Subgroups — #36472646

Order 22784.

510 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	C₂×#19151403	C₂	1	4	yes	maximal
3	#26020412	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₅₆	—	2	1	no	—
6	C₂²×Dic₃₅₆	C₂²	1	2	yes	—
7	C₂×#15668012	—	2	1	no	—
8	#19151978	—	2	2	no	—
9	#19151978	C₄	1	4	yes	complemented
10	C₂×#15667946	—	2	1	no	—
11	C₂×#15667946	C₂²	1	2	yes	—
12	#19151468	—	2	1	no	—
13	#19151403	—	2	8	no	—
14	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₈₉	—	2	2	no	—
17	C₂²×Dic₁₇₈	—	2	2	no	—
18	C₂×Dic₃₅₆	—	2	4	no	—
19	C₂×Dic₃₅₆	—	2	4	no	—
20	C₂×Dic₃₅₆	—	2	2	no	—
21	C₂×Dic₃₅₆	—	2	4	no	—
22	C₂×Dic₃₅₆	—	4	2	no	—
23	C₂×Dic₃₅₆	—	4	2	no	—
24	C₂×Dic₃₅₆	C₂×C₄	1	4	yes	—
25	#15668012	—	2	4	no	—
26	#15668012	—	4	2	no	—
27	C₂×C₄⋊Dic₈₉	—	2	2	no	—
28	C₂×C₄⋊Dic₈₉	—	4	2	no	—
29	C₂×C₄⋊Dic₈₉	—	4	2	no	—
30	#15667946	—	2	4	no	—
31	#15667946	—	4	2	no	—
32	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₈₉	—	2	2	no	—
36	C₂×C₁₆×C₈₉	—	2	2	no	—
37	C₂×C₁₆×C₈₉	D₄	1	4	yes	—
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	2	no	—
42	C₂×Dic₁₇₈	—	4	2	no	—
43	C₂×Dic₁₇₈	—	4	2	no	—
44	C₂×Dic₁₇₈	—	4	2	no	—
45	C₂×Dic₁₇₈	—	4	2	no	—
46	C₂²×Dic₈₉	—	4	2	no	—
47	C₂²×Dic₈₉	—	4	2	no	—
48	Dic₃₅₆	—	2	8	no	—
49	Dic₃₅₆	—	4	8	no	—
50	Dic₃₅₆	—	4	4	no	—
51	C₄⋊Dic₈₉	—	4	8	no	—
52	C₄⋊Dic₈₉	—	4	2	no	—
53	C₄⋊Dic₈₉	—	4	2	no	—
54	C₄⋊Dic₈₉	—	8	4	no	—
55	C₂²×C₄×C₈₉	D₈	1	2	yes	—
56	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₈₉	D₈	1	2	yes	—
59	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
60	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₈₉	#41	1	2	yes	—
64	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₈₉	SD₁₆	1	2	yes	—
66	C₁₆×C₈₉	—	2	4	no	—
67	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₈₉	#102	1	4	yes	—
69	C₂×C₄×C₈₉	SD₃₂	1	4	yes	—
70	C₂×C₄×C₈₉	Q₃₂	1	4	yes	—
71	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
73	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×Dic₈₉	—	4	4	no	—
78	C₂×Dic₈₉	—	8	2	no	—
79	C₂×Dic₈₉	—	8	2	no	—
80	C₂×Dic₈₉	—	8	2	no	—
81	C₂×Dic₈₉	—	8	4	no	—
82	Dic₁₇₈	—	4	4	no	—
83	Dic₁₇₈	—	4	4	no	—
84	Dic₁₇₈	—	8	2	no	—
85	Dic₁₇₈	—	8	2	no	—
86	C₈×C₈₉	#102	1	4	yes	—
87	C₈×C₈₉	C₄×D₄	1	2	yes	—
88	C₈×C₈₉	#121	1	2	yes	—
89	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
90	C₂²×C₈₉	#486	1	1	yes	characteristic
91	C₂²×C₈₉	#493	1	1	yes	characteristic
92	C₂²×C₈₉	C₂×SD₃₂	1	2	yes	—
93	C₂²×C₈₉	C₂×Q₃₂	1	2	yes	—
94	C₄×C₈₉	#358	1	8	yes	complemented
95	C₄×C₈₉	#390	1	1	yes	characteristic
96	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
97	C₄×C₈₉	#486	1	1	yes	characteristic
98	C₄×C₈₉	#492	1	1	yes	characteristic
99	Dic₈₉	—	8	4	no	—
100	Dic₈₉	—	16	2	no	—
101	#13203	—	89	1	no	maximal, Sylow
102	C₂×C₈₉	#1957	1	1	yes	characteristic
103	C₂×C₈₉	C₂×#358	1	2	yes	—
104	C₂×C₈₉	#2241	1	2	yes	—
105	C₂×C₈₉	#2247	1	1	yes	characteristic
106	C₂×C₈₉	#2248	1	1	yes	characteristic
107	C₂×#494	—	89	1	no	—
108	C₂×#358	—	89	4	no	—
109	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic
110	#1845	—	89	1	no	—
111	C₈₉	#13203	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₁₆	—	89	2	no	—
113	C₂²×Q₁₆	—	178	1	no	—
114	C₂×C₄⋊Q₈	—	178	1	no	—
115	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
116	#494	—	89	4	no	—
117	#494	—	178	2	no	—
118	C₂×#107	—	89	2	no	—
119	C₂×#107	—	178	1	no	—
120	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
121	#384	—	178	1	no	—
122	#358	—	178	8	no	—
123	C₄×C₁₆	—	2	2	no	—
124	C₂²×Q₈	—	178	2	no	—
125	C₂×Q₁₆	—	89	4	no	—
126	C₂×Q₁₆	—	356	2	no	—
127	C₂×Q₁₆	—	356	2	no	—
128	C₂×Q₁₆	—	178	4	no	—
129	C₂×Q₁₆	—	178	2	no	—
130	C₂×Q₁₆	—	178	4	no	—
131	C₂×Q₁₆	—	178	4	no	—
132	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
133	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
134	C₄⋊Q₈	—	356	2	no	—
135	C₄⋊Q₈	—	178	4	no	—
136	C₂×C₄⋊C₄	—	356	2	no	—
137	C₂×C₄⋊C₄	—	356	2	no	—
138	C₂×C₄⋊C₄	—	178	2	no	—
139	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
140	C₂×C₁₆	—	2	2	no	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	D₃₅₆	1	4	yes	—
143	#107	—	356	2	no	—
144	#107	—	178	4	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	2	no	—
149	C₂×Q₈	—	356	2	no	—
150	C₂×Q₈	—	356	2	no	—
151	C₂×Q₈	—	356	2	no	—
152	C₂×Q₈	—	178	4	no	—
153	C₂²×C₄	—	356	2	no	—
154	C₂²×C₄	—	356	2	no	—
155	C₂²×C₄	D₇₁₂	1	2	yes	—
156	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
157	Q₁₆	—	356	4	no	—
158	Q₁₆	—	356	8	no	—
159	Q₁₆	—	178	8	no	—
160	C₂×C₈	#13083551	1	2	yes	—
161	C₂×C₈	D₇₁₂	1	2	yes	—
162	C₂×C₈	#13083563	1	2	yes	—
163	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
164	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
165	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
166	C₂×C₈	#13083574	1	2	yes	—
167	C₄⋊C₄	—	356	8	no	—
168	C₄⋊C₄	—	356	2	no	—
169	C₄⋊C₄	—	356	2	no	—
170	C₄⋊C₄	—	712	4	no	—
171	C₄²	#13083551	1	2	yes	—
172	C₄²	#13083563	1	2	yes	—
173	C₁₆	—	2	4	no	—
174	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
175	Q₈	—	356	4	no	—
176	Q₈	—	356	4	no	—
177	Q₈	—	712	2	no	—
178	Q₈	—	712	2	no	—
179	C₂×C₄	—	356	4	no	—
180	C₂×C₄	—	712	2	no	—
181	C₂×C₄	—	712	4	no	—
182	C₂×C₄	—	712	2	no	—
183	C₂×C₄	—	712	2	no	—
184	C₂×C₄	#15667934	1	4	yes	—
185	C₂×C₄	#15667957	1	4	yes	—
186	C₂×C₄	Dic₇₁₂	1	4	yes	—
187	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
188	C₂×C₄	#15668000	1	1	yes	characteristic
189	C₂×C₄	#15668009	1	1	yes	characteristic
190	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
191	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
192	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
193	C₈	#15667934	1	4	yes	—
194	C₈	C₄×D₃₅₆	1	2	yes	—
195	C₈	#15667986	1	2	yes	—
196	C₂²	C₂×#15667934	1	1	yes	characteristic
197	C₂²	#19151945	1	1	yes	characteristic
198	C₂²	#19151975	1	1	yes	characteristic
199	C₂²	C₂×#15667957	1	2	yes	—
200	C₂²	C₂×Dic₇₁₂	1	2	yes	—
201	C₄	—	1424	2	no	—
202	C₄	—	712	4	no	—
203	C₄	#19151403	1	8	yes	complemented
204	C₄	#19151500	1	1	yes	characteristic
205	C₄	C₂×#15667934	1	1	yes	characteristic
206	C₄	#19151945	1	1	yes	characteristic
207	C₄	#19151971	1	1	yes	characteristic
208	C₂	#26021058	1	1	yes	characteristic
209	C₂	C₂×#19151403	1	2	yes	—
210	C₂	#26022480	1	2	yes	—
211	C₂	#26022505	1	1	yes	characteristic
212	C₂	#26022508	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	#13203
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
9. #19151978	C₄	semidirect
94. C₄×C₈₉	#358	semidirect
111. C₈₉	#13203	semidirect
203. C₄	#19151403	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
C₂×#19151403	2
#26020412	2
C₂×C₄×C₁₆×C₈₉	2
#13203	89