Subgroups — C₂×C₂²⋊#15667918 #36474619

Order 22784.

342 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₈₉
commutator	C₂×C₈₉	C₂²×C₃₂
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₂³×C₁₆×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₁₆
prime radical	C₂⁴×C₈₉	C₁₆
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	C₂³×C₁₆	D₈₉
hypocenter	C₈₉	C₂×C₂²⋊C₃₂
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
1. C₂²×#15667918	C₂	semidirect
2. C₂²⋊#15667918	C₂	direct (N × H)
5. C₂×#15667918	C₂²	semidirect
48. C₂³×C₈₉	C₃₂	semidirect
69. C₂²×C₈₉	C₂×C₃₂	semidirect
78. C₂×C₈₉	C₂²⋊C₃₂	semidirect
85. C₈₉	C₂×C₂²⋊C₃₂	semidirect
130. C₂³	#15667918	semidirect
151. C₂²	C₂×#15667918	semidirect
159. C₂	C₂²⋊#15667918	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×#15667918	2
C₂²⋊#15667918	2
C₂³×C₁₆×C₈₉	2
C₂×C₂²⋊C₃₂	89

Subgroups — C2×C22⋊#15667918 #36474619

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊#15667918 #36474619