Subgroups — C₈₉×#33434 #36411282

Order 22784.

690 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504×C₈₉	C₂	1	2	yes	maximal, complemented
2	C₈₉×#2931	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2169	C₂	1	4	yes	maximal, complemented
4	D₄×C₁₆×C₈₉	C₂	1	4	yes	maximal, complemented
5	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2113	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₈₉	C₂²	1	2	yes	—
9	#532×C₈₉	C₄	1	1	yes	characteristic
10	#504×C₈₉	—	2	8	no	—
11	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
12	C₂×M₃₂×C₈₉	C₂²	1	4	yes	—
13	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₈₉	C₂²	1	4	yes	—
15	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	#436×C₈₉	C₄	1	4	yes	—
17	C₈×D₄×C₈₉	C₂²	1	4	yes	—
18	#423×C₈₉	C₄	1	1	yes	characteristic
19	#408×C₈₉	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₁₆×C₈₉	C₂²	1	4	yes	—
22	C₂²⋊C₁₆×C₈₉	C₂²	1	8	yes	—
23	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
24	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
25	C₂×#41×C₈₉	C₂×C₄	1	2	yes	—
26	#131×C₈₉	—	2	4	no	—
27	#131×C₈₉	—	2	4	no	—
28	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
32	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₈₉	C₈	1	1	yes	characteristic
34	#127×C₈₉	C₈	1	1	yes	characteristic
35	#124×C₈₉	C₈	1	2	yes	—
36	#121×C₈₉	C₈	1	4	yes	—
37	C₄×D₄×C₈₉	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
39	M₃₂×C₈₉	—	2	4	no	—
40	M₃₂×C₈₉	—	2	8	no	—
41	C₂×C₁₆×C₈₉	—	2	2	no	—
42	C₂×C₁₆×C₈₉	—	2	8	no	—
43	C₂×C₁₆×C₈₉	D₄	1	4	yes	complemented
44	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
45	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
46	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
47	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
49	C₂²⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
50	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₈₉	C₂³	1	2	yes	—
52	#41×C₈₉	—	2	4	no	—
53	#41×C₈₉	—	2	4	no	—
54	C₂×Q₈×C₈₉	C₂×C₈	1	2	yes	—
55	C₂×D₄×C₈₉	C₂×C₈	1	4	yes	—
56	C₂×D₄×C₈₉	C₂×C₈	1	4	yes	—
57	C₂×D₄×C₈₉	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
61	M₁₆×C₈₉	—	2	4	no	—
62	M₁₆×C₈₉	—	2	8	no	—
63	C₂×C₈×C₈₉	—	2	2	no	—
64	C₂×C₈×C₈₉	—	2	4	no	—
65	C₂×C₈×C₈₉	—	2	4	no	—
66	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₈₉	#41	1	2	yes	—
71	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
73	C₂²⋊C₄×C₈₉	C₂×C₈	1	8	yes	—
74	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
75	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
76	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
77	C₁₆×C₈₉	—	2	4	no	—
78	C₁₆×C₈₉	—	2	2	no	—
79	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
80	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
81	C₂³×C₈₉	C₂²×C₈	1	4	yes	—
82	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
83	Q₈×C₈₉	—	2	4	no	—
84	D₄×C₈₉	—	2	8	no	—
85	D₄×C₈₉	—	2	4	no	—
86	D₄×C₈₉	—	2	8	no	—
87	C₂×C₄×C₈₉	—	2	4	no	—
88	C₂×C₄×C₈₉	—	2	4	no	—
89	C₂×C₄×C₈₉	—	2	4	no	—
90	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
92	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
93	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₈₉	#131	1	2	yes	—
96	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
97	C₈×C₈₉	—	2	4	no	—
98	C₈×C₈₉	—	2	2	no	—
99	C₈×C₈₉	C₄×D₄	1	2	yes	—
100	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
102	C₂²×C₈₉	—	2	2	no	—
103	C₂²×C₈₉	—	2	8	no	—
104	C₂²×C₈₉	—	2	4	no	—
105	C₂²×C₈₉	C₈×D₄	1	2	yes	—
106	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
107	C₄×C₈₉	—	2	2	no	—
108	C₄×C₈₉	—	2	4	no	—
109	C₄×C₈₉	C₈×D₄	1	2	yes	—
110	C₄×C₈₉	#504	1	4	yes	complemented
111	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
112	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
113	#33434	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂×C₈₉	—	2	2	no	—
115	C₂×C₈₉	—	4	4	no	—
116	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
117	C₂×C₈₉	C₂×#504	1	2	yes	—
118	C₂×#504	C₂×C₈₉	1	2	yes	complemented
119	#2931	C₂×C₈₉	1	1	yes	characteristic
120	#2169	C₂×C₈₉	1	4	yes	complemented
121	D₄×C₁₆	C₂×C₈₉	1	4	yes	complemented
122	#2150	C₂×C₈₉	1	1	yes	characteristic, complemented
123	#2113	C₂×C₈₉	1	2	yes	complemented
124	C₂×C₄×C₁₆	C₂×C₈₉	1	1	yes	characteristic, complemented
125	C₈₉	#33434	1	1	yes	characteristic, direct product factor, Sylow
126	C₂×#131	C₂²×C₈₉	1	2	yes	—
127	#532	C₄×C₈₉	1	1	yes	characteristic
128	#504	—	2	8	no	—
129	C₂×M₃₂	C₂²×C₈₉	1	2	yes	—
130	C₂×M₃₂	C₂²×C₈₉	1	4	yes	—
131	C₂²×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
132	C₂²×C₁₆	C₂²×C₈₉	1	4	yes	—
133	C₂²×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
134	#436	C₄×C₈₉	1	4	yes	—
135	C₈×D₄	C₂²×C₈₉	1	4	yes	—
136	#423	C₄×C₈₉	1	1	yes	characteristic
137	#408	C₄×C₈₉	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
139	C₄⋊C₁₆	C₂²×C₈₉	1	4	yes	—
140	C₂²⋊C₁₆	C₂²×C₈₉	1	8	yes	—
141	C₄×C₁₆	C₂²×C₈₉	1	2	yes	—
142	C₄×C₁₆	C₂²×C₈₉	1	2	yes	—
143	C₂×#41	C₂×C₄×C₈₉	1	2	yes	—
144	#131	—	2	4	no	—
145	#131	—	2	4	no	—
146	C₂×M₁₆	C₂×C₄×C₈₉	1	2	yes	—
147	C₂×M₁₆	C₂×C₄×C₈₉	1	4	yes	—
148	C₂²×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
149	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
150	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
151	C₄⋊Q₈	C₈×C₈₉	1	1	yes	characteristic
152	#127	C₈×C₈₉	1	1	yes	characteristic
153	#124	C₈×C₈₉	1	2	yes	—
154	#121	C₈×C₈₉	1	4	yes	—
155	C₄×D₄	C₂×C₄×C₈₉	1	4	yes	—
156	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
157	M₃₂	—	2	4	no	—
158	M₃₂	—	2	8	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	2	8	no	—
161	C₂×C₁₆	D₄×C₈₉	1	4	yes	complemented
162	C₂×C₁₆	C₂³×C₈₉	1	2	yes	—
163	C₂×C₁₆	C₂³×C₈₉	1	4	yes	—
164	C₂×C₁₆	C₂³×C₈₉	1	1	yes	characteristic
165	C₂×C₁₆	C₂³×C₈₉	1	1	yes	characteristic
166	C₄⋊C₈	C₂×C₄×C₈₉	1	4	yes	—
167	C₂²⋊C₈	C₂×C₄×C₈₉	1	8	yes	—
168	C₄×C₈	C₂×C₄×C₈₉	1	2	yes	—
169	C₄×C₈	C₂³×C₈₉	1	2	yes	—
170	#41	—	2	4	no	—
171	#41	—	2	4	no	—
172	C₂×Q₈	C₂×C₈×C₈₉	1	2	yes	—
173	C₂×D₄	C₂×C₈×C₈₉	1	4	yes	—
174	C₂×D₄	C₂×C₈×C₈₉	1	4	yes	—
175	C₂×D₄	C₂×C₈×C₈₉	1	2	yes	—
176	C₂²×C₄	C₂×C₈×C₈₉	1	2	yes	—
177	C₂²×C₄	C₂²×C₄×C₈₉	1	4	yes	—
178	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
179	M₁₆	—	2	4	no	—
180	M₁₆	—	2	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂²×C₄×C₈₉	1	2	yes	—
185	C₂×C₈	C₂²×C₄×C₈₉	1	4	yes	—
186	C₂×C₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
187	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
188	C₂×C₈	#41×C₈₉	1	2	yes	—
189	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
190	C₄⋊C₄	C₂×C₈×C₈₉	1	4	yes	—
191	C₂²⋊C₄	C₂×C₈×C₈₉	1	8	yes	—
192	C₄²	C₂×C₈×C₈₉	1	1	yes	characteristic
193	C₄²	C₂×C₈×C₈₉	1	1	yes	characteristic
194	C₄²	C₂²×C₄×C₈₉	1	2	yes	—
195	C₁₆	—	2	4	no	—
196	C₁₆	—	2	2	no	—
197	C₁₆	C₂×D₄×C₈₉	1	2	yes	—
198	C₁₆	C₂×D₄×C₈₉	1	2	yes	—
199	C₂³	C₂²×C₈×C₈₉	1	4	yes	—
200	C₂³	C₂²×C₈×C₈₉	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	D₄	—	2	8	no	—
203	D₄	—	2	4	no	—
204	D₄	—	2	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	C₄×D₄×C₈₉	1	2	yes	—
209	C₂×C₄	C₂²×C₈×C₈₉	1	2	yes	—
210	C₂×C₄	C₂²×C₈×C₈₉	1	4	yes	—
211	C₂×C₄	C₂²×C₈×C₈₉	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₈×C₈₉	1	2	yes	—
213	C₂×C₄	#131×C₈₉	1	2	yes	—
214	C₂×C₄	C₂³×C₄×C₈₉	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₈	—	2	2	no	—
217	C₈	C₄×D₄×C₈₉	1	2	yes	—
218	C₈	C₂²×D₄×C₈₉	1	1	yes	characteristic
219	C₈	C₂×#41×C₈₉	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	2	8	no	—
222	C₂²	—	2	4	no	—
223	C₂²	C₈×D₄×C₈₉	1	2	yes	—
224	C₂²	C₂³×C₈×C₈₉	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	2	4	no	—
227	C₄	C₈×D₄×C₈₉	1	2	yes	—
228	C₄	#504×C₈₉	1	4	yes	complemented
229	C₄	C₂×C₄×D₄×C₈₉	1	1	yes	characteristic
230	C₄	C₂×#131×C₈₉	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	4	4	no	—
233	C₂	C₂×C₈×D₄×C₈₉	1	1	yes	characteristic
234	C₂	C₂×#504×C₈₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆×C₈₉	C₂³
commutator	C₂²	C₂³×C₈×C₈₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₈₉
socle	C₂²×C₈₉	C₂³×C₈
prime radical	#532×C₈₉	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#504×C₈₉	C₂	semidirect
3. C₈₉×#2169	C₂	semidirect
4. D₄×C₁₆×C₈₉	C₂	semidirect
5. C₈₉×#2150	C₂	semidirect
6. C₈₉×#2113	C₂	semidirect
7. C₂×C₄×C₁₆×C₈₉	C₂	semidirect
43. C₂×C₁₆×C₈₉	D₄	semidirect
110. C₄×C₈₉	#504	semidirect
113. #33434	C₈₉	direct (N × H)
118. C₂×#504	C₂×C₈₉	semidirect
120. #2169	C₂×C₈₉	semidirect
121. D₄×C₁₆	C₂×C₈₉	semidirect
122. #2150	C₂×C₈₉	semidirect
123. #2113	C₂×C₈₉	semidirect
124. C₂×C₄×C₁₆	C₂×C₈₉	semidirect
125. C₈₉	#33434	direct (N × H)
161. C₂×C₁₆	D₄×C₈₉	semidirect
228. C₄	#504×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#504×C₈₉	2
C₈₉×#2931	2
C₈₉×#2169	2
D₄×C₁₆×C₈₉	2
C₈₉×#2150	2
C₈₉×#2113	2
C₂×C₄×C₁₆×C₈₉	2
#33434	89

Subgroups — C89×#33434 #36411282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#33434 #36411282