Subgroups — #36470807

Order 22784.

342 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic
2	#26020412	C₂	1	1	yes	maximal, characteristic
3	#26018343	C₂	1	4	yes	maximal
4	C₈₉×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083549	C₂²	1	2	yes	—
6	C₂×#15667946	—	2	1	no	—
7	C₂×#15667946	C₂²	1	2	yes	—
8	#19151468	—	2	1	no	—
9	C₄×#13083549	C₄	1	4	yes	complemented
10	C₂×#107×C₈₉	—	2	1	no	—
11	C₂×#107×C₈₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	#384×C₈₉	—	2	1	no	—
14	C₂×C₄⋊Dic₈₉	—	2	2	no	—
15	C₂×C₄⋊Dic₈₉	—	4	1	no	—
16	C₂×C₄⋊Dic₈₉	—	4	1	no	—
17	C₂×#13083549	—	2	2	no	—
18	C₂×#13083549	—	2	2	no	—
19	C₂×#13083549	C₂×C₄	1	4	yes	—
20	#15667946	—	2	4	no	—
21	#15667946	—	4	2	no	—
22	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
26	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
27	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
28	#107×C₈₉	—	2	4	no	—
29	#107×C₈₉	—	4	2	no	—
30	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₈₉	Q₈	1	2	yes	—
32	C₂²×Dic₈₉	—	4	2	no	—
33	C₄⋊Dic₈₉	—	4	2	no	—
34	C₄⋊Dic₈₉	—	4	2	no	—
35	C₄⋊Dic₈₉	—	8	2	no	—
36	C₄⋊Dic₈₉	—	8	2	no	—
37	#13083549	—	2	4	no	—
38	C₂²×C₄×C₈₉	—	4	2	no	—
39	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
44	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
45	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₈₉	#41	1	2	yes	—
48	C₂×C₈×C₈₉	#41	1	2	yes	—
49	C₄⋊C₄×C₈₉	—	4	2	no	—
50	C₄⋊C₄×C₈₉	—	4	2	no	—
51	C₄⋊C₄×C₈₉	—	8	2	no	—
52	C₄⋊C₄×C₈₉	—	8	2	no	—
53	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
54	C₄²×C₈₉	Q₁₆	1	2	yes	—
55	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₈₉	—	8	2	no	—
57	C₂×C₄×C₈₉	—	8	2	no	—
58	C₂×C₄×C₈₉	—	8	2	no	—
59	C₂×C₄×C₈₉	#107	1	2	yes	—
60	C₂×C₄×C₈₉	#107	1	2	yes	—
61	C₂×C₄×C₈₉	D₁₆	1	2	yes	—
62	C₂×C₄×C₈₉	SD₃₂	1	2	yes	—
63	C₂×C₄×C₈₉	SD₃₂	1	2	yes	—
64	C₂×C₄×C₈₉	Q₃₂	1	2	yes	—
65	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×Dic₈₉	—	8	2	no	—
72	C₂×Dic₈₉	—	8	2	no	—
73	C₂×Dic₈₉	—	8	2	no	—
74	C₈×C₈₉	#106	1	4	yes	—
75	C₈×C₈₉	#117	1	2	yes	—
76	C₈×C₈₉	#125	1	2	yes	—
77	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
78	C₂²×C₈₉	#486	1	1	yes	characteristic
79	C₂²×C₈₉	#486	1	1	yes	characteristic
80	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
81	C₂²×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
82	C₂²×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
83	C₂²×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
84	C₄×C₈₉	—	16	2	no	—
85	C₄×C₈₉	#366	1	4	yes	—
86	C₄×C₈₉	#367	1	4	yes	—
87	C₄×C₈₉	#382	1	1	yes	characteristic
88	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
89	C₄×C₈₉	#487	1	1	yes	characteristic
90	C₄×C₈₉	#487	1	1	yes	characteristic
91	Dic₈₉	—	16	2	no	—
92	#12888	—	89	1	no	maximal, Sylow
93	C₂×C₈₉	#1834	1	1	yes	characteristic
94	C₂×C₈₉	C₂×#366	1	1	yes	characteristic
95	C₂×C₈₉	C₂×#367	1	1	yes	characteristic
96	C₂×C₈₉	#2240	1	1	yes	characteristic
97	C₂×C₈₉	#2240	1	1	yes	characteristic
98	C₂×C₈₉	#2241	1	1	yes	characteristic
99	C₂×C₈₉	#2241	1	1	yes	characteristic
100	C₂×C₄×C₁₆	—	89	1	no	—
101	#1845	—	89	1	no	—
102	#1845	D₈₉	1	1	yes	characteristic
103	#1380	—	89	4	no	—
104	C₈₉	#12888	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₁₆	—	89	2	no	—
106	C₂×#107	—	2	1	no	—
107	C₂×#107	—	89	2	no	—
108	C₂×#107	—	178	1	no	—
109	C₂×#107	C₂×D₈₉	1	2	yes	—
110	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
111	#384	—	2	1	no	—
112	#384	—	178	1	no	—
113	C₄×C₁₆	—	89	4	no	—
114	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	4	1	no	—
118	C₂×C₄⋊C₄	—	4	1	no	—
119	C₂×C₄⋊C₄	—	356	1	no	—
120	C₂×C₄⋊C₄	—	356	1	no	—
121	C₂×C₄⋊C₄	—	178	2	no	—
122	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
123	C₂×C₁₆	—	89	4	no	—
124	C₂×C₁₆	—	178	2	no	—
125	C₂×C₁₆	—	178	2	no	—
126	#107	—	2	4	no	—
127	#107	—	4	2	no	—
128	#107	—	356	2	no	—
129	#107	—	178	4	no	—
130	C₄×C₈	Dic₁₇₈	1	2	yes	—
131	C₄×C₈	C₄×D₈₉	1	2	yes	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	—	356	2	no	—
134	C₂²×C₄	#13083559	1	1	yes	characteristic
135	C₂²×C₄	#13083559	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
137	C₂×C₈	#13083553	1	2	yes	—
138	C₂×C₈	#13083561	1	2	yes	—
139	C₂×C₈	#13083562	1	2	yes	—
140	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
142	C₂×C₈	#13083573	1	2	yes	—
143	C₂×C₈	#13083575	1	2	yes	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	8	2	no	—
147	C₄⋊C₄	—	8	2	no	—
148	C₄⋊C₄	—	356	2	no	—
149	C₄⋊C₄	—	356	2	no	—
150	C₄⋊C₄	—	712	2	no	—
151	C₄⋊C₄	—	712	2	no	—
152	C₄²	#13083553	1	2	yes	—
153	C₄²	#13083564	1	2	yes	—
154	C₁₆	—	178	4	no	—
155	C₂³	C₂×#13083559	1	1	yes	characteristic
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	712	2	no	—
160	C₂×C₄	—	712	2	no	—
161	C₂×C₄	—	712	2	no	—
162	C₂×C₄	#15667945	1	2	yes	—
163	C₂×C₄	#15667945	1	2	yes	—
164	C₂×C₄	#15667953	1	2	yes	—
165	C₂×C₄	#15667955	1	2	yes	—
166	C₂×C₄	#15667956	1	2	yes	—
167	C₂×C₄	#15667958	1	2	yes	—
168	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
169	C₂×C₄	#15667999	1	1	yes	characteristic
170	C₂×C₄	#15668001	1	1	yes	characteristic
171	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
172	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
173	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
174	C₈	#15667943	1	4	yes	—
175	C₈	#15667970	1	2	yes	—
176	C₈	#15668003	1	2	yes	—
177	C₂²	C₂×#15667945	1	1	yes	characteristic
178	C₂²	#19151944	1	1	yes	characteristic
179	C₂²	#19151946	1	1	yes	characteristic
180	C₂²	C₂×#15667953	1	1	yes	characteristic
181	C₂²	C₂×#15667955	1	1	yes	characteristic
182	C₂²	C₂×#15667956	1	1	yes	characteristic
183	C₂²	C₂×#15667958	1	1	yes	characteristic
184	C₄	—	16	2	no	—
185	C₄	—	1424	2	no	—
186	C₄	#19151423	1	4	yes	—
187	C₄	#19151425	1	4	yes	—
188	C₄	#19151463	1	1	yes	characteristic
189	C₄	C₂×#15667943	1	1	yes	characteristic
190	C₄	#19151948	1	1	yes	characteristic
191	C₄	#19151950	1	1	yes	characteristic
192	C₂	#26020379	1	1	yes	characteristic
193	C₂	C₂×#19151423	1	1	yes	characteristic
194	C₂	C₂×#19151425	1	1	yes	characteristic
195	C₂	#26022475	1	1	yes	characteristic
196	C₂	#26022477	1	1	yes	characteristic
197	C₂	#26022479	1	1	yes	characteristic
198	C₂	#26022481	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083559
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1845	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1845	D₈₉
hypocenter	C₈₉	#12888
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
9. C₄×#13083549	C₄	semidirect
104. C₈₉	#12888	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#13083549	2
#26020412	2
#26018343	2
C₈₉×#1845	2
#12888	89