Finite Group Databasebeta

Subgroups — #44509342

Order 30848.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29508155	C₂	1	1	yes	maximal, characteristic
2	#29507838	C₂	1	1	yes	maximal, characteristic, complemented
3	#29507828	C₂	1	1	yes	maximal, characteristic, complemented
4	#29507529	C₂	1	1	yes	maximal, characteristic, complemented
5	#29507526	C₂	1	1	yes	maximal, characteristic, complemented
6	#29507494	C₂	1	1	yes	maximal, characteristic, complemented
7	#365×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#21440664	C₂²	1	1	yes	characteristic
9	C₂×D₁₉₂₈	C₂²	1	1	yes	characteristic
10	#21440650	—	2	1	no	—
11	#21440604	—	2	1	no	—
12	#21440587	C₄	1	2	yes	complemented
13	D₃₈₅₆	C₄	1	2	yes	complemented
14	#21440582	C₂²	1	1	yes	characteristic
15	#21440568	—	2	1	no	—
16	#21440562	—	2	1	no	—
17	#21440552	C₂²	1	1	yes	characteristic
18	M₃₂×C₂₄₁	C₂²	1	1	yes	characteristic
19	#108×C₂₄₁	C₂²	1	1	yes	characteristic
20	#106×C₂₄₁	C₂²	1	1	yes	characteristic
21	#16838770	—	2	1	no	—
22	C₂×D₉₆₄	—	2	1	no	—
23	C₂×C₄×D₂₄₁	—	4	1	no	—
24	Dic₉₆₄	C₂×C₄	1	1	yes	characteristic, complemented
25	#16838759	—	2	1	no	—
26	D₁₉₂₈	—	2	1	no	—
27	D₁₉₂₈	C₂×C₄	1	2	yes	—
28	D₁₉₂₈	C₂×C₄	1	1	yes	characteristic, complemented
29	#16838753	—	4	1	no	—
30	C₂×#14515349	D₄	1	1	yes	characteristic
31	C₈×D₂₄₁	—	4	1	no	—
32	#16838747	—	4	1	no	—
33	C₄×Dic₂₄₁	D₄	1	1	yes	characteristic
34	#16838745	—	2	1	no	—
35	M₁₆×C₂₄₁	—	2	1	no	—
36	C₂×C₈×C₂₄₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₂₄₁	—	2	1	no	—
38	C₁₆×C₂₄₁	C₂×C₄	1	2	yes	complemented
39	C₂²×D₂₄₁	—	4	1	no	—
40	Dic₄₈₂	—	2	1	no	—
41	D₉₆₄	—	2	2	no	—
42	D₉₆₄	—	2	1	no	—
43	D₉₆₄	—	4	1	no	—
44	C₂²⋊D₂₄₁	—	4	1	no	—
45	C₂×Dic₂₄₁	—	2	1	no	—
46	C₄×D₂₄₁	—	4	1	no	—
47	C₄×D₂₄₁	—	8	2	no	—
48	#14515349	—	2	1	no	—
49	C₂×C₄×C₂₄₁	—	4	1	no	—
50	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
51	C₈×C₂₄₁	—	4	1	no	—
52	C₈×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
53	C₈×C₂₄₁	#41	1	1	yes	characteristic
54	C₂²×C₂₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×D₂₄₁	—	4	1	no	—
56	C₂×D₂₄₁	—	4	2	no	—
57	C₂×D₂₄₁	—	8	1	no	—
58	C₂×D₂₄₁	—	8	1	no	—
59	C₄×C₂₄₁	—	8	1	no	—
60	C₄×C₂₄₁	C₄×D₄	1	1	yes	characteristic
61	C₄×C₂₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₂₄₁	—	4	1	no	—
63	Dic₂₄₁	—	4	1	no	—
64	C₂×C₂₄₁	—	2	1	no	—
65	C₂×C₂₄₁	#442	1	1	yes	characteristic
66	D₂₄₁	—	8	1	no	—
67	D₂₄₁	—	8	2	no	—
68	C₂₄₁	AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
69	AGL(1,ℤ/16)	—	241	1	no	maximal, Sylow
70	#509	—	241	1	no	—
71	#444	—	241	1	no	—
72	#442	—	241	1	no	—
73	#365	D₂₄₁	1	1	yes	characteristic, complemented
74	#361	—	241	1	no	—
75	#360	—	241	1	no	—
76	#347	—	241	1	no	—
77	#135	—	241	1	no	—
78	C₂×D₈	—	241	1	no	—
79	#131	—	482	1	no	—
80	C₄×D₄	—	482	1	no	—
81	SD₃₂	—	241	2	no	—
82	D₁₆	—	241	2	no	—
83	M₃₂	—	241	1	no	—
84	M₃₂	C₂×D₂₄₁	1	1	yes	characteristic
85	#108	C₂×D₂₄₁	1	1	yes	characteristic
86	#106	C₂×D₂₄₁	1	1	yes	characteristic
87	#104	—	482	1	no	—
88	#102	—	482	1	no	—
89	#97	—	241	1	no	—
90	#41	—	482	1	no	—
91	C₂×D₄	—	482	1	no	—
92	C₂²×C₄	—	964	1	no	—
93	Q₁₆	—	241	1	no	—
94	SD₁₆	—	482	1	no	—
95	D₈	—	482	1	no	—
96	D₈	—	241	2	no	—
97	D₈	—	241	1	no	—
98	M₁₆	—	2	1	no	—
99	M₁₆	—	964	1	no	—
100	C₂×C₈	—	964	1	no	—
101	C₂×C₈	—	241	1	no	—
102	C₂×C₈	C₂²×D₂₄₁	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₂²⋊C₄	—	964	1	no	—
105	C₄²	—	241	1	no	—
106	C₁₆	—	482	1	no	—
107	C₁₆	C₄×D₂₄₁	1	2	yes	complemented
108	C₂³	—	964	1	no	—
109	Q₈	—	482	1	no	—
110	D₄	—	964	1	no	—
111	D₄	—	964	1	no	—
112	D₄	—	482	2	no	—
113	D₄	—	482	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	1928	2	no	—
116	C₂×C₄	—	964	1	no	—
117	C₂×C₄	—	482	1	no	—
118	C₂×C₄	D₄×D₂₄₁	1	1	yes	characteristic
119	C₈	—	4	1	no	—
120	C₈	—	482	1	no	—
121	C₈	C₂×C₄×D₂₄₁	1	1	yes	characteristic
122	C₈	#16838771	1	1	yes	characteristic
123	C₂²	—	1928	1	no	—
124	C₂²	—	1928	1	no	—
125	C₂²	—	964	1	no	—
126	C₂²	—	964	2	no	—
127	C₂²	#21440669	1	1	yes	characteristic
128	C₄	—	8	1	no	—
129	C₄	—	964	1	no	—
130	C₄	—	964	1	no	—
131	C₄	#21440604	1	1	yes	characteristic
132	C₄	#21440669	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	1928	2	no	—
135	C₂	—	1928	1	no	—
136	C₂	#29507828	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#29507828
commutator	C₈×C₂₄₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₄₁
Fitting	#365×C₂₄₁	C₂
socle	C₂×C₂₄₁	#442
prime radical	#29508155	C₂
cosocle	C₂×C₈×C₂₄₁	C₂³
hypercenter	#365	D₂₄₁
hypocenter	C₂₄₁	AGL(1,ℤ/16)
LCS	G ▷ C₈×C₂₄₁ ▷ C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

Complements

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

N	H ≅ G/N	type
2. #29507838	C₂	semidirect
3. #29507828	C₂	semidirect
4. #29507529	C₂	semidirect
5. #29507526	C₂	semidirect
6. #29507494	C₂	semidirect
7. #365×C₂₄₁	C₂	semidirect
12. #21440587	C₄	semidirect
13. D₃₈₅₆	C₄	semidirect
24. Dic₉₆₄	C₂×C₄	semidirect
28. D₁₉₂₈	C₂×C₄	semidirect
38. C₁₆×C₂₄₁	C₂×C₄	semidirect
68. C₂₄₁	AGL(1,ℤ/16)	semidirect
73. #365	D₂₄₁	semidirect
107. C₁₆	C₄×D₂₄₁	semidirect

Maximal subgroups

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

subgroup	index
#29508155	2
#29507838	2
#29507828	2
#29507529	2
#29507526	2
#29507494	2
#365×C₂₄₁	2
AGL(1,ℤ/16)	241