Subgroups — C₂²×#550×C₈₉ #36440548

Order 22784.

12918 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	C₂³×#41×C₈₉	C₂	1	2	yes	maximal, complemented
2	C₂×#550×C₈₉	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#127×C₈₉	C₂	1	3	yes	maximal, complemented
4	C₂²×#121×C₈₉	C₂	1	6	yes	maximal, complemented
5	C₂²×C₄×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₄×C₈₉	C₂	1	3	yes	maximal, complemented
7	C₂²×#41×C₈₉	—	2	32	no	—
8	C₂²×#41×C₈₉	C₂²	1	48	yes	complemented
9	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₈₉	C₂²	1	6	yes	—
11	C₂³×D₄×C₈₉	C₂²	1	3	yes	—
12	C₂³×D₄×C₈₉	C₂²	1	6	yes	—
13	C₂⁴×C₄×C₈₉	C₂²	1	6	yes	—
14	#550×C₈₉	C₂²	1	256	yes	direct product factor
15	C₂×#127×C₈₉	C₂²	1	72	yes	complemented
16	C₂×#121×C₈₉	C₂²	1	144	yes	complemented
17	C₂×C₄×Q₈×C₈₉	C₂²	1	24	yes	complemented
18	C₂×C₄×D₄×C₈₉	C₂²	1	72	yes	complemented
19	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
21	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	6	yes	—
22	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
23	C₂⁵×C₈₉	C₂³	1	6	yes	—
24	C₂×#41×C₈₉	—	2	384	no	—
25	C₂×#41×C₈₉	C₂³	1	128	yes	complemented
26	C₂²×Q₈×C₈₉	D₄	1	16	yes	complemented
27	C₂²×Q₈×C₈₉	C₂³	1	12	yes	—
28	C₂²×D₄×C₈₉	—	2	48	no	—
29	C₂²×D₄×C₈₉	—	2	24	no	—
30	C₂²×D₄×C₈₉	—	4	24	no	—
31	C₂²×D₄×C₈₉	C₂³	1	36	yes	—
32	C₂²×D₄×C₈₉	C₂³	1	72	yes	—
33	C₂²×D₄×C₈₉	C₂³	1	72	yes	—
34	C₂³×C₄×C₈₉	—	2	48	no	—
35	C₂³×C₄×C₈₉	C₂³	1	72	yes	—
36	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
37	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
38	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
39	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	#127×C₈₉	C₂³	1	192	yes	complemented
41	#121×C₈₉	C₂³	1	384	yes	complemented
42	C₄×Q₈×C₈₉	C₂³	1	64	yes	complemented
43	C₄×D₄×C₈₉	C₂³	1	192	yes	complemented
44	C₂×C₄⋊C₄×C₈₉	C₂³	1	12	yes	—
45	C₂×C₄⋊C₄×C₈₉	C₂³	1	36	yes	—
46	C₂×C₂²⋊C₄×C₈₉	C₂³	1	72	yes	—
47	C₂×C₄²×C₈₉	C₂³	1	36	yes	—
48	C₂⁴×C₈₉	—	2	24	no	—
49	C₂⁴×C₈₉	—	2	24	no	—
50	C₂⁴×C₈₉	—	2	24	no	—
51	C₂⁴×C₈₉	C₂⁴	1	36	yes	—
52	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
53	#41×C₈₉	—	2	512	no	—
54	C₂×Q₈×C₈₉	C₂×D₄	1	96	yes	complemented
55	C₂×Q₈×C₈₉	C₂⁴	1	16	yes	—
56	C₂×D₄×C₈₉	—	2	288	no	—
57	C₂×D₄×C₈₉	—	2	144	no	—
58	C₂×D₄×C₈₉	—	4	144	no	—
59	C₂×D₄×C₈₉	C₂⁴	1	48	yes	—
60	C₂×D₄×C₈₉	C₂⁴	1	96	yes	—
61	C₂×D₄×C₈₉	C₂⁴	1	96	yes	—
62	C₂²×C₄×C₈₉	—	2	8	no	—
63	C₂²×C₄×C₈₉	—	2	12	no	—
64	C₂²×C₄×C₈₉	—	2	288	no	—
65	C₂²×C₄×C₈₉	C₂×D₄	1	24	yes	—
66	C₂²×C₄×C₈₉	#41	1	8	yes	complemented
67	C₂²×C₄×C₈₉	C₂⁴	1	18	yes	—
68	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
69	C₂²×C₄×C₈₉	C₂⁴	1	96	yes	—
70	C₂²×C₄×C₈₉	C₂⁴	1	18	yes	—
71	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
72	C₄⋊C₄×C₈₉	C₂⁴	1	16	yes	—
73	C₄⋊C₄×C₈₉	C₂⁴	1	48	yes	—
74	C₂²⋊C₄×C₈₉	C₂⁴	1	96	yes	—
75	C₄²×C₈₉	C₂⁴	1	48	yes	—
76	C₂³×C₈₉	—	2	72	no	—
77	C₂³×C₈₉	—	2	72	no	—
78	C₂³×C₈₉	—	2	72	no	—
79	C₂³×C₈₉	—	4	96	no	—
80	C₂³×C₈₉	C₂²×D₄	1	4	yes	—
81	C₂³×C₈₉	C₂×#41	1	4	yes	—
82	C₂³×C₈₉	H(5,𝔽₂)	1	4	yes	—
83	C₂³×C₈₉	C₂⁵	1	24	yes	—
84	C₂³×C₈₉	C₂⁵	1	3	yes	—
85	Q₈×C₈₉	C₂²×D₄	1	64	yes	complemented
86	D₄×C₈₉	—	2	192	no	—
87	D₄×C₈₉	—	2	96	no	—
88	D₄×C₈₉	—	4	96	no	—
89	C₂×C₄×C₈₉	—	2	24	no	—
90	C₂×C₄×C₈₉	—	2	36	no	—
91	C₂×C₄×C₈₉	—	2	192	no	—
92	C₂×C₄×C₈₉	C₂²×D₄	1	72	yes	—
93	C₂×C₄×C₈₉	C₂×#41	1	24	yes	complemented
94	C₂×C₄×C₈₉	C₂⁵	1	12	yes	—
95	C₂×C₄×C₈₉	C₂⁵	1	8	yes	—
96	C₂×C₄×C₈₉	C₂⁵	1	12	yes	—
97	C₂×C₄×C₈₉	C₂⁵	1	4	yes	—
98	C₂²×C₈₉	—	2	24	no	—
99	C₂²×C₈₉	—	2	24	no	—
100	C₂²×C₈₉	—	2	24	no	—
101	C₂²×C₈₉	—	4	144	no	—
102	C₂²×C₈₉	#550	1	16	yes	direct product factor
103	C₂²×C₈₉	C₂³×D₄	1	6	yes	—
104	C₂²×C₈₉	C₂²×#41	1	6	yes	—
105	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	6	yes	—
106	C₂²×C₈₉	C₂⁶	1	1	yes	characteristic
107	C₄×C₈₉	—	2	8	no	—
108	C₄×C₈₉	—	2	12	no	—
109	C₄×C₈₉	C₂³×D₄	1	24	yes	—
110	C₄×C₈₉	C₂²×#41	1	8	yes	complemented
111	C₂²×#550	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₈₉	—	4	24	no	—
113	C₂×C₈₉	C₂×#550	1	12	yes	direct product factor
114	C₂×C₈₉	C₂⁴×D₄	1	1	yes	characteristic
115	C₂×C₈₉	C₂³×#41	1	1	yes	characteristic
116	C₂×C₈₉	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
117	C₂³×#41	C₂×C₈₉	1	2	yes	complemented
118	C₂×#550	C₂×C₈₉	1	48	yes	direct product factor
119	C₂²×#127	C₂×C₈₉	1	3	yes	complemented
120	C₂²×#121	C₂×C₈₉	1	6	yes	complemented
121	C₂²×C₄×Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
122	C₂²×C₄×D₄	C₂×C₈₉	1	3	yes	complemented
123	C₈₉	C₂²×#550	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²×#41	—	2	32	no	—
125	C₂²×#41	C₂²×C₈₉	1	48	yes	complemented
126	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic
127	C₂³×D₄	C₂²×C₈₉	1	6	yes	—
128	C₂³×D₄	C₂²×C₈₉	1	3	yes	—
129	C₂³×D₄	C₂²×C₈₉	1	6	yes	—
130	C₂⁴×C₄	C₂²×C₈₉	1	6	yes	—
131	#550	C₂²×C₈₉	1	256	yes	direct product factor
132	C₂×#127	C₂²×C₈₉	1	72	yes	complemented
133	C₂×#121	C₂²×C₈₉	1	144	yes	complemented
134	C₂×C₄×Q₈	C₂²×C₈₉	1	24	yes	complemented
135	C₂×C₄×D₄	C₂²×C₈₉	1	72	yes	complemented
136	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
137	C₂²×C₄⋊C₄	C₂²×C₈₉	1	3	yes	—
138	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	6	yes	—
139	C₂²×C₄²	C₂²×C₈₉	1	3	yes	—
140	C₂⁵	C₂³×C₈₉	1	6	yes	—
141	C₂×#41	—	2	384	no	—
142	C₂×#41	C₂³×C₈₉	1	128	yes	complemented
143	C₂²×Q₈	D₄×C₈₉	1	16	yes	complemented
144	C₂²×Q₈	C₂³×C₈₉	1	12	yes	—
145	C₂²×D₄	—	2	48	no	—
146	C₂²×D₄	—	2	24	no	—
147	C₂²×D₄	—	4	24	no	—
148	C₂²×D₄	C₂³×C₈₉	1	36	yes	—
149	C₂²×D₄	C₂³×C₈₉	1	72	yes	—
150	C₂²×D₄	C₂³×C₈₉	1	72	yes	—
151	C₂³×C₄	—	2	48	no	—
152	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
153	C₂³×C₄	C₂³×C₈₉	1	3	yes	—
154	C₂³×C₄	C₂³×C₈₉	1	72	yes	—
155	C₂³×C₄	C₂³×C₈₉	1	3	yes	—
156	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
157	#127	C₂³×C₈₉	1	192	yes	complemented
158	#121	C₂³×C₈₉	1	384	yes	complemented
159	C₄×Q₈	C₂³×C₈₉	1	64	yes	complemented
160	C₄×D₄	C₂³×C₈₉	1	192	yes	complemented
161	C₂×C₄⋊C₄	C₂³×C₈₉	1	12	yes	—
162	C₂×C₄⋊C₄	C₂³×C₈₉	1	36	yes	—
163	C₂×C₂²⋊C₄	C₂³×C₈₉	1	72	yes	—
164	C₂×C₄²	C₂³×C₈₉	1	36	yes	—
165	C₂⁴	—	2	24	no	—
166	C₂⁴	—	2	24	no	—
167	C₂⁴	—	2	24	no	—
168	C₂⁴	C₂⁴×C₈₉	1	36	yes	—
169	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
170	#41	—	2	512	no	—
171	C₂×Q₈	C₂×D₄×C₈₉	1	96	yes	complemented
172	C₂×Q₈	C₂⁴×C₈₉	1	16	yes	—
173	C₂×D₄	—	2	288	no	—
174	C₂×D₄	—	2	144	no	—
175	C₂×D₄	—	4	144	no	—
176	C₂×D₄	C₂⁴×C₈₉	1	48	yes	—
177	C₂×D₄	C₂⁴×C₈₉	1	96	yes	—
178	C₂×D₄	C₂⁴×C₈₉	1	96	yes	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	12	no	—
181	C₂²×C₄	—	2	288	no	—
182	C₂²×C₄	C₂×D₄×C₈₉	1	24	yes	—
183	C₂²×C₄	#41×C₈₉	1	8	yes	complemented
184	C₂²×C₄	C₂⁴×C₈₉	1	18	yes	—
185	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
186	C₂²×C₄	C₂⁴×C₈₉	1	96	yes	—
187	C₂²×C₄	C₂⁴×C₈₉	1	18	yes	—
188	C₂²×C₄	C₂⁴×C₈₉	1	6	yes	—
189	C₄⋊C₄	C₂⁴×C₈₉	1	16	yes	—
190	C₄⋊C₄	C₂⁴×C₈₉	1	48	yes	—
191	C₂²⋊C₄	C₂⁴×C₈₉	1	96	yes	—
192	C₄²	C₂⁴×C₈₉	1	48	yes	—
193	C₂³	—	2	72	no	—
194	C₂³	—	2	72	no	—
195	C₂³	—	2	72	no	—
196	C₂³	—	4	96	no	—
197	C₂³	C₂²×D₄×C₈₉	1	4	yes	—
198	C₂³	C₂×#41×C₈₉	1	4	yes	—
199	C₂³	H(5,𝔽₂)×C₈₉	1	4	yes	—
200	C₂³	C₂⁵×C₈₉	1	24	yes	—
201	C₂³	C₂⁵×C₈₉	1	3	yes	—
202	Q₈	C₂²×D₄×C₈₉	1	64	yes	complemented
203	D₄	—	2	192	no	—
204	D₄	—	2	96	no	—
205	D₄	—	4	96	no	—
206	C₂×C₄	—	2	24	no	—
207	C₂×C₄	—	2	36	no	—
208	C₂×C₄	—	2	192	no	—
209	C₂×C₄	C₂²×D₄×C₈₉	1	72	yes	—
210	C₂×C₄	C₂×#41×C₈₉	1	24	yes	complemented
211	C₂×C₄	C₂⁵×C₈₉	1	12	yes	—
212	C₂×C₄	C₂⁵×C₈₉	1	8	yes	—
213	C₂×C₄	C₂⁵×C₈₉	1	12	yes	—
214	C₂×C₄	C₂⁵×C₈₉	1	4	yes	—
215	C₂²	—	2	24	no	—
216	C₂²	—	2	24	no	—
217	C₂²	—	2	24	no	—
218	C₂²	—	4	144	no	—
219	C₂²	#550×C₈₉	1	16	yes	direct product factor
220	C₂²	C₂³×D₄×C₈₉	1	6	yes	—
221	C₂²	C₂²×#41×C₈₉	1	6	yes	—
222	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	6	yes	—
223	C₂²	C₂⁶×C₈₉	1	1	yes	characteristic
224	C₄	—	2	8	no	—
225	C₄	—	2	12	no	—
226	C₄	C₂³×D₄×C₈₉	1	24	yes	—
227	C₄	C₂²×#41×C₈₉	1	8	yes	complemented
228	C₂	—	4	24	no	—
229	C₂	C₂×#550×C₈₉	1	12	yes	direct product factor
230	C₂	C₂⁴×D₄×C₈₉	1	1	yes	characteristic
231	C₂	C₂³×#41×C₈₉	1	1	yes	characteristic
232	C₂	C₂²×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₈₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₈₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×#41×C₈₉	C₂	semidirect
2. C₂×#550×C₈₉	C₂	direct (N × H)
3. C₂²×#127×C₈₉	C₂	semidirect
4. C₂²×#121×C₈₉	C₂	semidirect
5. C₂²×C₄×Q₈×C₈₉	C₂	semidirect
6. C₂²×C₄×D₄×C₈₉	C₂	semidirect
8. C₂²×#41×C₈₉	C₂²	semidirect
14. #550×C₈₉	C₂²	direct (N × H)
15. C₂×#127×C₈₉	C₂²	semidirect
16. C₂×#121×C₈₉	C₂²	semidirect
17. C₂×C₄×Q₈×C₈₉	C₂²	semidirect
18. C₂×C₄×D₄×C₈₉	C₂²	semidirect
25. C₂×#41×C₈₉	C₂³	semidirect
26. C₂²×Q₈×C₈₉	D₄	semidirect
40. #127×C₈₉	C₂³	semidirect
41. #121×C₈₉	C₂³	semidirect
42. C₄×Q₈×C₈₉	C₂³	semidirect
43. C₄×D₄×C₈₉	C₂³	semidirect
54. C₂×Q₈×C₈₉	C₂×D₄	semidirect
66. C₂²×C₄×C₈₉	#41	semidirect
85. Q₈×C₈₉	C₂²×D₄	semidirect
93. C₂×C₄×C₈₉	C₂×#41	semidirect
102. C₂²×C₈₉	#550	direct (N × H)
110. C₄×C₈₉	C₂²×#41	semidirect
111. C₂²×#550	C₈₉	direct (N × H)
113. C₂×C₈₉	C₂×#550	direct (N × H)
117. C₂³×#41	C₂×C₈₉	semidirect
118. C₂×#550	C₂×C₈₉	direct (N × H)
119. C₂²×#127	C₂×C₈₉	semidirect
120. C₂²×#121	C₂×C₈₉	semidirect
121. C₂²×C₄×Q₈	C₂×C₈₉	semidirect
122. C₂²×C₄×D₄	C₂×C₈₉	semidirect
123. C₈₉	C₂²×#550	direct (N × H)
125. C₂²×#41	C₂²×C₈₉	semidirect
131. #550	C₂²×C₈₉	direct (N × H)
132. C₂×#127	C₂²×C₈₉	semidirect
133. C₂×#121	C₂²×C₈₉	semidirect
134. C₂×C₄×Q₈	C₂²×C₈₉	semidirect
135. C₂×C₄×D₄	C₂²×C₈₉	semidirect
142. C₂×#41	C₂³×C₈₉	semidirect
143. C₂²×Q₈	D₄×C₈₉	semidirect
157. #127	C₂³×C₈₉	semidirect
158. #121	C₂³×C₈₉	semidirect
159. C₄×Q₈	C₂³×C₈₉	semidirect
160. C₄×D₄	C₂³×C₈₉	semidirect
171. C₂×Q₈	C₂×D₄×C₈₉	semidirect
183. C₂²×C₄	#41×C₈₉	semidirect
202. Q₈	C₂²×D₄×C₈₉	semidirect
210. C₂×C₄	C₂×#41×C₈₉	semidirect
219. C₂²	#550×C₈₉	direct (N × H)
227. C₄	C₂²×#41×C₈₉	semidirect
229. C₂	C₂×#550×C₈₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41×C₈₉	2
C₂×#550×C₈₉	2
C₂²×#127×C₈₉	2
C₂²×#121×C₈₉	2
C₂²×C₄×Q₈×C₈₉	2
C₂²×C₄×D₄×C₈₉	2
C₂²×#550	89

Subgroups — C22×#550×C89 #36440548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#550×C₈₉ #36440548