Subgroups — #36454185

Order 22784.

526 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152003	C₂	1	1	yes	maximal, characteristic
2	#26021212	C₂	1	2	yes	maximal
3	#26021053	C₂	1	2	yes	maximal
4	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668012	C₂²	1	2	yes	—
7	#19152003	—	2	2	no	—
8	#19152003	C₄	1	4	yes	complemented
9	C₂×#15667946	—	2	2	no	—
10	C₂×#15667935	—	2	4	no	—
11	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
12	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
13	#19151360	—	2	4	no	—
14	C₈×#10623494	—	2	2	no	—
15	C₂×#107×C₈₉	—	2	2	no	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#384×C₈₉	C₂²	1	2	yes	—
18	C₂²×Dic₁₇₈	—	2	2	no	—
19	C₂²×#10623494	D₄	1	2	yes	—
20	C₂²×#10623494	D₄	1	2	yes	—
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	#15667946	—	2	8	no	—
28	#15667935	—	4	8	no	—
29	C₄×#10623494	—	2	1	no	—
30	C₄×#10623494	—	2	1	no	—
31	C₄×#10623494	—	2	1	no	—
32	C₄×#10623494	—	2	1	no	—
33	C₂²×C₈×C₈₉	D₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
37	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
38	#107×C₈₉	—	4	4	no	—
39	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₈₉	D₄	1	2	yes	—
41	C₂×Dic₁₇₈	—	2	4	no	—
42	C₂×Dic₁₇₈	—	4	4	no	—
43	C₂×Dic₁₇₈	—	4	4	no	—
44	C₂×Dic₁₇₈	—	4	2	no	—
45	C₂²×Dic₈₉	—	4	2	no	—
46	C₂×#10623494	—	2	2	no	—
47	C₂×#10623494	—	2	2	no	—
48	C₂×#10623494	—	2	2	no	—
49	C₂×#10623494	—	2	2	no	—
50	C₂×#10623494	D₈	1	4	yes	—
51	C₂×#10623494	Q₁₆	1	4	yes	—
52	C₄⋊Dic₈₉	—	2	4	no	—
53	C₄⋊Dic₈₉	—	2	4	no	—
54	C₄⋊Dic₈₉	—	4	2	no	—
55	C₄⋊Dic₈₉	—	4	2	no	—
56	C₂²×C₄×C₈₉	—	4	2	no	—
57	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
61	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
62	C₂×C₈×C₈₉	Q₁₆	1	4	yes	—
63	C₄⋊C₄×C₈₉	—	4	4	no	—
64	C₄⋊C₄×C₈₉	—	4	4	no	—
65	C₄⋊C₄×C₈₉	—	4	4	no	—
66	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₈₉	#41	1	1	yes	characteristic
69	C₄²×C₈₉	#41	1	1	yes	characteristic
70	C₂³×C₈₉	#127	1	1	yes	characteristic
71	C₂×C₄×C₈₉	—	8	2	no	—
72	C₂×C₄×C₈₉	—	8	2	no	—
73	C₂×C₄×C₈₉	—	8	2	no	—
74	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
78	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
79	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
84	C₂×C₄×C₈₉	#135	1	2	yes	—
85	C₂×C₄×C₈₉	#135	1	2	yes	—
86	C₂×Dic₈₉	—	4	4	no	—
87	C₂×Dic₈₉	—	8	2	no	—
88	C₂×Dic₈₉	—	8	2	no	—
89	Dic₁₇₈	—	4	8	no	—
90	Dic₁₇₈	—	8	4	no	—
91	C₈×C₈₉	#103	1	8	yes	complemented
92	#10623494	—	2	4	no	—
93	#10623494	—	2	4	no	—
94	C₂²×C₈₉	#390	1	1	yes	characteristic
95	C₂²×C₈₉	#492	1	1	yes	characteristic
96	C₂²×C₈₉	#493	1	1	yes	characteristic
97	C₂²×C₈₉	#494	1	1	yes	characteristic
98	C₂²×C₈₉	#494	1	1	yes	characteristic
99	C₂²×C₈₉	#495	1	1	yes	characteristic
100	C₂²×C₈₉	#495	1	1	yes	characteristic
101	C₄×C₈₉	—	16	2	no	—
102	C₄×C₈₉	C₂×#103	1	2	yes	—
103	C₄×C₈₉	C₄×D₈	1	2	yes	—
104	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
105	C₄×C₈₉	#466	1	2	yes	—
106	C₄×C₈₉	#467	1	2	yes	—
107	C₄×C₈₉	#487	1	2	yes	—
108	Dic₈₉	—	8	4	no	—
109	#9823	—	89	1	no	maximal, Sylow
110	C₂×C₈₉	#1706	1	1	yes	characteristic
111	C₂×C₈₉	#1707	1	1	yes	characteristic
112	C₂×C₈₉	#1710	1	1	yes	characteristic
113	C₂×C₈₉	#1713	1	1	yes	characteristic
114	C₂×C₈₉	#1956	1	1	yes	characteristic
115	C₂×C₈₉	#1969	1	1	yes	characteristic
116	C₂×C₈₉	#1970	1	1	yes	characteristic
117	C₂×#500	—	89	1	no	—
118	#1990	—	89	2	no	—
119	#1956	—	89	2	no	—
120	#1845	D₈₉	1	1	yes	characteristic
121	C₂×C₈²	—	89	1	no	—
122	C₈₉	#9823	1	1	yes	characteristic, complemented, Sylow
123	C₂×C₄⋊Q₈	—	89	2	no	—
124	#500	—	89	4	no	—
125	#500	—	178	2	no	—
126	C₂×#107	—	2	2	no	—
127	C₂×#107	—	178	2	no	—
128	C₂×#103	—	178	4	no	—
129	C₂×C₄×C₈	—	89	1	no	—
130	C₂×C₄×C₈	—	89	1	no	—
131	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
132	#384	C₂×D₈₉	1	2	yes	—
133	#340	—	178	4	no	—
134	C₈²	—	178	2	no	—
135	C₂²×Q₈	—	178	2	no	—
136	C₂²×C₈	—	89	2	no	—
137	C₂²×C₈	—	89	2	no	—
138	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
139	C₄⋊Q₈	—	89	4	no	—
140	C₄⋊Q₈	—	178	2	no	—
141	C₄⋊Q₈	—	178	2	no	—
142	C₄⋊Q₈	—	178	2	no	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₄⋊C₄	—	178	2	no	—
147	C₂×C₄⋊C₄	—	178	2	no	—
148	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
149	#107	—	4	4	no	—
150	#107	—	178	8	no	—
151	#103	—	356	8	no	—
152	C₄×C₈	—	178	1	no	—
153	C₄×C₈	—	178	1	no	—
154	C₄×C₈	—	178	1	no	—
155	C₄×C₈	—	178	1	no	—
156	C₄×C₈	C₄×D₈₉	1	2	yes	—
157	C₄×C₈	D₃₅₆	1	2	yes	—
158	C₂×Q₈	—	356	2	no	—
159	C₂×Q₈	—	356	4	no	—
160	C₂×Q₈	—	356	4	no	—
161	C₂×Q₈	—	178	4	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	—	356	2	no	—
164	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
165	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
167	C₂×C₈	—	89	4	no	—
168	C₂×C₈	—	89	4	no	—
169	C₂×C₈	—	178	2	no	—
170	C₂×C₈	—	178	2	no	—
171	C₂×C₈	—	178	2	no	—
172	C₂×C₈	—	178	2	no	—
173	C₂×C₈	#13083551	1	4	yes	—
174	C₂×C₈	#13083562	1	4	yes	—
175	C₂×C₈	#13083564	1	4	yes	—
176	C₄⋊C₄	—	4	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	—	4	4	no	—
179	C₄⋊C₄	—	356	2	no	—
180	C₄⋊C₄	—	356	2	no	—
181	C₄⋊C₄	—	178	4	no	—
182	C₄⋊C₄	—	178	4	no	—
183	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
184	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
185	C₄²	#13083575	1	1	yes	characteristic
186	C₄²	#13083575	1	1	yes	characteristic
187	C₂³	#15668008	1	1	yes	characteristic
188	Q₈	—	356	8	no	—
189	Q₈	—	712	4	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	356	4	no	—
194	C₂×C₄	—	712	2	no	—
195	C₂×C₄	—	712	2	no	—
196	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
197	C₂×C₄	#15667974	1	1	yes	characteristic
198	C₂×C₄	#15667974	1	1	yes	characteristic
199	C₂×C₄	#15667987	1	1	yes	characteristic
200	C₂×C₄	#15667987	1	1	yes	characteristic
201	C₂×C₄	#15668001	1	1	yes	characteristic
202	C₂×C₄	D₈×D₈₉	1	2	yes	—
203	C₂×C₄	C₂×#13083562	1	2	yes	—
204	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
205	C₂×C₄	C₂×#13083564	1	2	yes	—
206	C₂×C₄	#15668031	1	2	yes	—
207	C₂×C₄	#15668035	1	2	yes	—
208	C₈	—	178	4	no	—
209	C₈	—	178	4	no	—
210	C₈	#15667935	1	8	yes	complemented
211	C₂²	#19151499	1	1	yes	characteristic
212	C₂²	#19151970	1	1	yes	characteristic
213	C₂²	#19151973	1	1	yes	characteristic
214	C₂²	#19151976	1	1	yes	characteristic
215	C₂²	#19151977	1	1	yes	characteristic
216	C₂²	#19151981	1	1	yes	characteristic
217	C₂²	#19151983	1	1	yes	characteristic
218	C₄	—	16	2	no	—
219	C₄	—	712	4	no	—
220	C₄	C₂×#15667935	1	2	yes	—
221	C₄	#19151680	1	2	yes	—
222	C₄	#19151692	1	2	yes	—
223	C₄	#19151835	1	2	yes	—
224	C₄	#19151840	1	2	yes	—
225	C₄	#19151950	1	2	yes	—
226	C₂	#26019841	1	1	yes	characteristic
227	C₂	#26019846	1	1	yes	characteristic
228	C₂	#26019861	1	1	yes	characteristic
229	C₂	#26019869	1	1	yes	characteristic
230	C₂	#26021053	1	1	yes	characteristic
231	C₂	#26021111	1	1	yes	characteristic
232	C₂	#26021116	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15668008
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1845	C₂
socle, prime radical	C₂³×C₈₉	#127
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1845	D₈₉
hypocenter	C₈₉	#9823
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
8. #19152003	C₄	semidirect
91. C₈×C₈₉	#103	semidirect
122. C₈₉	#9823	semidirect
210. C₈	#15667935	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152003	2
#26021212	2
#26021053	2
C₂×C₈×#10623494	2
C₈₉×#1845	2
#9823	89