Subgroups — #36463477

Order 22784.

248 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022055	C₂	1	1	yes	maximal, characteristic
2	#26018541	C₂	1	1	yes	maximal, characteristic
3	#26018174	C₂	1	2	yes	maximal, complemented
4	#26018172	C₂	1	2	yes	maximal, complemented
5	C₄×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₄	1	1	yes	characteristic
7	C₄×#13083558	C₄	1	1	yes	characteristic
8	C₄⋊#13083549	C₄	1	2	yes	—
9	C₂²⋊#13083549	C₄	1	2	yes	—
10	#19151374	C₂²	1	2	yes	—
11	#19151320	C₄	1	2	yes	—
12	#19151319	C₂²	1	2	yes	—
13	C₂×M₃₂×C₈₉	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	#346×C₈₉	C₂²	1	1	yes	characteristic
16	#346×C₈₉	C₂²	1	1	yes	characteristic
17	C₄×C₁₆×C₈₉	C₄	1	2	yes	—
18	C₂×#13083558	—	2	1	no	—
19	C₂×#13083549	C₂×C₄	1	4	yes	—
20	C₄⋊#10623494	—	2	1	no	—
21	C₂²⋊#10623494	—	2	1	no	—
22	#15667923	—	2	1	no	—
23	C₄×#10623494	C₂×C₄	1	2	yes	—
24	C₄×#10623494	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₈₉	—	2	4	no	—
29	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
35	#13083558	—	4	2	no	—
36	C₂×#10623494	—	2	2	no	—
37	C₂×#10623494	—	2	2	no	—
38	#13083549	—	2	4	no	—
39	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
40	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
42	C₂×C₈×C₈₉	—	2	2	no	—
43	C₂×C₈×C₈₉	—	2	2	no	—
44	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
46	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₈₉	C₄²	1	1	yes	characteristic
49	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
50	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
51	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
52	C₁₆×C₈₉	—	2	2	no	—
53	C₁₆×C₈₉	M₁₆	1	4	yes	complemented
54	C₂³×C₈₉	#97	1	1	yes	characteristic
55	C₂×C₄×C₈₉	—	2	2	no	—
56	C₂×C₄×C₈₉	—	2	1	no	—
57	C₂×C₄×C₈₉	—	2	1	no	—
58	C₂×C₄×C₈₉	—	4	1	no	—
59	C₂×C₄×C₈₉	#97	1	1	yes	characteristic
60	C₂×C₄×C₈₉	#97	1	1	yes	characteristic
61	C₂×C₄×C₈₉	#97	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
65	C₈×C₈₉	—	2	1	no	—
66	C₈×C₈₉	—	2	1	no	—
67	C₈×C₈₉	C₂×M₁₆	1	2	yes	—
68	C₈×C₈₉	#131	1	2	yes	—
69	#10623494	—	4	2	no	—
70	#10623494	—	4	2	no	—
71	C₂²×C₈₉	—	2	1	no	—
72	C₂²×C₈₉	—	2	1	no	—
73	C₂²×C₈₉	—	2	1	no	—
74	C₂²×C₈₉	C₂×#97	1	1	yes	characteristic
75	C₄×C₈₉	—	2	1	no	—
76	C₄×C₈₉	—	2	1	no	—
77	C₄×C₈₉	—	2	1	no	—
78	C₄×C₈₉	—	2	1	no	—
79	C₄×C₈₉	#347	1	2	yes	—
80	C₄×C₈₉	C₄×M₁₆	1	1	yes	characteristic
81	C₄×C₈₉	#405	1	1	yes	characteristic
82	#11414	—	89	1	no	maximal, Sylow
83	C₂×C₈₉	—	4	1	no	—
84	C₂×C₈₉	#1451	1	1	yes	characteristic
85	C₂×C₈₉	C₂×#347	1	1	yes	characteristic
86	C₂×C₈₉	#2110	1	1	yes	characteristic
87	#2163	—	89	1	no	—
88	C₄×M₃₂	D₈₉	1	1	yes	characteristic
89	#1451	—	89	1	no	—
90	#1313	—	89	2	no	—
91	#1313	—	89	2	no	—
92	C₈₉	#11414	1	1	yes	characteristic, complemented, Sylow
93	C₂×M₃₂	Dic₈₉	1	2	yes	—
94	#431	—	89	1	no	—
95	C₄×M₁₆	—	89	1	no	—
96	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
97	C₄⋊C₁₆	—	89	2	no	—
98	C₂²⋊C₁₆	—	89	2	no	—
99	#346	—	89	2	no	—
100	#346	C₂×D₈₉	1	1	yes	characteristic
101	#346	C₂×D₈₉	1	1	yes	characteristic
102	C₄×C₁₆	Dic₈₉	1	2	yes	—
103	#322	—	89	2	no	—
104	#322	—	89	2	no	—
105	C₂×M₁₆	—	178	1	no	—
106	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
108	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
109	M₃₂	—	2	4	no	—
110	C₂×C₁₆	—	89	4	no	—
111	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
112	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
113	C₄⋊C₈	—	178	1	no	—
114	C₂²⋊C₈	—	178	1	no	—
115	#97	—	178	1	no	—
116	C₄×C₈	—	89	2	no	—
117	C₄×C₈	—	89	2	no	—
118	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
119	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
122	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
123	C₂²×C₄	#13083557	1	1	yes	characteristic
124	C₂²×C₄	#13083557	1	1	yes	characteristic
125	M₁₆	—	356	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	178	2	no	—
129	C₂×C₈	—	178	2	no	—
130	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
131	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
133	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
134	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
135	C₄²	#13083557	1	1	yes	characteristic
136	C₄²	#13083557	1	1	yes	characteristic
137	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
138	C₁₆	—	2	2	no	—
139	C₁₆	—	178	4	no	—
140	C₁₆	#13083558	1	4	yes	complemented
141	C₂³	#15667922	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	#15667922	1	1	yes	characteristic
147	C₂×C₄	#15667922	1	1	yes	characteristic
148	C₂×C₄	#15667922	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
150	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
151	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	356	2	no	—
155	C₈	—	356	2	no	—
156	C₈	C₂×#13083558	1	2	yes	—
157	C₈	#15668018	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	C₂×#15667922	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	#19151375	1	2	yes	—
167	C₄	C₄×#13083558	1	1	yes	characteristic
168	C₄	#19151553	1	1	yes	characteristic
169	C₂	—	4	1	no	—
170	C₂	#26018541	1	1	yes	characteristic
171	C₂	C₂×#19151375	1	1	yes	characteristic
172	C₂	#26021856	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₄×M₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#97
prime radical	C₂³×C₈₉	#97
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₄×M₃₂	D₈₉
hypocenter	C₈₉	#11414
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #26018174	C₂	semidirect
4. #26018172	C₂	semidirect
53. C₁₆×C₈₉	M₁₆	semidirect
92. C₈₉	#11414	semidirect
140. C₁₆	#13083558	semidirect

Maximal subgroups

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

subgroup	index
#26022055	2
#26018541	2
#26018174	2
#26018172	2
C₄×M₃₂×C₈₉	2
#11414	89