Subgroups — #36448186

Order 22784.

362 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020369	C₂	1	1	yes	maximal, characteristic
2	#26019998	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151350	C₂	1	2	yes	maximal
4	C₂×#19151348	C₂	1	2	yes	maximal
5	C₈₉×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
8	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
9	#19151353	—	2	2	no	—
10	#19151353	C₄	1	2	yes	—
11	#19151353	C₄	1	2	yes	—
12	#19151350	—	2	4	no	—
13	#19151348	—	2	4	no	—
14	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
15	C₄³×C₈₉	C₂²	1	1	yes	characteristic
16	#336×C₈₉	—	2	2	no	—
17	C₂²×#10623494	—	2	2	no	—
18	C₂²×#10623494	C₂×C₄	1	2	yes	—
19	C₂²×#10623494	C₂×C₄	1	2	yes	—
20	C₄⋊#10623494	—	2	2	no	—
21	C₄⋊#10623494	—	2	2	no	—
22	C₄×#10623494	D₄	1	2	yes	—
23	C₄×#10623494	D₄	1	2	yes	—
24	C₄×#10623494	Q₈	1	2	yes	—
25	C₄×#10623494	Q₈	1	2	yes	—
26	C₂²×C₈×C₈₉	—	2	2	no	—
27	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	D₄	1	2	yes	—
30	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
31	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₈₉	—	2	2	no	—
33	C₄⋊C₈×C₈₉	—	2	2	no	—
34	C₂×#10623494	—	2	2	no	—
35	C₂×#10623494	—	2	2	no	—
36	C₂×#10623494	—	2	2	no	—
37	C₂×#10623494	—	2	2	no	—
38	C₂×#10623494	—	4	2	no	—
39	C₂×#10623494	—	4	4	no	—
40	C₂×#10623494	C₄⋊C₄	1	4	yes	—
41	C₂×#10623494	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₈₉	—	4	2	no	—
48	C₂×C₈×C₈₉	—	4	4	no	—
49	C₄²×C₈₉	—	2	2	no	—
50	C₄²×C₈₉	—	2	2	no	—
51	C₄²×C₈₉	—	2	4	no	—
52	C₄²×C₈₉	—	4	2	no	—
53	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
57	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	—	2	2	no	—
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	—	2	2	no	—
64	C₂×C₄×C₈₉	—	2	2	no	—
65	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₈₉	#131	1	2	yes	—
72	C₂×C₄×C₈₉	#131	1	2	yes	—
73	C₂×C₄×C₈₉	#135	1	4	yes	—
74	C₈×C₈₉	—	8	2	no	—
75	#10623494	—	2	4	no	—
76	#10623494	—	2	4	no	—
77	#10623494	—	8	2	no	—
78	C₂²×C₈₉	#384	1	1	yes	characteristic
79	C₂²×C₈₉	#424	1	1	yes	characteristic
80	C₂²×C₈₉	#424	1	1	yes	characteristic
81	C₂²×C₈₉	#427	1	2	yes	—
82	C₂²×C₈₉	#430	1	2	yes	—
83	C₄×C₈₉	—	2	1	no	—
84	C₄×C₈₉	—	2	1	no	—
85	C₄×C₈₉	—	4	2	no	—
86	C₄×C₈₉	—	4	2	no	—
87	C₄×C₈₉	#424	1	2	yes	—
88	C₄×C₈₉	#424	1	2	yes	—
89	C₄×C₈₉	#495	1	2	yes	—
90	C₄×C₈₉	#499	1	2	yes	—
91	#8811	—	89	1	no	maximal, Sylow
92	C₂×C₈₉	#1572	1	2	yes	—
93	C₂×C₈₉	#1572	1	2	yes	—
94	C₂×C₈₉	#1843	1	1	yes	characteristic
95	C₂×C₈₉	#1846	1	1	yes	characteristic
96	C₂×C₈₉	#1849	1	1	yes	characteristic
97	#1831	—	89	1	no	—
98	#1831	D₈₉	1	1	yes	characteristic
99	#1744	—	89	1	no	—
100	C₂×#335	—	89	2	no	—
101	C₂×#334	—	89	2	no	—
102	C₈₉	#8811	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊C₈	—	89	2	no	—
104	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
105	C₂×C₄×C₈	—	89	1	no	—
106	C₂×C₄×C₈	—	89	1	no	—
107	C₄³	C₂×D₈₉	1	1	yes	characteristic
108	#336	—	2	2	no	—
109	#336	—	89	2	no	—
110	#336	—	89	2	no	—
111	#336	—	178	2	no	—
112	#335	—	178	4	no	—
113	#334	—	178	4	no	—
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	—	89	2	no	—
116	C₂²×C₈	—	89	2	no	—
117	C₂²×C₈	—	178	2	no	—
118	C₂×C₄²	Dic₁₇₈	1	2	yes	—
119	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
120	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
121	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
122	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	178	2	no	—
126	C₄⋊C₈	—	178	2	no	—
127	C₄×C₈	—	89	2	no	—
128	C₄×C₈	—	89	2	no	—
129	C₄×C₈	—	89	2	no	—
130	C₄×C₈	—	89	2	no	—
131	C₂²×C₄	#13083553	1	2	yes	—
132	C₂²×C₄	#13083553	1	2	yes	—
133	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
135	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	4	no	—
138	C₂×C₈	—	89	4	no	—
139	C₂×C₈	—	89	4	no	—
140	C₂×C₈	—	356	2	no	—
141	C₂×C₈	—	356	4	no	—
142	C₂×C₈	—	178	2	no	—
143	C₂×C₈	—	178	2	no	—
144	C₂×C₈	—	178	2	no	—
145	C₂×C₈	—	178	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	4	2	no	—
150	C₄²	D₄×D₈₉	1	1	yes	characteristic
151	C₄²	D₄×D₈₉	1	1	yes	characteristic
152	C₄²	Q₈×D₈₉	1	1	yes	characteristic
153	C₄²	Q₈×D₈₉	1	1	yes	characteristic
154	C₂³	C₂×#13083553	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
163	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
164	C₂×C₄	#15668008	1	1	yes	characteristic
165	C₂×C₄	#15668010	1	1	yes	characteristic
166	C₂×C₄	#15668011	1	1	yes	characteristic
167	C₂×C₄	#15668011	1	1	yes	characteristic
168	C₂×C₄	#15668018	1	2	yes	—
169	C₂×C₄	#15668020	1	2	yes	—
170	C₂×C₄	#15668032	1	4	yes	—
171	C₈	—	8	2	no	—
172	C₈	—	178	4	no	—
173	C₈	—	178	4	no	—
174	C₈	—	712	2	no	—
175	C₂²	#19151467	1	1	yes	characteristic
176	C₂²	#19151624	1	1	yes	characteristic
177	C₂²	#19151626	1	1	yes	characteristic
178	C₂²	#19151634	1	2	yes	—
179	C₂²	#19151646	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	#19151623	1	2	yes	—
185	C₄	#19151623	1	2	yes	—
186	C₄	#19151980	1	2	yes	—
187	C₄	#19151997	1	2	yes	—
188	C₂	#26019030	1	2	yes	—
189	C₂	#26019033	1	2	yes	—
190	C₂	#26020405	1	1	yes	characteristic
191	C₂	#26020414	1	1	yes	characteristic
192	C₂	#26020426	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1831	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1831	D₈₉
hypocenter	C₈₉	#8811
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
102. C₈₉	#8811	semidirect

Maximal subgroups

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

subgroup	index
#26020369	2
#26019998	2
C₂×#19151350	2
C₂×#19151348	2
C₈₉×#1831	2
#8811	89