Subgroups — C₂×#26018328 #36466021

Order 22784.

230 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151419	C₂	1	2	yes	maximal
2	#26018328	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667952	—	2	2	no	—
5	#19151419	—	2	2	no	—
6	#19151419	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₈₉	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₈₉	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₈₉	C₄	1	2	yes	—
11	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
12	#15667952	—	2	4	no	—
13	#15667952	—	4	2	no	—
14	C₂×#13083549	—	4	2	no	—
15	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
18	C₂×C₁₆×C₈₉	—	2	1	no	—
19	C₂×C₁₆×C₈₉	—	2	1	no	—
20	C₂×C₁₆×C₈₉	C₈	1	2	yes	—
21	C₂×C₁₆×C₈₉	C₈	1	2	yes	—
22	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
26	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
28	#13083549	—	4	4	no	—
29	C₂²×C₄×C₈₉	—	2	1	no	—
30	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
38	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
39	C₂×C₈×C₈₉	Q₁₆	1	1	yes	characteristic
40	C₂×C₈×C₈₉	Q₁₆	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₈₉	—	2	1	no	—
44	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
46	C₁₆×C₈₉	—	2	2	no	—
47	C₁₆×C₈₉	C₂×C₈	1	4	yes	—
48	C₂³×C₈₉	#108	1	1	yes	characteristic
49	C₂×C₄×C₈₉	—	2	4	no	—
50	C₂×C₄×C₈₉	—	2	1	no	—
51	C₂×C₄×C₈₉	—	2	1	no	—
52	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
55	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#108	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
59	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₈₉	#107	1	1	yes	characteristic
61	C₈×C₈₉	#107	1	1	yes	characteristic
62	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
63	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
64	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
65	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
66	C₂²×C₈₉	—	2	2	no	—
67	C₂²×C₈₉	#335	1	1	yes	characteristic
68	C₂²×C₈₉	#335	1	1	yes	characteristic
69	C₂²×C₈₉	C₂×#108	1	1	yes	characteristic
70	C₄×C₈₉	—	2	2	no	—
71	C₄×C₈₉	—	2	2	no	—
72	C₄×C₈₉	#335	1	1	yes	characteristic
73	C₄×C₈₉	#335	1	1	yes	characteristic
74	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
75	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
76	C₂×#1373	—	89	1	no	maximal, Sylow
77	C₂×C₈₉	—	2	1	no	—
78	C₂×C₈₉	—	2	1	no	—
79	C₂×C₈₉	#1373	1	2	yes	complemented
80	C₂×C₈₉	C₂×#335	1	1	yes	characteristic
81	C₂×#364	—	89	2	no	—
82	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic
83	#1373	—	89	4	no	—
84	C₈₉	C₂×#1373	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₃₂	—	178	2	no	—
86	C₂²×C₁₆	Dic₈₉	1	1	yes	characteristic
87	C₂²×C₁₆	Dic₈₉	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
89	#364	—	89	4	no	—
90	#364	—	178	2	no	—
91	C₄×C₁₆	Dic₈₉	1	2	yes	—
92	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
93	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
94	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
95	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
96	M₃₂	—	356	2	no	—
97	M₃₂	—	178	4	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	356	2	no	—
101	C₂×C₁₆	#10623494	1	2	yes	—
102	C₂×C₁₆	#10623494	1	2	yes	—
103	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
104	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
105	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
106	C₄×C₈	D₃₅₆	1	1	yes	characteristic
107	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
111	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
112	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
113	C₂×C₈	C₂×#10623494	1	1	yes	characteristic
114	C₂×C₈	C₂×#10623494	1	1	yes	characteristic
115	C₂×C₈	#13083558	1	1	yes	characteristic
116	C₂×C₈	#13083558	1	1	yes	characteristic
117	C₂×C₈	D₇₁₂	1	1	yes	characteristic
118	C₂×C₈	D₇₁₂	1	1	yes	characteristic
119	C₂×C₈	Dic₃₅₆	1	1	yes	characteristic
120	C₂×C₈	Dic₃₅₆	1	1	yes	characteristic
121	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
122	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
123	C₄²	—	2	1	no	—
124	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
125	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
126	C₁₆	—	2	2	no	—
127	C₁₆	—	356	4	no	—
128	C₁₆	C₂×#10623494	1	4	yes	—
129	C₂³	#15667948	1	1	yes	characteristic
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	C₄⋊#10623494	1	1	yes	characteristic
134	C₂×C₄	C₄⋊#10623494	1	1	yes	characteristic
135	C₂×C₄	#15667946	1	1	yes	characteristic
136	C₂×C₄	#15667946	1	1	yes	characteristic
137	C₂×C₄	#15667948	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
139	C₈	C₄⋊#10623494	1	1	yes	characteristic
140	C₈	C₄⋊#10623494	1	1	yes	characteristic
141	C₈	#15667946	1	1	yes	characteristic
142	C₈	#15667946	1	1	yes	characteristic
143	C₈	C₂²×#10623494	1	1	yes	characteristic
144	C₈	C₂×#13083558	1	1	yes	characteristic
145	C₈	C₂×D₇₁₂	1	1	yes	characteristic
146	C₈	C₂×Dic₃₅₆	1	1	yes	characteristic
147	C₂²	—	2	2	no	—
148	C₂²	#19151351	1	1	yes	characteristic
149	C₂²	#19151351	1	1	yes	characteristic
150	C₂²	C₂×#15667948	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	#19151351	1	1	yes	characteristic
154	C₄	#19151351	1	1	yes	characteristic
155	C₄	C₂×C₄⋊#10623494	1	1	yes	characteristic
156	C₄	C₂×#15667946	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	#26018328	1	2	yes	direct product factor
160	C₂	C₂×#19151351	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₇₁₂
commutator	C₈×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	#335
prime radical	C₂³×C₈₉	#108
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	C₂×#1373
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #26018328	C₂	direct (N × H)
79. C₂×C₈₉	#1373	semidirect
84. C₈₉	C₂×#1373	semidirect
159. C₂	#26018328	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×#19151419	2
#26018328	2
C₂×C₄×C₁₆×C₈₉	2
C₂×#1373	89

Subgroups — C2×#26018328 #36466021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018328 #36466021