Subgroups — #31913832

Order 18176.

342 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588500	C₂	1	2	yes	maximal
2	#22586432	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556062	—	2	2	no	—
5	C₂×#14556062	C₂²	1	4	yes	—
6	#17985869	—	2	2	no	—
7	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₇₁	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₇₁	—	2	4	no	—
11	C₂×C₄⋊Dic₇₁	—	4	2	no	—
12	C₂×C₄⋊Dic₇₁	—	4	2	no	—
13	#14556062	—	2	8	no	—
14	#14556062	—	4	4	no	—
15	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₁	—	2	4	no	—
19	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
20	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₇₁	Q₈	1	2	yes	—
22	C₂²×Dic₇₁	—	4	4	no	—
23	C₄⋊Dic₇₁	—	4	4	no	—
24	C₄⋊Dic₇₁	—	4	4	no	—
25	C₄⋊Dic₇₁	—	8	4	no	—
26	C₄⋊Dic₇₁	—	8	4	no	—
27	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
28	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
30	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₇₁	SD₁₆	1	4	yes	—
32	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₇₁	#41	1	4	yes	—
35	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
36	C₄²×C₇₁	Q₁₆	1	2	yes	—
37	C₁₆×C₇₁	—	2	4	no	—
38	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
39	C₂×C₄×C₇₁	#107	1	2	yes	—
40	C₂×C₄×C₇₁	#107	1	2	yes	—
41	C₂×C₄×C₇₁	D₁₆	1	2	yes	—
42	C₂×C₄×C₇₁	SD₃₂	1	4	yes	—
43	C₂×C₄×C₇₁	Q₃₂	1	2	yes	—
44	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₁	#124	1	2	yes	—
46	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
47	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
48	C₂×Dic₇₁	—	8	4	no	—
49	C₂×Dic₇₁	—	8	4	no	—
50	C₂×Dic₇₁	—	8	4	no	—
51	C₈×C₇₁	#106	1	4	yes	—
52	C₈×C₇₁	#117	1	2	yes	—
53	C₈×C₇₁	#125	1	2	yes	—
54	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
55	C₂²×C₇₁	#486	1	2	yes	—
56	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
57	C₂²×C₇₁	C₂×SD₃₂	1	2	yes	—
58	C₂²×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
59	C₄×C₇₁	#366	1	4	yes	—
60	C₄×C₇₁	#367	1	4	yes	—
61	C₄×C₇₁	#382	1	1	yes	characteristic
62	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
63	C₄×C₇₁	#487	1	2	yes	—
64	Dic₇₁	—	16	4	no	—
65	#12888	—	71	1	no	maximal, Sylow
66	C₂×C₇₁	#1834	1	1	yes	characteristic
67	C₂×C₇₁	C₂×#366	1	1	yes	characteristic
68	C₂×C₇₁	C₂×#367	1	1	yes	characteristic
69	C₂×C₇₁	#2240	1	2	yes	—
70	C₂×C₇₁	#2241	1	2	yes	—
71	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
72	#1845	—	71	2	no	—
73	#1380	—	71	4	no	—
74	C₇₁	#12888	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
76	C₂×#107	—	71	4	no	—
77	C₂×#107	—	142	2	no	—
78	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
79	#384	—	142	2	no	—
80	C₄×C₁₆	Dic₇₁	1	4	yes	complemented
81	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
82	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
83	C₂×C₄⋊C₄	—	284	2	no	—
84	C₂×C₄⋊C₄	—	284	2	no	—
85	C₂×C₄⋊C₄	—	142	4	no	—
86	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
87	C₂×C₁₆	—	2	4	no	—
88	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	—
89	#107	—	284	4	no	—
90	#107	—	142	8	no	—
91	C₄×C₈	Dic₁₄₂	1	2	yes	—
92	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
93	C₂²×C₄	—	284	4	no	—
94	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
95	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
96	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
97	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
98	C₂×C₈	#11765854	1	4	yes	—
99	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
100	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
101	C₂×C₈	#11765865	1	4	yes	—
102	C₄⋊C₄	—	284	4	no	—
103	C₄⋊C₄	—	284	4	no	—
104	C₄⋊C₄	—	568	4	no	—
105	C₄⋊C₄	—	568	4	no	—
106	C₄²	C₄⋊Dic₇₁	1	2	yes	—
107	C₄²	Dic₂₈₄	1	2	yes	—
108	C₁₆	—	2	4	no	—
109	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
110	C₂×C₄	—	568	4	no	—
111	C₂×C₄	—	568	4	no	—
112	C₂×C₄	—	568	4	no	—
113	C₂×C₄	#14556062	1	2	yes	—
114	C₂×C₄	#14556062	1	2	yes	—
115	C₂×C₄	D₁₁₃₆	1	2	yes	—
116	C₂×C₄	#14556073	1	4	yes	—
117	C₂×C₄	Dic₅₆₈	1	2	yes	—
118	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
119	C₂×C₄	#14556116	1	2	yes	—
120	C₂×C₄	C₂×#11765854	1	2	yes	—
121	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
122	C₈	#14556060	1	4	yes	—
123	C₈	#14556087	1	2	yes	—
124	C₈	#14556120	1	2	yes	—
125	C₂²	C₂×#14556062	1	1	yes	characteristic
126	C₂²	#17986346	1	2	yes	—
127	C₂²	C₂×D₁₁₃₆	1	1	yes	characteristic
128	C₂²	C₂×#14556073	1	2	yes	—
129	C₂²	C₂×Dic₅₆₈	1	1	yes	characteristic
130	C₄	—	1136	4	no	—
131	C₄	#17985825	1	4	yes	—
132	C₄	#17985827	1	4	yes	—
133	C₄	#17985865	1	1	yes	characteristic
134	C₄	C₂×#14556060	1	1	yes	characteristic
135	C₄	#17986350	1	2	yes	—
136	C₂	#22588468	1	1	yes	characteristic
137	C₂	C₂×#17985825	1	1	yes	characteristic
138	C₂	C₂×#17985827	1	1	yes	characteristic
139	C₂	#22590564	1	2	yes	—
140	C₂	#22590568	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₆₈
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	#12888
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
9. C₄×C₁₆×C₇₁	C₄	semidirect
74. C₇₁	#12888	semidirect
80. C₄×C₁₆	Dic₇₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22588500	2
#22586432	2
C₂×C₄×C₁₆×C₇₁	2
#12888	71