Subgroups — C₂×#26018535 #36520006

Order 22784.

982 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151651	C₂	1	2	yes	maximal
2	#26018535	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151319	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×#10623494	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₄	1	8	yes	—
7	#19151651	C₂²	1	8	yes	—
8	C₂×C₄⋊#10623494	C₄	1	4	yes	—
9	C₂×C₂²⋊#10623494	C₄	1	4	yes	—
10	C₂×C₄×#10623494	C₂²	1	4	yes	—
11	#19151319	C₂²	1	8	yes	complemented
12	C₈×#10623494	C₂²	1	8	yes	complemented
13	C₂³×C₈×C₈₉	C₄	1	2	yes	—
14	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₄	1	4	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₈₉	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×#10623494	C₂×C₄	1	8	yes	—
21	C₄⋊#10623494	C₂×C₄	1	16	yes	—
22	C₂²⋊#10623494	C₂×C₄	1	16	yes	—
23	C₄×#10623494	C₂×C₄	1	8	yes	—
24	C₄×#10623494	C₂³	1	8	yes	—
25	C₂³×C₄×C₈₉	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	C₈	1	16	yes	complemented
28	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
36	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
37	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₈₉	C₂×C₄	1	8	yes	—
39	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₈₉	C₂³	1	2	yes	—
41	C₄×C₈×C₈₉	C₂³	1	2	yes	—
42	C₂×#10623494	—	2	16	no	—
43	C₂×#10623494	C₂²×C₄	1	16	yes	—
44	C₂⁴×C₈₉	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	—	2	8	no	—
46	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
47	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
48	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
49	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	C₂×C₈	1	8	yes	—
51	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₈₉	C₂×C₈	1	8	yes	—
56	C₂×C₈×C₈₉	C₂×C₈	1	8	yes	—
57	C₂×C₈×C₈₉	C₂×C₈	1	32	yes	complemented
58	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
59	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
60	C₄²×C₈₉	C₄²	1	2	yes	—
61	C₄²×C₈₉	C₄²	1	4	yes	—
62	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
63	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
64	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
65	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
67	C₂³×C₈₉	—	2	4	no	—
68	C₂³×C₈₉	C₄×C₈	1	4	yes	—
69	C₂³×C₈₉	C₂×C₄²	1	2	yes	—
70	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₈₉	—	2	4	no	—
72	C₂×C₄×C₈₉	—	2	8	no	—
73	C₂×C₄×C₈₉	—	2	8	no	—
74	C₂×C₄×C₈₉	C₄×C₈	1	8	yes	—
75	C₂×C₄×C₈₉	C₄×C₈	1	4	yes	—
76	C₂×C₄×C₈₉	C₂×C₄²	1	4	yes	—
77	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
79	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
80	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
81	C₂×C₄×C₈₉	C₂²×C₈	1	8	yes	—
82	C₂×C₄×C₈₉	#131	1	8	yes	—
83	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
84	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
85	C₈×C₈₉	C₂²×C₈	1	8	yes	—
86	C₈×C₈₉	C₂²×C₈	1	8	yes	—
87	#10623494	—	2	16	no	—
88	C₂²×C₈₉	—	2	4	no	—
89	C₂²×C₈₉	—	2	8	no	—
90	C₂²×C₈₉	C₂×C₄×C₈	1	4	yes	—
91	C₂²×C₈₉	C₂×C₄×C₈	1	2	yes	—
92	C₂²×C₈₉	#405	1	4	yes	—
93	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
94	C₄×C₈₉	—	2	4	no	—
95	C₄×C₈₉	C₂×C₄×C₈	1	2	yes	—
96	C₄×C₈₉	C₂×C₄×C₈	1	4	yes	—
97	C₄×C₈₉	#405	1	4	yes	—
98	C₄×C₈₉	C₂³×C₈	1	2	yes	—
99	C₄×C₈₉	C₂×#131	1	4	yes	—
100	C₂×#1450	—	89	1	no	maximal, Sylow
101	C₂×C₈₉	—	2	4	no	—
102	C₂×C₈₉	#1450	1	4	yes	complemented
103	C₂×C₈₉	C₂²×C₄×C₈	1	1	yes	characteristic
104	C₂×C₈₉	C₂×#405	1	2	yes	—
105	C₂×#431	—	89	2	no	—
106	C₂²×C₄×C₈	D₈₉	1	1	yes	characteristic
107	#1450	—	89	8	no	—
108	C₂×#322	—	89	2	no	—
109	C₂×C₈²	—	89	2	no	—
110	C₈₉	C₂×#1450	1	1	yes	characteristic, complemented, Sylow
111	C₂³×C₈	Dic₈₉	1	2	yes	—
112	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
113	#431	—	89	8	no	—
114	#431	—	89	8	no	—
115	C₂×C₄⋊C₈	—	89	4	no	—
116	C₂×C₂²⋊C₈	—	89	4	no	—
117	C₂×C₄×C₈	—	89	4	no	—
118	C₂×C₄×C₈	Dic₈₉	1	2	yes	—
119	C₂×C₄×C₈	Dic₈₉	1	4	yes	—
120	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂×D₈₉	1	4	yes	—
122	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
123	#322	—	89	8	no	—
124	C₈²	—	89	8	no	—
125	C₂³×C₄	C₄×D₈₉	1	2	yes	—
126	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
127	C₂²×C₈	—	89	8	no	—
128	C₂²×C₈	#10623494	1	16	yes	complemented
129	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
130	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
131	C₂²×C₈	C₂×Dic₈₉	1	8	yes	—
132	C₂×C₄²	C₄×D₈₉	1	2	yes	—
133	C₂×C₄²	C₄×D₈₉	1	4	yes	—
134	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
135	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
136	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
137	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
138	C₄⋊C₈	—	89	16	no	—
139	C₂²⋊C₈	—	89	16	no	—
140	C₄×C₈	—	89	8	no	—
141	C₄×C₈	—	89	8	no	—
142	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
143	C₄×C₈	C₂×Dic₈₉	1	8	yes	—
144	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
145	C₄×C₈	C₂²×D₈₉	1	2	yes	—
146	C₄×C₈	C₂²×D₈₉	1	2	yes	—
147	C₂⁴	C₄×Dic₈₉	1	1	yes	characteristic
148	C₂²×C₄	—	2	8	no	—
149	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
150	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
151	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
152	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
153	C₂²×C₄	C₂×#10623494	1	8	yes	—
154	C₂²×C₄	C₂×C₄×D₈₉	1	4	yes	—
155	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
156	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
157	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
158	C₂×C₈	—	89	16	no	—
159	C₂×C₈	—	178	16	no	—
160	C₂×C₈	C₂×#10623494	1	8	yes	—
161	C₂×C₈	C₂×#10623494	1	8	yes	—
162	C₂×C₈	C₂×#10623494	1	32	yes	complemented
163	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
164	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
165	C₄²	C₄×Dic₈₉	1	2	yes	—
166	C₄²	C₄×Dic₈₉	1	4	yes	—
167	C₄²	C₂×C₄×D₈₉	1	2	yes	—
168	C₄²	C₂×C₄×D₈₉	1	4	yes	—
169	C₄²	C₂²×Dic₈₉	1	2	yes	—
170	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
171	C₄²	C₂³×D₈₉	1	1	yes	characteristic
172	C₂³	—	2	4	no	—
173	C₂³	C₄×#10623494	1	4	yes	—
174	C₂³	C₂×C₄×Dic₈₉	1	2	yes	—
175	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	C₄×#10623494	1	8	yes	—
180	C₂×C₄	C₄×#10623494	1	4	yes	—
181	C₂×C₄	C₂×C₄×Dic₈₉	1	4	yes	—
182	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
184	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
185	C₂×C₄	C₂²×#10623494	1	4	yes	—
186	C₂×C₄	C₂²×#10623494	1	8	yes	—
187	C₂×C₄	#15668018	1	8	yes	—
188	C₂×C₄	C₂²×C₄×D₈₉	1	2	yes	—
189	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
190	C₈	—	178	16	no	—
191	C₈	C₂²×#10623494	1	8	yes	—
192	C₈	C₂²×#10623494	1	8	yes	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	8	no	—
195	C₂²	C₂×C₄×#10623494	1	4	yes	—
196	C₂²	C₂×C₄×#10623494	1	2	yes	—
197	C₂²	#19151553	1	4	yes	—
198	C₂²	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	C₂×C₄×#10623494	1	2	yes	—
201	C₄	C₂×C₄×#10623494	1	4	yes	—
202	C₄	#19151553	1	4	yes	—
203	C₄	C₂³×#10623494	1	2	yes	—
204	C₄	C₂×#15668018	1	4	yes	—
205	C₂	—	2	4	no	—
206	C₂	#26018535	1	4	yes	direct product factor
207	C₂	C₂²×C₄×#10623494	1	1	yes	characteristic
208	C₂	C₂×#19151553	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₈₉
commutator	C₂×C₈₉	C₂²×C₄×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂²×C₄×C₈×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₄²
prime radical	C₂⁴×C₈₉	C₄²
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂²×C₄×C₈	D₈₉
hypocenter	C₈₉	C₂×#1450
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #26018535	C₂	direct (N × H)
3. C₂×#19151319	C₂	semidirect
4. C₂×C₈×#10623494	C₂	semidirect
11. #19151319	C₂²	semidirect
12. C₈×#10623494	C₂²	semidirect
27. C₂²×C₈×C₈₉	C₈	semidirect
57. C₂×C₈×C₈₉	C₂×C₈	semidirect
102. C₂×C₈₉	#1450	semidirect
110. C₈₉	C₂×#1450	semidirect
128. C₂²×C₈	#10623494	semidirect
162. C₂×C₈	C₂×#10623494	semidirect
206. C₂	#26018535	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×#19151651	2
#26018535	2
C₂×#19151319	2
C₂×C₈×#10623494	2
C₂²×C₄×C₈×C₈₉	2
C₂×#1450	89

Subgroups — C2×#26018535 #36520006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018535 #36520006