Subgroups — C₈₉×#13185 #36391033

Order 22784.

306 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#365×C₈₉	C₂	1	2	yes	maximal
2	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1848	C₂	1	1	yes	maximal, characteristic
4	C₄×#106×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1383	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
7	C₂×#108×C₈₉	—	2	1	no	—
8	C₂×#108×C₈₉	C₂²	1	2	yes	—
9	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
11	#423×C₈₉	—	2	1	no	—
12	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
14	#365×C₈₉	—	2	4	no	—
15	C₄⋊C₁₆×C₈₉	C₄	1	2	yes	complemented
16	C₄⋊C₁₆×C₈₉	C₄	1	2	yes	complemented
17	#340×C₈₉	—	2	1	no	—
18	C₂×M₁₆×C₈₉	—	2	2	no	—
19	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
22	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
23	C₂×C₄²×C₈₉	—	4	1	no	—
24	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
25	M₃₂×C₈₉	—	2	2	no	—
26	M₃₂×C₈₉	—	2	2	no	—
27	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
28	#108×C₈₉	—	2	2	no	—
29	#108×C₈₉	—	2	2	no	—
30	#108×C₈₉	—	4	2	no	—
31	#106×C₈₉	—	2	1	no	—
32	#106×C₈₉	—	2	1	no	—
33	#106×C₈₉	D₄	1	2	yes	—
34	#106×C₈₉	Q₈	1	2	yes	—
35	C₄⋊C₈×C₈₉	—	4	1	no	—
36	C₄⋊C₈×C₈₉	—	4	1	no	—
37	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
40	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
41	#95×C₈₉	—	4	1	no	—
42	C₂²×C₄×C₈₉	—	4	1	no	—
43	C₂²×C₄×C₈₉	—	4	1	no	—
44	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	M₁₆×C₈₉	—	4	2	no	—
48	M₁₆×C₈₉	—	4	2	no	—
49	C₂×C₈×C₈₉	—	2	1	no	—
50	C₂×C₈×C₈₉	—	2	1	no	—
51	C₂×C₈×C₈₉	—	4	2	no	—
52	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
59	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₈₉	—	2	2	no	—
61	C₄⋊C₄×C₈₉	—	2	2	no	—
62	C₄⋊C₄×C₈₉	—	4	1	no	—
63	C₄⋊C₄×C₈₉	—	4	1	no	—
64	C₄²×C₈₉	—	8	2	no	—
65	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
66	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
68	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
69	C₁₆×C₈₉	—	2	4	no	—
70	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₈₉	—	2	2	no	—
72	C₂×C₄×C₈₉	—	4	1	no	—
73	C₂×C₄×C₈₉	—	4	2	no	—
74	C₂×C₄×C₈₉	—	4	2	no	—
75	C₂×C₄×C₈₉	—	8	1	no	—
76	C₂×C₄×C₈₉	—	8	1	no	—
77	C₂×C₄×C₈₉	—	8	1	no	—
78	C₂×C₄×C₈₉	—	8	1	no	—
79	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
80	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
81	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
82	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
83	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
89	C₈×C₈₉	—	2	1	no	—
90	C₈×C₈₉	—	2	1	no	—
91	C₈×C₈₉	—	8	2	no	—
92	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
93	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
94	C₈×C₈₉	#125	1	1	yes	characteristic
95	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
96	C₂²×C₈₉	—	2	1	no	—
97	C₂²×C₈₉	—	2	1	no	—
98	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
99	C₂²×C₈₉	#461	1	1	yes	characteristic
100	C₂²×C₈₉	#476	1	1	yes	characteristic
101	C₄×C₈₉	—	2	1	no	—
102	C₄×C₈₉	—	2	1	no	—
103	C₄×C₈₉	—	2	1	no	—
104	C₄×C₈₉	—	8	2	no	—
105	C₄×C₈₉	—	8	2	no	—
106	C₄×C₈₉	#365	1	2	yes	complemented
107	C₄×C₈₉	#389	1	1	yes	characteristic
108	C₄×C₈₉	#428	1	1	yes	characteristic
109	C₄×C₈₉	#460	1	1	yes	characteristic
110	C₄×C₈₉	#475	1	1	yes	characteristic
111	#13185	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₈₉	—	2	1	no	—
113	C₂×C₈₉	—	2	1	no	—
114	C₂×C₈₉	#1948	1	1	yes	characteristic
115	C₂×C₈₉	C₂×#365	1	1	yes	characteristic
116	C₂×C₈₉	#2155	1	1	yes	characteristic
117	C₂×#365	C₂×C₈₉	1	2	yes	—
118	#2150	C₂×C₈₉	1	1	yes	characteristic
119	#1848	C₂×C₈₉	1	1	yes	characteristic
120	C₄×#106	C₂×C₈₉	1	1	yes	characteristic
121	#1383	C₂×C₈₉	1	2	yes	—
122	C₈₉	#13185	1	1	yes	characteristic, direct product factor, Sylow
123	C₂×M₃₂	C₂²×C₈₉	1	2	yes	—
124	C₂×#108	—	2	1	no	—
125	C₂×#108	C₂²×C₈₉	1	2	yes	—
126	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
127	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
128	#423	—	2	1	no	—
129	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
130	C₄×C₄⋊C₄	—	2	1	no	—
131	#365	—	2	4	no	—
132	C₄⋊C₁₆	C₄×C₈₉	1	2	yes	complemented
133	C₄⋊C₁₆	C₄×C₈₉	1	2	yes	complemented
134	#340	—	2	1	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
137	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	2	1	no	—
140	C₂×C₄²	—	4	1	no	—
141	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
142	M₃₂	—	2	2	no	—
143	M₃₂	—	2	2	no	—
144	C₂×C₁₆	C₂×C₄×C₈₉	1	4	yes	—
145	#108	—	2	2	no	—
146	#108	—	2	2	no	—
147	#108	—	4	2	no	—
148	#106	—	2	1	no	—
149	#106	—	2	1	no	—
150	#106	D₄×C₈₉	1	2	yes	—
151	#106	Q₈×C₈₉	1	2	yes	—
152	C₄⋊C₈	—	4	1	no	—
153	C₄⋊C₈	—	4	1	no	—
154	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
155	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
156	C₄×C₈	Q₈×C₈₉	1	1	yes	characteristic
157	C₄×C₈	Q₈×C₈₉	1	1	yes	characteristic
158	#95	—	4	1	no	—
159	C₂²×C₄	—	4	1	no	—
160	C₂²×C₄	—	4	1	no	—
161	C₂²×C₄	SD₁₆×C₈₉	1	1	yes	characteristic
162	C₂²×C₄	SD₁₆×C₈₉	1	1	yes	characteristic
163	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
164	M₁₆	—	4	2	no	—
165	M₁₆	—	4	2	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	C₄⋊C₄×C₈₉	1	1	yes	characteristic
170	C₂×C₈	C₄⋊C₄×C₈₉	1	1	yes	characteristic
171	C₂×C₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
172	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
173	C₂×C₈	C₂×Q₈×C₈₉	1	1	yes	characteristic
174	C₂×C₈	C₂×Q₈×C₈₉	1	1	yes	characteristic
175	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
176	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	4	1	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₄²	—	8	2	no	—
182	C₄²	C₄⋊C₄×C₈₉	1	1	yes	characteristic
183	C₄²	C₄⋊C₄×C₈₉	1	1	yes	characteristic
184	C₄²	SD₁₆×C₈₉	1	1	yes	characteristic
185	C₄²	SD₁₆×C₈₉	1	1	yes	characteristic
186	C₁₆	—	2	4	no	—
187	C₂³	C₂×SD₁₆×C₈₉	1	1	yes	characteristic
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	8	1	no	—
193	C₂×C₄	—	8	1	no	—
194	C₂×C₄	—	8	1	no	—
195	C₂×C₄	—	8	1	no	—
196	C₂×C₄	#106×C₈₉	1	1	yes	characteristic
197	C₂×C₄	#106×C₈₉	1	1	yes	characteristic
198	C₂×C₄	#106×C₈₉	1	1	yes	characteristic
199	C₂×C₄	#106×C₈₉	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	1	yes	characteristic
201	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
202	C₂×C₄	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
203	C₂×C₄	C₂×SD₁₆×C₈₉	1	1	yes	characteristic
204	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
205	C₂×C₄	#137×C₈₉	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	8	2	no	—
209	C₈	C₄×D₄×C₈₉	1	1	yes	characteristic
210	C₈	C₄×Q₈×C₈₉	1	1	yes	characteristic
211	C₈	#125×C₈₉	1	1	yes	characteristic
212	C₈	C₄⋊Q₈×C₈₉	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	C₂×#106×C₈₉	1	1	yes	characteristic
216	C₂²	#461×C₈₉	1	1	yes	characteristic
217	C₂²	#476×C₈₉	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	8	2	no	—
222	C₄	—	8	2	no	—
223	C₄	#365×C₈₉	1	2	yes	complemented
224	C₄	#389×C₈₉	1	1	yes	characteristic
225	C₄	#428×C₈₉	1	1	yes	characteristic
226	C₄	#460×C₈₉	1	1	yes	characteristic
227	C₄	#475×C₈₉	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	C₈₉×#1948	1	1	yes	characteristic
231	C₂	C₂×#365×C₈₉	1	1	yes	characteristic
232	C₂	C₈₉×#2155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₈₉
Frattini, cosocle	C₂²×C₈	C₂³×C₈₉
prime radical	C₂³×C₈₉	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ C₂×C₄×C₈×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₄⋊C₁₆×C₈₉	C₄	semidirect
16. C₄⋊C₁₆×C₈₉	C₄	semidirect
106. C₄×C₈₉	#365	semidirect
111. #13185	C₈₉	direct (N × H)
122. C₈₉	#13185	direct (N × H)
132. C₄⋊C₁₆	C₄×C₈₉	semidirect
133. C₄⋊C₁₆	C₄×C₈₉	semidirect
223. C₄	#365×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#365×C₈₉	2
C₈₉×#2150	2
C₈₉×#1848	2
C₄×#106×C₈₉	2
C₈₉×#1383	2
#13185	89

Subgroups — C89×#13185 #36391033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#13185 #36391033