Subgroups — #36470843

Order 22784.

382 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151419	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151418	C₂	1	2	yes	maximal
3	#26022007	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#364×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667952	C₂²	1	2	yes	—
7	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
12	C₄²×Dic₈₉	C₄	1	1	yes	characteristic
13	#19151419	—	2	2	no	—
14	#19151418	—	2	4	no	—
15	#19151416	—	2	4	no	—
16	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	#364×C₈₉	—	2	2	no	—
19	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
20	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₈₉	—	2	2	no	—
22	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
24	#15667952	—	2	2	no	—
25	#15667952	—	2	2	no	—
26	C₂×#13083549	—	2	2	no	—
27	C₄×#10623494	—	2	2	no	—
28	C₈×Dic₈₉	—	2	2	no	—
29	C₈×Dic₈₉	C₈	1	2	yes	—
30	C₈×Dic₈₉	C₈	1	2	yes	—
31	C₈×Dic₈₉	D₄	1	2	yes	—
32	C₈×Dic₈₉	D₄	1	2	yes	—
33	C₈×Dic₈₉	Q₈	1	2	yes	—
34	C₈×Dic₈₉	Q₈	1	2	yes	—
35	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
38	M₃₂×C₈₉	—	2	2	no	—
39	M₃₂×C₈₉	—	2	2	no	—
40	C₂×C₁₆×C₈₉	—	2	2	no	—
41	C₄×C₈×C₈₉	D₄	1	2	yes	—
42	C₄×C₈×C₈₉	Q₈	1	2	yes	—
43	C₂²×Dic₈₉	—	2	1	no	—
44	C₂²×Dic₈₉	M₁₆	1	2	yes	—
45	C₂×#10623494	—	2	1	no	—
46	C₂×#10623494	—	2	1	no	—
47	C₂×#10623494	C₄⋊C₄	1	2	yes	—
48	C₂×#10623494	C₄⋊C₄	1	2	yes	—
49	C₂×#10623494	C₂×C₈	1	2	yes	—
50	C₂×#10623494	M₁₆	1	2	yes	—
51	C₄×Dic₈₉	—	2	8	no	—
52	C₄×Dic₈₉	—	2	1	no	—
53	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
54	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
55	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
56	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
57	#13083549	—	4	2	no	—
58	C₂²×C₄×C₈₉	—	2	1	no	—
59	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
63	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
69	C₄²×C₈₉	—	2	1	no	—
70	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
71	C₁₆×C₈₉	—	4	2	no	—
72	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₈₉	—	2	4	no	—
74	C₂×C₄×C₈₉	—	2	2	no	—
75	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
81	C₂×Dic₈₉	—	2	4	no	—
82	C₂×Dic₈₉	—	2	2	no	—
83	C₂×Dic₈₉	—	2	4	no	—
84	C₂×Dic₈₉	C₄⋊C₈	1	2	yes	—
85	C₂×Dic₈₉	C₄⋊C₈	1	2	yes	—
86	C₈×C₈₉	C₄⋊C₈	1	4	yes	—
87	C₈×C₈₉	#127	1	1	yes	characteristic
88	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
89	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
90	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
91	#10623494	—	2	2	no	—
92	#10623494	C₄⋊C₈	1	4	yes	—
93	C₂²×C₈₉	—	2	2	no	—
94	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₂²×C₈₉	#423	1	1	yes	characteristic
96	C₂²×C₈₉	#423	1	1	yes	characteristic
97	C₄×C₈₉	—	2	4	no	—
98	C₄×C₈₉	#384	1	1	yes	characteristic
99	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₄×C₈₉	#423	1	1	yes	characteristic
102	Dic₈₉	—	2	4	no	—
103	Dic₈₉	—	2	2	no	—
104	Dic₈₉	#364	1	4	yes	complemented
105	#12900	—	89	1	no	maximal, Sylow
106	C₂×C₈₉	—	2	1	no	—
107	C₂×C₈₉	—	2	1	no	—
108	C₂×C₈₉	#1842	1	1	yes	characteristic
109	C₂×C₈₉	C₂×#364	1	2	yes	—
110	C₂×#364	—	89	2	no	—
111	C₂×#364	—	89	1	no	—
112	C₂×#364	D₈₉	1	1	yes	characteristic
113	#2150	—	89	2	no	—
114	C₄²×C₈	—	89	1	no	—
115	C₈₉	#12900	1	1	yes	characteristic, complemented, Sylow
116	C₂×M₃₂	—	89	2	no	—
117	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
118	C₂×C₄×C₈	—	89	1	no	—
119	C₂×C₄×C₈	—	89	1	no	—
120	C₂×C₄×C₈	—	89	1	no	—
121	C₂×C₄×C₈	—	89	1	no	—
122	C₂×C₄×C₈	—	89	1	no	—
123	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
124	C₄³	—	89	1	no	—
125	#364	—	2	2	no	—
126	#364	—	178	4	no	—
127	#364	—	178	2	no	—
128	C₄⋊C₁₆	—	178	4	no	—
129	C₂²×C₈	—	89	1	no	—
130	C₂²×C₈	—	89	1	no	—
131	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
133	C₂×C₄²	—	89	1	no	—
134	C₂×C₄²	—	89	1	no	—
135	C₂×C₄²	—	178	2	no	—
136	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
137	M₃₂	—	2	2	no	—
138	M₃₂	—	2	2	no	—
139	M₃₂	—	178	2	no	—
140	M₃₂	—	178	2	no	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	178	2	no	—
143	C₄×C₈	—	89	2	no	—
144	C₄×C₈	—	89	2	no	—
145	C₄×C₈	—	89	2	no	—
146	C₄×C₈	—	89	2	no	—
147	C₄×C₈	—	89	2	no	—
148	C₄×C₈	—	89	2	no	—
149	C₄×C₈	—	178	2	no	—
150	C₄×C₈	—	178	2	no	—
151	C₄×C₈	Dic₁₇₈	1	2	yes	—
152	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	—	89	2	no	—
155	C₂²×C₄	—	178	1	no	—
156	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
157	C₂×C₈	—	89	2	no	—
158	C₂×C₈	—	89	2	no	—
159	C₂×C₈	—	89	2	no	—
160	C₂×C₈	—	89	2	no	—
161	C₂×C₈	—	178	1	no	—
162	C₂×C₈	—	178	1	no	—
163	C₂×C₈	#13083553	1	2	yes	—
164	C₂×C₈	C₈×D₈₉	1	2	yes	—
165	C₂×C₈	#13083557	1	2	yes	—
166	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
167	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
168	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
169	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
170	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
171	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
172	C₄²	—	2	1	no	—
173	C₄²	—	89	1	no	—
174	C₄²	—	89	1	no	—
175	C₄²	—	89	2	no	—
176	C₄²	—	89	2	no	—
177	C₄²	—	178	8	no	—
178	C₄²	—	178	1	no	—
179	C₄²	#13083553	1	2	yes	—
180	C₁₆	—	4	2	no	—
181	C₁₆	—	356	2	no	—
182	C₂³	M₁₆×D₈₉	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	89	2	no	—
186	C₂×C₄	—	89	2	no	—
187	C₂×C₄	—	178	4	no	—
188	C₂×C₄	—	178	4	no	—
189	C₂×C₄	—	178	2	no	—
190	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
191	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
192	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
193	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
194	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
195	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
196	C₈	—	89	4	no	—
197	C₈	—	178	2	no	—
198	C₈	#15667941	1	4	yes	—
199	C₈	#15668008	1	1	yes	characteristic
200	C₈	#15668010	1	1	yes	characteristic
201	C₈	#15668011	1	1	yes	characteristic
202	C₈	#15668011	1	1	yes	characteristic
203	C₂²	—	2	2	no	—
204	C₂²	C₄⋊C₈×D₈₉	1	1	yes	characteristic
205	C₂²	#19151620	1	1	yes	characteristic
206	C₂²	#19151621	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	89	4	no	—
209	C₄	—	178	4	no	—
210	C₄	—	178	2	no	—
211	C₄	#19151467	1	1	yes	characteristic
212	C₄	C₄⋊C₈×D₈₉	1	1	yes	characteristic
213	C₄	C₂×#15667941	1	1	yes	characteristic
214	C₄	#19151620	1	1	yes	characteristic
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	#26020403	1	1	yes	characteristic
218	C₂	#364×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#364×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄⋊C₈
prime radical	C₂³×C₈₉	C₂×M₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#364	D₈₉
hypocenter	C₈₉	#12900
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
104. Dic₈₉	#364	semidirect
115. C₈₉	#12900	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151419	2
C₂×#19151418	2
#26022007	2
C₄×C₈×Dic₈₉	2
C₂×#364×C₈₉	2
#12900	89