Finite Group Databasebeta

Subgroups — #49123156

Order 36224.

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	#31826900	C₂	1	1	yes	maximal, characteristic
2	#31826587	C₂	1	1	yes	maximal, characteristic, complemented
3	#31826576	C₂	1	1	yes	maximal, characteristic, complemented
4	#31826287	C₂	1	1	yes	maximal, characteristic, complemented
5	#31826279	C₂	1	1	yes	maximal, characteristic, complemented
6	#31826242	C₂	1	1	yes	maximal, characteristic, complemented
7	#360×C₂₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₂₂₆₄	C₂²	1	1	yes	characteristic
9	#22583617	—	2	1	no	—
10	#22583572	—	2	1	no	—
11	#22583553	C₄	1	2	yes	complemented
12	#22583551	C₄	1	2	yes	complemented
13	#22583550	C₂²	1	1	yes	characteristic
14	#22583546	C₂²	1	1	yes	characteristic
15	#22583537	—	2	1	no	—
16	#22583530	—	2	1	no	—
17	#22583520	C₂²	1	1	yes	characteristic
18	#135×C₂₈₃	C₂²	1	1	yes	characteristic
19	M₃₂×C₂₈₃	C₂²	1	1	yes	characteristic
20	#106×C₂₈₃	C₂²	1	1	yes	characteristic
21	C₂×D₁₁₃₂	—	2	1	no	—
22	C₂×C₄×D₂₈₃	—	4	1	no	—
23	D₂₂₆₄	—	2	1	no	—
24	D₂₂₆₄	C₂×C₄	1	2	yes	—
25	#17985376	—	2	1	no	—
26	#17985376	—	4	1	no	—
27	C₂×C₂₈₃⋊C₈	—	4	1	no	—
28	C₂×C₂₈₃⋊C₈	D₄	1	1	yes	characteristic
29	(C₂×D₂₈₃)⋊C₄	—	4	1	no	—
30	C₄×Dic₂₈₃	D₄	1	1	yes	characteristic
31	C₂₈₃⋊C₁₆	C₂×C₄	1	2	yes	complemented
32	#41×C₂₈₃	—	2	1	no	—
33	Q₁₆×C₂₈₃	C₂×C₄	1	1	yes	characteristic, complemented
34	SD₁₆×C₂₈₃	—	2	1	no	—
35	D₈×C₂₈₃	C₂×C₄	1	1	yes	characteristic, complemented
36	C₂×C₈×C₂₈₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₂₈₃	—	2	1	no	—
38	C₁₆×C₂₈₃	—	2	1	no	—
39	C₂²×D₂₈₃	—	4	1	no	—
40	D₁₁₃₂	—	2	2	no	—
41	D₁₁₃₂	—	4	1	no	—
42	C₂×Dic₂₈₃	—	2	1	no	—
43	C₄×D₂₈₃	—	8	2	no	—
44	C₂₈₃⋊C₈	—	2	1	no	—
45	C₂₈₃⋊C₈	—	4	1	no	—
46	Q₈×C₂₈₃	—	2	1	no	—
47	D₄×C₂₈₃	—	2	1	no	—
48	D₄×C₂₈₃	—	4	1	no	—
49	C₂×C₄×C₂₈₃	—	4	1	no	—
50	C₂×C₄×C₂₈₃	—	4	1	no	—
51	C₂×C₄×C₂₈₃	C₂×D₄	1	1	yes	characteristic
52	C₈×C₂₈₃	C₂²×C₄	1	1	yes	characteristic
53	C₈×C₂₈₃	#41	1	1	yes	characteristic
54	C₂²×C₂₈₃	—	4	1	no	—
55	C₂²×C₂₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
56	C₂×D₂₈₃	—	4	2	no	—
57	C₂×D₂₈₃	—	8	1	no	—
58	C₂×D₂₈₃	—	8	1	no	—
59	C₄×C₂₈₃	—	4	1	no	—
60	C₄×C₂₈₃	—	8	1	no	—
61	C₄×C₂₈₃	C₄×D₄	1	1	yes	characteristic
62	C₄×C₂₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
63	Dic₂₈₃	—	4	1	no	—
64	C₂×C₂₈₃	—	2	1	no	—
65	C₂×C₂₈₃	—	8	1	no	—
66	C₂×C₂₈₃	#442	1	1	yes	characteristic
67	D₂₈₃	—	8	2	no	—
68	C₂₈₃	AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
69	AGL(1,ℤ/16)	—	283	1	no	maximal, Sylow
70	#509	—	283	1	no	—
71	#444	—	283	1	no	—
72	#442	—	283	1	no	—
73	#365	—	283	1	no	—
74	#361	—	283	1	no	—
75	#360	D₂₈₃	1	1	yes	characteristic, complemented
76	#347	—	283	1	no	—
77	#135	C₂×D₂₈₃	1	1	yes	characteristic
78	C₂×D₈	—	283	1	no	—
79	#131	—	566	1	no	—
80	C₄×D₄	—	566	1	no	—
81	SD₃₂	—	283	2	no	—
82	D₁₆	—	283	2	no	—
83	M₃₂	—	283	1	no	—
84	M₃₂	C₂×D₂₈₃	1	1	yes	characteristic
85	#108	—	283	1	no	—
86	#106	C₂×D₂₈₃	1	1	yes	characteristic
87	#104	—	566	1	no	—
88	#102	—	566	1	no	—
89	#97	—	283	1	no	—
90	#41	—	2	1	no	—
91	C₂×D₄	—	566	1	no	—
92	C₂²×C₄	—	1132	1	no	—
93	Q₁₆	C₄×D₂₈₃	1	1	yes	characteristic, complemented
94	SD₁₆	—	2	1	no	—
95	D₈	—	283	2	no	—
96	D₈	—	566	1	no	—
97	D₈	C₄×D₂₈₃	1	1	yes	characteristic, complemented
98	M₁₆	—	566	1	no	—
99	M₁₆	—	1132	1	no	—
100	C₂×C₈	—	283	1	no	—
101	C₂×C₈	—	1132	1	no	—
102	C₂×C₈	C₂²×D₂₈₃	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₂²⋊C₄	—	1132	1	no	—
105	C₄²	—	283	1	no	—
106	C₁₆	—	2	1	no	—
107	C₁₆	—	283	2	no	—
108	C₂³	—	1132	1	no	—
109	Q₈	—	2	1	no	—
110	D₄	—	2	1	no	—
111	D₄	—	4	1	no	—
112	D₄	—	566	2	no	—
113	D₄	—	1132	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	566	1	no	—
117	C₂×C₄	—	2264	2	no	—
118	C₂×C₄	D₄×D₂₈₃	1	1	yes	characteristic
119	C₈	—	566	1	no	—
120	C₈	—	1132	1	no	—
121	C₈	C₂×C₄×D₂₈₃	1	1	yes	characteristic
122	C₈	#17985391	1	1	yes	characteristic
123	C₂²	—	4	1	no	—
124	C₂²	—	1132	2	no	—
125	C₂²	—	2264	1	no	—
126	C₂²	—	2264	1	no	—
127	C₂²	#22583634	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	8	1	no	—
130	C₄	—	1132	1	no	—
131	C₄	(C₂²⋊D₂₈₃)⋊C₄	1	1	yes	characteristic
132	C₄	#22583637	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	8	1	no	—
135	C₂	—	2264	2	no	—
136	C₂	#31826574	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#31826574
commutator	C₈×C₂₈₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₈₃
Fitting	#360×C₂₈₃	C₂
socle	C₂×C₂₈₃	#442
prime radical	#31826900	C₂
cosocle	C₂×C₈×C₂₈₃	C₂³
hypercenter	#360	D₂₈₃
hypocenter	C₂₈₃	AGL(1,ℤ/16)
LCS	G ▷ C₈×C₂₈₃ ▷ C₄×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

Complements

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

N	H ≅ G/N	type
2. #31826587	C₂	semidirect
3. #31826576	C₂	semidirect
4. #31826287	C₂	semidirect
5. #31826279	C₂	semidirect
6. #31826242	C₂	semidirect
7. #360×C₂₈₃	C₂	semidirect
11. #22583553	C₄	semidirect
12. #22583551	C₄	semidirect
31. C₂₈₃⋊C₁₆	C₂×C₄	semidirect
33. Q₁₆×C₂₈₃	C₂×C₄	semidirect
35. D₈×C₂₈₃	C₂×C₄	semidirect
68. C₂₈₃	AGL(1,ℤ/16)	semidirect
75. #360	D₂₈₃	semidirect
93. Q₁₆	C₄×D₂₈₃	semidirect
97. D₈	C₄×D₂₈₃	semidirect

Maximal subgroups

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

subgroup	index
#31826900	2
#31826587	2
#31826576	2
#31826287	2
#31826279	2
#31826242	2
#360×C₂₈₃	2
AGL(1,ℤ/16)	283