Finite Group Databasebeta

← #31805363

Subgroups — #31805363

Order 17792.

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

normal subgroups only

This group has 256 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#22581579C211yesmaximal, characteristic
2#22581111C212yesmaximal, complemented
3#22580865C211yesmaximal, characteristic
4#22580691C212yesmaximal, complemented
5#323×C139C211yesmaximal, characteristic
6C22×Dic278C2211yescharacteristic
7C2×Dic556—22no—
8#17984821—21no—
9#17984821C2212yes—
10#17984819—22no—
11#17984819—22no—
12C2×C22⋊Dic139—21no—
13C2×C22⋊Dic139—21no—
14#17984767—22no—
15C139⋊#95—21no—
16C139⋊#95C2211yescharacteristic
17C2×C22⋊C4×C139C2211yescharacteristic
18C22⋊C8×C139C2212yes—
19C2×Dic278—21no—
20C2×Dic278—22no—
21C2×Dic278—42no—
22C2×Dic278D412yes—
23C22×Dic139—21no—
24C22×Dic139—21no—
25C22×Dic139—21no—
26C22×Dic139—21no—
27Dic556—44no—
28C4⋊Dic139—21no—
29C4⋊Dic139D412yes—
30Dic139⋊C4—22no—
31Dic139⋊C4—22no—
32C22⋊Dic139—22no—
33C22⋊Dic139—22no—
34C22⋊Dic139—42no—
35C24×C139D411yescharacteristic
36C22×C4×C139D411yescharacteristic
37C22×C4×C139C2311yescharacteristic
38C2×C8×C139—22no—
39C22⋊C4×C139—22no—
40Dic278—24no—
41Dic278—42no—
42Dic278—42no—
43Dic278—81no—
44C2×Dic139—22no—
45C2×Dic139—22no—
46C2×Dic139—22no—
47C2×Dic139—41no—
48C2×Dic139—41no—
49C2×Dic139—41no—
50C2×Dic139—41no—
51C2×Dic139—41no—
52C2×Dic139—81no—
53C2×Dic139—81no—
54C23×C139—21no—
55C23×C139—21no—
56C23×C139—41no—
57C23×C139—41no—
58C23×C139Q1612yescomplemented
59C23×C139C2×D411yescharacteristic
60C2×C4×C139—21no—
61C2×C4×C139—41no—
62C2×C4×C139—41no—
63C2×C4×C139C2×D412yes—
64C8×C139—42no—
65C22×C139—21no—
66C22×C139—21no—
67C22×C139—22no—
68C22×C139—42no—
69C22×C139—41no—
70C22×C139—41no—
71C22×C139—81no—
72C22×C139#12011yescharacteristic
73C22×C139C2×Q1611yescharacteristic
74C22×C139#13711yescharacteristic
75C4×C139—22no—
76C4×C139—41no—
77Dic139—42no—
78Dic139—42no—
79Dic139—42no—
80Dic139—81no—
81C2×C139—21no—
82C2×C139—21no—
83C2×C139—42no—
84C2×C139#45111yescharacteristic
85C2×C139#45511yescharacteristic
86C2×C139B(4,𝔽2)11yescharacteristic
87C139#160211yescharacteristic, complemented, Sylow
88#1602—1391nomaximal, Sylow
89#543—1391no—
90#451—1392no—
91#393—1391no—
92#328—1392no—
93#323D13911yescharacteristic
94C22×Q8—1391no—
95C2×Q16—2782no—
96C22⋊Q8—2782no—
97C22⋊Q8—2782no—
98C22⋊Q8—2781no—
99C22⋊Q8—1392no—
100C2×C22⋊C4—2781no—
101C2×C22⋊C4—2781no—
102C2×C22⋊C4C2×D13911yescharacteristic
103#103—2782no—
104C22⋊C8C2×D13912yes—
105#95—2781no—
106#95—1391no—
107C24D55611yescharacteristic
108C2×Q8—2781no—
109C2×Q8—2782no—
110C2×Q8—5562no—
111C2×Q8—1392no—
112C22×C4—2781no—
113C22×C4—2781no—
114C22×C4—2781no—
115C22×C4—2781no—
116C22×C4D55611yescharacteristic
117C22×C4C22×D13911yescharacteristic
118Q16—5564no—
119C2×C8—22no—
120C4⋊C4—2782no—
121C4⋊C4—2781no—
122C4⋊C4—2782no—
123C4⋊C4—1392no—
124C22⋊C4—22no—
125C22⋊C4—2782no—
126C22⋊C4—2782no—
127C22⋊C4—5562no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic55612yescomplemented
133C23C2×D55611yescharacteristic
134Q8—2784no—
135Q8—5562no—
136Q8—5562no—
137Q8—11121no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—2782no—
142C2×C4—2782no—
143C2×C4—2782no—
144C2×C4—5561no—
145C2×C4—5561no—
146C2×C4—5561no—
147C2×C4—5561no—
148C2×C4—5561no—
149C2×C4—11121no—
150C2×C4—11121no—
151C2×C4D4×D13912yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1798481211yescharacteristic
161C22C2×Dic55611yescharacteristic
162C22#1798487511yescharacteristic
163C4—22no—
164C4—41no—
165C4—5562no—
166C4—5562no—
167C4—5562no—
168C4—11121no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2258111111yescharacteristic
173C2#2258113311yescharacteristic
174C2#2258114111yescharacteristic
subgroupG/H
centerC22#17984812
commutator, cosocleC22×C4×C139C23
FrattiniC22×C4C22×D139
Fitting#323×C139C2
socleC22×C139#120
prime radicalC24×C139D4
hypercenter#323D139
hypocenterC139#1602
LCSG ▷ C22×C4×C139 ▷ C22×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #22581111C2semidirect
4. #22580691C2semidirect
58. C23×C139Q16semidirect
87. C139#1602semidirect
132. C23Dic556semidirect

Maximal subgroups

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

subgroupindex
#225815792
#225811112
#225808652
#225806912
#323×C1392
#1602139