Subgroups — #36709920

Order 22784.

490 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030766	C₂	1	2	yes	maximal
2	#26022534	C₂	1	2	yes	maximal, complemented
3	#26022531	C₂	1	4	yes	maximal, complemented
4	#26022527	C₂	1	2	yes	maximal, complemented
5	#26022036	C₂	1	4	yes	maximal
6	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
7	#19152155	—	2	2	no	—
8	#19152003	C₂²	1	4	yes	—
9	#19152000	—	2	2	no	—
10	#19152000	C₂²	1	4	yes	—
11	#19151996	—	2	2	no	—
12	#19151636	—	2	2	no	—
13	#19151636	C₂²	1	4	yes	—
14	#19151426	C₂²	1	8	yes	—
15	#19151424	C₂²	1	8	yes	—
16	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
19	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
20	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	#15668012	—	2	4	no	—
22	#15668004	—	2	4	no	—
23	#15667991	—	4	4	no	—
24	C₄×Dic₁₇₈	—	4	4	no	—
25	#15667971	—	2	4	no	—
26	#15667946	—	2	4	no	—
27	#15667946	C₂³	1	8	yes	—
28	#15667944	—	2	4	no	—
29	#15667944	—	2	4	no	—
30	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₈₉	Q₈	1	8	yes	complemented
34	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
35	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
36	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
37	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
38	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
40	C₂×Dic₁₇₈	—	4	4	no	—
41	C₄⋊Dic₈₉	—	2	8	no	—
42	C₄⋊Dic₈₉	—	4	4	no	—
43	#13083553	—	4	8	no	—
44	C₂²⋊Dic₈₉	—	4	4	no	—
45	C₄×Dic₈₉	—	4	4	no	—
46	C₂²×C₄×C₈₉	D₈	1	2	yes	—
47	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₈₉	Q₁₆	1	4	yes	—
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×D₄	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
53	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
54	C₄²×C₈₉	D₈	1	2	yes	—
55	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₁₆×C₈₉	C₂×Q₈	1	4	yes	—
58	C₁₆×C₈₉	C₂×Q₈	1	4	yes	—
59	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₈₉	—	2	4	no	—
61	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
62	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
63	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
64	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
66	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
67	C₂×Dic₈₉	—	4	8	no	—
68	Dic₁₇₈	—	8	8	no	—
69	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
70	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
71	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
72	C₈×C₈₉	C₂²×Q₈	1	2	yes	—
73	C₂²×C₈₉	—	2	2	no	—
74	C₂²×C₈₉	#500	1	2	yes	—
75	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
76	C₄×C₈₉	—	2	2	no	—
77	C₄×C₈₉	#500	1	2	yes	—
78	C₄×C₈₉	#508	1	4	yes	—
79	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
80	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
81	Dic₈₉	—	8	8	no	—
82	#33726	—	89	1	no	maximal, Sylow
83	C₂×C₈₉	—	2	2	no	—
84	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
85	C₂×C₈₉	C₂×#508	1	2	yes	—
86	#3160	—	89	2	no	—
87	#2254	—	89	2	no	—
88	#2253	—	89	4	no	—
89	#2252	—	89	2	no	—
90	#2158	—	89	4	no	—
91	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic
92	C₈₉	#33726	1	1	yes	characteristic, complemented, Sylow
93	#533	—	178	2	no	—
94	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
95	#500	—	89	4	no	—
96	#499	—	89	4	no	—
97	#499	—	178	2	no	—
98	#498	—	178	2	no	—
99	#427	—	89	4	no	—
100	#427	—	178	2	no	—
101	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
102	#367	—	89	8	no	—
103	#366	—	89	8	no	—
104	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
105	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
106	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
107	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
109	C₄⋊Q₈	—	178	4	no	—
110	#125	—	178	4	no	—
111	C₂²⋊Q₈	—	356	4	no	—
112	C₄×Q₈	—	356	4	no	—
113	#117	—	178	4	no	—
114	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
115	C₂×C₁₆	Dic₁₇₈	1	8	yes	complemented
116	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
117	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
118	#107	—	89	8	no	—
119	#107	—	178	4	no	—
120	#106	—	178	4	no	—
121	#106	—	178	4	no	—
122	C₄×C₈	D₃₅₆	1	1	yes	characteristic
123	C₄×C₈	D₃₅₆	1	1	yes	characteristic
124	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
125	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
126	C₂×Q₈	—	356	4	no	—
127	C₂²×C₄	D₇₁₂	1	2	yes	—
128	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
129	C₂×C₈	Dic₃₅₆	1	4	yes	—
130	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
131	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
132	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
133	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
134	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
135	C₄⋊C₄	—	356	8	no	—
136	C₄⋊C₄	—	356	4	no	—
137	C₄⋊C₄	—	178	8	no	—
138	C₂²⋊C₄	—	356	4	no	—
139	C₄²	—	356	4	no	—
140	C₄²	D₇₁₂	1	2	yes	—
141	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
142	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
143	C₁₆	C₂×Dic₁₇₈	1	4	yes	—
144	C₁₆	C₂×Dic₁₇₈	1	4	yes	—
145	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
146	Q₈	—	712	8	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	356	8	no	—
149	C₂×C₄	#15668012	1	2	yes	—
150	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
151	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
152	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
153	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
154	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
155	C₈	#15668012	1	2	yes	—
156	C₈	C₂×Dic₃₅₆	1	2	yes	—
157	C₈	C₂×Dic₃₅₆	1	2	yes	—
158	C₈	C₂²×Dic₁₇₈	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	#19152003	1	2	yes	—
161	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	712	8	no	—
164	C₄	#19152003	1	2	yes	—
165	C₄	#19152030	1	4	yes	—
166	C₄	C₂×#15668012	1	1	yes	characteristic
167	C₄	C₂²×Dic₃₅₆	1	1	yes	characteristic
168	C₂	—	2	2	no	—
169	C₂	C₂×#19152003	1	1	yes	characteristic
170	C₂	C₂×#19152030	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	#33726
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #26022534	C₂	semidirect
3. #26022531	C₂	semidirect
4. #26022527	C₂	semidirect
33. C₂×C₁₆×C₈₉	Q₈	semidirect
92. C₈₉	#33726	semidirect
115. C₂×C₁₆	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26030766	2
#26022534	2
#26022531	2
#26022527	2
#26022036	2
C₂×C₄×C₁₆×C₈₉	2
#33726	89