Subgroups — C₂×#26022545 #36712536

Order 22784.

470 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152014	C₂	1	1	yes	maximal, characteristic
2	#26022545	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151432	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151431	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₂×D₈₉	C₂	1	2	yes	maximal, complemented
6	C₂²×C₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15668018	C₄	1	1	yes	characteristic
8	#19152014	C₄	1	4	yes	—
9	#19152014	C₂²	1	4	yes	—
10	C₂×#15667952	C₄	1	1	yes	characteristic
11	C₂×#15667951	C₄	1	2	yes	—
12	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
13	#19151432	C₂²	1	4	yes	complemented
14	#19151431	C₂²	1	8	yes	complemented
15	C₂×#15667918	C₂²	1	2	yes	—
16	C₃₂×D₈₉	C₂²	1	8	yes	complemented
17	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₃₂×C₈₉	C₂²	1	4	yes	complemented
21	C₂×#13083574	C₈	1	1	yes	characteristic
22	#15668018	C₈	1	4	yes	—
23	#15668018	C₂×C₄	1	4	yes	—
24	C₂×#13083558	C₈	1	1	yes	characteristic
25	C₂×#13083557	C₈	1	2	yes	—
26	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
27	#15667952	C₂×C₄	1	4	yes	—
28	#15667951	C₂×C₄	1	8	yes	—
29	C₂×#13083549	C₂×C₄	1	2	yes	—
30	C₁₆×D₈₉	C₂×C₄	1	4	yes	—
31	C₁₆×D₈₉	C₂³	1	4	yes	—
32	#15667918	C₂³	1	4	yes	—
33	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
36	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
38	C₃₂×C₈₉	C₂³	1	2	yes	—
39	C₃₂×C₈₉	C₂³	1	2	yes	—
40	#13083574	C₁₆	1	4	yes	—
41	#13083574	C₂×C₈	1	4	yes	—
42	C₂×Dic₁₇₈	C₁₆	1	1	yes	characteristic
43	C₂×D₃₅₆	C₁₆	1	1	yes	characteristic
44	C₂×C₂²⋊D₈₉	C₁₆	1	2	yes	—
45	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
46	#13083558	C₂×C₈	1	4	yes	—
47	#13083557	C₂×C₈	1	8	yes	—
48	C₂×#10623494	C₂×C₈	1	2	yes	—
49	C₈×D₈₉	C₂×C₈	1	4	yes	—
50	C₈×D₈₉	C₂²×C₄	1	4	yes	—
51	#13083549	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
57	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
59	C₁₆×C₈₉	C₂⁴	1	1	yes	characteristic
60	C₂³×C₈₉	C₂×C₁₆	1	1	yes	characteristic
61	C₂²×D₈₉	C₂×C₁₆	1	2	yes	—
62	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₂×C₁₆	1	2	yes	—
64	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
66	C₂²⋊D₈₉	C₂×C₁₆	1	8	yes	—
67	C₂×Dic₈₉	C₂×C₁₆	1	2	yes	—
68	D₃₅₆	C₂×C₁₆	1	4	yes	—
69	C₄×D₈₉	C₂×C₁₆	1	4	yes	—
70	C₄×D₈₉	C₂²×C₈	1	4	yes	—
71	Dic₁₇₈	C₂×C₁₆	1	4	yes	—
72	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
73	C₈×C₈₉	C₂²×C₈	1	2	yes	—
74	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
75	#10623494	C₂²×C₈	1	4	yes	—
76	C₂²×C₈₉	—	2	2	no	—
77	C₂²×C₈₉	C₂²×C₁₆	1	2	yes	—
78	C₂²×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
79	C₂×D₈₉	—	2	4	no	—
80	C₂×D₈₉	C₂²×C₁₆	1	4	yes	—
81	C₄×C₈₉	C₂²×C₁₆	1	2	yes	—
82	C₄×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
83	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
84	Dic₈₉	C₂²×C₁₆	1	4	yes	—
85	C₂×#2258	—	89	1	no	maximal, Sylow
86	C₂×C₈₉	—	2	2	no	—
87	C₂×C₈₉	#2258	1	2	yes	complemented
88	C₂×C₈₉	C₂³×C₁₆	1	1	yes	characteristic
89	D₈₉	—	2	4	no	—
90	C₂×#504	—	89	1	no	—
91	#2258	—	89	8	no	—
92	C₂×M₆₄	—	89	1	no	—
93	C₂×M₆₄	—	89	2	no	—
94	C₂²×C₃₂	—	89	2	no	—
95	C₂²×C₃₂	D₈₉	1	1	yes	characteristic, complemented
96	C₈₉	C₂×#2258	1	1	yes	characteristic, complemented, Sylow
97	C₂×#131	—	89	1	no	—
98	#504	—	89	4	no	—
99	#504	—	89	4	no	—
100	C₂×M₃₂	—	89	1	no	—
101	C₂×M₃₂	—	89	2	no	—
102	C₂²×C₁₆	—	89	2	no	—
103	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
104	M₆₄	—	89	4	no	—
105	M₆₄	—	89	8	no	—
106	C₂×C₃₂	—	89	2	no	—
107	C₂×C₃₂	—	89	8	no	—
108	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
109	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
110	C₂×C₃₂	C₂×D₈₉	1	4	yes	complemented
111	C₂×#41	—	89	1	no	—
112	#131	—	89	4	no	—
113	#131	—	89	4	no	—
114	C₂×M₁₆	—	89	1	no	—
115	C₂×M₁₆	—	89	2	no	—
116	C₂²×C₈	—	89	2	no	—
117	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
118	M₃₂	—	89	4	no	—
119	M₃₂	—	89	8	no	—
120	C₂×C₁₆	—	89	2	no	—
121	C₂×C₁₆	—	89	4	no	—
122	C₂×C₁₆	—	89	4	no	—
123	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
124	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
125	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
126	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
127	C₃₂	—	89	4	no	—
128	C₃₂	C₂²×D₈₉	1	2	yes	—
129	C₃₂	C₂²×D₈₉	1	2	yes	—
130	#41	—	89	4	no	—
131	#41	—	89	4	no	—
132	C₂×Q₈	—	89	1	no	—
133	C₂×D₄	—	89	2	no	—
134	C₂×D₄	—	89	1	no	—
135	C₂²×C₄	—	89	2	no	—
136	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
137	M₁₆	—	89	4	no	—
138	M₁₆	—	89	8	no	—
139	C₂×C₈	—	89	2	no	—
140	C₂×C₈	—	89	4	no	—
141	C₂×C₈	—	89	4	no	—
142	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
143	C₂×C₈	C₈×D₈₉	1	2	yes	—
144	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
145	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
146	C₁₆	—	89	4	no	—
147	C₁₆	C₂×C₄×D₈₉	1	1	yes	characteristic
148	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
149	C₁₆	C₂³×D₈₉	1	1	yes	characteristic
150	C₂³	—	89	2	no	—
151	C₂³	C₁₆×D₈₉	1	1	yes	characteristic
152	Q₈	—	89	4	no	—
153	D₄	—	89	8	no	—
154	D₄	—	89	4	no	—
155	C₂×C₄	—	89	2	no	—
156	C₂×C₄	—	89	4	no	—
157	C₂×C₄	—	89	4	no	—
158	C₂×C₄	C₁₆×D₈₉	1	1	yes	characteristic
159	C₂×C₄	C₁₆×D₈₉	1	2	yes	—
160	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
161	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
162	C₈	—	89	4	no	—
163	C₈	C₂×C₈×D₈₉	1	1	yes	characteristic
164	C₈	C₂×C₈×D₈₉	1	2	yes	—
165	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
166	C₂²	—	2	2	no	—
167	C₂²	—	89	4	no	—
168	C₂²	—	178	4	no	—
169	C₂²	C₂×C₁₆×D₈₉	1	2	yes	—
170	C₂²	C₂×C₁₆×D₈₉	1	1	yes	characteristic
171	C₄	—	89	4	no	—
172	C₄	C₂×C₁₆×D₈₉	1	1	yes	characteristic
173	C₄	C₂×C₁₆×D₈₉	1	2	yes	—
174	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	—	178	4	no	—
177	C₂	#26022545	1	2	yes	direct product factor
178	C₂	C₂²×C₁₆×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃₂	C₂×D₈₉
commutator	C₂×C₈₉	C₂³×C₁₆
Frattini	C₁₆	C₂³×D₈₉
Fitting	C₂²×C₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₁₆
prime radical	C₂×#13083574	C₈
cosocle	C₁₆×C₈₉	C₂⁴
hypercenter	C₂²×C₃₂	D₈₉
hypocenter	C₈₉	C₂×#2258
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂²×C₃₂

Complements

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

N	H ≅ G/N	type
2. #26022545	C₂	direct (N × H)
3. C₂×#19151432	C₂	semidirect
4. C₂×#19151431	C₂	semidirect
5. C₂×C₃₂×D₈₉	C₂	semidirect
6. C₂²×C₃₂×C₈₉	C₂	semidirect
13. #19151432	C₂²	semidirect
14. #19151431	C₂²	semidirect
16. C₃₂×D₈₉	C₂²	semidirect
20. C₂×C₃₂×C₈₉	C₂²	semidirect
87. C₂×C₈₉	#2258	semidirect
95. C₂²×C₃₂	D₈₉	semidirect
96. C₈₉	C₂×#2258	semidirect
110. C₂×C₃₂	C₂×D₈₉	semidirect
177. C₂	#26022545	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×#19152014	2
#26022545	2
C₂×#19151432	2
C₂×#19151431	2
C₂×C₃₂×D₈₉	2
C₂²×C₃₂×C₈₉	2
C₂×#2258	89

Subgroups — C2×#26022545 #36712536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022545 #36712536