Subgroups — #36712657

Order 22784.

3598 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×Dic₈₉	C₂	1	2	yes	maximal
2	#26022617	C₂	1	4	yes	maximal, complemented
3	#26022613	C₂	1	12	yes	maximal, complemented
4	#117×Dic₈₉	C₂	1	6	yes	maximal, complemented
5	C₄×#15667971	C₂	1	3	yes	maximal, complemented
6	C₂×#19151444	C₂	1	3	yes	maximal, complemented
7	C₂×C₄×#41×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#41×Dic₈₉	C₄	1	32	yes	complemented
9	C₂×Q₈×Dic₈₉	C₂²	1	4	yes	—
10	C₂×D₄×Dic₈₉	C₂²	1	12	yes	—
11	C₂×#15667971	C₂²	1	6	yes	—
12	C₂²×C₄×Dic₈₉	C₂²	1	6	yes	—
13	C₄⋊C₄×Dic₈₉	C₂²	1	24	yes	—
14	C₄×C₄⋊Dic₈₉	C₂²	1	12	yes	—
15	C₂²⋊C₄×Dic₈₉	C₂²	1	24	yes	—
16	C₄×C₂²⋊Dic₈₉	C₂²	1	12	yes	—
17	#19151444	C₂²	1	1	yes	characteristic
18	#19151444	C₂²	1	3	yes	—
19	#19151444	C₂²	1	6	yes	—
20	#19151444	C₂²	1	12	yes	—
21	C₂×#15667919	C₂²	1	12	yes	—
22	C₄²×Dic₈₉	C₂²	1	6	yes	—
23	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
24	C₄×#41×C₈₉	C₄	1	16	yes	complemented
25	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₈₉	C₂²	1	6	yes	—
27	C₂×#117×C₈₉	C₂²	1	3	yes	—
28	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
29	C₂³×Dic₈₉	C₂³	1	12	yes	—
30	Q₈×Dic₈₉	C₂×C₄	1	32	yes	complemented
31	D₄×Dic₈₉	C₂×C₄	1	96	yes	complemented
32	#15667971	C₂×C₄	1	48	yes	complemented
33	C₂×C₄⋊Dic₈₉	C₂³	1	12	yes	—
34	C₂×C₂²⋊Dic₈₉	C₂³	1	12	yes	—
35	C₂×C₄×Dic₈₉	C₂×C₄	1	48	yes	complemented
36	C₂×C₄×Dic₈₉	C₂³	1	24	yes	—
37	C₂×C₄×Dic₈₉	C₂³	1	6	yes	—
38	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
39	C₂×C₄×Dic₈₉	C₂³	1	12	yes	—
40	#15667919	C₂³	1	4	yes	—
41	#15667919	C₂³	1	12	yes	—
42	#15667919	C₂³	1	24	yes	—
43	C₂×#41×C₈₉	C₂×C₄	1	16	yes	—
44	C₂×#41×C₈₉	C₂×C₄	1	8	yes	—
45	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₈₉	C₂³	1	3	yes	—
47	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
48	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
49	C₄×Q₈×C₈₉	C₂×C₄	1	16	yes	complemented
50	C₄×D₄×C₈₉	C₂×C₄	1	48	yes	complemented
51	#117×C₈₉	C₂×C₄	1	24	yes	complemented
52	C₂×C₄⋊C₄×C₈₉	C₂³	1	6	yes	—
53	C₂×C₂²⋊C₄×C₈₉	C₂³	1	6	yes	—
54	C₂×C₄²×C₈₉	C₂×C₄	1	24	yes	complemented
55	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
56	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	C₂³	1	6	yes	—
58	C₂²×Dic₈₉	—	4	24	no	—
59	C₂²×Dic₈₉	C₂²×C₄	1	48	yes	—
60	C₂²×Dic₈₉	C₂⁴	1	4	yes	—
61	C₂²×Dic₈₉	C₂⁴	1	12	yes	—
62	C₄⋊Dic₈₉	C₂²×C₄	1	48	yes	—
63	C₂²⋊Dic₈₉	C₂²×C₄	1	48	yes	—
64	C₄×Dic₈₉	—	4	24	no	—
65	C₄×Dic₈₉	C₂²×C₄	1	24	yes	—
66	C₄×Dic₈₉	C₂²×C₄	1	48	yes	—
67	C₄×Dic₈₉	C₂²×C₄	1	8	yes	—
68	C₂⁴×C₈₉	C₂⁴	1	3	yes	—
69	#41×C₈₉	C₄²	1	64	yes	complemented
70	C₂×Q₈×C₈₉	C₂²×C₄	1	8	yes	—
71	C₂×Q₈×C₈₉	C₂²×C₄	1	4	yes	—
72	C₂×D₄×C₈₉	C₂²×C₄	1	24	yes	—
73	C₂×D₄×C₈₉	C₂²×C₄	1	12	yes	—
74	C₂²×C₄×C₈₉	—	4	12	no	—
75	C₂²×C₄×C₈₉	C₂²×C₄	1	24	yes	—
76	C₂²×C₄×C₈₉	C₂²×C₄	1	24	yes	—
77	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
78	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
79	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
80	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
81	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₈₉	C₂²×C₄	1	24	yes	—
83	C₂²⋊C₄×C₈₉	C₂²×C₄	1	24	yes	—
84	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
85	C₄²×C₈₉	C₂²×C₄	1	12	yes	—
86	C₄²×C₈₉	C₂²×C₄	1	24	yes	—
87	C₂³×C₈₉	—	2	12	no	—
88	C₂³×C₈₉	C₂³×C₄	1	12	yes	—
89	C₂³×C₈₉	C₂³×C₄	1	6	yes	—
90	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
91	Q₈×C₈₉	C₂×C₄²	1	16	yes	—
92	D₄×C₈₉	C₂×C₄²	1	48	yes	—
93	C₂×C₄×C₈₉	—	2	4	no	—
94	C₂×C₄×C₈₉	—	2	12	no	—
95	C₂×C₄×C₈₉	—	4	24	no	—
96	C₂×C₄×C₈₉	C₂×C₄²	1	48	yes	—
97	C₂×C₄×C₈₉	C₂³×C₄	1	4	yes	—
98	C₂×C₄×C₈₉	C₂³×C₄	1	12	yes	—
99	C₂×C₄×C₈₉	C₂³×C₄	1	4	yes	—
100	C₂×C₄×C₈₉	C₂³×C₄	1	12	yes	—
101	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
102	C₂×C₄×C₈₉	C₂³×C₄	1	6	yes	—
103	C₂×Dic₈₉	—	2	24	no	—
104	C₂×Dic₈₉	—	2	8	no	—
105	C₂×Dic₈₉	—	4	48	no	—
106	C₂×Dic₈₉	C₂³×C₄	1	8	yes	—
107	C₂×Dic₈₉	C₂³×C₄	1	24	yes	—
108	C₂²×C₈₉	—	2	24	no	—
109	C₂²×C₈₉	—	2	12	no	—
110	C₂²×C₈₉	C₂²×C₄²	1	12	yes	—
111	C₂²×C₈₉	C₂⁴×C₄	1	2	yes	—
112	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
113	C₂²×C₈₉	#585	1	4	yes	—
114	C₄×C₈₉	—	2	4	no	—
115	C₄×C₈₉	—	2	12	no	—
116	C₄×C₈₉	C₂²×C₄²	1	4	yes	—
117	C₄×C₈₉	C₂²×C₄²	1	12	yes	—
118	Dic₈₉	—	2	8	no	—
119	Dic₈₉	—	2	24	no	—
120	#33999	—	89	1	no	maximal, Sylow
121	C₂×C₈₉	—	2	12	no	—
122	C₂×C₈₉	C₂³×C₄²	1	1	yes	characteristic
123	C₂×C₈₉	#3428	1	4	yes	—
124	C₂×C₈₉	#3428	1	2	yes	—
125	C₂×C₄×#41	—	89	2	no	—
126	C₂×C₄×#41	D₈₉	1	1	yes	characteristic
127	#2276	—	89	4	no	—
128	#2275	—	89	12	no	—
129	C₄×#117	—	89	3	no	—
130	C₄×#117	—	89	6	no	—
131	C₂×#376	—	89	3	no	—
132	C₈₉	#33999	1	1	yes	characteristic, complemented, Sylow
133	C₂²×#41	C₂×D₈₉	1	1	yes	characteristic
134	C₄×#41	—	89	32	no	—
135	C₄×#41	Dic₈₉	1	16	yes	complemented
136	C₂×C₄×Q₈	—	89	4	no	—
137	C₂×C₄×Q₈	C₂×D₈₉	1	2	yes	—
138	C₂×C₄×D₄	—	89	12	no	—
139	C₂×C₄×D₄	C₂×D₈₉	1	6	yes	—
140	C₂×#117	—	89	6	no	—
141	C₂×#117	C₂×D₈₉	1	3	yes	—
142	C₂²×C₄²	—	89	6	no	—
143	C₂²×C₄²	C₂×D₈₉	1	3	yes	—
144	C₄×C₄⋊C₄	—	89	24	no	—
145	C₄×C₄⋊C₄	—	89	12	no	—
146	C₄×C₂²⋊C₄	—	89	12	no	—
147	C₄×C₂²⋊C₄	—	89	24	no	—
148	#376	—	89	1	no	—
149	#376	—	89	3	no	—
150	#376	—	89	6	no	—
151	#376	—	89	12	no	—
152	C₂×#95	—	89	12	no	—
153	C₄³	—	89	6	no	—
154	C₂×#41	C₄×D₈₉	1	16	yes	—
155	C₂×#41	C₂×Dic₈₉	1	8	yes	—
156	C₂²×Q₈	C₂²×D₈₉	1	1	yes	characteristic
157	C₂²×D₄	C₂²×D₈₉	1	3	yes	—
158	C₂³×C₄	—	89	12	no	—
159	C₂³×C₄	C₂²×D₈₉	1	6	yes	—
160	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
161	C₄×Q₈	—	89	32	no	—
162	C₄×Q₈	C₂×Dic₈₉	1	16	yes	complemented
163	C₄×D₄	—	89	96	no	—
164	C₄×D₄	C₂×Dic₈₉	1	48	yes	complemented
165	#117	—	89	48	no	—
166	#117	C₂×Dic₈₉	1	24	yes	complemented
167	C₂×C₄⋊C₄	—	89	12	no	—
168	C₂×C₄⋊C₄	C₂²×D₈₉	1	6	yes	—
169	C₂×C₂²⋊C₄	—	89	12	no	—
170	C₂×C₂²⋊C₄	C₂²×D₈₉	1	6	yes	—
171	C₂×C₄²	—	89	48	no	—
172	C₂×C₄²	—	89	12	no	—
173	C₂×C₄²	—	89	6	no	—
174	C₂×C₄²	—	89	2	no	—
175	C₂×C₄²	—	89	24	no	—
176	C₂×C₄²	C₂×Dic₈₉	1	24	yes	complemented
177	C₂×C₄²	C₂²×D₈₉	1	6	yes	—
178	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
179	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
180	#95	—	89	4	no	—
181	#95	—	89	24	no	—
182	#95	—	89	12	no	—
183	C₂⁴	C₂³×D₈₉	1	3	yes	—
184	#41	C₄×Dic₈₉	1	64	yes	complemented
185	C₂×Q₈	C₂×C₄×D₈₉	1	8	yes	—
186	C₂×Q₈	C₂²×Dic₈₉	1	4	yes	—
187	C₂×D₄	C₂×C₄×D₈₉	1	24	yes	—
188	C₂×D₄	C₂²×Dic₈₉	1	12	yes	—
189	C₂²×C₄	—	4	12	no	—
190	C₂²×C₄	—	89	48	no	—
191	C₂²×C₄	—	89	4	no	—
192	C₂²×C₄	—	89	12	no	—
193	C₂²×C₄	—	356	24	no	—
194	C₂²×C₄	C₂×C₄×D₈₉	1	24	yes	—
195	C₂²×C₄	C₂²×Dic₈₉	1	24	yes	—
196	C₂²×C₄	C₂²×Dic₈₉	1	12	yes	—
197	C₂²×C₄	C₂³×D₈₉	1	2	yes	—
198	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
199	C₂²×C₄	C₂³×D₈₉	1	3	yes	—
200	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
201	C₄⋊C₄	—	89	48	no	—
202	C₄⋊C₄	C₂²×Dic₈₉	1	24	yes	—
203	C₂²⋊C₄	—	89	48	no	—
204	C₂²⋊C₄	C₂²×Dic₈₉	1	24	yes	—
205	C₄²	—	89	8	no	—
206	C₄²	—	89	24	no	—
207	C₄²	—	89	48	no	—
208	C₄²	—	356	24	no	—
209	C₄²	C₂²×Dic₈₉	1	4	yes	—
210	C₄²	C₂²×Dic₈₉	1	12	yes	—
211	C₄²	C₂²×Dic₈₉	1	24	yes	—
212	C₂³	—	2	12	no	—
213	C₂³	C₂²×C₄×D₈₉	1	12	yes	—
214	C₂³	C₂³×Dic₈₉	1	6	yes	—
215	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
216	Q₈	C₂×C₄×Dic₈₉	1	16	yes	—
217	D₄	C₂×C₄×Dic₈₉	1	48	yes	—
218	C₂×C₄	—	2	12	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	24	no	—
221	C₂×C₄	—	89	8	no	—
222	C₂×C₄	—	89	24	no	—
223	C₂×C₄	—	356	48	no	—
224	C₂×C₄	—	178	24	no	—
225	C₂×C₄	—	178	8	no	—
226	C₂×C₄	C₂×C₄×Dic₈₉	1	48	yes	—
227	C₂×C₄	C₂²×C₄×D₈₉	1	4	yes	—
228	C₂×C₄	C₂²×C₄×D₈₉	1	12	yes	—
229	C₂×C₄	C₂³×Dic₈₉	1	4	yes	—
230	C₂×C₄	C₂³×Dic₈₉	1	12	yes	—
231	C₂×C₄	C₂³×Dic₈₉	1	2	yes	—
232	C₂×C₄	C₂³×Dic₈₉	1	6	yes	—
233	C₂²	—	2	24	no	—
234	C₂²	—	2	12	no	—
235	C₂²	C₂²×C₄×Dic₈₉	1	12	yes	—
236	C₂²	C₂³×C₄×D₈₉	1	2	yes	—
237	C₂²	C₂⁴×Dic₈₉	1	1	yes	characteristic
238	C₂²	#19152434	1	4	yes	—
239	C₄	—	2	4	no	—
240	C₄	—	2	12	no	—
241	C₄	—	178	8	no	—
242	C₄	—	178	24	no	—
243	C₄	C₂²×C₄×Dic₈₉	1	4	yes	—
244	C₄	C₂²×C₄×Dic₈₉	1	12	yes	—
245	C₂	—	2	12	no	—
246	C₂	C₂³×C₄×Dic₈₉	1	1	yes	characteristic
247	C₂	#26033451	1	4	yes	—
248	C₂	#26033455	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₈₉
commutator	C₂×C₈₉	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂×C₄×#41×C₈₉	C₂
socle, cosocle	C₂³×C₈₉	C₂⁵
prime radical	C₂²×#41×C₈₉	C₂²
hypercenter	C₂×C₄×#41	D₈₉
hypocenter	C₈₉	#33999
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41

Complements

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

N	H ≅ G/N	type
2. #26022617	C₂	semidirect
3. #26022613	C₂	semidirect
4. #117×Dic₈₉	C₂	semidirect
5. C₄×#15667971	C₂	semidirect
6. C₂×#19151444	C₂	semidirect
8. #41×Dic₈₉	C₄	semidirect
24. C₄×#41×C₈₉	C₄	semidirect
30. Q₈×Dic₈₉	C₂×C₄	semidirect
31. D₄×Dic₈₉	C₂×C₄	semidirect
32. #15667971	C₂×C₄	semidirect
35. C₂×C₄×Dic₈₉	C₂×C₄	semidirect
49. C₄×Q₈×C₈₉	C₂×C₄	semidirect
50. C₄×D₄×C₈₉	C₂×C₄	semidirect
51. #117×C₈₉	C₂×C₄	semidirect
54. C₂×C₄²×C₈₉	C₂×C₄	semidirect
69. #41×C₈₉	C₄²	semidirect
132. C₈₉	#33999	semidirect
135. C₄×#41	Dic₈₉	semidirect
162. C₄×Q₈	C₂×Dic₈₉	semidirect
164. C₄×D₄	C₂×Dic₈₉	semidirect
166. #117	C₂×Dic₈₉	semidirect
176. C₂×C₄²	C₂×Dic₈₉	semidirect
184. #41	C₄×Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×Dic₈₉	2
#26022617	2
#26022613	2
#117×Dic₈₉	2
C₄×#15667971	2
C₂×#19151444	2
C₂×C₄×#41×C₈₉	2
#33999	89