Subgroups — #36469756

Order 22784.

334 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022008	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151375	C₂	1	2	yes	maximal
3	#26019938	C₂	1	1	yes	maximal, characteristic
4	#26018331	C₂	1	2	yes	maximal
5	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667952	C₂²	1	2	yes	—
7	C₂×#15667923	C₄	1	1	yes	characteristic
8	C₂×#15667922	C₂²	1	1	yes	characteristic
9	C₂×#15667922	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
12	#19151444	C₄	1	1	yes	characteristic
13	C₄⋊#13083549	C₄	1	4	yes	complemented
14	#19151375	—	2	4	no	—
15	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₈₉	C₄	1	4	yes	complemented
18	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
19	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
22	#15667952	—	2	2	no	—
23	#15667952	—	2	2	no	—
24	C₂×#13083549	C₂×C₄	1	4	yes	—
25	#15667923	—	2	2	no	—
26	#15667922	—	2	1	no	—
27	#15667922	—	2	1	no	—
28	#15667922	D₄	1	2	yes	—
29	#15667922	Q₈	1	2	yes	—
30	C₈×Dic₈₉	—	2	2	no	—
31	C₈×Dic₈₉	—	2	2	no	—
32	#15667919	—	2	2	no	—
33	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₈₉	—	2	2	no	—
37	M₃₂×C₈₉	—	2	2	no	—
38	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
41	C₂²×Dic₈₉	—	2	1	no	—
42	C₂²×Dic₈₉	—	2	1	no	—
43	C₂×#10623494	—	2	1	no	—
44	C₂×#10623494	—	2	1	no	—
45	C₂×#10623494	—	2	1	no	—
46	C₂×#10623494	—	2	1	no	—
47	C₂×#10623494	C₄⋊C₄	1	2	yes	—
48	C₂×#10623494	M₁₆	1	2	yes	—
49	C₄×Dic₈₉	—	2	1	no	—
50	C₄×Dic₈₉	—	2	1	no	—
51	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
52	C₄×Dic₈₉	M₁₆	1	2	yes	—
53	#13083549	—	2	4	no	—
54	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
55	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₈₉	—	2	1	no	—
57	C₂×C₈×C₈₉	—	2	1	no	—
58	C₂×C₈×C₈₉	C₄²	1	2	yes	—
59	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
64	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
65	C₄²×C₈₉	C₄²	1	2	yes	—
66	C₄²×C₈₉	M₁₆	1	2	yes	—
67	C₁₆×C₈₉	—	2	4	no	—
68	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₈₉	—	4	2	no	—
70	C₂×C₄×C₈₉	#97	1	4	yes	—
71	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
77	C₂×Dic₈₉	—	2	2	no	—
78	C₂×Dic₈₉	—	2	2	no	—
79	C₂×Dic₈₉	—	4	2	no	—
80	C₂×Dic₈₉	—	4	2	no	—
81	C₈×C₈₉	—	2	2	no	—
82	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
83	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
84	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
85	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
86	#10623494	—	2	2	no	—
87	#10623494	—	2	2	no	—
88	C₂²×C₈₉	—	2	2	no	—
89	C₂²×C₈₉	C₂×#97	1	1	yes	characteristic
90	C₂²×C₈₉	#423	1	1	yes	characteristic
91	C₂²×C₈₉	#432	1	1	yes	characteristic
92	C₄×C₈₉	—	2	4	no	—
93	C₄×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
94	C₄×C₈₉	C₄×M₁₆	1	1	yes	characteristic
95	C₄×C₈₉	#423	1	1	yes	characteristic
96	C₄×C₈₉	#432	1	1	yes	characteristic
97	Dic₈₉	—	4	2	no	—
98	Dic₈₉	—	4	2	no	—
99	#12657	—	89	1	no	maximal, Sylow
100	C₂×C₈₉	—	2	1	no	—
101	C₂×C₈₉	—	2	1	no	—
102	C₂×C₈₉	#1767	1	1	yes	characteristic
103	C₂×C₈₉	#2110	1	2	yes	—
104	#2150	—	89	1	no	—
105	#2150	D₈₉	1	1	yes	characteristic
106	C₂×#347	—	89	2	no	—
107	#1726	—	89	1	no	—
108	#1375	—	89	2	no	—
109	C₈₉	#12657	1	1	yes	characteristic, complemented, Sylow
110	C₂×M₃₂	—	89	2	no	—
111	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
112	C₂×#97	—	89	1	no	—
113	C₂×#97	—	89	1	no	—
114	C₂×#97	—	89	1	no	—
115	C₂×C₄×C₈	—	89	1	no	—
116	C₂×C₄×C₈	—	89	1	no	—
117	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
118	#376	—	89	1	no	—
119	C₄⋊C₁₆	—	89	4	no	—
120	C₄⋊C₁₆	Dic₈₉	1	4	yes	complemented
121	#347	—	178	4	no	—
122	C₂²×C₈	—	89	1	no	—
123	C₂²×C₈	—	89	1	no	—
124	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
126	C₂×C₄²	—	89	1	no	—
127	C₂×C₄²	—	89	1	no	—
128	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
129	M₃₂	—	2	2	no	—
130	M₃₂	—	2	2	no	—
131	M₃₂	—	178	2	no	—
132	M₃₂	—	178	2	no	—
133	C₂×C₁₆	—	89	4	no	—
134	C₂×C₁₆	C₂×Dic₈₉	1	4	yes	—
135	#97	—	89	2	no	—
136	#97	—	89	2	no	—
137	#97	—	178	1	no	—
138	#97	—	178	1	no	—
139	#97	—	178	2	no	—
140	C₄×C₈	—	178	2	no	—
141	C₄×C₈	—	178	2	no	—
142	C₄×C₈	C₄×D₈₉	1	2	yes	—
143	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
144	#95	—	178	2	no	—
145	C₂²×C₄	—	178	1	no	—
146	C₂²×C₄	—	178	1	no	—
147	C₂²×C₄	#13083557	1	2	yes	—
148	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	89	2	no	—
152	C₂×C₈	—	89	2	no	—
153	C₂×C₈	—	178	1	no	—
154	C₂×C₈	—	178	1	no	—
155	C₂×C₈	—	178	1	no	—
156	C₂×C₈	—	178	1	no	—
157	C₂×C₈	C₄×Dic₈₉	1	2	yes	—
158	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
159	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
160	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
161	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
162	C₂×C₈	#13083573	1	1	yes	characteristic
163	C₂×C₈	#13083575	1	1	yes	characteristic
164	C₄²	—	89	2	no	—
165	C₄²	—	89	2	no	—
166	C₄²	—	178	1	no	—
167	C₄²	—	178	1	no	—
168	C₄²	C₄×Dic₈₉	1	2	yes	—
169	C₄²	#13083557	1	2	yes	—
170	C₁₆	—	2	4	no	—
171	C₁₆	—	178	4	no	—
172	C₂³	C₂×#13083557	1	1	yes	characteristic
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	356	2	no	—
175	C₂×C₄	—	356	2	no	—
176	C₂×C₄	—	178	2	no	—
177	C₂×C₄	—	178	2	no	—
178	C₂×C₄	#15667922	1	4	yes	—
179	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
180	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
181	C₂×C₄	#15667970	1	1	yes	characteristic
182	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
183	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
184	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	178	2	no	—
187	C₈	—	178	2	no	—
188	C₈	#15667974	1	1	yes	characteristic
189	C₈	D₄×Dic₈₉	1	1	yes	characteristic
190	C₈	#15667977	1	1	yes	characteristic
191	C₈	Q₈×Dic₈₉	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	C₂×#15667922	1	1	yes	characteristic
194	C₂²	#19151620	1	1	yes	characteristic
195	C₂²	#19151654	1	1	yes	characteristic
196	C₄	—	2	4	no	—
197	C₄	—	356	2	no	—
198	C₄	—	356	2	no	—
199	C₄	C₄⋊C₄×Dic₈₉	1	1	yes	characteristic
200	C₄	M₁₆×Dic₈₉	1	1	yes	characteristic
201	C₄	#19151620	1	1	yes	characteristic
202	C₄	#19151654	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	#26020107	1	1	yes	characteristic
206	C₂	#26021856	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2150	C₂
socle	C₂²×C₈₉	C₂×#97
prime radical	C₂³×C₈₉	C₂×M₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2150	D₈₉
hypocenter	C₈₉	#12657
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #2150

Complements

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

N	H ≅ G/N	type
13. C₄⋊#13083549	C₄	semidirect
17. C₄⋊C₁₆×C₈₉	C₄	semidirect
109. C₈₉	#12657	semidirect
120. C₄⋊C₁₆	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26022008	2
C₂×#19151375	2
#26019938	2
#26018331	2
C₈₉×#2150	2
#12657	89