Subgroups — #26131110

Order 12032.

350 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244993	C₂	1	1	yes	maximal, characteristic
2	#20244992	C₂	1	2	yes	maximal
3	#20244984	C₂	1	2	yes	maximal
4	#20243143	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	2	yes	—
7	#15689078	C₂²	1	2	yes	—
8	#15688901	C₄	1	4	yes	complemented
9	#15688900	C₄	1	8	yes	complemented
10	#15688898	C₄	1	4	yes	complemented
11	C₁₆×Dic₄₇	C₄	1	4	yes	complemented
12	C₄₇⋊#338	C₄	1	1	yes	characteristic
13	#15688880	C₄	1	1	yes	characteristic
14	#15688879	C₄	1	2	yes	—
15	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#405	C₂²	1	1	yes	characteristic
18	C₄₇×#346	C₄	1	2	yes	complemented
19	C₄×C₁₆×C₄₇	C₄	1	2	yes	complemented
20	C₂×#10625329	C₂×C₄	1	2	yes	—
21	#13093983	C₂×C₄	1	2	yes	—
22	#13093964	C₂×C₄	1	4	yes	—
23	#13093964	C₂×C₄	1	4	yes	—
24	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
25	#13093934	C₂×C₄	1	4	yes	—
26	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
30	M₃₂×C₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
36	#97×C₄₇	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
39	M₁₆×C₄₇	—	2	2	no	—
40	C₂×C₈×C₄₇	C₄²	1	2	yes	—
41	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
42	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
43	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₄⋊C₄×C₄₇	—	2	1	no	—
46	C₂²⋊C₄×C₄₇	—	2	1	no	—
47	C₄²×C₄₇	C₄²	1	2	yes	—
48	C₂²⋊Dic₄₇	—	2	2	no	—
49	C₄⋊Dic₄₇	—	2	2	no	—
50	C₄×Dic₄₇	C₄²	1	4	yes	—
51	#10625329	—	2	2	no	—
52	#10625329	—	2	2	no	—
53	C₂×C₄₇⋊C₈	C₄²	1	4	yes	—
54	C₁₆×C₄₇	C₄²	1	2	yes	—
55	C₁₆×C₄₇	C₄²	1	1	yes	characteristic
56	C₁₆×C₄₇	C₄²	1	4	yes	—
57	C₁₆×C₄₇	C₄²	1	1	yes	characteristic
58	C₄₇⋊C₁₆	C₄²	1	8	yes	—
59	C₂³×C₄₇	#131	1	1	yes	characteristic
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
62	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
64	C₂×C₄×C₄₇	#131	1	2	yes	—
65	C₂×Dic₄₇	—	2	4	no	—
66	C₈×C₄₇	—	2	2	no	—
67	C₈×C₄₇	C₂×C₄²	1	2	yes	—
68	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
70	C₄₇⋊C₈	—	2	4	no	—
71	#12542	—	47	1	no	maximal, Sylow
72	C₂²×C₄₇	—	4	1	no	—
73	C₂²×C₄₇	#405	1	2	yes	—
74	C₂²×C₄₇	#405	1	1	yes	characteristic
75	C₄×C₄₇	—	4	2	no	—
76	C₄×C₄₇	C₄³	1	1	yes	characteristic
77	C₄×C₄₇	#405	1	2	yes	—
78	C₄×C₄₇	#405	1	1	yes	characteristic
79	Dic₄₇	—	4	4	no	—
80	#2110	—	47	2	no	—
81	#2110	—	47	1	no	—
82	#2108	—	47	2	no	—
83	#2108	D₄₇	1	1	yes	characteristic
84	#1747	—	47	1	no	—
85	C₂×C₄₇	—	2	2	no	—
86	C₂×C₄₇	—	2	1	no	—
87	C₂×C₄₇	#1726	1	1	yes	characteristic
88	C₂×M₃₂	—	47	2	no	—
89	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
90	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
91	#405	—	47	2	no	—
92	#405	C₂×D₄₇	1	1	yes	characteristic
93	#347	—	47	8	no	—
94	#347	—	47	4	no	—
95	#346	—	47	4	no	—
96	#346	Dic₄₇	1	2	yes	complemented
97	C₄×C₁₆	—	47	4	no	—
98	C₄×C₁₆	Dic₄₇	1	2	yes	complemented
99	#338	—	47	1	no	—
100	H(3,ℤ/4)	—	47	1	no	—
101	H(3,ℤ/4)	—	47	2	no	—
102	C₄₇	#12542	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₁₆	—	47	2	no	—
104	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
105	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
106	#117	—	47	2	no	—
107	#117	C₄×D₄₇	1	1	yes	characteristic
108	M₃₂	—	47	4	no	—
109	M₃₂	—	47	4	no	—
110	M₃₂	C₄×D₄₇	1	4	yes	—
111	C₂×C₁₆	—	47	4	no	—
112	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
113	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
114	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
115	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
116	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
117	#97	—	47	4	no	—
118	#97	C₄×D₄₇	1	2	yes	—
119	C₄×C₈	—	47	4	no	—
120	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
121	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
122	M₁₆	—	2	2	no	—
123	M₁₆	—	94	2	no	—
124	M₁₆	—	94	2	no	—
125	C₂×C₈	—	47	4	no	—
126	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
127	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
128	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
129	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
130	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	94	2	no	—
133	C₂²⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	94	2	no	—
135	C₄²	—	47	4	no	—
136	C₄²	C₄×Dic₄₇	1	2	yes	—
137	C₁₆	—	47	8	no	—
138	C₁₆	C₄×Dic₄₇	1	1	yes	characteristic
139	C₁₆	C₄×Dic₄₇	1	2	yes	—
140	C₁₆	C₄×Dic₄₇	1	4	yes	—
141	C₁₆	C₄×Dic₄₇	1	1	yes	characteristic
142	C₂³	#13094030	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	94	4	no	—
145	C₂×C₄	C₄²×D₄₇	1	2	yes	—
146	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
147	C₂×C₄	#13094030	1	1	yes	characteristic
148	C₂×C₄	#13094030	1	2	yes	—
149	C₈	—	2	2	no	—
150	C₈	—	94	4	no	—
151	C₈	C₄²×D₄₇	1	2	yes	—
152	C₈	C₂×C₄×Dic₄₇	1	1	yes	characteristic
153	C₈	C₂×C₄×Dic₄₇	1	1	yes	characteristic
154	C₂²	—	4	1	no	—
155	C₂²	#15689077	1	2	yes	—
156	C₂²	#15689078	1	1	yes	characteristic
157	C₄	—	4	2	no	—
158	C₄	—	188	4	no	—
159	C₄	C₄²×Dic₄₇	1	1	yes	characteristic
160	C₄	#15689077	1	2	yes	—
161	C₄	#15689078	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	2	2	no	—
164	C₂	#20243074	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₄×Dic₄₇
commutator	C₄×C₄₇	C₄³
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#2108	C₂
socle	C₂×C₄₇	#1726
prime radical	C₂³×C₄₇	#131
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#2108	D₄₇
hypocenter	C₄₇	#12542
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
8. #15688901	C₄	semidirect
9. #15688900	C₄	semidirect
10. #15688898	C₄	semidirect
11. C₁₆×Dic₄₇	C₄	semidirect
18. C₄₇×#346	C₄	semidirect
19. C₄×C₁₆×C₄₇	C₄	semidirect
96. #346	Dic₄₇	semidirect
98. C₄×C₁₆	Dic₄₇	semidirect
102. C₄₇	#12542	semidirect

Maximal subgroups

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

subgroup	index
#20244993	2
#20244992	2
#20244984	2
#20243143	2
C₄₇×#2108	2
#12542	47