Subgroups — #36707514

Order 22784.

590 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151552	C₂	1	1	yes	maximal, characteristic
2	#26022058	C₂	1	8	yes	maximal, complemented
3	#26022008	C₂	1	1	yes	maximal, characteristic, complemented
4	#26021862	C₂	1	2	yes	maximal
5	M₃₂×Dic₈₉	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#13083557	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
9	C₂×#15667969	C₄	1	1	yes	characteristic
10	C₂×#15667952	C₂²	1	2	yes	—
11	#19151552	C₄	1	4	yes	—
12	#19151552	C₂²	1	4	yes	—
13	C₂×#15667923	C₄	1	1	yes	characteristic
14	C₂×#15667922	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
16	C₄⋊#13083549	C₂²	1	4	yes	complemented
17	#19151377	C₂²	1	8	yes	—
18	#19151373	C₂²	1	4	yes	complemented
19	C₁₆×Dic₈₉	C₂²	1	4	yes	complemented
20	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆×C₈₉	C₂²	1	4	yes	complemented
23	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂×#13083557	—	2	4	no	—
25	C₂×#13083557	C₂×C₄	1	4	yes	—
26	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
27	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₈₉	—	2	2	no	—
29	C₂×C₈×D₈₉	—	2	2	no	—
30	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₈₉	C₂³	1	2	yes	—
32	#15667969	C₈	1	4	yes	—
33	#15667969	C₂×C₄	1	4	yes	—
34	C₂×#13083553	C₈	1	2	yes	—
35	C₂×#13083551	C₈	1	2	yes	—
36	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
37	#15667952	—	2	4	no	—
38	C₂×#13083549	C₂³	1	4	yes	—
39	#15667923	C₂×C₄	1	4	yes	—
40	#15667922	C₂×C₄	1	4	yes	—
41	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
42	C₈×Dic₈₉	C₂³	1	2	yes	—
43	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
46	M₃₂×C₈₉	—	2	4	no	—
47	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
48	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₈₉	C₂³	1	2	yes	—
50	C₂³×D₈₉	C₂×C₈	1	1	yes	characteristic
51	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
52	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
53	C₂×C₄×D₈₉	—	2	2	no	—
54	C₂×C₄×D₈₉	—	2	2	no	—
55	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
56	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
57	#13083557	—	2	8	no	—
58	C₂×#10623494	—	2	2	no	—
59	C₂×#10623494	M₁₆	1	4	yes	—
60	C₂×#10623494	C₂²×C₄	1	2	yes	—
61	C₂×#10623494	C₂²×C₄	1	2	yes	—
62	C₈×D₈₉	—	2	4	no	—
63	C₈×D₈₉	—	2	2	no	—
64	C₈×D₈₉	—	2	2	no	—
65	#13083553	C₂×C₈	1	8	yes	—
66	#13083551	C₂×C₈	1	8	yes	—
67	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
68	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
69	#13083549	—	2	4	no	—
70	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	—	2	2	no	—
73	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₈₉	#41	1	2	yes	—
76	C₂×C₈×C₈₉	#41	1	2	yes	—
77	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₄²×C₈₉	C₂×C₈	1	2	yes	—
79	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
80	C₁₆×C₈₉	—	2	4	no	—
81	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
82	C₂²×D₈₉	—	2	2	no	—
83	C₂²×D₈₉	—	4	2	no	—
84	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₈₉	—	4	2	no	—
86	C₂×C₄×C₈₉	#117	1	2	yes	—
87	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
88	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
90	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
91	C₂×Dic₈₉	—	2	2	no	—
92	C₂×Dic₈₉	—	4	2	no	—
93	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
94	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
95	C₄×D₈₉	—	2	2	no	—
96	C₄×D₈₉	—	2	2	no	—
97	C₄×D₈₉	—	2	4	no	—
98	C₈×C₈₉	—	2	2	no	—
99	C₈×C₈₉	#117	1	2	yes	—
100	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
101	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
102	#10623494	—	2	2	no	—
103	#10623494	C₂×M₁₆	1	4	yes	—
104	C₂²×C₈₉	—	2	2	no	—
105	C₂²×C₈₉	#431	1	2	yes	—
106	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
107	C₂×D₈₉	—	2	4	no	—
108	C₂×D₈₉	—	2	2	no	—
109	C₂×D₈₉	—	4	4	no	—
110	C₄×C₈₉	—	2	4	no	—
111	C₄×C₈₉	#431	1	2	yes	—
112	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
113	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
114	Dic₈₉	—	2	4	no	—
115	Dic₈₉	—	2	2	no	—
116	#33419	—	89	1	no	maximal, Sylow
117	C₂×C₈₉	—	2	2	no	—
118	C₂×C₈₉	C₂×#431	1	1	yes	characteristic
119	C₂×C₈₉	#3407	1	2	yes	—
120	D₈₉	—	4	2	no	—
121	C₂×#405	—	89	1	no	—
122	#2164	—	89	8	no	—
123	#2150	—	89	1	no	—
124	#2150	D₈₉	1	1	yes	characteristic, complemented
125	#2112	—	89	2	no	—
126	C₄×M₃₂	—	89	2	no	—
127	C₈₉	#33419	1	1	yes	characteristic, complemented, Sylow
128	C₂²×M₁₆	—	89	1	no	—
129	C₂³×C₈	—	89	1	no	—
130	C₂×#117	—	89	1	no	—
131	C₂×M₃₂	—	89	2	no	—
132	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
133	#405	—	89	4	no	—
134	#405	—	89	4	no	—
135	C₂×#97	—	89	1	no	—
136	C₂×#97	—	89	1	no	—
137	C₂×C₄×C₈	—	89	1	no	—
138	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
139	C₄⋊C₁₆	—	89	4	no	—
140	C₄⋊C₁₆	C₂×D₈₉	1	4	yes	complemented
141	C₂²⋊C₁₆	—	89	8	no	—
142	#346	—	89	4	no	—
143	C₄×C₁₆	—	89	4	no	—
144	C₂³×C₄	—	89	1	no	—
145	C₂×M₁₆	—	89	4	no	—
146	C₂×M₁₆	—	178	4	no	—
147	C₂²×C₈	—	89	1	no	—
148	C₂²×C₈	—	89	1	no	—
149	C₂²×C₈	—	89	2	no	—
150	C₂²×C₈	—	89	2	no	—
151	C₂²×C₈	—	178	2	no	—
152	C₂²×C₈	—	178	2	no	—
153	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
154	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
155	#117	—	89	4	no	—
156	#117	—	89	4	no	—
157	C₂×C₄⋊C₄	—	89	2	no	—
158	C₂×C₂²⋊C₄	—	89	2	no	—
159	C₂×C₄²	—	89	1	no	—
160	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
161	M₃₂	—	2	4	no	—
162	M₃₂	—	178	4	no	—
163	C₂×C₁₆	—	89	4	no	—
164	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
165	#97	—	89	4	no	—
166	#97	—	89	4	no	—
167	C₄×C₈	—	89	2	no	—
168	C₄×C₈	—	89	2	no	—
169	C₄×C₈	C₄×D₈₉	1	2	yes	—
170	C₄×C₈	C₂²×D₈₉	1	2	yes	—
171	C₂⁴	—	89	1	no	—
172	C₂²×C₄	—	89	2	no	—
173	C₂²×C₄	—	89	1	no	—
174	C₂²×C₄	—	89	2	no	—
175	C₂²×C₄	—	89	2	no	—
176	C₂²×C₄	—	178	2	no	—
177	C₂²×C₄	—	178	2	no	—
178	C₂²×C₄	C₈×D₈₉	1	2	yes	—
179	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
180	M₁₆	—	178	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	89	4	no	—
183	C₂×C₈	—	89	2	no	—
184	C₂×C₈	—	89	2	no	—
185	C₂×C₈	—	178	2	no	—
186	C₂×C₈	—	178	4	no	—
187	C₂×C₈	—	178	2	no	—
188	C₂×C₈	—	178	2	no	—
189	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
190	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
191	C₂×C₈	#13083573	1	2	yes	—
192	C₂×C₈	#13083575	1	2	yes	—
193	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
194	C₄⋊C₄	—	89	8	no	—
195	C₂²⋊C₄	—	89	8	no	—
196	C₄²	—	89	2	no	—
197	C₄²	—	89	2	no	—
198	C₄²	C₈×D₈₉	1	2	yes	—
199	C₄²	C₂×C₄×D₈₉	1	2	yes	—
200	C₁₆	—	2	4	no	—
201	C₁₆	—	178	4	no	—
202	C₂³	—	89	2	no	—
203	C₂³	—	356	2	no	—
204	C₂³	—	178	2	no	—
205	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	89	4	no	—
208	C₂×C₄	—	89	2	no	—
209	C₂×C₄	—	356	2	no	—
210	C₂×C₄	—	178	2	no	—
211	C₂×C₄	—	178	4	no	—
212	C₂×C₄	—	178	2	no	—
213	C₂×C₄	—	178	2	no	—
214	C₂×C₄	#15667970	1	2	yes	—
215	C₂×C₄	C₂×C₈×D₈₉	1	4	yes	—
216	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
217	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
218	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	—	89	4	no	—
221	C₈	—	178	2	no	—
222	C₈	#15667970	1	2	yes	—
223	C₈	C₂×#13083573	1	1	yes	characteristic
224	C₈	C₂×#13083575	1	1	yes	characteristic
225	C₂²	—	2	2	no	—
226	C₂²	—	356	4	no	—
227	C₂²	—	178	2	no	—
228	C₂²	—	178	4	no	—
229	C₂²	#19151650	1	2	yes	—
230	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
231	C₄	—	2	4	no	—
232	C₄	—	178	2	no	—
233	C₄	—	178	4	no	—
234	C₄	#19151650	1	2	yes	—
235	C₄	C₂×#15667970	1	1	yes	characteristic
236	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	—	356	2	no	—
239	C₂	C₂×#19151650	1	1	yes	characteristic
240	C₂	#26033354	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2150	C₂
prime radical	C₂³×D₈₉	C₂×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2150	D₈₉
hypocenter	C₈₉	#33419
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #26022058	C₂	semidirect
3. #26022008	C₂	semidirect
5. M₃₂×Dic₈₉	C₂	semidirect
6. C₈₉×#2150	C₂	semidirect
16. C₄⋊#13083549	C₂²	semidirect
18. #19151373	C₂²	semidirect
19. C₁₆×Dic₈₉	C₂²	semidirect
22. C₄⋊C₁₆×C₈₉	C₂²	semidirect
124. #2150	D₈₉	semidirect
127. C₈₉	#33419	semidirect
140. C₄⋊C₁₆	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151552	2
#26022058	2
#26022008	2
#26021862	2
M₃₂×Dic₈₉	2
C₈₉×#2150	2
#33419	89