Subgroups — C₈₉×#33393 #36411241

Order 22784.

1102 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#578×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#3407	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2916	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2148	C₂	1	6	yes	maximal, complemented
5	C₂×#362×C₈₉	C₂	1	6	yes	maximal, complemented
6	C₂×2₋¹⁺⁴×C₈₉	—	2	1	no	—
7	#578×C₈₉	—	2	4	no	—
8	#578×C₈₉	C₄	1	8	yes	—
9	C₂×#137×C₈₉	—	2	3	no	—
10	C₂×#135×C₈₉	C₂²	1	3	yes	—
11	C₂²×Q₁₆×C₈₉	C₂²	1	3	yes	—
12	#568×C₈₉	—	2	1	no	—
13	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
14	#504×C₈₉	—	2	4	no	—
15	C₂×M₃₂×C₈₉	C₂²	1	6	yes	—
16	#430×C₈₉	—	2	3	no	—
17	C₂×#108×C₈₉	C₂²	1	6	yes	—
18	#362×C₈₉	C₂²	1	12	yes	—
19	#361×C₈₉	C₂²	1	4	yes	—
20	2₋¹⁺⁴×C₈₉	—	2	8	no	—
21	2₋¹⁺⁴×C₈₉	—	4	4	no	—
22	C₂×#41×C₈₉	—	2	3	no	—
23	C₂×#41×C₈₉	—	4	3	no	—
24	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
25	C₂²×Q₈×C₈₉	—	2	3	no	—
26	C₂²×Q₈×C₈₉	—	4	1	no	—
27	#137×C₈₉	—	2	24	no	—
28	#135×C₈₉	—	2	6	no	—
29	#135×C₈₉	C₂×C₄	1	12	yes	—
30	C₂×Q₁₆×C₈₉	—	2	3	no	—
31	C₂×Q₁₆×C₈₉	—	2	6	no	—
32	C₂×Q₁₆×C₈₉	C₂×C₄	1	12	yes	—
33	C₂×Q₁₆×C₈₉	C₂³	1	3	yes	—
34	C₂×SD₁₆×C₈₉	—	2	3	no	—
35	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
36	#131×C₈₉	—	4	4	no	—
37	#131×C₈₉	C₂×C₄	1	4	yes	—
38	#131×C₈₉	D₄	1	4	yes	—
39	C₂×M₁₆×C₈₉	—	2	6	no	—
40	C₂×M₁₆×C₈₉	D₄	1	3	yes	—
41	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
42	M₃₂×C₈₉	—	2	6	no	—
43	M₃₂×C₈₉	C₂³	1	4	yes	—
44	C₂×C₁₆×C₈₉	—	2	6	no	—
45	#108×C₈₉	—	2	6	no	—
46	#108×C₈₉	C₂³	1	4	yes	—
47	#41×C₈₉	—	2	12	no	—
48	#41×C₈₉	—	4	24	no	—
49	#41×C₈₉	—	4	6	no	—
50	#41×C₈₉	C₂²⋊C₄	1	4	yes	—
51	#41×C₈₉	D₈	1	4	yes	—
52	C₂×Q₈×C₈₉	—	2	12	no	—
53	C₂×Q₈×C₈₉	—	2	3	no	—
54	C₂×Q₈×C₈₉	—	4	8	no	—
55	C₂×Q₈×C₈₉	—	4	6	no	—
56	C₂×Q₈×C₈₉	—	4	3	no	—
57	C₂×Q₈×C₈₉	SD₁₆	1	1	yes	characteristic
58	C₂×D₄×C₈₉	—	2	1	no	—
59	C₂×D₄×C₈₉	—	4	3	no	—
60	C₂×D₄×C₈₉	SD₁₆	1	3	yes	—
61	C₂²×C₄×C₈₉	—	4	3	no	—
62	C₂²×C₄×C₈₉	—	4	3	no	—
63	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
64	Q₁₆×C₈₉	—	2	12	no	—
65	Q₁₆×C₈₉	—	2	3	no	—
66	Q₁₆×C₈₉	C₂²×C₄	1	6	yes	—
67	SD₁₆×C₈₉	—	2	12	no	—
68	D₈×C₈₉	—	2	1	no	—
69	D₈×C₈₉	C₂²×C₄	1	2	yes	—
70	M₁₆×C₈₉	—	2	4	no	—
71	M₁₆×C₈₉	—	4	6	no	—
72	M₁₆×C₈₉	C₂²⋊C₄	1	12	yes	—
73	C₂×C₈×C₈₉	—	4	6	no	—
74	C₂×C₈×C₈₉	C₂²×C₄	1	6	yes	—
75	C₂×C₈×C₈₉	C₂×D₄	1	3	yes	—
76	C₂×C₈×C₈₉	C₂×D₄	1	6	yes	—
77	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₁₆×C₈₉	—	2	4	no	—
79	C₂³×C₈₉	—	4	1	no	—
80	C₂³×C₈₉	C₂×SD₁₆	1	3	yes	—
81	Q₈×C₈₉	—	2	6	no	—
82	Q₈×C₈₉	—	4	3	no	—
83	Q₈×C₈₉	—	4	12	no	—
84	Q₈×C₈₉	#102	1	4	yes	—
85	D₄×C₈₉	—	2	2	no	—
86	D₄×C₈₉	—	4	12	no	—
87	D₄×C₈₉	—	4	1	no	—
88	D₄×C₈₉	#102	1	12	yes	—
89	C₂×C₄×C₈₉	—	4	3	no	—
90	C₂×C₄×C₈₉	—	4	12	no	—
91	C₂×C₄×C₈₉	—	4	12	no	—
92	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	6	yes	—
93	C₂×C₄×C₈₉	C₂×D₈	1	6	yes	—
94	C₂×C₄×C₈₉	C₂×SD₁₆	1	3	yes	—
95	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
96	C₈×C₈₉	—	4	4	no	—
97	C₈×C₈₉	C₂×C₂²⋊C₄	1	6	yes	—
98	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
99	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
100	C₂²×C₈₉	—	4	3	no	—
101	C₂²×C₈₉	—	4	2	no	—
102	C₂²×C₈₉	—	8	2	no	—
103	C₂²×C₈₉	C₂×#102	1	6	yes	—
104	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
105	C₄×C₈₉	—	4	6	no	—
106	C₄×C₈₉	C₂×#102	1	6	yes	—
107	C₄×C₈₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
108	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
109	#33393	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂×C₈₉	—	2	1	no	—
111	C₂×C₈₉	—	2	6	no	—
112	C₂×C₈₉	—	8	2	no	—
113	C₂×C₈₉	C₂²×#102	1	1	yes	characteristic
114	C₂×#578	C₂×C₈₉	1	1	yes	characteristic
115	#3407	C₂×C₈₉	1	1	yes	characteristic, complemented
116	#2916	C₂×C₈₉	1	1	yes	characteristic, complemented
117	#2148	C₂×C₈₉	1	6	yes	complemented
118	C₂×#362	C₂×C₈₉	1	6	yes	complemented
119	C₈₉	#33393	1	1	yes	characteristic, direct product factor, Sylow
120	C₂×2₋¹⁺⁴	—	2	1	no	—
121	#578	—	2	4	no	—
122	#578	C₄×C₈₉	1	8	yes	—
123	C₂×#137	—	2	3	no	—
124	C₂×#135	C₂²×C₈₉	1	3	yes	—
125	C₂²×Q₁₆	C₂²×C₈₉	1	3	yes	—
126	#568	—	2	1	no	—
127	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
128	#504	—	2	4	no	—
129	C₂×M₃₂	C₂²×C₈₉	1	6	yes	—
130	#430	—	2	3	no	—
131	C₂×#108	C₂²×C₈₉	1	6	yes	—
132	#362	C₂²×C₈₉	1	12	yes	—
133	#361	C₂²×C₈₉	1	4	yes	—
134	2₋¹⁺⁴	—	2	8	no	—
135	2₋¹⁺⁴	—	4	4	no	—
136	C₂×#41	—	2	3	no	—
137	C₂×#41	—	4	3	no	—
138	C₂×#41	D₄×C₈₉	1	1	yes	characteristic
139	C₂²×Q₈	—	2	3	no	—
140	C₂²×Q₈	—	4	1	no	—
141	#137	—	2	24	no	—
142	#135	—	2	6	no	—
143	#135	C₂×C₄×C₈₉	1	12	yes	—
144	C₂×Q₁₆	—	2	3	no	—
145	C₂×Q₁₆	—	2	6	no	—
146	C₂×Q₁₆	C₂×C₄×C₈₉	1	12	yes	—
147	C₂×Q₁₆	C₂³×C₈₉	1	3	yes	—
148	C₂×SD₁₆	—	2	3	no	—
149	C₂×D₈	C₂³×C₈₉	1	1	yes	characteristic
150	#131	—	4	4	no	—
151	#131	C₂×C₄×C₈₉	1	4	yes	—
152	#131	D₄×C₈₉	1	4	yes	—
153	C₂×M₁₆	—	2	6	no	—
154	C₂×M₁₆	D₄×C₈₉	1	3	yes	—
155	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
156	M₃₂	—	2	6	no	—
157	M₃₂	C₂³×C₈₉	1	4	yes	—
158	C₂×C₁₆	—	2	6	no	—
159	#108	—	2	6	no	—
160	#108	C₂³×C₈₉	1	4	yes	—
161	#41	—	2	12	no	—
162	#41	—	4	24	no	—
163	#41	—	4	6	no	—
164	#41	C₂²⋊C₄×C₈₉	1	4	yes	—
165	#41	D₈×C₈₉	1	4	yes	—
166	C₂×Q₈	—	2	12	no	—
167	C₂×Q₈	—	2	3	no	—
168	C₂×Q₈	—	4	8	no	—
169	C₂×Q₈	—	4	6	no	—
170	C₂×Q₈	—	4	3	no	—
171	C₂×Q₈	SD₁₆×C₈₉	1	1	yes	characteristic
172	C₂×D₄	—	2	1	no	—
173	C₂×D₄	—	4	3	no	—
174	C₂×D₄	SD₁₆×C₈₉	1	3	yes	—
175	C₂²×C₄	—	4	3	no	—
176	C₂²×C₄	—	4	3	no	—
177	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
178	Q₁₆	—	2	12	no	—
179	Q₁₆	—	2	3	no	—
180	Q₁₆	C₂²×C₄×C₈₉	1	6	yes	—
181	SD₁₆	—	2	12	no	—
182	D₈	—	2	1	no	—
183	D₈	C₂²×C₄×C₈₉	1	2	yes	—
184	M₁₆	—	2	4	no	—
185	M₁₆	—	4	6	no	—
186	M₁₆	C₂²⋊C₄×C₈₉	1	12	yes	—
187	C₂×C₈	—	4	6	no	—
188	C₂×C₈	C₂²×C₄×C₈₉	1	6	yes	—
189	C₂×C₈	C₂×D₄×C₈₉	1	3	yes	—
190	C₂×C₈	C₂×D₄×C₈₉	1	6	yes	—
191	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
192	C₁₆	—	2	4	no	—
193	C₂³	—	4	1	no	—
194	C₂³	C₂×SD₁₆×C₈₉	1	3	yes	—
195	Q₈	—	2	6	no	—
196	Q₈	—	4	12	no	—
197	Q₈	—	4	3	no	—
198	Q₈	#102×C₈₉	1	4	yes	—
199	D₄	—	2	2	no	—
200	D₄	—	4	12	no	—
201	D₄	—	4	1	no	—
202	D₄	#102×C₈₉	1	12	yes	—
203	C₂×C₄	—	4	12	no	—
204	C₂×C₄	—	4	3	no	—
205	C₂×C₄	—	4	12	no	—
206	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	6	yes	—
207	C₂×C₄	C₂×D₈×C₈₉	1	6	yes	—
208	C₂×C₄	C₂×SD₁₆×C₈₉	1	3	yes	—
209	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
210	C₈	—	4	4	no	—
211	C₈	C₂×C₂²⋊C₄×C₈₉	1	6	yes	—
212	C₈	C₂³×C₄×C₈₉	1	1	yes	characteristic
213	C₈	C₂²×D₄×C₈₉	1	1	yes	characteristic
214	C₂²	—	4	3	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	8	2	no	—
217	C₂²	C₂×#102×C₈₉	1	6	yes	—
218	C₂²	C₂²×SD₁₆×C₈₉	1	1	yes	characteristic
219	C₄	—	4	6	no	—
220	C₄	C₂×#102×C₈₉	1	6	yes	—
221	C₄	C₂²×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
222	C₄	C₂²×D₈×C₈₉	1	1	yes	characteristic
223	C₂	—	2	1	no	—
224	C₂	—	2	6	no	—
225	C₂	—	8	2	no	—
226	C₂	C₂²×#102×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₈₉	C₂²×#102
commutator	C₈	C₂³×C₄×C₈₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₈₉
prime radical	C₂×#578×C₈₉	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₉ ◁ C₂×#41×C₈₉ ◁ C₂×#131×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#3407	C₂	semidirect
3. C₈₉×#2916	C₂	semidirect
4. C₈₉×#2148	C₂	semidirect
5. C₂×#362×C₈₉	C₂	semidirect
109. #33393	C₈₉	direct (N × H)
115. #3407	C₂×C₈₉	semidirect
116. #2916	C₂×C₈₉	semidirect
117. #2148	C₂×C₈₉	semidirect
118. C₂×#362	C₂×C₈₉	semidirect
119. C₈₉	#33393	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#578×C₈₉	2
C₈₉×#3407	2
C₈₉×#2916	2
C₈₉×#2148	2
C₂×#362×C₈₉	2
#33393	89

Subgroups — C89×#33393 #36411241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#33393 #36411241