Finite Group Databasebeta

Subgroups — #31805987

Order 17792.

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

normal subgroups only

This group has 140 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	#22581304	C₂	1	1	yes	maximal, characteristic
2	#22581249	C₂	1	1	yes	maximal, characteristic
3	#22581239	C₂	1	1	yes	maximal, characteristic
4	#22581083	C₂	1	1	yes	maximal, characteristic
5	#22580701	C₂	1	1	yes	maximal, characteristic
6	#22580698	C₂	1	1	yes	maximal, characteristic
7	#334×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	#17984842	C₂²	1	1	yes	characteristic
9	#17984842	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
11	#17984776	—	2	1	no	—
12	#17984774	—	2	1	no	—
13	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
14	Dic₁₃₉⋊C₈	—	2	1	no	—
15	#17984767	—	2	1	no	—
16	#17984767	—	2	1	no	—
17	#17984767	—	2	1	no	—
18	#17984767	—	2	1	no	—
19	#17984752	—	2	1	no	—
20	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₇₈	—	2	1	no	—
24	C₂×Dic₂₇₈	—	2	1	no	—
25	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
26	C₂×C₁₃₉⋊C₈	—	2	1	no	—
27	C₄⋊Dic₁₃₉	—	2	1	no	—
28	C₄⋊Dic₁₃₉	—	2	1	no	—
29	C₄⋊Dic₁₃₉	—	2	1	no	—
30	C₄⋊Dic₁₃₉	—	2	1	no	—
31	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
32	Dic₁₃₉⋊C₄	—	2	1	no	—
33	C₄×Dic₁₃₉	—	2	1	no	—
34	C₂×C₈×C₁₃₉	—	2	1	no	—
35	C₂×C₈×C₁₃₉	—	2	1	no	—
36	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
38	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
39	Dic₂₇₈	—	2	1	no	—
40	Dic₂₇₈	—	2	1	no	—
41	Dic₂₇₈	—	4	1	no	—
42	Dic₂₇₈	—	4	1	no	—
43	Dic₂₇₈	—	8	1	no	—
44	C₂×Dic₁₃₉	—	2	1	no	—
45	C₂×Dic₁₃₉	—	2	1	no	—
46	C₂×Dic₁₃₉	—	4	1	no	—
47	C₂×Dic₁₃₉	—	4	1	no	—
48	C₁₃₉⋊C₈	—	4	1	no	—
49	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
52	C₈×C₁₃₉	—	2	1	no	—
53	C₈×C₁₃₉	—	2	1	no	—
54	C₈×C₁₃₉	—	4	1	no	—
55	C₈×C₁₃₉	—	4	1	no	—
56	C₂²×C₁₃₉	#121	1	1	yes	characteristic
57	C₄×C₁₃₉	—	2	1	no	—
58	C₄×C₁₃₉	#135	1	1	yes	characteristic
59	C₄×C₁₃₉	#137	1	1	yes	characteristic
60	C₄×C₁₃₉	#137	1	1	yes	characteristic
61	C₄×C₁₃₉	#137	1	1	yes	characteristic
62	Dic₁₃₉	—	4	1	no	—
63	Dic₁₃₉	—	4	1	no	—
64	Dic₁₃₉	—	8	1	no	—
65	Dic₁₃₉	—	8	1	no	—
66	C₂×C₁₃₉	#464	1	1	yes	characteristic
67	C₂×C₁₃₉	#470	1	1	yes	characteristic
68	C₂×C₁₃₉	#471	1	1	yes	characteristic
69	C₁₃₉	#1704	1	1	yes	characteristic, complemented, Sylow
70	#1704	—	139	1	no	maximal, Sylow
71	#487	—	139	1	no	—
72	#477	—	139	1	no	—
73	#475	—	139	1	no	—
74	#446	—	139	1	no	—
75	#334	D₁₃₉	1	1	yes	characteristic
76	#333	—	139	1	no	—
77	#332	—	139	1	no	—
78	C₄⋊Q₈	—	139	1	no	—
79	C₄⋊Q₈	—	139	1	no	—
80	C₄×Q₈	—	139	1	no	—
81	#107	—	278	1	no	—
82	#106	—	278	1	no	—
83	C₄⋊C₈	—	278	1	no	—
84	C₄⋊C₈	—	139	1	no	—
85	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
86	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
87	#103	—	278	1	no	—
88	#103	—	278	1	no	—
89	#103	—	278	1	no	—
90	#103	—	278	1	no	—
91	#97	—	278	1	no	—
92	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
93	C₂×Q₈	—	278	1	no	—
94	C₂×Q₈	—	278	1	no	—
95	C₂×Q₈	—	139	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	278	1	no	—
99	C₂×C₈	D₅₅₆	1	1	yes	characteristic
100	C₂×C₈	D₅₅₆	1	1	yes	characteristic
101	C₄⋊C₄	—	278	1	no	—
102	C₄⋊C₄	—	278	1	no	—
103	C₄⋊C₄	—	278	1	no	—
104	C₄⋊C₄	—	278	1	no	—
105	C₄⋊C₄	—	278	1	no	—
106	C₄⋊C₄	—	139	1	no	—
107	C₄²	—	278	1	no	—
108	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
109	Q₈	—	278	1	no	—
110	Q₈	—	278	1	no	—
111	Q₈	—	556	1	no	—
112	Q₈	—	556	1	no	—
113	Q₈	—	1112	1	no	—
114	C₂×C₄	—	278	1	no	—
115	C₂×C₄	—	278	1	no	—
116	C₂×C₄	—	556	1	no	—
117	C₂×C₄	—	556	1	no	—
118	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
119	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
120	C₂×C₄	#14555786	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	556	1	no	—
126	C₂²	#17984817	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	556	1	no	—
129	C₄	—	556	1	no	—
130	C₄	—	1112	1	no	—
131	C₄	—	1112	1	no	—
132	C₄	#17984864	1	1	yes	characteristic
133	C₄	#17984871	1	1	yes	characteristic
134	C₄	#17984871	1	1	yes	characteristic
135	C₄	#17984875	1	1	yes	characteristic
136	C₂	#22581176	1	1	yes	characteristic
137	C₂	#22581212	1	1	yes	characteristic
138	C₂	#22581217	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17984817
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#334×C₁₃₉	C₂
socle, prime radical	C₂²×C₁₃₉	#121
hypercenter	#334	D₁₃₉
hypocenter	C₁₃₉	#1704
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

Complements

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

N	H ≅ G/N	type
69. C₁₃₉	#1704	semidirect

Maximal subgroups

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

subgroup	index
#22581304	2
#22581249	2
#22581239	2
#22581083	2
#22580701	2
#22580698	2
#334×C₁₃₉	2
#1704	139