Subgroups — C₂×#26021848 #36707062

Order 22784.

854 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667952	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151553	C₂	1	1	yes	maximal, characteristic
3	#26021848	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151373	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₈₉	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#13083558	C₄	1	1	yes	characteristic
8	C₂×#15667971	C₄	1	1	yes	characteristic
9	C₂×#15667952	C₄	1	8	yes	complemented
10	C₂×#15667952	C₂²	1	4	yes	—
11	C₂²×#13083549	C₂²	1	2	yes	—
12	#19151553	C₄	1	4	yes	—
13	#19151553	C₂²	1	4	yes	—
14	C₂×#15667922	C₄	1	2	yes	—
15	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
16	#19151373	C₂²	1	8	yes	complemented
17	C₁₆×Dic₈₉	C₂²	1	8	yes	complemented
18	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆×C₈₉	C₄	1	8	yes	complemented
20	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₈₉	C₂²	1	4	yes	—
23	C₂×#13083558	C₈	1	8	yes	—
24	C₂×#13083558	C₂×C₄	1	4	yes	—
25	C₂²×#10623494	C₂×C₄	1	2	yes	—
26	#15667971	C₈	1	4	yes	—
27	#15667971	C₂×C₄	1	4	yes	—
28	C₂×C₄⋊Dic₈₉	C₈	1	2	yes	—
29	C₂×C₂²⋊Dic₈₉	C₈	1	2	yes	—
30	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
31	#15667952	C₂×C₄	1	16	yes	complemented
32	C₂×#13083549	C₂×C₄	1	8	yes	—
33	C₂×#13083549	C₂³	1	4	yes	—
34	#15667922	C₂×C₄	1	8	yes	—
35	C₈×Dic₈₉	C₂×C₄	1	4	yes	—
36	C₈×Dic₈₉	C₂³	1	4	yes	—
37	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
44	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
45	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
46	C₂×C₁₆×C₈₉	C₂×C₄	1	16	yes	complemented
47	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
48	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
49	C₂²×Dic₈₉	C₂×C₈	1	4	yes	—
50	#13083558	C₂×C₈	1	16	yes	—
51	C₂×#10623494	C₂×C₈	1	8	yes	—
52	C₂×#10623494	C₂²×C₄	1	4	yes	—
53	C₄⋊Dic₈₉	C₂×C₈	1	8	yes	—
54	C₂²⋊Dic₈₉	C₂×C₈	1	8	yes	—
55	C₄×Dic₈₉	C₂×C₈	1	4	yes	—
56	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
57	#13083549	C₂²×C₄	1	8	yes	—
58	C₂⁴×C₈₉	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
60	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₈₉	C₄²	1	4	yes	—
66	C₂×C₈×C₈₉	C₄²	1	8	yes	—
67	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
74	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
75	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
76	C₂³×C₈₉	—	2	4	no	—
77	C₂³×C₈₉	C₄×C₈	1	4	yes	—
78	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
79	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₄×C₈	1	8	yes	—
81	C₂×C₄×C₈₉	C₄×C₈	1	4	yes	—
82	C₂×C₄×C₈₉	C₂×C₄²	1	4	yes	—
83	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
84	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
86	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
89	C₂×Dic₈₉	—	2	8	no	—
90	C₂×Dic₈₉	C₂²×C₈	1	8	yes	—
91	C₈×C₈₉	C₂×C₄²	1	2	yes	—
92	C₈×C₈₉	C₂×C₄²	1	4	yes	—
93	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
94	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
95	#10623494	C₂²×C₈	1	8	yes	—
96	C₂²×C₈₉	—	2	8	no	—
97	C₂²×C₈₉	—	2	4	no	—
98	C₂²×C₈₉	C₂×C₄×C₈	1	4	yes	—
99	C₂²×C₈₉	C₂×C₄×C₈	1	2	yes	—
100	C₂²×C₈₉	#504	1	4	yes	—
101	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
102	C₄×C₈₉	C₂×C₄×C₈	1	2	yes	—
103	C₄×C₈₉	C₂×C₄×C₈	1	4	yes	—
104	C₄×C₈₉	C₂²×C₄²	1	1	yes	characteristic
105	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
106	Dic₈₉	—	2	8	no	—
107	C₂×#2108	—	89	1	no	maximal, Sylow
108	C₂×C₈₉	—	2	4	no	—
109	C₂×C₈₉	#2108	1	4	yes	complemented
110	C₂×C₈₉	C₂²×C₄×C₈	1	1	yes	characteristic
111	C₂×C₈₉	C₂×#504	1	2	yes	—
112	C₂²×M₃₂	—	89	1	no	—
113	C₂³×C₁₆	D₈₉	1	1	yes	characteristic
114	C₂×#405	—	89	1	no	—
115	#2108	—	89	8	no	—
116	C₂×#346	—	89	2	no	—
117	C₂×C₄×C₁₆	—	89	2	no	—
118	C₈₉	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
119	C₂²×M₁₆	—	89	1	no	—
120	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
121	C₂×#117	—	89	1	no	—
122	C₂×M₃₂	—	89	8	no	—
123	C₂×M₃₂	—	89	4	no	—
124	C₂²×C₁₆	—	89	2	no	—
125	C₂²×C₁₆	Dic₈₉	1	8	yes	complemented
126	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
127	C₂²×C₁₆	C₂×D₈₉	1	4	yes	—
128	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
129	#405	—	89	4	no	—
130	#405	—	89	4	no	—
131	C₂×#97	—	89	2	no	—
132	C₂×C₄×C₈	—	89	2	no	—
133	#346	—	89	8	no	—
134	C₄×C₁₆	—	89	8	no	—
135	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
136	C₂×M₁₆	—	89	8	no	—
137	C₂×M₁₆	—	89	4	no	—
138	C₂²×C₈	—	89	2	no	—
139	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
140	C₂²×C₈	C₄×D₈₉	1	2	yes	—
141	C₂²×C₈	C₄×D₈₉	1	4	yes	—
142	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
143	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
144	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
145	#117	—	89	4	no	—
146	#117	—	89	4	no	—
147	C₂×C₄⋊C₄	—	89	2	no	—
148	C₂×C₂²⋊C₄	—	89	2	no	—
149	C₂×C₄²	—	89	2	no	—
150	M₃₂	—	89	16	no	—
151	C₂×C₁₆	—	89	8	no	—
152	C₂×C₁₆	—	89	4	no	—
153	C₂×C₁₆	C₂×Dic₈₉	1	4	yes	—
154	C₂×C₁₆	C₂×Dic₈₉	1	4	yes	—
155	C₂×C₁₆	C₂×Dic₈₉	1	16	yes	complemented
156	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
157	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
158	#97	—	89	8	no	—
159	C₄×C₈	—	89	4	no	—
160	C₄×C₈	—	89	4	no	—
161	C₂⁴	C₈×D₈₉	1	1	yes	characteristic
162	C₂²×C₄	—	89	4	no	—
163	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
164	C₂²×C₄	C₈×D₈₉	1	4	yes	—
165	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
166	C₂²×C₄	C₈×D₈₉	1	2	yes	—
167	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
169	M₁₆	—	89	16	no	—
170	C₂×C₈	—	89	8	no	—
171	C₂×C₈	—	89	4	no	—
172	C₂×C₈	C₄×Dic₈₉	1	4	yes	—
173	C₂×C₈	C₄×Dic₈₉	1	8	yes	—
174	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
175	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
176	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
177	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
178	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
179	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
180	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
181	C₄⋊C₄	—	89	8	no	—
182	C₂²⋊C₄	—	89	8	no	—
183	C₄²	—	89	4	no	—
184	C₄²	—	89	4	no	—
185	C₁₆	—	89	8	no	—
186	C₁₆	C₂²×Dic₈₉	1	4	yes	—
187	C₁₆	C₂²×Dic₈₉	1	4	yes	—
188	C₂³	—	2	4	no	—
189	C₂³	C₈×Dic₈₉	1	4	yes	—
190	C₂³	C₂×C₈×D₈₉	1	2	yes	—
191	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
192	C₂×C₄	—	89	8	no	—
193	C₂×C₄	—	178	8	no	—
194	C₂×C₄	C₈×Dic₈₉	1	8	yes	—
195	C₂×C₄	C₈×Dic₈₉	1	4	yes	—
196	C₂×C₄	C₂×C₄×Dic₈₉	1	4	yes	—
197	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
198	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
199	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
200	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
201	C₂×C₄	C₂×C₈×D₈₉	1	4	yes	—
202	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
203	C₈	—	89	8	no	—
204	C₈	C₂×C₄×Dic₈₉	1	2	yes	—
205	C₈	C₂×C₄×Dic₈₉	1	4	yes	—
206	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
207	C₈	C₂³×Dic₈₉	1	1	yes	characteristic
208	C₂²	—	2	4	no	—
209	C₂²	—	2	8	no	—
210	C₂²	C₂×C₈×Dic₈₉	1	4	yes	—
211	C₂²	C₂×C₈×Dic₈₉	1	2	yes	—
212	C₂²	#19152014	1	4	yes	—
213	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
214	C₄	—	178	8	no	—
215	C₄	C₂×C₈×Dic₈₉	1	2	yes	—
216	C₄	C₂×C₈×Dic₈₉	1	4	yes	—
217	C₄	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
218	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
219	C₂	—	2	4	no	—
220	C₂	#26021848	1	4	yes	direct product factor
221	C₂	C₂²×C₈×Dic₈₉	1	1	yes	characteristic
222	C₂	C₂×#19152014	1	2	yes	—

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₈₉
commutator	C₂×C₈₉	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂³×C₁₆×C₈₉	C₂
socle	C₂³×C₈₉	C₂²×C₈
prime radical	C₂⁴×C₈₉	C₂×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂³×C₁₆	D₈₉
hypocenter	C₈₉	C₂×#2108
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. #26021848	C₂	direct (N × H)
4. C₂×#19151373	C₂	semidirect
5. C₂×C₁₆×Dic₈₉	C₂	semidirect
9. C₂×#15667952	C₄	semidirect
16. #19151373	C₂²	semidirect
17. C₁₆×Dic₈₉	C₂²	semidirect
19. C₂²×C₁₆×C₈₉	C₄	semidirect
31. #15667952	C₂×C₄	semidirect
46. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
109. C₂×C₈₉	#2108	semidirect
118. C₈₉	C₂×#2108	semidirect
125. C₂²×C₁₆	Dic₈₉	semidirect
155. C₂×C₁₆	C₂×Dic₈₉	semidirect
220. C₂	#26021848	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#15667952	2
C₂×#19151553	2
#26021848	2
C₂×#19151373	2
C₂×C₁₆×Dic₈₉	2
C₂³×C₁₆×C₈₉	2
C₂×#2108	89

Subgroups — C2×#26021848 #36707062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26021848 #36707062