Subgroups — #31891223

Order 18176.

454 conjugacy classes of subgroups.

The subgroup lattice has 162 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	#22588497	C₂	1	2	yes	maximal
3	#22588468	C₂	1	2	yes	maximal
4	C₂×C₈²×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556062	—	2	4	no	—
6	C₂×#14556060	—	2	4	no	—
7	#17985869	C₂²	1	4	yes	—
8	#17985762	—	2	4	no	—
9	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₈²×C₇₁	C₄	1	4	yes	complemented
13	C₂×C₄⋊Dic₇₁	—	2	4	no	—
14	C₂×C₄⋊Dic₇₁	—	2	4	no	—
15	C₂×C₄⋊Dic₇₁	—	2	4	no	—
16	#14556062	—	4	8	no	—
17	#14556060	—	4	8	no	—
18	C₂²×C₈×C₇₁	D₄	1	2	yes	—
19	C₂²×C₈×C₇₁	D₄	1	2	yes	—
20	C₂²×C₈×C₇₁	D₄	1	2	yes	—
21	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₇₁	Q₈	1	2	yes	—
26	C₄×C₈×C₇₁	Q₈	1	2	yes	—
27	C₄×C₈×C₇₁	Q₈	1	2	yes	—
28	C₂²×Dic₇₁	—	4	4	no	—
29	C₄⋊Dic₇₁	—	4	8	no	—
30	C₄⋊Dic₇₁	—	4	8	no	—
31	C₄⋊Dic₇₁	—	4	8	no	—
32	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₇₁	—	2	4	no	—
36	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
37	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
38	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
39	C₂×C₈×C₇₁	D₈	1	4	yes	—
40	C₂×C₈×C₇₁	SD₁₆	1	8	yes	—
41	C₂×C₈×C₇₁	Q₁₆	1	4	yes	—
42	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
46	C₂³×C₇₁	#127	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
54	C₂×C₄×C₇₁	C₂×SD₁₆	1	4	yes	—
55	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
56	C₂×C₄×C₇₁	#135	1	4	yes	—
57	C₂×Dic₇₁	—	8	4	no	—
58	C₂×Dic₇₁	—	8	4	no	—
59	C₂×Dic₇₁	—	8	4	no	—
60	C₈×C₇₁	—	2	4	no	—
61	C₈×C₇₁	#106	1	8	yes	complemented
62	C₈×C₇₁	#107	1	8	yes	complemented
63	C₂²×C₇₁	#384	1	1	yes	characteristic
64	C₂²×C₇₁	#492	1	2	yes	—
65	C₂²×C₇₁	#493	1	1	yes	characteristic
66	C₂²×C₇₁	#494	1	1	yes	characteristic
67	C₂²×C₇₁	#495	1	2	yes	—
68	C₄×C₇₁	C₂×#106	1	2	yes	—
69	C₄×C₇₁	C₂×#107	1	2	yes	—
70	C₄×C₇₁	#427	1	2	yes	—
71	C₄×C₇₁	#498	1	2	yes	—
72	C₄×C₇₁	#499	1	2	yes	—
73	C₄×C₇₁	#500	1	2	yes	—
74	Dic₇₁	—	16	4	no	—
75	#8814	—	71	1	no	maximal, Sylow
76	C₂×C₇₁	#1706	1	2	yes	—
77	C₂×C₇₁	#1707	1	2	yes	—
78	C₂×C₇₁	#1844	1	1	yes	characteristic
79	C₂×C₇₁	#1845	1	1	yes	characteristic
80	C₂×C₇₁	#1846	1	1	yes	characteristic
81	#1845	—	71	2	no	—
82	#1844	—	71	2	no	—
83	#1834	—	71	2	no	—
84	C₂×C₈²	D₇₁	1	1	yes	characteristic
85	C₇₁	#8814	1	1	yes	characteristic, complemented, Sylow
86	C₂×#107	—	142	4	no	—
87	C₂×#106	—	142	4	no	—
88	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
90	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
91	#384	—	71	4	no	—
92	#340	—	142	4	no	—
93	C₈²	Dic₇₁	1	4	yes	complemented
94	C₂²×C₈	D₂₈₄	1	2	yes	—
95	C₂²×C₈	D₂₈₄	1	2	yes	—
96	C₂²×C₈	D₂₈₄	1	2	yes	—
97	C₂×C₄⋊C₄	—	142	4	no	—
98	C₂×C₄⋊C₄	—	142	4	no	—
99	C₂×C₄⋊C₄	—	142	4	no	—
100	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
101	#107	—	284	8	no	—
102	#106	—	284	8	no	—
103	C₄×C₈	Dic₁₄₂	1	2	yes	—
104	C₄×C₈	Dic₁₄₂	1	2	yes	—
105	C₄×C₈	Dic₁₄₂	1	2	yes	—
106	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
107	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
108	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
109	C₂²×C₄	—	284	4	no	—
110	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	C₄⋊Dic₇₁	1	4	yes	—
115	C₂×C₈	C₄⋊Dic₇₁	1	4	yes	—
116	C₂×C₈	C₄⋊Dic₇₁	1	4	yes	—
117	C₂×C₈	D₅₆₈	1	4	yes	—
118	C₂×C₈	#11765854	1	8	yes	—
119	C₂×C₈	Dic₂₈₄	1	4	yes	—
120	C₄⋊C₄	—	284	8	no	—
121	C₄⋊C₄	—	284	8	no	—
122	C₄⋊C₄	—	284	8	no	—
123	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
124	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
125	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
126	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
127	C₂³	#14556125	1	1	yes	characteristic
128	C₂×C₄	—	568	4	no	—
129	C₂×C₄	—	568	4	no	—
130	C₂×C₄	—	568	4	no	—
131	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
134	C₂×C₄	#14556128	1	1	yes	characteristic
135	C₂×C₄	#14556128	1	1	yes	characteristic
136	C₂×C₄	#14556128	1	1	yes	characteristic
137	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
138	C₂×C₄	C₂×#11765854	1	4	yes	—
139	C₂×C₄	C₂×Dic₂₈₄	1	2	yes	—
140	C₂×C₄	#14556150	1	4	yes	—
141	C₈	—	2	4	no	—
142	C₈	#14556060	1	8	yes	complemented
143	C₈	#14556062	1	8	yes	complemented
144	C₂²	#17985869	1	1	yes	characteristic
145	C₂²	#17986372	1	2	yes	—
146	C₂²	#17986376	1	1	yes	characteristic
147	C₂²	#17986379	1	1	yes	characteristic
148	C₂²	#17986383	1	2	yes	—
149	C₄	—	1136	4	no	—
150	C₄	C₂×#14556060	1	2	yes	—
151	C₄	C₂×#14556062	1	2	yes	—
152	C₄	#17986037	1	2	yes	—
153	C₄	#17986397	1	2	yes	—
154	C₄	#17986401	1	2	yes	—
155	C₄	#17986404	1	2	yes	—
156	C₂	#22587932	1	2	yes	—
157	C₂	#22587937	1	2	yes	—
158	C₂	#22588497	1	1	yes	characteristic
159	C₂	#22588500	1	1	yes	characteristic
160	C₂	#22588504	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556125
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×C₈²×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	#127
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×C₈²	D₇₁
hypocenter	C₇₁	#8814
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
12. C₈²×C₇₁	C₄	semidirect
61. C₈×C₇₁	#106	semidirect
62. C₈×C₇₁	#107	semidirect
85. C₇₁	#8814	semidirect
93. C₈²	Dic₇₁	semidirect
142. C₈	#14556060	semidirect
143. C₈	#14556062	semidirect

Maximal subgroups

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

subgroup	index
#22588500	2
#22588497	2
#22588468	2
C₂×C₈²×C₇₁	2
#8814	71