Subgroups — #36445001

Order 22784.

430 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15667931	C₂	1	1	yes	maximal, characteristic
2	#26018576	C₂	1	1	yes	maximal, characteristic
3	#26018541	C₂	1	2	yes	maximal
4	#26018102	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₂²	1	2	yes	—
7	C₂×#15667931	C₄	1	2	yes	—
8	C₄×#13083558	C₂²	1	2	yes	—
9	#19151367	C₄	1	2	yes	—
10	#19151324	C₄	1	4	yes	—
11	#19151320	—	2	2	no	—
12	#19151319	—	2	2	no	—
13	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₈₉	—	2	2	no	—
15	C₈×Q₈×C₈₉	—	2	2	no	—
16	C₂×C₄⋊C₈×C₈₉	C₄	1	2	yes	—
17	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#13083558	C₂×C₄	1	4	yes	—
21	C₄⋊#10623494	—	2	2	no	—
22	#15667931	—	2	4	no	—
23	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
24	#15667923	—	2	2	no	—
25	C₄×#10623494	—	2	2	no	—
26	C₄×#10623494	—	2	2	no	—
27	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₈₉	—	2	2	no	—
32	C₄×Q₈×C₈₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₈₉	—	2	4	no	—
38	C₄⋊C₈×C₈₉	—	2	2	no	—
39	C₄×C₈×C₈₉	—	2	2	no	—
40	C₄×C₈×C₈₉	—	2	2	no	—
41	C₄×C₈×C₈₉	D₄	1	2	yes	—
42	C₄×C₈×C₈₉	D₄	1	2	yes	—
43	#13083558	—	2	4	no	—
44	#13083558	—	2	4	no	—
45	C₂×#10623494	—	2	4	no	—
46	C₂×#10623494	—	2	4	no	—
47	C₂×Q₈×C₈₉	—	2	2	no	—
48	C₂×Q₈×C₈₉	—	2	2	no	—
49	C₂×Q₈×C₈₉	M₁₆	1	4	yes	—
50	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
51	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
52	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₈₉	—	2	2	no	—
56	C₂×C₈×C₈₉	—	2	2	no	—
57	C₂×C₈×C₈₉	—	4	1	no	—
58	C₂×C₈×C₈₉	—	4	2	no	—
59	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₄⋊C₄×C₈₉	—	2	4	no	—
63	C₄⋊C₄×C₈₉	—	2	2	no	—
64	C₄²×C₈₉	—	2	2	no	—
65	C₄²×C₈₉	—	2	2	no	—
66	C₄²×C₈₉	C₂×D₄	1	2	yes	—
67	C₄²×C₈₉	#41	1	2	yes	—
68	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
69	Q₈×C₈₉	—	2	8	no	—
70	C₂×C₄×C₈₉	—	2	2	no	—
71	C₂×C₄×C₈₉	—	2	2	no	—
72	C₂×C₄×C₈₉	—	2	1	no	—
73	C₂×C₄×C₈₉	—	2	1	no	—
74	C₂×C₄×C₈₉	—	2	2	no	—
75	C₂×C₄×C₈₉	—	4	1	no	—
76	C₂×C₄×C₈₉	—	4	2	no	—
77	C₂×C₄×C₈₉	#97	1	4	yes	—
78	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₈₉	#131	1	2	yes	—
84	C₈×C₈₉	—	2	1	no	—
85	C₈×C₈₉	—	2	2	no	—
86	C₈×C₈₉	—	4	2	no	—
87	C₈×C₈₉	#101	1	2	yes	complemented
88	#10623494	—	4	4	no	—
89	#10623494	—	4	4	no	—
90	C₂²×C₈₉	—	2	1	no	—
91	C₂²×C₈₉	—	2	1	no	—
92	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₈₉	C₂×#97	1	1	yes	characteristic
94	C₂²×C₈₉	#405	1	1	yes	characteristic
95	C₄×C₈₉	—	2	4	no	—
96	C₄×C₈₉	—	2	1	no	—
97	C₄×C₈₉	—	2	2	no	—
98	C₄×C₈₉	—	4	2	no	—
99	C₄×C₈₉	C₂×#101	1	1	yes	characteristic
100	C₄×C₈₉	#413	1	1	yes	characteristic
101	C₄×C₈₉	#435	1	4	yes	—
102	#8165	—	89	1	no	maximal, Sylow
103	C₂×C₈₉	—	2	1	no	—
104	C₂×C₈₉	—	2	1	no	—
105	C₂×C₈₉	#1752	1	1	yes	characteristic
106	C₂×C₈₉	C₄×#101	1	1	yes	characteristic
107	C₂×C₈₉	#1757	1	1	yes	characteristic
108	C₂×C₈×Q₈	D₈₉	1	1	yes	characteristic
109	C₄×#101	—	89	1	no	—
110	#1461	—	89	1	no	—
111	#1451	—	89	2	no	—
112	#1283	—	89	2	no	—
113	C₈₉	#8165	1	1	yes	characteristic, complemented, Sylow
114	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
115	C₈×Q₈	—	2	2	no	—
116	C₈×Q₈	—	2	2	no	—
117	#431	—	89	2	no	—
118	C₂×C₄⋊C₈	Dic₈₉	1	2	yes	—
119	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
120	C₂×#101	—	89	2	no	—
121	C₄×M₁₆	—	89	2	no	—
122	C₂×C₄×C₈	Dic₈₉	1	2	yes	—
123	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
124	#343	—	89	2	no	—
125	#324	—	89	4	no	—
126	#322	—	178	2	no	—
127	#322	—	178	2	no	—
128	C₂²×Q₈	C₄×D₈₉	1	1	yes	characteristic
129	C₂×M₁₆	—	89	4	no	—
130	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
131	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
132	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
133	C₄×Q₈	—	2	2	no	—
134	C₄×Q₈	—	2	2	no	—
135	C₂×C₄⋊C₄	C₄×D₈₉	1	2	yes	—
136	C₂×C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
137	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
138	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
139	C₄⋊C₈	—	2	4	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	178	2	no	—
142	#101	—	178	4	no	—
143	C₂²⋊C₈	—	89	4	no	—
144	#97	—	178	2	no	—
145	C₄×C₈	—	2	2	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	—	178	2	no	—
148	C₄×C₈	—	178	2	no	—
149	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
150	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
151	C₂×Q₈	—	2	2	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	#13083557	1	4	yes	—
154	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
155	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
156	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
158	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
159	M₁₆	—	178	4	no	—
160	M₁₆	—	178	4	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	178	4	no	—
166	C₂×C₈	—	178	4	no	—
167	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
168	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
169	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
175	C₄²	#13083574	1	2	yes	—
176	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
177	Q₈	—	2	8	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	#15667922	1	4	yes	—
186	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
187	C₂×C₄	#15667971	1	1	yes	characteristic
188	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
189	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
190	C₂×C₄	C₂×#13083557	1	2	yes	—
191	C₂×C₄	#15668018	1	2	yes	—
192	C₈	—	2	1	no	—
193	C₈	—	2	2	no	—
194	C₈	—	4	2	no	—
195	C₈	—	356	4	no	—
196	C₈	—	356	4	no	—
197	C₈	#15667931	1	2	yes	complemented
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	C₄×C₂²⋊Dic₈₉	1	1	yes	characteristic
201	C₂²	C₂×#15667922	1	1	yes	characteristic
202	C₂²	#19151553	1	1	yes	characteristic
203	C₄	—	2	4	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	2	no	—
206	C₄	—	4	2	no	—
207	C₄	C₂×#15667931	1	1	yes	characteristic
208	C₄	#19151582	1	1	yes	characteristic
209	C₄	#19151669	1	4	yes	—
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	#26020028	1	1	yes	characteristic
213	C₂	C₄×#15667931	1	1	yes	characteristic
214	C₂	#26020052	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₈₉
commutator, prime radical	C₂³×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×C₈×Q₈×C₈₉	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×C₈×Q₈	D₈₉
hypocenter	C₈₉	#8165
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
87. C₈×C₈₉	#101	semidirect
113. C₈₉	#8165	semidirect
197. C₈	#15667931	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15667931	2
#26018576	2
#26018541	2
#26018102	2
C₂×C₈×Q₈×C₈₉	2
#8165	89