Subgroups — C₂×#26022531 #36709910

Order 22784.

494 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152000	C₂	1	2	yes	maximal
2	#26022531	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151426	C₂	1	2	yes	maximal
4	C₂×#19151424	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668004	—	2	2	no	—
7	#19152000	—	2	4	no	—
8	#19152000	C₂²	1	8	yes	—
9	C₂×#15667946	C₂²	1	4	yes	—
10	C₂×#15667944	—	2	2	no	—
11	#19151426	C₂²	1	8	yes	—
12	#19151424	C₂²	1	8	yes	—
13	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₈₉	C₂²	1	4	yes	—
17	#15668004	—	2	8	no	—
18	#15668004	—	4	4	no	—
19	C₂×C₄⋊Dic₈₉	—	2	4	no	—
20	C₂×#13083553	—	4	4	no	—
21	#15667946	—	2	4	no	—
22	#15667946	C₂³	1	8	yes	—
23	#15667944	—	2	8	no	—
24	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₈₉	Q₈	1	4	yes	—
28	C₂×C₁₆×C₈₉	Q₈	1	4	yes	—
29	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
30	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
31	C₄×C₈×C₈₉	D₄	1	2	yes	—
32	C₄×C₈×C₈₉	C₂³	1	2	yes	—
33	C₂²×Dic₈₉	—	4	4	no	—
34	C₄⋊Dic₈₉	—	2	8	no	—
35	C₄⋊Dic₈₉	—	4	4	no	—
36	#13083553	—	4	16	no	—
37	C₂²×C₄×C₈₉	D₈	1	2	yes	—
38	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	Q₁₆	1	4	yes	—
40	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
42	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
43	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
44	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
45	C₄²×C₈₉	D₈	1	2	yes	—
46	C₄²×C₈₉	C₂×D₄	1	2	yes	—
47	C₁₆×C₈₉	C₂×Q₈	1	4	yes	—
48	C₁₆×C₈₉	C₂×Q₈	1	4	yes	—
49	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₈₉	—	2	4	no	—
51	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
53	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
55	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
56	C₂×Dic₈₉	—	4	8	no	—
57	C₂×Dic₈₉	—	8	8	no	—
58	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
59	C₈×C₈₉	C₂×Q₁₆	1	4	yes	—
60	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
61	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
62	C₂²×C₈₉	#500	1	2	yes	—
63	C₂²×C₈₉	#508	1	4	yes	—
64	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
65	C₄×C₈₉	—	2	4	no	—
66	C₄×C₈₉	#500	1	2	yes	—
67	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
69	Dic₈₉	—	8	8	no	—
70	C₂×#2253	—	89	1	no	maximal, Sylow
71	C₂×C₈₉	#2253	1	4	yes	complemented
72	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
73	C₂×C₈₉	C₂×#508	1	2	yes	—
74	C₂×#499	—	89	2	no	—
75	#2253	—	89	8	no	—
76	C₂×#367	—	89	2	no	—
77	C₂×#366	—	89	2	no	—
78	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic
79	C₈₉	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
80	C₂×#125	—	178	2	no	—
81	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
82	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
83	#499	—	89	8	no	—
84	#499	—	178	4	no	—
85	C₂×#107	—	89	4	no	—
86	C₂×#106	—	178	2	no	—
87	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
88	#367	—	89	8	no	—
89	#366	—	89	8	no	—
90	C₄×C₁₆	C₂×D₈₉	1	4	yes	—
91	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
93	#125	—	356	4	no	—
94	#125	—	178	8	no	—
95	C₂×C₄⋊C₄	—	356	4	no	—
96	C₂×C₄⋊C₄	—	178	4	no	—
97	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
98	C₂×C₁₆	Dic₁₇₈	1	4	yes	—
99	C₂×C₁₆	Dic₁₇₈	1	4	yes	—
100	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
101	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
102	#107	—	89	8	no	—
103	#107	—	178	4	no	—
104	#106	—	178	8	no	—
105	C₄×C₈	D₃₅₆	1	2	yes	—
106	C₄×C₈	C₂²×D₈₉	1	2	yes	—
107	C₂²×C₄	—	356	4	no	—
108	C₂²×C₄	D₇₁₂	1	2	yes	—
109	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
110	C₂×C₈	Dic₃₅₆	1	4	yes	—
111	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
112	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
113	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
114	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
115	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
116	C₄⋊C₄	—	356	16	no	—
117	C₄⋊C₄	—	356	4	no	—
118	C₄⋊C₄	—	178	8	no	—
119	C₄²	D₇₁₂	1	2	yes	—
120	C₄²	C₂×D₃₅₆	1	2	yes	—
121	C₁₆	C₂×Dic₁₇₈	1	4	yes	—
122	C₁₆	C₂×Dic₁₇₈	1	4	yes	—
123	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	356	8	no	—
126	C₂×C₄	—	712	8	no	—
127	C₂×C₄	#15668012	1	2	yes	—
128	C₂×C₄	C₂×D₇₁₂	1	4	yes	—
129	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
130	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
131	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
132	C₈	#15668012	1	2	yes	—
133	C₈	C₂×Dic₃₅₆	1	4	yes	—
134	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
135	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
136	C₂²	#19152003	1	2	yes	—
137	C₂²	#19152030	1	4	yes	—
138	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	712	8	no	—
141	C₄	#19152003	1	2	yes	—
142	C₄	C₂×#15668012	1	1	yes	characteristic
143	C₄	C₂²×Dic₃₅₆	1	1	yes	characteristic
144	C₂	#26022531	1	4	yes	direct product factor
145	C₂	C₂×#19152003	1	1	yes	characteristic
146	C₂	C₂×#19152030	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	C₂×#2253
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #26022531	C₂	direct (N × H)
71. C₂×C₈₉	#2253	semidirect
79. C₈₉	C₂×#2253	semidirect
144. C₂	#26022531	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×#19152000	2
#26022531	2
C₂×#19151426	2
C₂×#19151424	2
C₂×C₄×C₁₆×C₈₉	2
C₂×#2253	89

Subgroups — C2×#26022531 #36709910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022531 #36709910