Subgroups — C₂×#22586412 #31909030

Order 18176.

262 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985752	C₂	1	2	yes	maximal
2	#22586412	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
5	#17985752	—	2	2	no	—
6	#17985752	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₇₁	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₇₁	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₇₁	C₄	1	2	yes	—
11	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
12	C₂²×C₇₁⋊C₈	—	4	2	no	—
13	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
14	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
15	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₁	—	2	2	no	—
19	C₂×C₁₆×C₇₁	C₈	1	4	yes	complemented
20	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
24	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
26	C₂×C₇₁⋊C₈	—	4	4	no	—
27	C₂×C₇₁⋊C₈	—	8	4	no	—
28	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
31	C₂×C₈×C₇₁	—	2	2	no	—
32	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
36	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
38	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₇₁	D₈	1	1	yes	characteristic
40	C₄²×C₇₁	Q₁₆	1	1	yes	characteristic
41	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₁	—	2	2	no	—
43	C₁₆×C₇₁	C₂×C₈	1	4	yes	complemented
44	C₂³×C₇₁	#107	1	1	yes	characteristic
45	C₂×C₄×C₇₁	—	2	2	no	—
46	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
47	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
48	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
49	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
50	C₂×C₄×C₇₁	#108	1	2	yes	—
51	C₂×C₄×C₇₁	D₁₆	1	2	yes	—
52	C₂×C₄×C₇₁	Q₃₂	1	2	yes	—
53	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
56	C₈×C₇₁	—	2	1	no	—
57	C₈×C₇₁	—	2	1	no	—
58	C₈×C₇₁	C₄⋊C₈	1	2	yes	—
59	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
60	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
61	C₇₁⋊C₈	—	8	4	no	—
62	C₂²×C₇₁	#335	1	2	yes	—
63	C₂²×C₇₁	#366	1	2	yes	—
64	C₂²×C₇₁	#368	1	2	yes	—
65	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
66	C₄×C₇₁	—	2	1	no	—
67	C₄×C₇₁	—	2	1	no	—
68	C₄×C₇₁	#335	1	2	yes	—
69	C₄×C₇₁	#366	1	2	yes	—
70	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₄×C₇₁	C₂×#108	1	1	yes	characteristic
72	C₄×C₇₁	C₂×D₁₆	1	1	yes	characteristic
73	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
74	C₂×#1371	—	71	1	no	maximal, Sylow
75	C₂×C₇₁	#1371	1	4	yes	complemented
76	C₂×C₇₁	C₂×#335	1	1	yes	characteristic
77	C₂×C₇₁	C₂×#366	1	1	yes	characteristic
78	C₂×C₇₁	C₂×#368	1	1	yes	characteristic
79	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
80	C₂×#335	—	71	2	no	—
81	#1371	—	71	4	no	—
82	C₇₁	C₂×#1371	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₁₆	Dic₇₁	1	1	yes	characteristic
84	C₂²×C₁₆	Dic₇₁	1	1	yes	characteristic
85	C₂×C₄⋊C₈	—	142	2	no	—
86	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
87	C₄×C₁₆	Dic₇₁	1	2	yes	—
88	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
89	#335	—	71	4	no	—
90	#335	—	142	2	no	—
91	C₂²×C₈	—	284	2	no	—
92	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
93	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
94	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	C₇₁⋊C₈	1	4	yes	complemented
97	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
98	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
99	C₄⋊C₈	—	284	2	no	—
100	C₄⋊C₈	—	142	4	no	—
101	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
102	C₄×C₈	D₂₈₄	1	1	yes	characteristic
103	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
104	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
105	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
106	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
107	C₂²×C₄	Dic₂₈₄	1	1	yes	characteristic
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	284	4	no	—
110	C₂×C₈	—	568	4	no	—
111	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
112	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
113	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
114	C₂×C₈	#11765849	1	2	yes	—
115	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
116	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
117	C₄²	C₄⋊Dic₇₁	1	1	yes	characteristic
118	C₄²	D₅₆₈	1	1	yes	characteristic
119	C₄²	Dic₂₈₄	1	1	yes	characteristic
120	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
121	C₁₆	—	2	2	no	—
122	C₁₆	C₂×C₇₁⋊C₈	1	4	yes	complemented
123	C₂³	#14556062	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
126	C₂×C₄	#14556062	1	1	yes	characteristic
127	C₂×C₄	#14556062	1	1	yes	characteristic
128	C₂×C₄	#14556062	1	1	yes	characteristic
129	C₂×C₄	#14556064	1	2	yes	—
130	C₂×C₄	D₁₁₃₆	1	2	yes	—
131	C₂×C₄	Dic₅₆₈	1	2	yes	—
132	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
133	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
134	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	568	4	no	—
138	C₈	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
139	C₈	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
140	C₈	C₂×#11765849	1	1	yes	characteristic
141	C₂²	#17985752	1	2	yes	—
142	C₂²	#17985825	1	2	yes	—
143	C₂²	#17985829	1	2	yes	—
144	C₂²	C₂×#14556062	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	#17985752	1	2	yes	—
148	C₄	#17985825	1	2	yes	—
149	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
150	C₄	C₂×#14556064	1	1	yes	characteristic
151	C₄	C₂×D₁₁₃₆	1	1	yes	characteristic
152	C₄	C₂×Dic₅₆₈	1	1	yes	characteristic
153	C₂	#22586412	1	4	yes	direct product factor
154	C₂	C₂×#17985752	1	1	yes	characteristic
155	C₂	C₂×#17985825	1	1	yes	characteristic
156	C₂	C₂×#17985829	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₅₆₈
commutator	C₈×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	#107
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	C₂×#1371
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ 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. #22586412	C₂	direct (N × H)
19. C₂×C₁₆×C₇₁	C₈	semidirect
43. C₁₆×C₇₁	C₂×C₈	semidirect
75. C₂×C₇₁	#1371	semidirect
82. C₇₁	C₂×#1371	semidirect
96. C₂×C₁₆	C₇₁⋊C₈	semidirect
122. C₁₆	C₂×C₇₁⋊C₈	semidirect
153. C₂	#22586412	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×#17985752	2
#22586412	2
C₂×C₄×C₁₆×C₇₁	2
C₂×#1371	71

Subgroups — C2×#22586412 #31909030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22586412 #31909030