Subgroups — C₂²×#562×C₈₉ #36440537

Order 22784.

8102 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×#562×C₈₉	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#125×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#124×C₈₉	C₂	1	2	yes	maximal, complemented
4	C₂²×#123×C₈₉	C₂	1	4	yes	maximal, complemented
5	C₂²×C₂²⋊Q₈×C₈₉	C₂	1	4	yes	maximal, complemented
6	C₂²×#121×C₈₉	C₂	1	4	yes	maximal, complemented
7	C₂³×Q₈×C₈₉	C₂²	1	2	yes	—
8	C₂³×D₄×C₈₉	C₂²	1	4	yes	—
9	C₂³×D₄×C₈₉	C₂²	1	2	yes	—
10	C₂⁴×C₄×C₈₉	C₂²	1	4	yes	—
11	#562×C₈₉	C₂²	1	256	yes	direct product factor
12	C₂×#125×C₈₉	C₂²	1	24	yes	complemented
13	C₂×#124×C₈₉	C₂²	1	48	yes	complemented
14	C₂×#123×C₈₉	C₂²	1	96	yes	complemented
15	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	96	yes	complemented
16	C₂×#121×C₈₉	C₂²	1	96	yes	complemented
17	C₂²×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
18	C₂²×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
19	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
20	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
21	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
22	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
23	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
24	C₂⁵×C₈₉	C₂³	1	4	yes	—
25	C₂²×Q₈×C₈₉	—	4	8	no	—
26	C₂²×Q₈×C₈₉	C₂³	1	24	yes	—
27	C₂²×D₄×C₈₉	—	4	16	no	—
28	C₂²×D₄×C₈₉	—	4	8	no	—
29	C₂²×D₄×C₈₉	C₂³	1	48	yes	—
30	C₂²×D₄×C₈₉	C₂³	1	24	yes	—
31	C₂³×C₄×C₈₉	—	4	16	no	—
32	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
34	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
35	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
36	C₂³×C₄×C₈₉	C₂³	1	48	yes	—
37	#125×C₈₉	C₂³	1	64	yes	complemented
38	#124×C₈₉	C₂³	1	128	yes	complemented
39	#123×C₈₉	C₂³	1	256	yes	complemented
40	C₂²⋊Q₈×C₈₉	C₂³	1	256	yes	complemented
41	#121×C₈₉	C₂³	1	256	yes	complemented
42	C₂×C₄⋊C₄×C₈₉	C₂³	1	48	yes	—
43	C₂×C₄⋊C₄×C₈₉	C₂³	1	48	yes	—
44	C₂×C₄⋊C₄×C₈₉	C₂³	1	24	yes	—
45	C₂×C₂²⋊C₄×C₈₉	C₂³	1	48	yes	—
46	C₂×C₂²⋊C₄×C₈₉	C₂³	1	48	yes	—
47	C₂×C₂²⋊C₄×C₈₉	C₂³	1	48	yes	—
48	C₂×C₄²×C₈₉	C₂³	1	12	yes	—
49	C₂⁴×C₈₉	—	2	16	no	—
50	C₂⁴×C₈₉	—	2	16	no	—
51	C₂⁴×C₈₉	—	2	16	no	—
52	C₂⁴×C₈₉	C₂⁴	1	24	yes	—
53	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
54	C₂×Q₈×C₈₉	—	4	48	no	—
55	C₂×Q₈×C₈₉	C₂⁴	1	32	yes	—
56	C₂×D₄×C₈₉	—	4	96	no	—
57	C₂×D₄×C₈₉	—	4	48	no	—
58	C₂×D₄×C₈₉	C₂⁴	1	64	yes	—
59	C₂×D₄×C₈₉	C₂⁴	1	32	yes	—
60	C₂²×C₄×C₈₉	—	2	16	no	—
61	C₂²×C₄×C₈₉	—	2	16	no	—
62	C₂²×C₄×C₈₉	—	2	8	no	—
63	C₂²×C₄×C₈₉	—	2	4	no	—
64	C₂²×C₄×C₈₉	—	4	96	no	—
65	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
66	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
67	C₂²×C₄×C₈₉	C₂⁴	1	24	yes	—
68	C₂²×C₄×C₈₉	C₂⁴	1	24	yes	—
69	C₂²×C₄×C₈₉	C₂⁴	1	64	yes	—
70	C₄⋊C₄×C₈₉	C₂⁴	1	64	yes	—
71	C₄⋊C₄×C₈₉	C₂⁴	1	64	yes	—
72	C₄⋊C₄×C₈₉	C₂⁴	1	32	yes	—
73	C₂²⋊C₄×C₈₉	C₂⁴	1	64	yes	—
74	C₂²⋊C₄×C₈₉	C₂⁴	1	64	yes	—
75	C₂²⋊C₄×C₈₉	C₂⁴	1	64	yes	—
76	C₄²×C₈₉	C₂⁴	1	16	yes	—
77	C₂³×C₈₉	—	2	48	no	—
78	C₂³×C₈₉	—	2	48	no	—
79	C₂³×C₈₉	—	2	48	no	—
80	C₂³×C₈₉	—	4	64	no	—
81	C₂³×C₈₉	H(5,𝔽₂)	1	8	yes	—
82	C₂³×C₈₉	2₋¹⁺⁴	1	4	yes	—
83	C₂³×C₈₉	C₂⁵	1	16	yes	—
84	C₂³×C₈₉	C₂⁵	1	3	yes	—
85	Q₈×C₈₉	—	4	32	no	—
86	D₄×C₈₉	—	4	64	no	—
87	D₄×C₈₉	—	4	32	no	—
88	C₂×C₄×C₈₉	—	2	48	no	—
89	C₂×C₄×C₈₉	—	2	48	no	—
90	C₂×C₄×C₈₉	—	2	24	no	—
91	C₂×C₄×C₈₉	—	2	12	no	—
92	C₂×C₄×C₈₉	—	4	64	no	—
93	C₂×C₄×C₈₉	C₂⁵	1	4	yes	—
94	C₂×C₄×C₈₉	C₂⁵	1	8	yes	—
95	C₂×C₄×C₈₉	C₂⁵	1	16	yes	—
96	C₂×C₄×C₈₉	C₂⁵	1	16	yes	—
97	C₂²×C₈₉	—	2	16	no	—
98	C₂²×C₈₉	—	2	16	no	—
99	C₂²×C₈₉	—	2	16	no	—
100	C₂²×C₈₉	—	4	96	no	—
101	C₂²×C₈₉	#562	1	16	yes	direct product factor
102	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	12	yes	—
103	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	6	yes	—
104	C₂²×C₈₉	C₂⁶	1	1	yes	characteristic
105	C₄×C₈₉	—	2	16	no	—
106	C₄×C₈₉	—	2	16	no	—
107	C₄×C₈₉	—	2	8	no	—
108	C₄×C₈₉	—	2	4	no	—
109	C₂²×#562	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂×C₈₉	—	4	16	no	—
111	C₂×C₈₉	C₂×#562	1	12	yes	direct product factor
112	C₂×C₈₉	C₂²×H(5,𝔽₂)	1	2	yes	—
113	C₂×C₈₉	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
114	C₂×#562	C₂×C₈₉	1	48	yes	direct product factor
115	C₂²×#125	C₂×C₈₉	1	1	yes	characteristic, complemented
116	C₂²×#124	C₂×C₈₉	1	2	yes	complemented
117	C₂²×#123	C₂×C₈₉	1	4	yes	complemented
118	C₂²×C₂²⋊Q₈	C₂×C₈₉	1	4	yes	complemented
119	C₂²×#121	C₂×C₈₉	1	4	yes	complemented
120	C₈₉	C₂²×#562	1	1	yes	characteristic, direct product factor, Sylow
121	C₂³×Q₈	C₂²×C₈₉	1	2	yes	—
122	C₂³×D₄	C₂²×C₈₉	1	4	yes	—
123	C₂³×D₄	C₂²×C₈₉	1	2	yes	—
124	C₂⁴×C₄	C₂²×C₈₉	1	4	yes	—
125	#562	C₂²×C₈₉	1	256	yes	direct product factor
126	C₂×#125	C₂²×C₈₉	1	24	yes	complemented
127	C₂×#124	C₂²×C₈₉	1	48	yes	complemented
128	C₂×#123	C₂²×C₈₉	1	96	yes	complemented
129	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	96	yes	complemented
130	C₂×#121	C₂²×C₈₉	1	96	yes	complemented
131	C₂²×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
132	C₂²×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
133	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
134	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
135	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
136	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
137	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
138	C₂⁵	C₂³×C₈₉	1	4	yes	—
139	C₂²×Q₈	—	4	8	no	—
140	C₂²×Q₈	C₂³×C₈₉	1	24	yes	—
141	C₂²×D₄	—	4	16	no	—
142	C₂²×D₄	—	4	8	no	—
143	C₂²×D₄	C₂³×C₈₉	1	48	yes	—
144	C₂²×D₄	C₂³×C₈₉	1	24	yes	—
145	C₂³×C₄	—	4	16	no	—
146	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
147	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
148	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
149	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
150	C₂³×C₄	C₂³×C₈₉	1	48	yes	—
151	#125	C₂³×C₈₉	1	64	yes	complemented
152	#124	C₂³×C₈₉	1	128	yes	complemented
153	#123	C₂³×C₈₉	1	256	yes	complemented
154	C₂²⋊Q₈	C₂³×C₈₉	1	256	yes	complemented
155	#121	C₂³×C₈₉	1	256	yes	complemented
156	C₂×C₄⋊C₄	C₂³×C₈₉	1	48	yes	—
157	C₂×C₄⋊C₄	C₂³×C₈₉	1	48	yes	—
158	C₂×C₄⋊C₄	C₂³×C₈₉	1	24	yes	—
159	C₂×C₂²⋊C₄	C₂³×C₈₉	1	48	yes	—
160	C₂×C₂²⋊C₄	C₂³×C₈₉	1	48	yes	—
161	C₂×C₂²⋊C₄	C₂³×C₈₉	1	48	yes	—
162	C₂×C₄²	C₂³×C₈₉	1	12	yes	—
163	C₂⁴	—	2	16	no	—
164	C₂⁴	—	2	16	no	—
165	C₂⁴	—	2	16	no	—
166	C₂⁴	C₂⁴×C₈₉	1	24	yes	—
167	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
168	C₂×Q₈	—	4	48	no	—
169	C₂×Q₈	C₂⁴×C₈₉	1	32	yes	—
170	C₂×D₄	—	4	96	no	—
171	C₂×D₄	—	4	48	no	—
172	C₂×D₄	C₂⁴×C₈₉	1	64	yes	—
173	C₂×D₄	C₂⁴×C₈₉	1	32	yes	—
174	C₂²×C₄	—	2	16	no	—
175	C₂²×C₄	—	2	16	no	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	4	96	no	—
179	C₂²×C₄	C₂⁴×C₈₉	1	6	yes	—
180	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
181	C₂²×C₄	C₂⁴×C₈₉	1	24	yes	—
182	C₂²×C₄	C₂⁴×C₈₉	1	24	yes	—
183	C₂²×C₄	C₂⁴×C₈₉	1	64	yes	—
184	C₄⋊C₄	C₂⁴×C₈₉	1	64	yes	—
185	C₄⋊C₄	C₂⁴×C₈₉	1	64	yes	—
186	C₄⋊C₄	C₂⁴×C₈₉	1	32	yes	—
187	C₂²⋊C₄	C₂⁴×C₈₉	1	64	yes	—
188	C₂²⋊C₄	C₂⁴×C₈₉	1	64	yes	—
189	C₂²⋊C₄	C₂⁴×C₈₉	1	64	yes	—
190	C₄²	C₂⁴×C₈₉	1	16	yes	—
191	C₂³	—	2	48	no	—
192	C₂³	—	2	48	no	—
193	C₂³	—	2	48	no	—
194	C₂³	—	4	64	no	—
195	C₂³	H(5,𝔽₂)×C₈₉	1	8	yes	—
196	C₂³	2₋¹⁺⁴×C₈₉	1	4	yes	—
197	C₂³	C₂⁵×C₈₉	1	16	yes	—
198	C₂³	C₂⁵×C₈₉	1	3	yes	—
199	Q₈	—	4	32	no	—
200	D₄	—	4	64	no	—
201	D₄	—	4	32	no	—
202	C₂×C₄	—	2	48	no	—
203	C₂×C₄	—	2	48	no	—
204	C₂×C₄	—	2	24	no	—
205	C₂×C₄	—	2	12	no	—
206	C₂×C₄	—	4	64	no	—
207	C₂×C₄	C₂⁵×C₈₉	1	4	yes	—
208	C₂×C₄	C₂⁵×C₈₉	1	8	yes	—
209	C₂×C₄	C₂⁵×C₈₉	1	16	yes	—
210	C₂×C₄	C₂⁵×C₈₉	1	16	yes	—
211	C₂²	—	2	16	no	—
212	C₂²	—	2	16	no	—
213	C₂²	—	2	16	no	—
214	C₂²	—	4	96	no	—
215	C₂²	#562×C₈₉	1	16	yes	direct product factor
216	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	12	yes	—
217	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	6	yes	—
218	C₂²	C₂⁶×C₈₉	1	1	yes	characteristic
219	C₄	—	2	16	no	—
220	C₄	—	2	16	no	—
221	C₄	—	2	8	no	—
222	C₄	—	2	4	no	—
223	C₂	—	4	16	no	—
224	C₂	C₂×#562×C₈₉	1	12	yes	direct product factor
225	C₂	C₂²×H(5,𝔽₂)×C₈₉	1	2	yes	—
226	C₂	C₂²×2₋¹⁺⁴×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. C₂×#562×C₈₉	C₂	direct (N × H)
2. C₂²×#125×C₈₉	C₂	semidirect
3. C₂²×#124×C₈₉	C₂	semidirect
4. C₂²×#123×C₈₉	C₂	semidirect
5. C₂²×C₂²⋊Q₈×C₈₉	C₂	semidirect
6. C₂²×#121×C₈₉	C₂	semidirect
11. #562×C₈₉	C₂²	direct (N × H)
12. C₂×#125×C₈₉	C₂²	semidirect
13. C₂×#124×C₈₉	C₂²	semidirect
14. C₂×#123×C₈₉	C₂²	semidirect
15. C₂×C₂²⋊Q₈×C₈₉	C₂²	semidirect
16. C₂×#121×C₈₉	C₂²	semidirect
37. #125×C₈₉	C₂³	semidirect
38. #124×C₈₉	C₂³	semidirect
39. #123×C₈₉	C₂³	semidirect
40. C₂²⋊Q₈×C₈₉	C₂³	semidirect
41. #121×C₈₉	C₂³	semidirect
101. C₂²×C₈₉	#562	direct (N × H)
109. C₂²×#562	C₈₉	direct (N × H)
111. C₂×C₈₉	C₂×#562	direct (N × H)
114. C₂×#562	C₂×C₈₉	direct (N × H)
115. C₂²×#125	C₂×C₈₉	semidirect
116. C₂²×#124	C₂×C₈₉	semidirect
117. C₂²×#123	C₂×C₈₉	semidirect
118. C₂²×C₂²⋊Q₈	C₂×C₈₉	semidirect
119. C₂²×#121	C₂×C₈₉	semidirect
120. C₈₉	C₂²×#562	direct (N × H)
125. #562	C₂²×C₈₉	direct (N × H)
126. C₂×#125	C₂²×C₈₉	semidirect
127. C₂×#124	C₂²×C₈₉	semidirect
128. C₂×#123	C₂²×C₈₉	semidirect
129. C₂×C₂²⋊Q₈	C₂²×C₈₉	semidirect
130. C₂×#121	C₂²×C₈₉	semidirect
151. #125	C₂³×C₈₉	semidirect
152. #124	C₂³×C₈₉	semidirect
153. #123	C₂³×C₈₉	semidirect
154. C₂²⋊Q₈	C₂³×C₈₉	semidirect
155. #121	C₂³×C₈₉	semidirect
215. C₂²	#562×C₈₉	direct (N × H)
224. C₂	C₂×#562×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
C₂×#562×C₈₉	2
C₂²×#125×C₈₉	2
C₂²×#124×C₈₉	2
C₂²×#123×C₈₉	2
C₂²×C₂²⋊Q₈×C₈₉	2
C₂²×#121×C₈₉	2
C₂²×#562	89

Subgroups — C22×#562×C89 #36440537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#562×C₈₉ #36440537