1	1	yes	characteristic
129	C₄²	C₂×#10623494	1	1	yes	characteristic
130	C₄²	C₂²×Dic₈₉	1	2	yes	—
131	C₄²	C₂²×Dic₈₉	1	4	yes	—
132	C₁₆	—	178	8	no	—
133	C₂³	C₂²×#10623494	1	1	yes	characteristic
134	Q₈	—	4	4	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
139	C₂×C₄	C₂²×#10623494	1	2	yes	—
140	C₂×C₄	C₂²×#10623494	1	2	yes	—
141	C₂×C₄	C₂²×#10623494	1	8	yes	—
142	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	2	2	no	—
145	C₈	—	2	4	no	—
146	C₈	#15668060	1	2	yes	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	2	no	—
149	C₂²	C₂³×#10623494	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	2	8	no	—
154	C₄	#19152076	1	1	yes	characteristic
155	C₄	#19152365	1	1	yes	characteristic
156	C₂	—	4	1	no	—
157	C₂	#26029333	1	1	yes	characteristic
158	C₂	#26033355	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2960	C₂
prime radical	C₂³×C₈₉	C₂²×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2960	D₈₉
hypocenter	C₈₉	#33501
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #26022161	C₂	semidirect
84. C₈₉	#33501	semidirect

Maximal subgroups

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

subgroup	index
#26022161	2
#26022059	2
#26022055	2
C₈₉×#2960	2
#33501	89