Finite Group Databasebeta

Subgroups — #31808271

Order 17792.

160 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₂×#17984785	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581035	C₂	1	1	yes	maximal, characteristic
4	#22580778	C₂	1	1	yes	maximal, characteristic, complemented
5	#22580755	C₂	1	1	yes	maximal, characteristic
6	#22580728	C₂	1	1	yes	maximal, characteristic, complemented
7	#358×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₁₁₂	C₂²	1	1	yes	characteristic
9	Q₈×Dic₁₃₉	—	2	1	no	—
10	#17984804	—	2	1	no	—
11	#17984785	C₄	1	4	yes	complemented
12	C₂×C₁₃₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	#17984776	C₂²	1	1	yes	characteristic
14	#17984766	—	2	1	no	—
15	#17984762	—	2	1	no	—
16	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
19	#107×C₁₃₉	C₂²	1	1	yes	characteristic
20	C₂×D₅₅₆	—	2	1	no	—
21	C₂×C₄×D₁₃₉	—	4	1	no	—
22	D₁₁₁₂	—	2	1	no	—
23	D₁₁₁₂	C₂×C₄	1	2	yes	—
24	C₂×C₁₃₉⋊C₈	D₄	1	1	yes	characteristic
25	C₄⋊Dic₁₃₉	—	2	1	no	—
26	C₄⋊Dic₁₃₉	—	4	1	no	—
27	(C₂×D₁₃₉)⋊C₄	—	4	1	no	—
28	C₄×Dic₁₃₉	—	4	1	no	—
29	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
30	C₁₃₉⋊C₁₆	C₂×C₄	1	2	yes	complemented
31	C₂×Q₈×C₁₃₉	—	2	1	no	—
32	Q₁₆×C₁₃₉	—	2	1	no	—
33	Q₁₆×C₁₃₉	C₂×C₄	1	2	yes	complemented
34	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₃₉	—	2	1	no	—
36	C₁₆×C₁₃₉	—	2	1	no	—
37	C₂²×D₁₃₉	—	4	1	no	—
38	D₅₅₆	—	2	2	no	—
39	D₅₅₆	—	4	1	no	—
40	C₂×Dic₁₃₉	—	4	1	no	—
41	C₂×Dic₁₃₉	D₈	1	1	yes	characteristic
42	C₂×Dic₁₃₉	D₈	1	1	yes	characteristic
43	C₄×D₁₃₉	—	8	2	no	—
44	C₁₃₉⋊C₈	—	2	1	no	—
45	Q₈×C₁₃₉	—	2	2	no	—
46	Q₈×C₁₃₉	—	4	1	no	—
47	C₂×C₄×C₁₃₉	—	4	1	no	—
48	C₂×C₄×C₁₃₉	—	4	1	no	—
49	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
50	C₈×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
51	C₈×C₁₃₉	#41	1	1	yes	characteristic
52	C₂²×C₁₃₉	C₂×D₈	1	1	yes	characteristic
53	C₂×D₁₃₉	—	4	2	no	—
54	C₂×D₁₃₉	—	8	1	no	—
55	C₂×D₁₃₉	—	8	1	no	—
56	C₄×C₁₃₉	—	4	2	no	—
57	C₄×C₁₃₉	—	8	1	no	—
58	C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
59	C₄×C₁₃₉	#135	1	1	yes	characteristic
60	Dic₁₃₉	—	2	1	no	—
61	Dic₁₃₉	—	2	1	no	—
62	Dic₁₃₉	—	8	1	no	—
63	C₂×C₁₃₉	C₄×D₈	1	1	yes	characteristic
64	C₂×C₁₃₉	#509	1	1	yes	characteristic
65	C₂×C₁₃₉	#510	1	1	yes	characteristic
66	D₁₃₉	—	8	2	no	—
67	C₁₃₉	#2175	1	1	yes	characteristic, complemented, Sylow
68	#2175	—	139	1	no	maximal, Sylow
69	C₂×SD₃₂	—	139	1	no	—
70	C₄×Q₁₆	—	139	1	no	—
71	C₄×D₈	—	139	1	no	—
72	#366	—	139	1	no	—
73	#358	D₁₃₉	1	1	yes	characteristic, complemented
74	#357	—	139	1	no	—
75	#346	—	139	1	no	—
76	C₂×Q₁₆	C₂×D₁₃₉	1	1	yes	characteristic
77	C₂×D₈	—	139	1	no	—
78	C₄×Q₈	—	278	1	no	—
79	C₄×D₄	—	278	1	no	—
80	SD₃₂	—	139	4	no	—
81	C₂×C₁₆	—	139	1	no	—
82	C₂×C₁₆	C₂×D₁₃₉	1	1	yes	characteristic
83	#107	—	139	1	no	—
84	#107	C₂×D₁₃₉	1	1	yes	characteristic
85	#103	—	278	1	no	—
86	#102	—	278	1	no	—
87	C₄×C₈	—	139	1	no	—
88	C₂×Q₈	—	2	1	no	—
89	C₂×D₄	—	278	1	no	—
90	C₂²×C₄	—	556	1	no	—
91	Q₁₆	—	2	1	no	—
92	Q₁₆	C₄×D₁₃₉	1	2	yes	complemented
93	D₈	—	278	1	no	—
94	D₈	—	139	2	no	—
95	C₂×C₈	—	139	1	no	—
96	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	278	1	no	—
99	C₄⋊C₄	—	556	1	no	—
100	C₂²⋊C₄	—	556	1	no	—
101	C₄²	—	556	1	no	—
102	C₄²	—	139	1	no	—
103	C₁₆	—	2	1	no	—
104	C₁₆	—	139	2	no	—
105	C₂³	—	556	1	no	—
106	Q₈	—	2	2	no	—
107	Q₈	—	4	1	no	—
108	D₄	—	278	2	no	—
109	D₄	—	556	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	556	1	no	—
113	C₂×C₄	—	1112	2	no	—
114	C₂×C₄	—	139	1	no	—
115	C₂×C₄	—	139	1	no	—
116	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
117	C₈	—	278	1	no	—
118	C₈	C₂×C₄×D₁₃₉	1	1	yes	characteristic
119	C₈	#14555784	1	1	yes	characteristic
120	C₂²	—	556	2	no	—
121	C₂²	—	1112	1	no	—
122	C₂²	—	1112	1	no	—
123	C₂²	D₈×D₁₃₉	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	—	8	1	no	—
126	C₄	—	278	1	no	—
127	C₄	—	278	1	no	—
128	C₄	—	1112	1	no	—
129	C₄	(C₂²⋊D₁₃₉)⋊C₄	1	1	yes	characteristic
130	C₄	#17984863	1	1	yes	characteristic
131	C₂	—	1112	2	no	—
132	C₂	#22581033	1	1	yes	characteristic
133	C₂	#22581390	1	1	yes	characteristic
134	C₂	#22581395	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₃₉
commutator	C₈×C₁₃₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₃₉
Fitting	#358×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×D₈
prime radical	C₂×D₁₁₁₂	C₂²
cosocle	C₂×C₈×C₁₃₉	C₂³
hypercenter	#358	D₁₃₉
hypocenter	C₁₃₉	#2175
LCS	G ▷ C₈×C₁₃₉ ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

Complements

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

N	H ≅ G/N	type
2. Q₁₆×Dic₁₃₉	C₂	semidirect
4. #22580778	C₂	semidirect
6. #22580728	C₂	semidirect
7. #358×C₁₃₉	C₂	semidirect
11. #17984785	C₄	semidirect
30. C₁₃₉⋊C₁₆	C₂×C₄	semidirect
33. Q₁₆×C₁₃₉	C₂×C₄	semidirect
67. C₁₃₉	#2175	semidirect
73. #358	D₁₃₉	semidirect
92. Q₁₆	C₄×D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984785	2
Q₁₆×Dic₁₃₉	2
#22581035	2
#22580778	2
#22580755	2
#22580728	2
#358×C₁₃₉	2
#2175	139