Subgroups — C₂×#22590615 #32152929

Order 18176.

558 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986404	C₂	1	2	yes	maximal
2	#22590615	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17985825	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556128	—	2	2	no	—
6	#17986404	—	2	4	no	—
7	#17986404	C₂²	1	8	yes	—
8	C₂×#14556062	—	2	2	no	—
9	C₂×#14556062	C₂²	1	4	yes	—
10	#17985825	C₂²	1	16	yes	—
11	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₇₁	C₂²	1	4	yes	—
14	C₂²×Dic₁₄₂	—	4	2	no	—
15	#14556128	—	2	8	no	—
16	#14556128	—	4	4	no	—
17	C₂×C₄⋊Dic₇₁	—	2	4	no	—
18	C₂×C₄⋊Dic₇₁	—	4	2	no	—
19	#14556062	—	2	4	no	—
20	#14556062	—	2	8	no	—
21	#14556062	C₂³	1	8	yes	—
22	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₁	Q₈	1	8	yes	complemented
26	C₂×C₁₆×C₇₁	C₂³	1	4	yes	—
27	C₄×C₈×C₇₁	D₄	1	2	yes	—
28	C₄×C₈×C₇₁	C₂³	1	2	yes	—
29	C₂×Dic₁₄₂	—	4	8	no	—
30	C₂×Dic₁₄₂	—	8	8	no	—
31	C₂²×Dic₇₁	—	4	4	no	—
32	C₄⋊Dic₇₁	—	2	8	no	—
33	C₄⋊Dic₇₁	—	4	4	no	—
34	C₄⋊Dic₇₁	—	4	8	no	—
35	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₇₁	Q₁₆	1	4	yes	—
39	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
41	C₂×C₈×C₇₁	C₂×Q₈	1	4	yes	—
42	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
43	C₄²×C₇₁	D₈	1	2	yes	—
44	C₄²×C₇₁	C₂×D₄	1	2	yes	—
45	C₁₆×C₇₁	C₂×Q₈	1	8	yes	complemented
46	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₇₁	D₁₆	1	4	yes	—
48	C₂×C₄×C₇₁	Q₃₂	1	4	yes	—
49	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
50	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
52	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
53	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
54	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
55	C₂×Dic₇₁	—	4	8	no	—
56	C₂×Dic₇₁	—	8	8	no	—
57	Dic₁₄₂	—	8	16	no	—
58	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
59	C₈×C₇₁	C₂×Q₁₆	1	4	yes	—
60	C₈×C₇₁	C₂²×Q₈	1	2	yes	—
61	C₂²×C₇₁	#500	1	2	yes	—
62	C₂²×C₇₁	C₂×D₁₆	1	2	yes	—
63	C₂²×C₇₁	C₂×Q₃₂	1	2	yes	—
64	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
65	C₄×C₇₁	#500	1	2	yes	—
66	C₄×C₇₁	C₂×D₁₆	1	4	yes	—
67	C₄×C₇₁	C₂×Q₃₂	1	4	yes	—
68	C₄×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
69	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
70	Dic₇₁	—	8	8	no	—
71	C₂×#2252	—	71	1	no	maximal, Sylow
72	C₂×C₇₁	#2252	1	4	yes	complemented
73	C₂×C₇₁	C₂×#500	1	1	yes	characteristic
74	C₂×C₇₁	C₂²×D₁₆	1	1	yes	characteristic
75	C₂×C₇₁	C₂²×Q₃₂	1	1	yes	characteristic
76	C₂×#500	—	71	2	no	—
77	#2252	—	71	8	no	—
78	C₂×#366	—	71	4	no	—
79	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
80	C₇₁	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₄⋊Q₈	—	142	2	no	—
82	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
83	#500	—	71	8	no	—
84	#500	—	142	4	no	—
85	C₂×#107	—	71	4	no	—
86	C₂×#107	—	142	2	no	—
87	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
88	#366	—	71	16	no	—
89	C₄×C₁₆	C₂×D₇₁	1	4	yes	—
90	C₂²×Q₈	—	284	2	no	—
91	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
93	C₄⋊Q₈	—	284	4	no	—
94	C₄⋊Q₈	—	142	8	no	—
95	C₂×C₄⋊C₄	—	284	2	no	—
96	C₂×C₄⋊C₄	—	142	4	no	—
97	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
98	C₂×C₁₆	Dic₁₄₂	1	8	yes	complemented
99	C₂×C₁₆	C₂²×D₇₁	1	4	yes	—
100	#107	—	71	8	no	—
101	#107	—	142	4	no	—
102	#107	—	142	8	no	—
103	C₄×C₈	D₂₈₄	1	2	yes	—
104	C₄×C₈	C₂²×D₇₁	1	2	yes	—
105	C₂×Q₈	—	284	8	no	—
106	C₂×Q₈	—	568	8	no	—
107	C₂²×C₄	—	284	4	no	—
108	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
109	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
110	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
111	C₂×C₈	Dic₂₈₄	1	4	yes	—
112	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
113	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
114	C₂×C₈	C₂×Dic₁₄₂	1	4	yes	—
115	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
116	C₄⋊C₄	—	284	8	no	—
117	C₄⋊C₄	—	284	4	no	—
118	C₄⋊C₄	—	142	8	no	—
119	C₄²	D₅₆₈	1	2	yes	—
120	C₄²	C₂×D₂₈₄	1	2	yes	—
121	C₁₆	C₂×Dic₁₄₂	1	8	yes	complemented
122	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
123	Q₈	—	568	16	no	—
124	C₂×C₄	—	284	8	no	—
125	C₂×C₄	—	568	8	no	—
126	C₂×C₄	D₁₁₃₆	1	4	yes	—
127	C₂×C₄	Dic₅₆₈	1	4	yes	—
128	C₂×C₄	#14556128	1	2	yes	—
129	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
130	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
131	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
132	C₂×C₄	C₂×Dic₂₈₄	1	2	yes	—
133	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
134	C₈	#14556128	1	2	yes	—
135	C₈	C₂×Dic₂₈₄	1	4	yes	—
136	C₈	C₂²×Dic₁₄₂	1	2	yes	—
137	C₂²	#17986404	1	2	yes	—
138	C₂²	C₂×D₁₁₃₆	1	2	yes	—
139	C₂²	C₂×Dic₅₆₈	1	2	yes	—
140	C₂²	C₂²×D₅₆₈	1	1	yes	characteristic
141	C₄	—	568	8	no	—
142	C₄	#17986404	1	2	yes	—
143	C₄	C₂×D₁₁₃₆	1	4	yes	—
144	C₄	C₂×Dic₅₆₈	1	4	yes	—
145	C₄	C₂×#14556128	1	1	yes	characteristic
146	C₄	C₂²×Dic₂₈₄	1	1	yes	characteristic
147	C₂	#22590615	1	4	yes	direct product factor
148	C₂	C₂×#17986404	1	1	yes	characteristic
149	C₂	C₂²×D₁₁₃₆	1	1	yes	characteristic
150	C₂	C₂²×Dic₅₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₆₈
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	C₂×#2252
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
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. #22590615	C₂	direct (N × H)
25. C₂×C₁₆×C₇₁	Q₈	semidirect
45. C₁₆×C₇₁	C₂×Q₈	semidirect
72. C₂×C₇₁	#2252	semidirect
80. C₇₁	C₂×#2252	semidirect
98. C₂×C₁₆	Dic₁₄₂	semidirect
121. C₁₆	C₂×Dic₁₄₂	semidirect
147. C₂	#22590615	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×#17986404	2
#22590615	2
C₂×#17985825	2
C₂×C₄×C₁₆×C₇₁	2
C₂×#2252	71

Subgroups — C2×#22590615 #32152929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22590615 #32152929