Subgroups — #26136770

Order 12032.

198 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688957	C₂	1	1	yes	maximal, characteristic
2	#20244984	C₂	1	1	yes	maximal, characteristic
3	#20241585	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×M₆₄	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093964	C₄	1	1	yes	characteristic
6	#15689078	C₄	1	1	yes	characteristic
7	#15688957	—	2	1	no	—
8	#15688957	C₂²	1	2	yes	—
9	C₂×C₄₇⋊C₃₂	—	2	1	no	—
10	#15688898	C₄	1	2	yes	—
11	C₁₆×Dic₄₇	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×M₆₄	—	2	1	no	—
14	C₄₇×M₆₄	C₂²	1	2	yes	—
15	C₂×C₃₂×C₄₇	—	2	1	no	—
16	C₂×#10625329	C₈	1	1	yes	characteristic
17	#13093983	C₈	1	1	yes	characteristic
18	#13093964	—	2	1	no	—
19	#13093964	—	2	1	no	—
20	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
21	#13093934	C₈	1	2	yes	—
22	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
23	C₄₇⋊C₃₂	—	2	2	no	—
24	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₄₇	Q₈	1	1	yes	characteristic
29	C₂×C₁₆×C₄₇	Q₈	1	1	yes	characteristic
30	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
31	C₃₂×C₄₇	—	2	2	no	—
32	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²⋊Dic₄₇	—	2	1	no	—
40	C₄⋊Dic₄₇	—	2	1	no	—
41	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
42	#10625329	C₁₆	1	2	yes	—
43	#10625329	C₁₆	1	2	yes	—
44	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
45	C₁₆×C₄₇	C₄⋊C₄	1	1	yes	characteristic
46	C₁₆×C₄₇	C₄⋊C₄	1	1	yes	characteristic
47	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₄₇⋊C₁₆	—	2	2	no	—
50	C₂³×C₄₇	M₃₂	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₄₇	M₃₂	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
57	C₂×Dic₄₇	—	2	2	no	—
58	C₈×C₄₇	C₄⋊C₈	1	1	yes	characteristic
59	C₈×C₄₇	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₈×C₄₇	C₂×M₁₆	1	1	yes	characteristic
62	C₄₇⋊C₈	C₂×C₁₆	1	4	yes	—
63	#13630	—	47	1	no	maximal, Sylow
64	C₂²×C₄₇	—	4	1	no	—
65	C₂²×C₄₇	C₄⋊C₁₆	1	1	yes	characteristic
66	C₂²×C₄₇	C₄⋊C₁₆	1	1	yes	characteristic
67	C₂²×C₄₇	C₂×M₃₂	1	1	yes	characteristic
68	C₄×C₄₇	C₄⋊C₁₆	1	1	yes	characteristic
69	C₄×C₄₇	C₄⋊C₁₆	1	1	yes	characteristic
70	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₄×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
72	Dic₄₇	—	4	2	no	—
73	C₂×M₆₄	—	47	1	no	—
74	C₂×M₆₄	D₄₇	1	1	yes	characteristic
75	#2108	—	47	1	no	—
76	#1422	—	47	4	no	—
77	C₂×C₄₇	—	2	1	no	—
78	C₂×C₄₇	—	2	1	no	—
79	C₂×C₄₇	—	2	1	no	—
80	C₂×C₄₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
81	C₂×M₃₂	—	47	1	no	—
82	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
83	#405	—	47	1	no	—
84	M₆₄	—	2	1	no	—
85	M₆₄	—	47	2	no	—
86	M₆₄	—	94	1	no	—
87	M₆₄	C₂×D₄₇	1	2	yes	—
88	C₂×C₃₂	—	2	1	no	—
89	C₂×C₃₂	—	94	1	no	—
90	#346	—	47	2	no	—
91	C₄×C₁₆	—	47	2	no	—
92	C₄₇	#13630	1	1	yes	characteristic, complemented, Sylow
93	C₂×M₁₆	—	47	1	no	—
94	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
95	#117	—	47	1	no	—
96	M₃₂	—	94	1	no	—
97	M₃₂	—	94	1	no	—
98	C₂×C₁₆	—	47	2	no	—
99	C₂×C₁₆	Dic₉₄	1	1	yes	characteristic
100	C₂×C₁₆	Dic₉₄	1	1	yes	characteristic
101	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
103	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
104	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
105	#97	—	47	2	no	—
106	C₄×C₈	—	47	2	no	—
107	C₃₂	—	2	2	no	—
108	C₃₂	—	94	2	no	—
109	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
110	M₁₆	—	47	2	no	—
111	M₁₆	—	47	2	no	—
112	C₂×C₈	—	47	2	no	—
113	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
114	C₂×C₈	#10625324	1	1	yes	characteristic
115	C₂×C₈	#10625324	1	1	yes	characteristic
116	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
117	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
118	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
119	C₄⋊C₄	—	94	1	no	—
120	C₂²⋊C₄	—	94	1	no	—
121	C₄²	—	47	2	no	—
122	C₁₆	—	94	2	no	—
123	C₁₆	Dic₄₇⋊C₄	1	1	yes	characteristic
124	C₁₆	Dic₄₇⋊C₄	1	1	yes	characteristic
125	C₁₆	C₂×Dic₉₄	1	1	yes	characteristic
126	C₁₆	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
127	C₂³	#13093963	1	1	yes	characteristic
128	C₂×C₄	—	94	2	no	—
129	C₂×C₄	Dic₄₇⋊C₈	1	1	yes	characteristic
130	C₂×C₄	Dic₄₇⋊C₈	1	1	yes	characteristic
131	C₂×C₄	C₁₆×D₄₇	1	1	yes	characteristic
132	C₂×C₄	C₁₆×D₄₇	1	1	yes	characteristic
133	C₂×C₄	#13093963	1	1	yes	characteristic
134	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
135	C₈	—	47	4	no	—
136	C₈	Dic₄₇⋊C₈	1	1	yes	characteristic
137	C₈	Dic₄₇⋊C₈	1	1	yes	characteristic
138	C₈	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
139	C₈	C₂×#10625324	1	1	yes	characteristic
140	C₂²	—	4	1	no	—
141	C₂²	Dic₄₇⋊C₁₆	1	1	yes	characteristic
142	C₂²	Dic₄₇⋊C₁₆	1	1	yes	characteristic
143	C₂²	C₂×#13093963	1	1	yes	characteristic
144	C₄	—	188	2	no	—
145	C₄	Dic₄₇⋊C₁₆	1	1	yes	characteristic
146	C₄	Dic₄₇⋊C₁₆	1	1	yes	characteristic
147	C₄	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
148	C₄	C₂×C₁₆×D₄₇	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	—	2	1	no	—
152	C₂	C₂×Dic₄₇⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₂×C₄₇×M₆₄	C₂
socle	C₂×C₄₇	C₂×C₄⋊C₁₆
prime radical	C₂³×C₄₇	M₃₂
cosocle	C₂×C₁₆×C₄₇	C₂³
hypercenter	C₂×M₆₄	D₄₇
hypocenter	C₄₇	#13630
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
3. #20241585	C₂	semidirect
92. C₄₇	#13630	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688957	2
#20244984	2
#20241585	2
C₂×C₄₇×M₆₄	2
#13630	47