Subgroups — #36563216

Order 22784.

854 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×Dic₈₉	C₂	1	2	yes	maximal
2	#26021214	C₂	1	2	yes	maximal
3	#26020991	C₂	1	4	yes	maximal
4	#26020409	C₂	1	1	yes	maximal, characteristic
5	#26020151	C₂	1	4	yes	maximal
6	C₄×#15667922	C₂	1	1	yes	maximal, characteristic
7	C₂×#500×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₈₉	—	2	2	no	—
9	C₂×#15667944	C₂²	1	4	yes	—
10	C₂×#15667922	C₂²	1	2	yes	—
11	C₂×#15667922	C₂²	1	2	yes	—
12	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
13	#19151505	—	2	2	no	—
14	#19151497	—	2	2	no	—
15	#19151497	C₂²	1	4	yes	—
16	#19151468	C₂²	1	2	yes	—
17	C₄⋊C₄×Dic₈₉	—	2	2	no	—
18	C₄⋊C₄×Dic₈₉	C₂²	1	4	yes	—
19	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
20	#19151360	C₂²	1	8	yes	—
21	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
22	#500×C₈₉	C₄	1	8	yes	complemented
23	C₂×#107×C₈₉	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×#10623494	C₂³	1	2	yes	—
26	Q₈×Dic₈₉	—	4	8	no	—
27	C₂×C₄⋊Dic₈₉	—	2	4	no	—
28	C₂×C₄⋊Dic₈₉	—	2	2	no	—
29	C₂×C₄⋊Dic₈₉	—	2	2	no	—
30	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
31	C₂×C₄×Dic₈₉	—	2	4	no	—
32	C₂×C₄×Dic₈₉	D₄	1	2	yes	—
33	C₂×C₄×Dic₈₉	D₄	1	2	yes	—
34	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
35	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
36	#15667944	—	2	8	no	—
37	#15667922	—	2	4	no	—
38	#15667922	Q₈	1	8	yes	complemented
39	C₄×#10623494	—	2	2	no	—
40	#15667919	—	2	4	no	—
41	#15667919	—	2	4	no	—
42	C₂²×Q₈×C₈₉	—	2	2	no	—
43	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₈₉	—	2	4	no	—
45	C₄⋊Q₈×C₈₉	C₂×C₄	1	8	yes	—
46	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
48	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
49	#107×C₈₉	C₂×C₄	1	16	yes	—
50	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
51	C₂²×Dic₈₉	—	2	4	no	—
52	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
53	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
54	C₂×#10623494	—	2	2	no	—
55	C₂×#10623494	—	2	2	no	—
56	C₂×#10623494	C₂×Q₈	1	4	yes	—
57	C₄⋊Dic₈₉	—	2	8	no	—
58	C₄⋊Dic₈₉	—	4	8	no	—
59	C₄⋊Dic₈₉	—	4	4	no	—
60	C₄⋊Dic₈₉	—	4	4	no	—
61	C₄×Dic₈₉	—	2	4	no	—
62	C₄×Dic₈₉	—	2	4	no	—
63	C₄×Dic₈₉	—	2	2	no	—
64	C₄×Dic₈₉	—	4	8	no	—
65	C₄×Dic₈₉	C₂×Q₈	1	4	yes	—
66	C₂×Q₈×C₈₉	—	2	8	no	—
67	C₂×Q₈×C₈₉	—	4	4	no	—
68	C₂×Q₈×C₈₉	—	4	4	no	—
69	C₂²×C₄×C₈₉	—	2	4	no	—
70	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
75	C₂×C₈×C₈₉	#41	1	4	yes	—
76	C₄⋊C₄×C₈₉	—	2	8	no	—
77	C₄⋊C₄×C₈₉	—	2	4	no	—
78	C₄⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
79	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
80	C₄²×C₈₉	#41	1	2	yes	—
81	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
82	Q₈×C₈₉	—	4	16	no	—
83	C₂×C₄×C₈₉	—	2	8	no	—
84	C₂×C₄×C₈₉	—	4	4	no	—
85	C₂×C₄×C₈₉	—	4	4	no	—
86	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
89	C₂×C₄×C₈₉	#137	1	4	yes	—
90	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
92	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
94	C₂×Dic₈₉	—	2	2	no	—
95	C₂×Dic₈₉	—	2	2	no	—
96	C₂×Dic₈₉	—	2	2	no	—
97	C₂×Dic₈₉	—	2	2	no	—
98	C₂×Dic₈₉	—	2	2	no	—
99	C₂×Dic₈₉	—	4	4	no	—
100	C₂×Dic₈₉	—	4	4	no	—
101	C₂×Dic₈₉	—	4	4	no	—
102	C₂×Dic₈₉	C₄⋊Q₈	1	4	yes	—
103	C₈×C₈₉	C₄×Q₈	1	8	yes	complemented
104	#10623494	—	2	4	no	—
105	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
106	C₂²×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
107	C₂²×C₈₉	#532	1	1	yes	characteristic
108	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
109	C₂²×C₈₉	C₂×#137	1	2	yes	—
110	C₄×C₈₉	—	4	8	no	—
111	C₄×C₈₉	#441	1	4	yes	—
112	C₄×C₈₉	#442	1	4	yes	—
113	C₄×C₈₉	C₂×C₄×Q₈	1	2	yes	—
114	C₄×C₈₉	#533	1	2	yes	—
115	Dic₈₉	—	2	4	no	—
116	Dic₈₉	—	4	2	no	—
117	Dic₈₉	—	8	4	no	—
118	#21726	—	89	1	no	maximal, Sylow
119	C₂×C₈₉	C₄×C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₈₉	C₂×#441	1	1	yes	characteristic
121	C₂×C₈₉	C₂×#442	1	1	yes	characteristic
122	C₂×C₈₉	#3165	1	2	yes	—
123	C₂×C₈₉	#3182	1	2	yes	—
124	C₂×#500	D₈₉	1	1	yes	characteristic
125	C₄×C₄⋊Q₈	—	89	2	no	—
126	#1990	—	89	2	no	—
127	#1943	—	89	4	no	—
128	#1844	—	89	1	no	—
129	#1776	—	89	4	no	—
130	C₄×#97	—	89	1	no	—
131	C₈₉	#21726	1	1	yes	characteristic, complemented, Sylow
132	C₂×C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
133	C₂×C₄×Q₈	—	178	2	no	—
134	#500	Dic₈₉	1	8	yes	complemented
135	C₂×#107	C₂×D₈₉	1	4	yes	—
136	C₂×#106	—	89	4	no	—
137	C₂×#97	—	89	2	no	—
138	C₂×#97	—	89	2	no	—
139	C₂×C₄×C₈	—	89	1	no	—
140	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
141	#391	—	178	2	no	—
142	#389	—	89	4	no	—
143	#389	—	178	2	no	—
144	#384	—	89	2	no	—
145	C₄×C₄⋊C₄	—	89	4	no	—
146	C₄×C₄⋊C₄	—	178	2	no	—
147	C₄³	—	89	1	no	—
148	#340	—	89	8	no	—
149	C₂²×Q₈	—	2	2	no	—
150	C₂²×C₈	—	89	2	no	—
151	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
152	C₄⋊Q₈	—	2	4	no	—
153	C₄⋊Q₈	C₂×Dic₈₉	1	8	yes	—
154	C₄×Q₈	—	356	8	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	89	4	no	—
157	C₂×C₄⋊C₄	—	178	4	no	—
158	C₂×C₄⋊C₄	—	178	2	no	—
159	C₂×C₄⋊C₄	—	178	2	no	—
160	C₂×C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
161	C₂×C₄²	—	89	1	no	—
162	C₂×C₄²	—	89	2	no	—
163	C₂×C₄²	—	89	2	no	—
164	C₂×C₄²	—	89	1	no	—
165	C₂×C₄²	—	178	4	no	—
166	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
167	#107	C₂×Dic₈₉	1	16	yes	—
168	#106	—	178	8	no	—
169	#97	—	89	8	no	—
170	#97	—	178	4	no	—
171	C₄×C₈	—	178	2	no	—
172	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
173	#95	—	178	4	no	—
174	#95	—	178	4	no	—
175	C₂×Q₈	—	2	8	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂×Q₈	—	4	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	89	2	no	—
180	C₂²×C₄	—	89	2	no	—
181	C₂²×C₄	—	178	4	no	—
182	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
183	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
184	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
185	C₂×C₈	—	89	4	no	—
186	C₂×C₈	—	178	2	no	—
187	C₂×C₈	—	178	2	no	—
188	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
189	C₂×C₈	Q₈×D₈₉	1	4	yes	—
190	C₂×C₈	#13083575	1	4	yes	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	356	8	no	—
194	C₄⋊C₄	—	356	4	no	—
195	C₄⋊C₄	—	356	4	no	—
196	C₄⋊C₄	—	178	8	no	—
197	C₄⋊C₄	C₂²×Dic₈₉	1	8	yes	—
198	C₄²	—	89	4	no	—
199	C₄²	—	356	8	no	—
200	C₄²	—	178	4	no	—
201	C₄²	—	178	4	no	—
202	C₄²	—	178	2	no	—
203	C₄²	C₂²×Dic₈₉	1	2	yes	—
204	C₄²	#13083573	1	2	yes	—
205	C₂³	C₂×D₄×D₈₉	1	1	yes	characteristic
206	Q₈	—	4	16	no	—
207	C₂×C₄	—	2	8	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	89	4	no	—
211	C₂×C₄	—	356	4	no	—
212	C₂×C₄	—	356	4	no	—
213	C₂×C₄	—	356	4	no	—
214	C₂×C₄	—	178	2	no	—
215	C₂×C₄	—	178	2	no	—
216	C₂×C₄	—	178	2	no	—
217	C₂×C₄	—	178	2	no	—
218	C₂×C₄	—	178	2	no	—
219	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
220	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
221	C₂×C₄	#15668036	1	4	yes	—
222	C₂×C₄	#15668044	1	4	yes	—
223	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
224	C₂×C₄	C₂×Q₈×D₈₉	1	2	yes	—
225	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
226	C₂×C₄	C₂×#13083575	1	2	yes	—
227	C₈	—	178	4	no	—
228	C₈	Q₈×Dic₈₉	1	8	yes	complemented
229	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
230	C₂²	C₄⋊Q₈×D₈₉	1	1	yes	characteristic
231	C₂²	#19152149	1	1	yes	characteristic
232	C₂²	C₂×#15668036	1	2	yes	—
233	C₂²	C₂×#15668044	1	2	yes	—
234	C₄	—	4	8	no	—
235	C₄	—	356	2	no	—
236	C₄	—	178	4	no	—
237	C₄	—	712	4	no	—
238	C₄	#19151705	1	4	yes	—
239	C₄	#19151710	1	4	yes	—
240	C₄	C₂×Q₈×Dic₈₉	1	2	yes	—
241	C₄	#19152153	1	2	yes	—
242	C₂	C₄⋊Q₈×Dic₈₉	1	1	yes	characteristic
243	C₂	C₂×#19151705	1	1	yes	characteristic
244	C₂	C₂×#19151710	1	1	yes	characteristic
245	C₂	#26030795	1	2	yes	—
246	C₂	#26030972	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×#500×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×#500	D₈₉
hypocenter	C₈₉	#21726
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
22. #500×C₈₉	C₄	semidirect
38. #15667922	Q₈	semidirect
103. C₈×C₈₉	C₄×Q₈	semidirect
131. C₈₉	#21726	semidirect
134. #500	Dic₈₉	semidirect
228. C₈	Q₈×Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×Dic₈₉	2
#26021214	2
#26020991	2
#26020409	2
#26020151	2
C₄×#15667922	2
C₂×#500×C₈₉	2
#21726	89