Subgroups — #31913901

Order 18176.

314 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985829	C₂	1	4	yes	maximal
2	#22588512	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556064	—	2	2	no	—
5	C₂×#14556064	C₂²	1	4	yes	—
6	#17986023	—	2	2	no	—
7	#17985829	—	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	2	yes	—
11	C₄×C₁₆×C₇₁	C₄	1	2	yes	—
12	C₂×#11765849	—	2	4	no	—
13	#14556064	—	2	8	no	—
14	#14556064	—	4	4	no	—
15	C₄⋊(C₇₁⋊C₈)	—	4	4	no	—
16	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
20	C₂×C₁₆×C₇₁	D₄	1	4	yes	—
21	C₂×C₁₆×C₇₁	Q₈	1	4	yes	—
22	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
26	#11765849	—	4	4	no	—
27	#11765849	—	4	4	no	—
28	C₂×C₇₁⋊C₈	—	4	4	no	—
29	C₂²×C₄×C₇₁	Q₁₆	1	2	yes	—
30	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₇₁	D₈	1	2	yes	—
34	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
35	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
37	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
38	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
39	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₇₁	D₈	1	2	yes	—
42	C₁₆×C₇₁	C₄⋊C₄	1	8	yes	—
43	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
44	C₂×C₄×C₇₁	—	2	4	no	—
45	C₂×C₄×C₇₁	#107	1	2	yes	—
46	C₂×C₄×C₇₁	#107	1	2	yes	—
47	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
53	C₈×C₇₁	#107	1	2	yes	—
54	C₈×C₇₁	#107	1	2	yes	—
55	C₈×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
56	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
57	C₇₁⋊C₈	—	8	4	no	—
58	C₂²×C₇₁	—	2	2	no	—
59	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
60	C₂²×C₇₁	#494	1	1	yes	characteristic
61	C₂²×C₇₁	#500	1	1	yes	characteristic
62	C₄×C₇₁	—	2	2	no	—
63	C₄×C₇₁	#368	1	4	yes	complemented
64	C₄×C₇₁	#384	1	1	yes	characteristic
65	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
66	C₄×C₇₁	#493	1	1	yes	characteristic
67	C₄×C₇₁	#500	1	1	yes	characteristic
68	#12909	—	71	1	no	maximal, Sylow
69	C₂×C₇₁	—	2	1	no	—
70	C₂×C₇₁	—	2	1	no	—
71	C₂×C₇₁	#1845	1	1	yes	characteristic
72	C₂×C₇₁	C₂×#368	1	2	yes	—
73	C₂×#368	—	71	4	no	—
74	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
75	#1848	—	71	2	no	—
76	C₇₁	#12909	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
78	C₂×#108	—	71	4	no	—
79	C₂×#108	—	142	2	no	—
80	#423	—	142	2	no	—
81	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
82	#368	—	142	8	no	—
83	C₄×C₁₆	Dic₇₁	1	2	yes	—
84	C₄×C₁₆	Dic₇₁	1	2	yes	—
85	C₂×M₁₆	—	142	4	no	—
86	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
87	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
88	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
89	C₂×C₁₆	Dic₁₄₂	1	4	yes	—
90	C₂×C₁₆	D₂₈₄	1	4	yes	—
91	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	—
92	#108	—	284	4	no	—
93	#108	—	142	8	no	—
94	C₄⋊C₈	—	284	4	no	—
95	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
96	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
97	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
98	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
99	C₂²×C₄	Dic₂₈₄	1	2	yes	—
100	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
101	M₁₆	—	284	4	no	—
102	M₁₆	—	284	4	no	—
103	C₂×C₈	—	284	4	no	—
104	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
105	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
106	C₂×C₈	D₅₆₈	1	2	yes	—
107	C₂×C₈	Dic₂₈₄	1	2	yes	—
108	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
109	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
110	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
111	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
112	C₄²	C₄⋊Dic₇₁	1	1	yes	characteristic
113	C₄²	C₄⋊Dic₇₁	1	1	yes	characteristic
114	C₄²	D₅₆₈	1	2	yes	—
115	C₁₆	C₄⋊Dic₇₁	1	8	yes	—
116	C₂³	C₂×Dic₂₈₄	1	1	yes	characteristic
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	#14556062	1	2	yes	—
119	C₂×C₄	#14556062	1	2	yes	—
120	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
121	C₂×C₄	#14556125	1	1	yes	characteristic
122	C₂×C₄	#14556128	1	1	yes	characteristic
123	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
124	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
125	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
126	C₈	—	568	4	no	—
127	C₈	#14556062	1	2	yes	—
128	C₈	#14556062	1	2	yes	—
129	C₈	C₂×C₄⋊Dic₇₁	1	2	yes	—
130	C₈	#14556128	1	2	yes	—
131	C₂²	—	2	2	no	—
132	C₂²	C₂×#14556062	1	1	yes	characteristic
133	C₂²	#17986379	1	1	yes	characteristic
134	C₂²	#17986404	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	#17985829	1	4	yes	complemented
137	C₄	#17985869	1	1	yes	characteristic
138	C₄	C₂×#14556062	1	1	yes	characteristic
139	C₄	#17986376	1	1	yes	characteristic
140	C₄	#17986404	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	#22588500	1	1	yes	characteristic
144	C₂	C₂×#17985829	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₆₈
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#107
prime radical	C₂³×C₇₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	#12909
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×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
63. C₄×C₇₁	#368	semidirect
76. C₇₁	#12909	semidirect
136. C₄	#17985829	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985829	2
#22588512	2
C₂×C₄×C₁₆×C₇₁	2
#12909	71