Subgroups — C₄×#20294081 #42121764

Order 27712.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20294081	C₂	1	2	yes	maximal
2	#28357131	C₂	1	2	yes	maximal, complemented
3	C₄×#16812358	C₂	1	2	yes	maximal, complemented
4	C₂×C₄²×C₄₃₃	C₂	1	1	yes	maximal, characteristic
5	#20294081	C₄	1	8	yes	direct product factor
6	C₂×#16812358	C₂²	1	4	yes	—
7	C₂²×C₄×C₄₃₃	C₄	1	2	yes	—
8	C₂²×C₄×C₄₃₃	C₂²	1	1	yes	characteristic
9	C₄²×C₄₃₃	C₄	1	2	yes	—
10	C₄²×C₄₃₃	C₂²	1	1	yes	characteristic
11	C₄²×C₄₃₃	C₂²	1	1	yes	characteristic
12	#16812358	C₂×C₄	1	8	yes	complemented
13	C₂³×C₄₃₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₄₃₃	—	2	4	no	—
15	C₂×C₄×C₄₃₃	C₂×C₄	1	4	yes	—
16	C₂×C₄×C₄₃₃	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₄₃₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₄₃₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₄₃₃	C₂³	1	1	yes	characteristic
20	C₂²×C₄₃₃	—	2	2	no	—
21	C₂²×C₄₃₃	C₄²	1	2	yes	—
22	C₂²×C₄₃₃	C₂²×C₄	1	1	yes	characteristic
23	C₄×C₄₃₃	—	2	2	no	—
24	C₄×C₄₃₃	C₄²	1	2	yes	—
25	C₄×C₄₃₃	M₁₆	1	4	yes	complemented
26	C₄×C₄₃₃	C₂²×C₄	1	2	yes	—
27	C₂×C₄₃₃	—	2	2	no	—
28	C₂×C₄₃₃	C₂×C₄²	1	1	yes	characteristic
29	C₂×C₄₃₃	C₂×M₁₆	1	2	yes	—
30	C₄₃₃	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
31	C₄×M₁₆	—	433	1	no	maximal, Sylow
32	C₂×M₁₆	—	433	2	no	—
33	C₂×C₄²	D₄₃₃	1	1	yes	characteristic
34	#97	—	433	2	no	—
35	C₄×C₈	—	433	2	no	—
36	C₂²×C₄	Dic₄₃₃	1	2	yes	—
37	C₂²×C₄	C₂×D₄₃₃	1	1	yes	characteristic
38	M₁₆	—	433	8	no	—
39	C₂×C₈	—	433	4	no	—
40	C₄²	Dic₄₃₃	1	2	yes	—
41	C₄²	C₂×D₄₃₃	1	1	yes	characteristic
42	C₄²	C₂×D₄₃₃	1	1	yes	characteristic
43	C₂³	C₂×Dic₄₃₃	1	1	yes	characteristic
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	C₄×D₄₃₃	1	4	yes	—
46	C₂×C₄	C₂×Dic₄₃₃	1	2	yes	—
47	C₂×C₄	C₂×Dic₄₃₃	1	1	yes	characteristic
48	C₂×C₄	C₂×Dic₄₃₃	1	2	yes	—
49	C₂×C₄	C₂²×D₄₃₃	1	1	yes	characteristic
50	C₈	—	433	8	no	—
51	C₂²	—	2	2	no	—
52	C₂²	C₄×Dic₄₃₃	1	2	yes	—
53	C₂²	C₂²×Dic₄₃₃	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	C₄×Dic₄₃₃	1	2	yes	—
56	C₄	#20294081	1	4	yes	direct product factor
57	C₄	C₂×C₄×D₄₃₃	1	2	yes	—
58	C₂	—	2	2	no	—
59	C₂	C₂×C₄×Dic₄₃₃	1	1	yes	characteristic
60	C₂	C₂×#20294081	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₄₃₃
commutator	C₂×C₄₃₃	C₂×C₄²
Frattini	C₂×C₄	C₂²×D₄₃₃
Fitting	C₂×C₄²×C₄₃₃	C₂
socle	C₂²×C₄₃₃	C₂²×C₄
prime radical	C₂³×C₄₃₃	C₂×C₄
cosocle	C₂×C₄×C₄₃₃	C₂³
hypercenter	C₂×C₄²	D₄₃₃
hypocenter	C₄₃₃	C₄×M₁₆
LCS	G ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #28357131	C₂	semidirect
3. C₄×#16812358	C₂	semidirect
5. #20294081	C₄	direct (N × H)
12. #16812358	C₂×C₄	semidirect
25. C₄×C₄₃₃	M₁₆	semidirect
30. C₄₃₃	C₄×M₁₆	semidirect
56. C₄	#20294081	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#20294081	2
#28357131	2
C₄×#16812358	2
C₂×C₄²×C₄₃₃	2
C₄×M₁₆	433

Subgroups — C4×#20294081 #42121764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#20294081 #42121764