Finite Group Databasebeta

← #31805380

Subgroups — #31805380

Order 17792.

238 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22581584C211yesmaximal, characteristic
2#22581111C212yesmaximal
3#22580873C211yesmaximal, characteristic
4#22580691C212yesmaximal
5#324×C139C211yesmaximal, characteristic
6C22×Dic278C2211yescharacteristic
7C2×Dic556—22no—
8#17984841—22no—
9#17984833—22no—
10#17984821—21no—
11#17984821C2212yes—
12#17984801—21no—
13C2×Dic139⋊C4—21no—
14#17984767—22no—
15C139⋊#95—21no—
16C139⋊#95C2211yescharacteristic
17C2×C4⋊C4×C139C2211yescharacteristic
18C22⋊C8×C139C2212yes—
19C2×Dic278—21no—
20C2×Dic278—22no—
21C2×Dic278—42no—
22C2×Dic278D412yes—
23C22×Dic139—21no—
24C22×Dic139—21no—
25Dic556—44no—
26C4⋊Dic139—21no—
27C4⋊Dic139—21no—
28C4⋊Dic139—21no—
29C4⋊Dic139D412yes—
30Dic139⋊C4—22no—
31Dic139⋊C4—22no—
32Dic139⋊C4—42no—
33C22⋊Dic139—22no—
34C4×Dic139—22no—
35C22×C4×C139D411yescharacteristic
36C22×C4×C139D411yescharacteristic
37C22×C4×C139C2311yescharacteristic
38C2×C8×C139—22no—
39C4⋊C4×C139—22no—
40Dic278—24no—
41Dic278—42no—
42Dic278—42no—
43Dic278—81no—
44C2×Dic139—22no—
45C2×Dic139—22no—
46C2×Dic139—22no—
47C2×Dic139—22no—
48C2×Dic139—41no—
49C2×Dic139—41no—
50C2×Dic139—41no—
51C2×Dic139—41no—
52C23×C139C2×D411yescharacteristic
53C2×C4×C139—21no—
54C2×C4×C139—21no—
55C2×C4×C139—21no—
56C2×C4×C139—41no—
57C2×C4×C139—41no—
58C2×C4×C139Q1612yescomplemented
59C2×C4×C139C2×D412yes—
60C8×C139—42no—
61C22×C139—21no—
62C22×C139—21no—
63C22×C139#12011yescharacteristic
64C22×C139C2×Q1611yescharacteristic
65C22×C139#13711yescharacteristic
66C4×C139—22no—
67C4×C139—22no—
68C4×C139—41no—
69Dic139—42no—
70Dic139—42no—
71Dic139—42no—
72Dic139—42no—
73C2×C139—21no—
74C2×C139—21no—
75C2×C139#45111yescharacteristic
76C2×C139#45611yescharacteristic
77C2×C139B(4,𝔽2)11yescharacteristic
78C139#160511yescharacteristic, complemented, Sylow
79#1605—1391nomaximal, Sylow
80#544—1391no—
81#451—1392no—
82#395—1391no—
83#328—1392no—
84#324D13911yescharacteristic
85C22×Q8—1391no—
86C2×Q16—2782no—
87C4⋊Q8—2782no—
88#125—2782no—
89C22⋊Q8—2781no—
90C22⋊Q8—1392no—
91#117—2781no—
92C2×C4⋊C4—2781no—
93C2×C4⋊C4C2×D13911yescharacteristic
94#103—2782no—
95C22⋊C8C2×D13912yes—
96#95—2781no—
97#95—1391no—
98C2×Q8—2781no—
99C2×Q8—2782no—
100C2×Q8—5562no—
101C2×Q8—1392no—
102C22×C4—2781no—
103C22×C4—2781no—
104C22×C4D55611yescharacteristic
105C22×C4D55611yescharacteristic
106C22×C4C22×D13911yescharacteristic
107Q16—5564no—
108C2×C8—22no—
109C4⋊C4—22no—
110C4⋊C4—2782no—
111C4⋊C4—2781no—
112C4⋊C4—2781no—
113C4⋊C4—2781no—
114C4⋊C4—2782no—
115C4⋊C4—5562no—
116C4⋊C4—1392no—
117C22⋊C4—2782no—
118C42—2782no—
119C23C2×D55611yescharacteristic
120Q8—2784no—
121Q8—5562no—
122Q8—5562no—
123Q8—11121no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—2782no—
130C2×C4—2782no—
131C2×C4—2782no—
132C2×C4—2782no—
133C2×C4—5561no—
134C2×C4—5561no—
135C2×C4—5561no—
136C2×C4—5561no—
137C2×C4Dic55612yescomplemented
138C2×C4D4×D13912yes—
139C8—42no—
140C22—21no—
141C22—21no—
142C22#1798481211yescharacteristic
143C22C2×Dic55611yescharacteristic
144C22#1798487511yescharacteristic
145C4—22no—
146C4—22no—
147C4—41no—
148C4—5562no—
149C4—5562no—
150C4—5562no—
151C4—5562no—
152C2—21no—
153C2—21no—
154C2#2258111111yescharacteristic
155C2#2258113811yescharacteristic
156C2#2258114111yescharacteristic
subgroupG/H
centerC22#17984812
commutator, cosocleC22×C4×C139C23
FrattiniC22×C4C22×D139
Fitting#324×C139C2
socleC22×C139#120
prime radicalC23×C139C2×D4
hypercenter#324D139
hypocenterC139#1605
LCSG ▷ C22×C4×C139 ▷ C22×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
58. C2×C4×C139Q16semidirect
78. C139#1605semidirect
137. C2×C4Dic556semidirect

Maximal subgroups

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

subgroupindex
#225815842
#225811112
#225808732
#225806912
#324×C1392
#1605139