Subgroups — #36454257

Order 22784.

526 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151978	C₂	1	1	yes	maximal, characteristic
2	#26021213	C₂	1	2	yes	maximal
3	#26021213	C₂	1	2	yes	maximal
4	#26020380	C₂	1	1	yes	maximal, characteristic
5	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₅₆	—	2	2	no	—
7	C₂×#15668012	C₂²	1	2	yes	—
8	#19151978	—	2	2	no	—
9	#19151978	C₄	1	4	yes	complemented
10	#19151468	C₂²	1	2	yes	—
11	#19151361	—	2	2	no	—
12	#19151361	—	2	4	no	—
13	#19151361	—	2	4	no	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#322×C₈₉	—	2	2	no	—
18	C₂²×Dic₁₇₈	—	2	2	no	—
19	C₂×Dic₃₅₆	—	2	8	no	—
20	C₂×Dic₃₅₆	—	4	8	no	—
21	#15668012	—	2	2	no	—
22	#15668012	—	2	2	no	—
23	#15668012	—	2	2	no	—
24	#15668012	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₈₉	—	2	2	no	—
26	C₂×C₄⋊Dic₈₉	—	2	2	no	—
27	C₂×C₄⋊Dic₈₉	—	2	2	no	—
28	C₂×C₄⋊Dic₈₉	—	2	2	no	—
29	C₂×C₄⋊Dic₈₉	—	2	2	no	—
30	C₂²×C₈×C₈₉	D₄	1	2	yes	—
31	C₂²×C₈×C₈₉	D₄	1	2	yes	—
32	C₂²×C₈×C₈₉	D₄	1	2	yes	—
33	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₈₉	—	2	1	no	—
35	C₄×C₈×C₈₉	—	2	1	no	—
36	C₄×C₈×C₈₉	—	2	1	no	—
37	C₄×C₈×C₈₉	—	2	1	no	—
38	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₈₉	D₄	1	2	yes	—
40	C₂×Dic₁₇₈	—	2	4	no	—
41	C₂×Dic₁₇₈	—	4	8	no	—
42	C₂×Dic₁₇₈	—	4	2	no	—
43	C₂²×Dic₈₉	—	4	2	no	—
44	C₂²×Dic₈₉	—	4	2	no	—
45	Dic₃₅₆	—	4	16	no	—
46	C₄⋊Dic₈₉	—	2	4	no	—
47	C₄⋊Dic₈₉	—	2	4	no	—
48	C₄⋊Dic₈₉	—	4	2	no	—
49	C₄⋊Dic₈₉	—	4	2	no	—
50	C₄⋊Dic₈₉	—	4	4	no	—
51	C₄⋊Dic₈₉	—	4	4	no	—
52	C₄⋊Dic₈₉	—	4	4	no	—
53	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₈₉	—	2	2	no	—
57	C₂×C₈×C₈₉	—	2	2	no	—
58	C₂×C₈×C₈₉	—	2	4	no	—
59	C₂×C₈×C₈₉	—	2	2	no	—
60	C₂×C₈×C₈₉	—	2	4	no	—
61	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
62	C₂×C₈×C₈₉	Q₁₆	1	4	yes	complemented
63	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₈₉	#41	1	1	yes	characteristic
66	C₄²×C₈₉	#41	1	1	yes	characteristic
67	C₂³×C₈₉	#127	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
72	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
73	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
75	C₂×C₄×C₈₉	#135	1	2	yes	—
76	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
77	C₂×C₄×C₈₉	#137	1	4	yes	—
78	C₂×Dic₈₉	—	4	4	no	—
79	C₂×Dic₈₉	—	8	4	no	—
80	C₂×Dic₈₉	—	8	2	no	—
81	C₂×Dic₈₉	—	8	2	no	—
82	C₂×Dic₈₉	—	8	2	no	—
83	Dic₁₇₈	—	4	8	no	—
84	Dic₁₇₈	—	8	4	no	—
85	C₈×C₈₉	—	2	4	no	—
86	C₈×C₈₉	—	2	4	no	—
87	C₈×C₈₉	—	4	2	no	—
88	C₂²×C₈₉	#390	1	1	yes	characteristic
89	C₂²×C₈₉	#494	1	1	yes	characteristic
90	C₂²×C₈₉	#495	1	1	yes	characteristic
91	C₂²×C₈₉	#496	1	2	yes	—
92	C₂²×C₈₉	#497	1	2	yes	—
93	C₄×C₈₉	#418	1	2	yes	—
94	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
95	C₄×C₈₉	#441	1	2	yes	—
96	C₄×C₈₉	#467	1	2	yes	—
97	C₄×C₈₉	#470	1	2	yes	—
98	C₄×C₈₉	#490	1	2	yes	—
99	Dic₈₉	—	8	4	no	—
100	Dic₈₉	—	16	2	no	—
101	#9835	—	89	1	no	maximal, Sylow
102	C₂×C₈₉	#1720	1	2	yes	—
103	C₂×C₈₉	#1723	1	2	yes	—
104	C₂×C₈₉	#1961	1	1	yes	characteristic
105	C₂×C₈₉	#1969	1	1	yes	characteristic
106	C₂×C₈₉	#1971	1	1	yes	characteristic
107	C₂×#494	—	89	1	no	—
108	#1990	—	89	2	no	—
109	#1990	—	89	2	no	—
110	#1834	—	89	1	no	—
111	C₂×#322	D₈₉	1	1	yes	characteristic
112	C₈₉	#9835	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₁₆	—	178	2	no	—
114	C₂×C₄⋊Q₈	—	89	2	no	—
115	#494	—	89	4	no	—
116	#494	—	178	2	no	—
117	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
120	#384	—	89	2	no	—
121	#340	—	178	2	no	—
122	#340	—	178	4	no	—
123	#340	—	178	4	no	—
124	#322	—	2	2	no	—
125	C₂²×Q₈	—	178	2	no	—
126	C₂×Q₁₆	—	356	8	no	—
127	C₂×Q₁₆	—	178	8	no	—
128	C₂²×C₈	D₃₅₆	1	2	yes	—
129	C₂²×C₈	D₃₅₆	1	2	yes	—
130	C₂²×C₈	D₃₅₆	1	2	yes	—
131	C₄⋊Q₈	—	89	4	no	—
132	C₄⋊Q₈	—	178	2	no	—
133	C₄⋊Q₈	—	178	2	no	—
134	C₄⋊Q₈	—	178	2	no	—
135	C₂×C₄⋊C₄	—	178	2	no	—
136	C₂×C₄⋊C₄	—	178	2	no	—
137	C₂×C₄⋊C₄	—	178	2	no	—
138	C₂×C₄⋊C₄	—	178	2	no	—
139	C₂×C₄⋊C₄	—	178	2	no	—
140	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	C₄×D₈₉	1	2	yes	—
146	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
147	C₂×Q₈	—	356	2	no	—
148	C₂×Q₈	—	356	8	no	—
149	C₂×Q₈	—	178	4	no	—
150	C₂²×C₄	—	356	2	no	—
151	C₂²×C₄	—	356	2	no	—
152	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
155	Q₁₆	—	356	16	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	#13083551	1	4	yes	—
162	C₂×C₈	Dic₃₅₆	1	4	yes	complemented
163	C₄⋊C₄	—	356	2	no	—
164	C₄⋊C₄	—	356	2	no	—
165	C₄⋊C₄	—	356	4	no	—
166	C₄⋊C₄	—	356	4	no	—
167	C₄⋊C₄	—	356	4	no	—
168	C₄⋊C₄	—	178	4	no	—
169	C₄⋊C₄	—	178	4	no	—
170	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
171	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
172	C₄²	#13083574	1	1	yes	characteristic
173	C₄²	#13083574	1	1	yes	characteristic
174	C₂³	#15668009	1	1	yes	characteristic
175	Q₈	—	356	8	no	—
176	Q₈	—	712	4	no	—
177	C₂×C₄	—	356	4	no	—
178	C₂×C₄	—	712	2	no	—
179	C₂×C₄	—	712	2	no	—
180	C₂×C₄	—	712	2	no	—
181	C₂×C₄	—	712	4	no	—
182	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
183	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
184	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
185	C₂×C₄	#15667986	1	1	yes	characteristic
186	C₂×C₄	#15667986	1	1	yes	characteristic
187	C₂×C₄	#15668000	1	1	yes	characteristic
188	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
189	C₂×C₄	#15668034	1	2	yes	—
190	C₂×C₄	#15668040	1	4	yes	—
191	C₂×C₄	#15668045	1	4	yes	—
192	C₈	—	2	4	no	—
193	C₈	—	2	4	no	—
194	C₈	—	4	2	no	—
195	C₂²	#19151500	1	1	yes	characteristic
196	C₂²	#19151978	1	1	yes	characteristic
197	C₂²	#19151982	1	1	yes	characteristic
198	C₂²	#19151987	1	2	yes	—
199	C₂²	#19151992	1	2	yes	—
200	C₄	—	1424	2	no	—
201	C₄	—	712	4	no	—
202	C₄	#19151604	1	2	yes	—
203	C₄	C₄×Dic₃₅₆	1	2	yes	—
204	C₄	#19151706	1	2	yes	—
205	C₄	#19151839	1	2	yes	—
206	C₄	#19151856	1	2	yes	—
207	C₄	#19151963	1	2	yes	—
208	C₂	#26019912	1	2	yes	—
209	C₂	#26019929	1	2	yes	—
210	C₂	#26021076	1	1	yes	characteristic
211	C₂	#26021113	1	1	yes	characteristic
212	C₂	#26021123	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15668009
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#127
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#9835
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
9. #19151978	C₄	semidirect
62. C₂×C₈×C₈₉	Q₁₆	semidirect
112. C₈₉	#9835	semidirect
162. C₂×C₈	Dic₃₅₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151978	2
#26021213	2
#26020380	2
C₂×#322×C₈₉	2
#9835	89