Subgroups — #36473049

Order 22784.

370 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030766	C₂	1	1	yes	maximal, characteristic
2	#26020409	C₂	1	1	yes	maximal, characteristic
3	#26018358	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19152155	—	2	1	no	—
6	#19152003	C₄	1	2	yes	complemented
7	#19152000	—	2	1	no	—
8	#19152000	C₄	1	2	yes	complemented
9	#19151996	—	2	1	no	—
10	#19151636	—	2	1	no	—
11	#19151636	C₂²	1	2	yes	—
12	C₂×#15667944	—	2	1	no	—
13	C₂×#15667944	C₂²	1	2	yes	—
14	#19151468	—	2	1	no	—
15	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#346×C₈₉	—	2	1	no	—
18	C₄×C₁₆×C₈₉	—	2	1	no	—
19	#15668012	—	2	2	no	—
20	#15668004	—	2	2	no	—
21	#15667991	—	4	2	no	—
22	C₄×Dic₁₇₈	—	4	2	no	—
23	#15667971	—	2	2	no	—
24	C₂×C₄⋊Dic₈₉	—	2	2	no	—
25	C₂×C₄⋊Dic₈₉	—	4	2	no	—
26	#15667946	—	2	2	no	—
27	#15667946	C₂×C₄	1	4	yes	—
28	#15667944	—	2	2	no	—
29	#15667944	—	2	2	no	—
30	#15667944	—	2	4	no	—
31	#15667944	—	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	M₃₂×C₈₉	—	2	2	no	—
36	M₃₂×C₈₉	Q₈	1	4	yes	complemented
37	C₂×C₁₆×C₈₉	—	2	2	no	—
38	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
42	C₂×Dic₁₇₈	—	4	2	no	—
43	C₂²×Dic₈₉	—	4	2	no	—
44	C₄⋊Dic₈₉	—	2	4	no	—
45	C₄⋊Dic₈₉	—	4	2	no	—
46	C₄⋊Dic₈₉	—	4	2	no	—
47	C₄⋊Dic₈₉	—	4	2	no	—
48	C₄⋊Dic₈₉	—	8	4	no	—
49	#13083553	—	4	4	no	—
50	C₂²⋊Dic₈₉	—	4	2	no	—
51	C₄×Dic₈₉	—	4	2	no	—
52	C₂²×C₄×C₈₉	D₈	1	2	yes	—
53	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
57	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
58	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₈₉	#41	1	2	yes	—
62	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₈₉	SD₁₆	1	2	yes	—
65	C₁₆×C₈₉	—	2	4	no	—
66	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	—	2	4	no	—
68	C₂×C₄×C₈₉	#102	1	2	yes	—
69	C₂×C₄×C₈₉	#102	1	2	yes	—
70	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×Dic₈₉	—	4	4	no	—
77	C₂×Dic₈₉	—	8	4	no	—
78	C₂×Dic₈₉	—	8	2	no	—
79	Dic₁₇₈	—	8	4	no	—
80	C₈×C₈₉	#103	1	2	yes	—
81	C₈×C₈₉	#103	1	2	yes	—
82	C₈×C₈₉	C₄×Q₈	1	2	yes	—
83	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
84	C₂²×C₈₉	—	2	2	no	—
85	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
86	C₂²×C₈₉	#486	1	1	yes	characteristic
87	C₂²×C₈₉	#500	1	1	yes	characteristic
88	C₄×C₈₉	—	2	2	no	—
89	C₄×C₈₉	#360	1	4	yes	—
90	C₄×C₈₉	#391	1	1	yes	characteristic
91	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
92	C₄×C₈₉	#487	1	1	yes	characteristic
93	C₄×C₈₉	#498	1	1	yes	characteristic
94	Dic₈₉	—	8	4	no	—
95	Dic₈₉	—	16	2	no	—
96	#13287	—	89	1	no	maximal, Sylow
97	C₂×C₈₉	—	2	1	no	—
98	C₂×C₈₉	—	2	1	no	—
99	C₂×C₈₉	#1990	1	1	yes	characteristic
100	C₂×C₈₉	C₂×#360	1	2	yes	—
101	#3160	—	89	1	no	—
102	C₄×M₃₂	D₈₉	1	1	yes	characteristic
103	#1844	—	89	1	no	—
104	#1386	—	89	4	no	—
105	C₈₉	#13287	1	1	yes	characteristic, complemented, Sylow
106	#533	—	178	1	no	—
107	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
108	#500	—	89	2	no	—
109	#499	—	89	2	no	—
110	#499	—	178	1	no	—
111	#498	—	178	1	no	—
112	#427	—	89	2	no	—
113	#427	—	178	1	no	—
114	C₂×#106	—	89	2	no	—
115	C₂×#106	—	178	1	no	—
116	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
117	#384	—	178	1	no	—
118	#346	—	2	1	no	—
119	C₄×C₁₆	—	2	1	no	—
120	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
122	C₄⋊Q₈	—	178	2	no	—
123	#125	—	178	2	no	—
124	C₂²⋊Q₈	—	356	2	no	—
125	C₄×Q₈	—	356	2	no	—
126	#117	—	178	2	no	—
127	C₂×C₄⋊C₄	—	356	2	no	—
128	C₂×C₄⋊C₄	—	178	2	no	—
129	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
130	M₃₂	—	2	2	no	—
131	M₃₂	Dic₁₇₈	1	4	yes	complemented
132	C₂×C₁₆	—	2	2	no	—
133	#107	—	89	4	no	—
134	#107	—	178	2	no	—
135	#106	—	356	2	no	—
136	#106	—	178	4	no	—
137	#106	—	178	2	no	—
138	#106	—	178	2	no	—
139	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
140	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
141	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
143	C₂×Q₈	—	356	2	no	—
144	C₂²×C₄	—	356	2	no	—
145	C₂²×C₄	D₇₁₂	1	2	yes	—
146	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
147	C₂×C₈	#13083551	1	1	yes	characteristic
148	C₂×C₈	#13083551	1	1	yes	characteristic
149	C₂×C₈	#13083563	1	2	yes	—
150	C₂×C₈	Dic₃₅₆	1	2	yes	—
151	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
152	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
153	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
154	C₂×C₈	#13083574	1	2	yes	—
155	C₄⋊C₄	—	356	4	no	—
156	C₄⋊C₄	—	356	2	no	—
157	C₄⋊C₄	—	356	2	no	—
158	C₄⋊C₄	—	356	2	no	—
159	C₄⋊C₄	—	178	4	no	—
160	C₄⋊C₄	—	712	4	no	—
161	C₂²⋊C₄	—	356	2	no	—
162	C₄²	—	356	2	no	—
163	C₄²	#13083551	1	1	yes	characteristic
164	C₄²	#13083551	1	1	yes	characteristic
165	C₄²	#13083563	1	2	yes	—
166	C₁₆	—	2	4	no	—
167	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
168	Q₈	—	712	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	356	4	no	—
171	C₂×C₄	—	712	4	no	—
172	C₂×C₄	—	712	2	no	—
173	C₂×C₄	#15667934	1	2	yes	—
174	C₂×C₄	#15667934	1	2	yes	—
175	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
176	C₂×C₄	#15668000	1	1	yes	characteristic
177	C₂×C₄	#15668012	1	1	yes	characteristic
178	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
179	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
180	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
181	C₈	#15667937	1	2	yes	—
182	C₈	#15667937	1	2	yes	—
183	C₈	C₄×Dic₁₇₈	1	2	yes	—
184	C₈	#15667991	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×#15667934	1	1	yes	characteristic
187	C₂²	#19151945	1	1	yes	characteristic
188	C₂²	#19152003	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	1424	2	no	—
191	C₄	—	712	4	no	—
192	C₄	#19151409	1	4	yes	—
193	C₄	#19151504	1	1	yes	characteristic
194	C₄	C₂×#15667937	1	1	yes	characteristic
195	C₄	#19151949	1	1	yes	characteristic
196	C₄	#19151996	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#26021213	1	1	yes	characteristic
200	C₂	C₂×#19151409	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#15667934
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₄×M₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#102
prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄×M₃₂	D₈₉
hypocenter	C₈₉	#13287
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
6. #19152003	C₄	semidirect
8. #19152000	C₄	semidirect
36. M₃₂×C₈₉	Q₈	semidirect
105. C₈₉	#13287	semidirect
131. M₃₂	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26030766	2
#26020409	2
#26018358	2
C₄×M₃₂×C₈₉	2
#13287	89