Subgroups — #36553490

Order 22784.

914 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#10623494	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26020122	C₂	1	3	yes	maximal
4	#26020119	C₂	1	3	yes	maximal
5	#26020107	C₂	1	3	yes	maximal
6	#26019997	C₂	1	3	yes	maximal
7	C₂×C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₈₉	C₄	1	2	yes	—
9	Q₈×#10623494	C₄	1	8	yes	complemented
10	C₂×C₄⋊#10623494	C₂²	1	3	yes	—
11	C₂×#15667922	C₂²	1	6	yes	—
12	C₂×C₄×#10623494	C₂²	1	3	yes	—
13	C₄⋊C₄×Dic₈₉	C₄	1	6	yes	—
14	C₄×C₄⋊Dic₈₉	C₂²	1	3	yes	—
15	C₄²×Dic₈₉	C₂²	1	3	yes	—
16	#19151352	C₂²	1	3	yes	—
17	#19151352	C₂²	1	3	yes	—
18	#19151352	C₂²	1	3	yes	—
19	#19151352	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₈₉	C₄	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
23	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
24	C₂²×#10623494	C₂³	1	3	yes	—
25	C₂²×#10623494	C₂³	1	1	yes	characteristic
26	Q₈×Dic₈₉	—	2	8	no	—
27	C₂×C₄⋊Dic₈₉	C₂×C₄	1	6	yes	—
28	C₂×C₄×Dic₈₉	C₂×C₄	1	6	yes	—
29	C₂×C₄×Dic₈₉	C₂×C₄	1	6	yes	—
30	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
31	C₂×C₄×Dic₈₉	C₂³	1	3	yes	—
32	C₄⋊#10623494	C₂×C₄	1	12	yes	—
33	#15667922	—	4	6	no	—
34	C₄×#10623494	C₂×C₄	1	12	yes	—
35	#15667919	C₂×C₄	1	6	yes	—
36	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
39	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
40	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	3	yes	—
42	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
43	C₄⋊C₈×C₈₉	C₂×C₄	1	12	yes	—
44	C₄×C₈×C₈₉	C₂×C₄	1	12	yes	—
45	C₂²×Dic₈₉	C₂²×C₄	1	6	yes	—
46	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
47	C₂×#10623494	—	2	3	no	—
48	C₂×#10623494	—	2	1	no	—
49	C₂×#10623494	—	2	1	no	—
50	C₂×#10623494	—	2	3	no	—
51	C₂×#10623494	C₂²×C₄	1	2	yes	—
52	C₂×#10623494	C₂²×C₄	1	6	yes	—
53	C₄⋊Dic₈₉	—	2	12	no	—
54	C₄×Dic₈₉	—	2	12	no	—
55	C₄×Dic₈₉	—	2	3	no	—
56	C₄×Dic₈₉	—	2	1	no	—
57	C₄×Dic₈₉	—	2	1	no	—
58	C₄×Dic₈₉	—	2	3	no	—
59	C₄×Dic₈₉	—	2	12	no	—
60	C₂×Q₈×C₈₉	C₄²	1	4	yes	—
61	C₂×Q₈×C₈₉	M₁₆	1	8	yes	—
62	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
63	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
64	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
65	C₂×C₈×C₈₉	—	2	3	no	—
66	C₂×C₈×C₈₉	—	2	1	no	—
67	C₂×C₈×C₈₉	—	2	1	no	—
68	C₂×C₈×C₈₉	—	2	3	no	—
69	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₈₉	C₂²×C₄	1	6	yes	—
71	C₄⋊C₄×C₈₉	C₄²	1	12	yes	—
72	C₄²×C₈₉	C₂²×C₄	1	6	yes	—
73	C₄²×C₈₉	C₂²×C₄	1	6	yes	—
74	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
75	Q₈×C₈₉	#97	1	16	yes	complemented
76	C₂×C₄×C₈₉	—	2	6	no	—
77	C₂×C₄×C₈₉	C₂×C₄²	1	6	yes	—
78	C₂×C₄×C₈₉	C₂×C₄²	1	6	yes	—
79	C₂×C₄×C₈₉	C₂×M₁₆	1	12	yes	—
80	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
83	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
84	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
85	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
86	C₂×Dic₈₉	—	2	2	no	—
87	C₂×Dic₈₉	—	2	2	no	—
88	C₂×Dic₈₉	—	2	6	no	—
89	C₂×Dic₈₉	—	2	6	no	—
90	C₂×Dic₈₉	—	2	6	no	—
91	C₂×Dic₈₉	#131	1	4	yes	—
92	C₈×C₈₉	—	2	2	no	—
93	C₈×C₈₉	—	2	6	no	—
94	#10623494	—	2	2	no	—
95	#10623494	—	2	6	no	—
96	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
97	C₂²×C₈₉	#519	1	1	yes	characteristic
98	C₂²×C₈₉	#520	1	1	yes	characteristic
99	C₂²×C₈₉	C₂²×M₁₆	1	2	yes	—
100	C₂²×C₈₉	C₂×#131	1	2	yes	—
101	C₄×C₈₉	—	2	6	no	—
102	C₄×C₈₉	C₂×#97	1	12	yes	—
103	C₄×C₈₉	#519	1	1	yes	characteristic
104	C₄×C₈₉	#519	1	1	yes	characteristic
105	C₄×C₈₉	#520	1	1	yes	characteristic
106	C₄×C₈₉	#520	1	1	yes	characteristic
107	Dic₈₉	—	2	4	no	—
108	Dic₈₉	—	4	6	no	—
109	#20572	—	89	1	no	maximal, Sylow
110	C₂×C₈₉	#2276	1	1	yes	characteristic
111	C₂×C₈₉	C₂²×#97	1	1	yes	characteristic
112	C₂×C₈₉	C₄×#131	1	1	yes	characteristic
113	C₂×C₈₉	#2984	1	2	yes	—
114	C₂×C₈₉	#2991	1	2	yes	—
115	C₂×C₈×Q₈	—	89	1	no	—
116	C₂×C₈×Q₈	D₈₉	1	1	yes	characteristic
117	C₄²×Q₈	—	89	1	no	—
118	#1770	—	89	3	no	—
119	#1770	—	89	3	no	—
120	#1767	—	89	3	no	—
121	#1744	—	89	3	no	—
122	C₈₉	#20572	1	1	yes	characteristic, complemented, Sylow
123	C₂×C₄×Q₈	—	89	2	no	—
124	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
125	C₈×Q₈	—	89	8	no	—
126	C₈×Q₈	Dic₈₉	1	8	yes	complemented
127	C₂×C₄⋊C₈	—	89	3	no	—
128	C₂×C₄⋊C₈	C₂×D₈₉	1	3	yes	—
129	C₂×#97	—	89	6	no	—
130	C₂×C₄×C₈	—	89	3	no	—
131	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
132	C₄×C₄⋊C₄	—	89	6	no	—
133	C₄×C₄⋊C₄	—	89	3	no	—
134	C₄³	—	89	3	no	—
135	#336	—	89	1	no	—
136	#336	—	89	3	no	—
137	#336	—	89	3	no	—
138	#336	—	89	3	no	—
139	C₂²×Q₈	C₄×D₈₉	1	1	yes	characteristic
140	C₂²×C₈	—	89	1	no	—
141	C₂²×C₈	—	89	3	no	—
142	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₈₉	1	3	yes	—
144	C₄×Q₈	—	178	8	no	—
145	C₄×Q₈	C₄×D₈₉	1	4	yes	—
146	C₄×Q₈	C₂×Dic₈₉	1	4	yes	—
147	C₂×C₄⋊C₄	—	89	6	no	—
148	C₂×C₄⋊C₄	C₄×D₈₉	1	3	yes	—
149	C₂×C₄²	—	89	6	no	—
150	C₂×C₄²	—	89	6	no	—
151	C₂×C₄²	—	89	1	no	—
152	C₂×C₄²	—	89	3	no	—
153	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
154	C₄⋊C₈	—	89	12	no	—
155	C₄⋊C₈	C₂×Dic₈₉	1	12	yes	—
156	#97	—	356	6	no	—
157	C₄×C₈	—	89	12	no	—
158	C₄×C₈	C₂×Dic₈₉	1	12	yes	—
159	#95	—	89	6	no	—
160	C₂×Q₈	C₄×Dic₈₉	1	4	yes	—
161	C₂×Q₈	#13083557	1	8	yes	—
162	C₂²×C₄	—	89	2	no	—
163	C₂²×C₄	—	89	6	no	—
164	C₂²×C₄	C₂×C₄×D₈₉	1	3	yes	—
165	C₂²×C₄	C₂×C₄×D₈₉	1	3	yes	—
166	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
167	C₂×C₈	—	2	3	no	—
168	C₂×C₈	—	2	3	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	89	2	no	—
172	C₂×C₈	—	89	6	no	—
173	C₂×C₈	—	178	3	no	—
174	C₂×C₈	—	178	1	no	—
175	C₂×C₈	—	178	1	no	—
176	C₂×C₈	—	178	3	no	—
177	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
178	C₂×C₈	C₂²×Dic₈₉	1	6	yes	—
179	C₄⋊C₄	—	178	12	no	—
180	C₄⋊C₄	C₄×Dic₈₉	1	12	yes	—
181	C₄²	—	178	12	no	—
182	C₄²	—	178	12	no	—
183	C₄²	—	178	3	no	—
184	C₄²	—	178	1	no	—
185	C₄²	—	178	3	no	—
186	C₄²	—	178	1	no	—
187	C₄²	C₂×C₄×D₈₉	1	6	yes	—
188	C₄²	C₂²×Dic₈₉	1	6	yes	—
189	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
190	Q₈	#15667922	1	16	yes	complemented
191	C₂×C₄	—	2	6	no	—
192	C₂×C₄	—	89	4	no	—
193	C₂×C₄	—	178	6	no	—
194	C₂×C₄	—	178	6	no	—
195	C₂×C₄	—	178	6	no	—
196	C₂×C₄	—	178	2	no	—
197	C₂×C₄	—	178	2	no	—
198	C₂×C₄	C₂×C₄×Dic₈₉	1	6	yes	—
199	C₂×C₄	C₂×C₄×Dic₈₉	1	6	yes	—
200	C₂×C₄	C₂×#13083557	1	12	yes	—
201	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
202	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
203	C₂×C₄	#15668058	1	1	yes	characteristic
204	C₂×C₄	#15668059	1	1	yes	characteristic
205	C₂×C₄	#15668060	1	1	yes	characteristic
206	C₂×C₄	#15668060	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	2	6	no	—
209	C₈	—	178	2	no	—
210	C₈	—	178	6	no	—
211	C₂²	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
212	C₂²	#19152074	1	1	yes	characteristic
213	C₂²	#19152080	1	1	yes	characteristic
214	C₂²	C₂²×#13083557	1	2	yes	—
215	C₂²	#131×D₈₉	1	2	yes	—
216	C₄	—	2	6	no	—
217	C₄	—	356	6	no	—
218	C₄	—	178	4	no	—
219	C₄	C₂×#15667922	1	12	yes	—
220	C₄	#19152074	1	1	yes	characteristic
221	C₄	#19152076	1	1	yes	characteristic
222	C₄	#19152080	1	1	yes	characteristic
223	C₄	#19152081	1	1	yes	characteristic
224	C₂	#26022617	1	1	yes	characteristic
225	C₂	C₂²×#15667922	1	1	yes	characteristic
226	C₂	#131×Dic₈₉	1	1	yes	characteristic
227	C₂	#26029340	1	2	yes	—
228	C₂	#26029383	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator	C₂²×C₈₉	C₂²×C₄²
Fitting	C₂×C₈×Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂³×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×C₈×Q₈	D₈₉
hypocenter	C₈₉	#20572
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
9. Q₈×#10623494	C₄	semidirect
21. C₈×Q₈×C₈₉	C₄	semidirect
75. Q₈×C₈₉	#97	semidirect
122. C₈₉	#20572	semidirect
126. C₈×Q₈	Dic₈₉	semidirect
190. Q₈	#15667922	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#10623494	2
C₄×Q₈×Dic₈₉	2
#26020122	2
#26020119	2
#26020107	2
#26019997	2
C₂×C₈×Q₈×C₈₉	2
#20572	89