Subgroups — C₂×#22590619 #32152934

Order 18176.

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×#17986401	C₂	1	2	yes	maximal
2	#22590619	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17985827	C₂	1	2	yes	maximal
4	C₂×#17985825	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556120	—	2	2	no	—
7	#17986401	—	2	4	no	—
8	#17986401	C₂²	1	8	yes	—
9	C₂×#14556062	C₂²	1	4	yes	—
10	C₂×#14556060	—	2	2	no	—
11	#17985827	C₂²	1	8	yes	—
12	#17985825	C₂²	1	8	yes	—
13	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₇₁	C₂²	1	4	yes	—
17	#14556120	—	2	8	no	—
18	#14556120	—	4	4	no	—
19	C₂×C₄⋊Dic₇₁	—	2	4	no	—
20	C₂×Dic₇₁⋊C₄	—	4	4	no	—
21	#14556062	—	2	4	no	—
22	#14556062	C₂³	1	8	yes	—
23	#14556060	—	2	8	no	—
24	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₁	Q₈	1	4	yes	—
28	C₂×C₁₆×C₇₁	Q₈	1	4	yes	—
29	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
30	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
31	C₄×C₈×C₇₁	D₄	1	2	yes	—
32	C₄×C₈×C₇₁	C₂³	1	2	yes	—
33	C₂²×Dic₇₁	—	4	4	no	—
34	C₄⋊Dic₇₁	—	2	8	no	—
35	C₄⋊Dic₇₁	—	4	4	no	—
36	Dic₇₁⋊C₄	—	4	16	no	—
37	C₂²×C₄×C₇₁	D₈	1	2	yes	—
38	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₁	Q₁₆	1	4	yes	—
40	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
42	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
43	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
44	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
45	C₄²×C₇₁	D₈	1	2	yes	—
46	C₄²×C₇₁	C₂×D₄	1	2	yes	—
47	C₁₆×C₇₁	C₂×Q₈	1	4	yes	—
48	C₁₆×C₇₁	C₂×Q₈	1	4	yes	—
49	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₇₁	—	2	4	no	—
51	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₇₁	C₂×D₈	1	4	yes	—
53	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
55	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
56	C₂×Dic₇₁	—	4	8	no	—
57	C₂×Dic₇₁	—	8	8	no	—
58	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
59	C₈×C₇₁	C₂×Q₁₆	1	4	yes	—
60	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
61	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
62	C₂²×C₇₁	#500	1	2	yes	—
63	C₂²×C₇₁	#508	1	4	yes	—
64	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
65	C₄×C₇₁	—	2	4	no	—
66	C₄×C₇₁	#500	1	2	yes	—
67	C₄×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
69	Dic₇₁	—	8	8	no	—
70	C₂×#2253	—	71	1	no	maximal, Sylow
71	C₂×C₇₁	#2253	1	4	yes	complemented
72	C₂×C₇₁	C₂×#500	1	1	yes	characteristic
73	C₂×C₇₁	C₂×#508	1	2	yes	—
74	C₂×#499	—	71	2	no	—
75	#2253	—	71	8	no	—
76	C₂×#367	—	71	2	no	—
77	C₂×#366	—	71	2	no	—
78	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
79	C₇₁	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
80	C₂×#125	—	142	2	no	—
81	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
82	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
83	#499	—	71	8	no	—
84	#499	—	142	4	no	—
85	C₂×#107	—	71	4	no	—
86	C₂×#106	—	142	2	no	—
87	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
88	#367	—	71	8	no	—
89	#366	—	71	8	no	—
90	C₄×C₁₆	C₂×D₇₁	1	4	yes	—
91	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
93	#125	—	284	4	no	—
94	#125	—	142	8	no	—
95	C₂×C₄⋊C₄	—	284	4	no	—
96	C₂×C₄⋊C₄	—	142	4	no	—
97	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
98	C₂×C₁₆	Dic₁₄₂	1	4	yes	—
99	C₂×C₁₆	Dic₁₄₂	1	4	yes	—
100	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
101	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
102	#107	—	71	8	no	—
103	#107	—	142	4	no	—
104	#106	—	142	8	no	—
105	C₄×C₈	D₂₈₄	1	2	yes	—
106	C₄×C₈	C₂²×D₇₁	1	2	yes	—
107	C₂²×C₄	—	284	4	no	—
108	C₂²×C₄	D₅₆₈	1	2	yes	—
109	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
110	C₂×C₈	Dic₂₈₄	1	4	yes	—
111	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
112	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
113	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
114	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
115	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
116	C₄⋊C₄	—	284	16	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	4	yes	—
122	C₁₆	C₂×Dic₁₄₂	1	4	yes	—
123	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	284	8	no	—
126	C₂×C₄	—	568	8	no	—
127	C₂×C₄	#14556128	1	2	yes	—
128	C₂×C₄	C₂×D₅₆₈	1	4	yes	—
129	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
130	C₂×C₄	C₂×Dic₂₈₄	1	2	yes	—
131	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
132	C₈	#14556128	1	2	yes	—
133	C₈	C₂×Dic₂₈₄	1	4	yes	—
134	C₈	C₂²×Dic₁₄₂	1	1	yes	characteristic
135	C₈	C₂²×Dic₁₄₂	1	1	yes	characteristic
136	C₂²	#17986404	1	2	yes	—
137	C₂²	#17986431	1	4	yes	—
138	C₂²	C₂²×D₅₆₈	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	568	8	no	—
141	C₄	#17986404	1	2	yes	—
142	C₄	C₂×#14556128	1	1	yes	characteristic
143	C₄	C₂²×Dic₂₈₄	1	1	yes	characteristic
144	C₂	#22590619	1	4	yes	direct product factor
145	C₂	C₂×#17986404	1	1	yes	characteristic
146	C₂	C₂×#17986431	1	2	yes	—

	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×#2253
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. #22590619	C₂	direct (N × H)
71. C₂×C₇₁	#2253	semidirect
79. C₇₁	C₂×#2253	semidirect
144. C₂	#22590619	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×#17986401	2
#22590619	2
C₂×#17985827	2
C₂×#17985825	2
C₂×C₄×C₁₆×C₇₁	2
C₂×#2253	71

Subgroups — C2×#22590619 #32152934

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22590619 #32152934