Subgroups — #2972×C₂₄₁ #44504639

Order 30848.

500 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₂₄₁	C₂	1	1	yes	maximal, characteristic
2	C₈×D₄×C₂₄₁	C₂	1	8	yes	maximal, complemented
3	#431×C₂₄₁	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊C₈×C₂₄₁	C₂	1	4	yes	maximal, complemented
5	C₂²×D₄×C₂₄₁	C₄	1	1	yes	characteristic
6	C₂³×C₄×C₂₄₁	C₂²	1	2	yes	—
7	C₂²×C₈×C₂₄₁	C₂²	1	8	yes	complemented
8	C₄×D₄×C₂₄₁	C₄	1	4	yes	—
9	C₄×D₄×C₂₄₁	C₂²	1	4	yes	—
10	C₂×C₄⋊C₄×C₂₄₁	C₄	1	1	yes	characteristic
11	C₂×C₂²⋊C₄×C₂₄₁	C₄	1	2	yes	—
12	C₂×C₄²×C₂₄₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₂₄₁	C₂²	1	4	yes	complemented
14	C₂²⋊C₈×C₂₄₁	C₂²	1	4	yes	—
15	C₂²⋊C₈×C₂₄₁	C₂²	1	8	yes	complemented
16	C₄×C₈×C₂₄₁	C₂²	1	4	yes	complemented
17	C₂⁴×C₂₄₁	C₂×C₄	1	2	yes	—
18	C₂×D₄×C₂₄₁	C₈	1	8	yes	complemented
19	C₂×D₄×C₂₄₁	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₂₄₁	—	4	4	no	—
21	C₂²×C₄×C₂₄₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₄₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₄₁	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₄₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₂₄₁	C₂³	1	4	yes	—
27	C₂×C₈×C₂₄₁	—	4	8	no	—
28	C₂×C₈×C₂₄₁	C₂³	1	8	yes	—
29	C₄⋊C₄×C₂₄₁	C₂×C₄	1	4	yes	—
30	C₂²⋊C₄×C₂₄₁	C₂×C₄	1	8	yes	—
31	C₄²×C₂₄₁	C₂×C₄	1	2	yes	—
32	C₄²×C₂₄₁	C₂³	1	2	yes	—
33	C₂³×C₂₄₁	—	4	2	no	—
34	C₂³×C₂₄₁	—	4	2	no	—
35	C₂³×C₂₄₁	C₂×C₈	1	8	yes	complemented
36	C₂³×C₂₄₁	C₂²×C₄	1	4	yes	—
37	C₂³×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
38	D₄×C₂₄₁	C₂×C₈	1	16	yes	complemented
39	C₂×C₄×C₂₄₁	—	2	8	no	—
40	C₂×C₄×C₂₄₁	—	2	2	no	—
41	C₂×C₄×C₂₄₁	—	4	4	no	—
42	C₂×C₄×C₂₄₁	—	4	2	no	—
43	C₂×C₄×C₂₄₁	—	4	1	no	—
44	C₂×C₄×C₂₄₁	C₂×C₈	1	4	yes	complemented
45	C₂×C₄×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₄₁	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₂₄₁	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₂₄₁	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₂₄₁	C₂⁴	1	1	yes	characteristic
50	C₈×C₂₄₁	—	2	8	no	—
51	C₂²×C₂₄₁	—	2	1	no	—
52	C₂²×C₂₄₁	—	2	4	no	—
53	C₂²×C₂₄₁	—	2	1	no	—
54	C₂²×C₂₄₁	—	2	4	no	—
55	C₂²×C₂₄₁	—	4	8	no	—
56	C₂²×C₂₄₁	C₂²×C₈	1	2	yes	—
57	C₂²×C₂₄₁	C₂²×C₈	1	8	yes	—
58	C₂²×C₂₄₁	C₂³×C₄	1	1	yes	characteristic
59	C₄×C₂₄₁	—	2	1	no	—
60	C₄×C₂₄₁	—	2	4	no	—
61	C₄×C₂₄₁	—	2	2	no	—
62	C₄×C₂₄₁	C₂²×C₈	1	4	yes	—
63	C₄×C₂₄₁	H(5,𝔽₂)	1	2	yes	—
64	C₂×C₂₄₁	—	2	2	no	—
65	C₂×C₂₄₁	—	2	8	no	—
66	C₂×C₂₄₁	#518	1	1	yes	characteristic
67	C₂×C₂₄₁	C₂³×C₈	1	1	yes	characteristic
68	C₂×C₂₄₁	#568	1	1	yes	characteristic
69	C₂₄₁	#2972	1	1	yes	characteristic, direct product factor, Sylow
70	#2972	C₂₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂×C₄×D₄	C₂×C₂₄₁	1	1	yes	characteristic
72	C₈×D₄	C₂×C₂₄₁	1	8	yes	complemented
73	#431	C₂×C₂₄₁	1	2	yes	complemented
74	C₂×C₂²⋊C₈	C₂×C₂₄₁	1	4	yes	complemented
75	C₂²×D₄	C₄×C₂₄₁	1	1	yes	characteristic
76	C₂³×C₄	C₂²×C₂₄₁	1	2	yes	—
77	C₂²×C₈	C₂²×C₂₄₁	1	8	yes	complemented
78	C₄×D₄	C₄×C₂₄₁	1	4	yes	—
79	C₄×D₄	C₂²×C₂₄₁	1	4	yes	—
80	C₂×C₄⋊C₄	C₄×C₂₄₁	1	1	yes	characteristic
81	C₂×C₂²⋊C₄	C₄×C₂₄₁	1	2	yes	—
82	C₂×C₄²	C₂²×C₂₄₁	1	1	yes	characteristic
83	C₄⋊C₈	C₂²×C₂₄₁	1	4	yes	complemented
84	C₂²⋊C₈	C₂²×C₂₄₁	1	4	yes	—
85	C₂²⋊C₈	C₂²×C₂₄₁	1	8	yes	complemented
86	C₄×C₈	C₂²×C₂₄₁	1	4	yes	complemented
87	C₂⁴	C₂×C₄×C₂₄₁	1	2	yes	—
88	C₂×D₄	C₈×C₂₄₁	1	8	yes	complemented
89	C₂×D₄	C₂×C₄×C₂₄₁	1	4	yes	—
90	C₂²×C₄	—	4	4	no	—
91	C₂²×C₄	C₂×C₄×C₂₄₁	1	1	yes	characteristic
92	C₂²×C₄	C₂×C₄×C₂₄₁	1	2	yes	—
93	C₂²×C₄	C₂×C₄×C₂₄₁	1	4	yes	—
94	C₂²×C₄	C₂×C₄×C₂₄₁	1	1	yes	characteristic
95	C₂²×C₄	C₂³×C₂₄₁	1	1	yes	characteristic
96	C₂²×C₄	C₂³×C₂₄₁	1	4	yes	—
97	C₂×C₈	—	4	8	no	—
98	C₂×C₈	C₂³×C₂₄₁	1	8	yes	—
99	C₄⋊C₄	C₂×C₄×C₂₄₁	1	4	yes	—
100	C₂²⋊C₄	C₂×C₄×C₂₄₁	1	8	yes	—
101	C₄²	C₂×C₄×C₂₄₁	1	2	yes	—
102	C₄²	C₂³×C₂₄₁	1	2	yes	—
103	C₂³	—	4	2	no	—
104	C₂³	—	4	2	no	—
105	C₂³	C₂×C₈×C₂₄₁	1	8	yes	complemented
106	C₂³	C₂²×C₄×C₂₄₁	1	4	yes	—
107	C₂³	C₂²×C₄×C₂₄₁	1	1	yes	characteristic
108	D₄	C₂×C₈×C₂₄₁	1	16	yes	complemented
109	C₂×C₄	—	2	8	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	C₂×C₈×C₂₄₁	1	4	yes	complemented
115	C₂×C₄	C₂²×C₄×C₂₄₁	1	1	yes	characteristic
116	C₂×C₄	C₂²×C₄×C₂₄₁	1	2	yes	—
117	C₂×C₄	C₂²×C₄×C₂₄₁	1	4	yes	—
118	C₂×C₄	C₂²×C₄×C₂₄₁	1	2	yes	—
119	C₂×C₄	C₂⁴×C₂₄₁	1	1	yes	characteristic
120	C₈	—	2	8	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	4	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	4	8	no	—
126	C₂²	C₂²×C₈×C₂₄₁	1	2	yes	—
127	C₂²	C₂²×C₈×C₂₄₁	1	8	yes	—
128	C₂²	C₂³×C₄×C₂₄₁	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	4	no	—
131	C₄	—	2	2	no	—
132	C₄	C₂²×C₈×C₂₄₁	1	4	yes	—
133	C₄	H(5,𝔽₂)×C₂₄₁	1	2	yes	—
134	C₂	—	2	2	no	—
135	C₂	—	2	8	no	—
136	C₂	#518×C₂₄₁	1	1	yes	characteristic
137	C₂	C₂³×C₈×C₂₄₁	1	1	yes	characteristic
138	C₂	#568×C₂₄₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₄₁	C₂⁴
commutator	C₂	C₂³×C₈×C₂₄₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₄₁
socle	C₂²×C₂₄₁	C₂³×C₄
prime radical	C₂²×D₄×C₂₄₁	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×D₄×C₂₄₁	C₂	semidirect
3. #431×C₂₄₁	C₂	semidirect
4. C₂×C₂²⋊C₈×C₂₄₁	C₂	semidirect
7. C₂²×C₈×C₂₄₁	C₂²	semidirect
13. C₄⋊C₈×C₂₄₁	C₂²	semidirect
15. C₂²⋊C₈×C₂₄₁	C₂²	semidirect
16. C₄×C₈×C₂₄₁	C₂²	semidirect
18. C₂×D₄×C₂₄₁	C₈	semidirect
35. C₂³×C₂₄₁	C₂×C₈	semidirect
38. D₄×C₂₄₁	C₂×C₈	semidirect
44. C₂×C₄×C₂₄₁	C₂×C₈	semidirect
69. C₂₄₁	#2972	direct (N × H)
70. #2972	C₂₄₁	direct (N × H)
72. C₈×D₄	C₂×C₂₄₁	semidirect
73. #431	C₂×C₂₄₁	semidirect
74. C₂×C₂²⋊C₈	C₂×C₂₄₁	semidirect
77. C₂²×C₈	C₂²×C₂₄₁	semidirect
83. C₄⋊C₈	C₂²×C₂₄₁	semidirect
85. C₂²⋊C₈	C₂²×C₂₄₁	semidirect
86. C₄×C₈	C₂²×C₂₄₁	semidirect
88. C₂×D₄	C₈×C₂₄₁	semidirect
105. C₂³	C₂×C₈×C₂₄₁	semidirect
108. D₄	C₂×C₈×C₂₄₁	semidirect
114. C₂×C₄	C₂×C₈×C₂₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₂₄₁	2
C₈×D₄×C₂₄₁	2
#431×C₂₄₁	2
C₂×C₂²⋊C₈×C₂₄₁	2
#2972	241

Subgroups — #2972×C241 #44504639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2972×C₂₄₁ #44504639