Subgroups — #36485620

Order 22784.

1710 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151468	C₂	1	2	yes	maximal
2	C₂×#19151464	C₂	1	4	yes	maximal
3	#26018482	C₂	1	8	yes	maximal
4	C₂×C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₈₉	C₂²	1	8	yes	—
6	C₂²×C₄⋊Dic₈₉	C₂²	1	4	yes	—
7	#19151468	—	2	8	no	—
8	#19151464	—	2	16	no	—
9	C₂×#15667919	C₂²	1	16	yes	—
10	C₂²×C₄²×C₈₉	C₂²	1	4	yes	—
11	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
12	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
13	C₄³×C₈₉	C₄	1	8	yes	complemented
14	C₂³×Dic₈₉	C₂³	1	8	yes	—
15	C₂×C₄⋊Dic₈₉	—	2	16	no	—
16	C₂×C₄⋊Dic₈₉	—	2	16	no	—
17	C₂×C₄⋊Dic₈₉	—	2	8	no	—
18	C₂×C₄⋊Dic₈₉	—	2	32	no	—
19	#15667919	—	2	32	no	—
20	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
21	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
22	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
23	C₂×C₄²×C₈₉	—	2	16	no	—
24	C₂×C₄²×C₈₉	C₂×C₄	1	16	yes	—
25	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₈₉	C₂×C₄	1	8	yes	—
27	C₂×C₄²×C₈₉	D₄	1	4	yes	—
28	C₂×C₄²×C₈₉	D₄	1	8	yes	—
29	C₂×C₄²×C₈₉	Q₈	1	4	yes	—
30	C₂×C₄²×C₈₉	Q₈	1	8	yes	—
31	C₂²×Dic₈₉	—	2	16	no	—
32	C₂²×Dic₈₉	—	2	8	no	—
33	C₂²×Dic₈₉	—	2	32	no	—
34	C₄⋊Dic₈₉	—	4	32	no	—
35	C₄⋊Dic₈₉	—	4	16	no	—
36	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	—	2	16	no	—
38	C₂²×C₄×C₈₉	C₂²×C₄	1	8	yes	—
39	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
41	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
42	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
43	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
45	C₂²×C₄×C₈₉	C₂×Q₈	1	8	yes	—
46	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
47	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
48	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
49	C₂²×C₄×C₈₉	#41	1	16	yes	—
50	C₄²×C₈₉	—	2	32	no	—
51	C₄²×C₈₉	C₄⋊C₄	1	16	yes	—
52	C₄²×C₈₉	C₄⋊C₄	1	32	yes	complemented
53	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
54	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
55	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
56	C₂³×C₈₉	C₂²×Q₈	1	1	yes	characteristic
57	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
58	C₂³×C₈₉	C₂×#41	1	8	yes	—
59	C₂×C₄×C₈₉	—	2	16	no	—
60	C₂×C₄×C₈₉	—	2	16	no	—
61	C₂×C₄×C₈₉	—	2	16	no	—
62	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	16	yes	—
63	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	8	yes	—
64	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
65	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	8	yes	—
66	C₂×C₄×C₈₉	#117	1	16	yes	—
67	C₂×C₄×C₈₉	#124	1	16	yes	—
68	C₂×C₄×C₈₉	#125	1	16	yes	—
69	C₂×C₄×C₈₉	#127	1	4	yes	—
70	C₂×C₄×C₈₉	C₄⋊Q₈	1	8	yes	—
71	C₂×C₄×C₈₉	C₄⋊Q₈	1	4	yes	—
72	C₂×Dic₈₉	—	4	16	no	—
73	C₂×Dic₈₉	—	4	8	no	—
74	C₂×Dic₈₉	—	4	32	no	—
75	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₈₉	C₂²×C₄⋊C₄	1	2	yes	—
77	C₂²×C₈₉	C₂×#117	1	4	yes	—
78	C₂²×C₈₉	C₂×#124	1	4	yes	—
79	C₂²×C₈₉	C₂×#125	1	4	yes	—
80	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
81	C₂²×C₈₉	C₂×C₄⋊Q₈	1	2	yes	—
82	C₂²×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
83	C₂²×C₈₉	#532	1	8	yes	—
84	C₂²×C₈₉	#533	1	8	yes	—
85	C₄×C₈₉	—	2	16	no	—
86	C₄×C₈₉	#382	1	16	yes	complemented
87	C₄×C₈₉	#384	1	8	yes	—
88	Dic₈₉	—	8	8	no	—
89	#14687	—	89	1	no	maximal, Sylow
90	C₂×C₈₉	C₂×#382	1	2	yes	—
91	C₂×C₈₉	C₂×#384	1	1	yes	characteristic
92	C₂×C₈₉	#2285	1	4	yes	—
93	C₂×C₈₉	#2858	1	2	yes	—
94	C₂×C₈₉	#2859	1	4	yes	—
95	C₂×C₈₉	#2862	1	2	yes	—
96	C₂×#384	—	89	2	no	—
97	C₂×#382	—	89	4	no	—
98	C₂×C₄³	D₈₉	1	1	yes	characteristic
99	C₄⋊#95	—	89	8	no	—
100	C₈₉	#14687	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₄⋊C₄	—	89	8	no	—
102	C₂²×C₄⋊C₄	—	89	4	no	—
103	C₂²×C₄²	C₂×D₈₉	1	4	yes	—
104	C₂²×C₄²	C₂×D₈₉	1	2	yes	—
105	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
106	#384	—	178	8	no	—
107	#382	—	178	16	no	—
108	C₂×#95	—	89	16	no	—
109	C₄³	Dic₈₉	1	8	yes	complemented
110	C₂³×C₄	—	89	8	no	—
111	C₂³×C₄	C₂²×D₈₉	1	4	yes	—
112	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
113	C₂³×C₄	C₂²×D₈₉	1	2	yes	—
114	C₂×C₄⋊C₄	—	178	16	no	—
115	C₂×C₄⋊C₄	—	178	16	no	—
116	C₂×C₄⋊C₄	—	178	8	no	—
117	C₂×C₄⋊C₄	—	178	32	no	—
118	C₂×C₄²	—	2	16	no	—
119	C₂×C₄²	Dic₁₇₈	1	4	yes	—
120	C₂×C₄²	Dic₁₇₈	1	8	yes	—
121	C₂×C₄²	D₃₅₆	1	4	yes	—
122	C₂×C₄²	D₃₅₆	1	8	yes	—
123	C₂×C₄²	C₂×Dic₈₉	1	16	yes	—
124	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
125	C₂×C₄²	C₂×Dic₈₉	1	8	yes	—
126	#95	—	178	32	no	—
127	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
128	C₂²×C₄	—	2	16	no	—
129	C₂²×C₄	—	178	16	no	—
130	C₂²×C₄	—	178	8	no	—
131	C₂²×C₄	—	178	32	no	—
132	C₂²×C₄	C₂²×Dic₈₉	1	8	yes	—
133	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
134	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
135	C₂²×C₄	C₂×D₃₅₆	1	8	yes	—
136	C₂²×C₄	C₂×D₃₅₆	1	4	yes	—
137	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
138	C₂²×C₄	C₂×D₃₅₆	1	4	yes	—
139	C₂²×C₄	C₂×Dic₁₇₈	1	8	yes	—
140	C₂²×C₄	C₂×Dic₁₇₈	1	4	yes	—
141	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
142	C₂²×C₄	C₂×Dic₁₇₈	1	4	yes	—
143	C₂²×C₄	#13083574	1	16	yes	—
144	C₄⋊C₄	—	356	32	no	—
145	C₄⋊C₄	—	356	16	no	—
146	C₄²	—	2	32	no	—
147	C₄²	C₄⋊Dic₈₉	1	16	yes	—
148	C₄²	C₄⋊Dic₈₉	1	32	yes	complemented
149	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
150	C₂³	C₂²×D₃₅₆	1	2	yes	—
151	C₂³	C₂²×D₃₅₆	1	1	yes	characteristic
152	C₂³	C₂²×Dic₁₇₈	1	2	yes	—
153	C₂³	C₂²×Dic₁₇₈	1	1	yes	characteristic
154	C₂³	C₂×#13083574	1	8	yes	—
155	C₂×C₄	—	2	16	no	—
156	C₂×C₄	—	2	16	no	—
157	C₂×C₄	—	2	16	no	—
158	C₂×C₄	—	356	8	no	—
159	C₂×C₄	—	356	32	no	—
160	C₂×C₄	—	356	16	no	—
161	C₂×C₄	C₂×C₄⋊Dic₈₉	1	16	yes	—
162	C₂×C₄	C₂×C₄⋊Dic₈₉	1	8	yes	—
163	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
164	C₂×C₄	C₂×C₄⋊Dic₈₉	1	8	yes	—
165	C₂×C₄	#15667971	1	16	yes	—
166	C₂×C₄	#15668000	1	16	yes	—
167	C₂×C₄	#15668004	1	16	yes	—
168	C₂×C₄	#15668009	1	4	yes	—
169	C₂×C₄	#15668012	1	8	yes	—
170	C₂×C₄	#15668012	1	4	yes	—
171	C₂²	C₂²×C₄⋊Dic₈₉	1	2	yes	—
172	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
173	C₂²	C₂×#15667971	1	4	yes	—
174	C₂²	C₂×#15668000	1	4	yes	—
175	C₂²	C₂×#15668004	1	4	yes	—
176	C₂²	C₂×#15668009	1	1	yes	characteristic
177	C₂²	C₂×#15668012	1	2	yes	—
178	C₂²	C₂×#15668012	1	1	yes	characteristic
179	C₂²	#19152148	1	8	yes	—
180	C₂²	#19152155	1	8	yes	—
181	C₄	—	2	16	no	—
182	C₄	—	712	8	no	—
183	C₄	#19151464	1	16	yes	complemented
184	C₄	#19151468	1	8	yes	—
185	C₂	C₂×#19151464	1	2	yes	—
186	C₂	C₂×#19151468	1	1	yes	characteristic
187	C₂	#26022659	1	4	yes	—
188	C₂	#26028609	1	2	yes	—
189	C₂	#26028615	1	4	yes	—
190	C₂	#26028630	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₈₉
commutator	C₂³×C₈₉	C₂³×C₄
Fitting	C₂×C₄³×C₈₉	C₂
socle, prime radical, cosocle	C₂⁴×C₈₉	C₂⁴
hypercenter	C₂×C₄³	D₈₉
hypocenter	C₈₉	#14687
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
13. C₄³×C₈₉	C₄	semidirect
52. C₄²×C₈₉	C₄⋊C₄	semidirect
86. C₄×C₈₉	#382	semidirect
100. C₈₉	#14687	semidirect
109. C₄³	Dic₈₉	semidirect
148. C₄²	C₄⋊Dic₈₉	semidirect
183. C₄	#19151464	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151468	2
C₂×#19151464	2
#26018482	2
C₂×C₄³×C₈₉	2
#14687	89