Subgroups — #31913886

Order 18176.

374 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	C₂×#17985825	C₂	1	4	yes	maximal
2	#22588500	C₂	1	2	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	#17985825	—	2	8	no	—
8	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₇₁	C₄	1	4	yes	complemented
11	C₂×C₄⋊Dic₇₁	—	2	4	no	—
12	C₂×C₄⋊Dic₇₁	—	4	4	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	C₂×C₄	1	4	yes	—
19	C₂×C₁₆×C₇₁	D₄	1	4	yes	—
20	C₂×C₁₆×C₇₁	Q₈	1	4	yes	—
21	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₇₁	Q₈	1	2	yes	—
23	C₂²×Dic₇₁	—	4	4	no	—
24	C₄⋊Dic₇₁	—	4	4	no	—
25	C₄⋊Dic₇₁	—	4	4	no	—
26	C₄⋊Dic₇₁	—	8	8	no	—
27	C₂²×C₄×C₇₁	D₈	1	2	yes	—
28	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₇₁	D₈	1	2	yes	—
31	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
32	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
34	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
35	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
37	C₄²×C₇₁	Q₁₆	1	2	yes	—
38	C₁₆×C₇₁	C₄⋊C₄	1	8	yes	complemented
39	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
40	C₂×C₄×C₇₁	#107	1	4	yes	—
41	C₂×C₄×C₇₁	D₁₆	1	4	yes	—
42	C₂×C₄×C₇₁	Q₃₂	1	4	yes	—
43	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
49	C₂×Dic₇₁	—	8	4	no	—
50	C₂×Dic₇₁	—	8	8	no	—
51	C₈×C₇₁	#107	1	4	yes	—
52	C₈×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
53	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
54	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
55	C₂²×C₇₁	#493	1	1	yes	characteristic
56	C₂²×C₇₁	#500	1	1	yes	characteristic
57	C₂²×C₇₁	C₂×D₁₆	1	2	yes	—
58	C₂²×C₇₁	C₂×Q₃₂	1	2	yes	—
59	C₄×C₇₁	#366	1	8	yes	complemented
60	C₄×C₇₁	#384	1	1	yes	characteristic
61	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
62	C₄×C₇₁	#494	1	1	yes	characteristic
63	C₄×C₇₁	#500	1	1	yes	characteristic
64	Dic₇₁	—	16	4	no	—
65	#12905	—	71	1	no	maximal, Sylow
66	C₂×C₇₁	#1845	1	1	yes	characteristic
67	C₂×C₇₁	C₂×#366	1	2	yes	—
68	C₂×C₇₁	#2246	1	1	yes	characteristic
69	C₂×C₇₁	#2247	1	1	yes	characteristic
70	C₂×C₇₁	#2252	1	2	yes	—
71	C₂×#366	—	71	4	no	—
72	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
73	#1845	—	71	2	no	—
74	C₇₁	#12905	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	#366	—	142	8	no	—
81	C₄×C₁₆	Dic₇₁	1	4	yes	complemented
82	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
83	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
84	C₂×C₄⋊C₄	—	284	4	no	—
85	C₂×C₄⋊C₄	—	142	4	no	—
86	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
87	C₂×C₁₆	Dic₁₄₂	1	4	yes	—
88	C₂×C₁₆	D₂₈₄	1	4	yes	—
89	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	—
90	#107	—	284	4	no	—
91	#107	—	142	8	no	—
92	C₄×C₈	Dic₁₄₂	1	2	yes	—
93	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
94	C₂²×C₄	—	284	4	no	—
95	C₂²×C₄	D₅₆₈	1	2	yes	—
96	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
97	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
98	C₂×C₈	D₅₆₈	1	2	yes	—
99	C₂×C₈	Dic₂₈₄	1	2	yes	—
100	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
101	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
102	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
103	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
104	C₄⋊C₄	—	284	4	no	—
105	C₄⋊C₄	—	284	4	no	—
106	C₄⋊C₄	—	568	8	no	—
107	C₄²	C₄⋊Dic₇₁	1	2	yes	—
108	C₄²	Dic₂₈₄	1	2	yes	—
109	C₁₆	C₄⋊Dic₇₁	1	8	yes	complemented
110	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
111	C₂×C₄	—	568	4	no	—
112	C₂×C₄	—	568	8	no	—
113	C₂×C₄	#14556062	1	4	yes	—
114	C₂×C₄	D₁₁₃₆	1	4	yes	—
115	C₂×C₄	Dic₅₆₈	1	4	yes	—
116	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
117	C₂×C₄	#14556125	1	1	yes	characteristic
118	C₂×C₄	#14556128	1	1	yes	characteristic
119	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
121	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
122	C₈	#14556062	1	4	yes	—
123	C₈	C₂×C₄⋊Dic₇₁	1	2	yes	—
124	C₈	#14556128	1	2	yes	—
125	C₂²	C₂×#14556062	1	1	yes	characteristic
126	C₂²	#17986376	1	1	yes	characteristic
127	C₂²	#17986404	1	1	yes	characteristic
128	C₂²	C₂×D₁₁₃₆	1	2	yes	—
129	C₂²	C₂×Dic₅₆₈	1	2	yes	—
130	C₄	—	1136	4	no	—
131	C₄	#17985825	1	8	yes	complemented
132	C₄	#17985869	1	1	yes	characteristic
133	C₄	C₂×#14556062	1	1	yes	characteristic
134	C₄	#17986379	1	1	yes	characteristic
135	C₄	#17986404	1	1	yes	characteristic
136	C₂	#22588500	1	1	yes	characteristic
137	C₂	C₂×#17985825	1	2	yes	—
138	C₂	#22590590	1	1	yes	characteristic
139	C₂	#22590593	1	1	yes	characteristic
140	C₂	#22590615	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	#12905
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
10. C₄×C₁₆×C₇₁	C₄	semidirect
38. C₁₆×C₇₁	C₄⋊C₄	semidirect
59. C₄×C₇₁	#366	semidirect
74. C₇₁	#12905	semidirect
81. C₄×C₁₆	Dic₇₁	semidirect
109. C₁₆	C₄⋊Dic₇₁	semidirect
131. C₄	#17985825	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985825	2
#22588500	2
C₂×C₄×C₁₆×C₇₁	2
#12905	71