Finite Group Databasebeta

← #31806926

Subgroups — #31806926

Order 17792.

312 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 89 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#22581524C211yesmaximal, characteristic
2#22580963C212yesmaximal, complemented
3#22580927C211yesmaximal, characteristic
4#22580710C212yesmaximal, complemented
5#568×C139C211yesmaximal, characteristic
6C22×Dic278C2211yescharacteristic
7#17984840C412yescomplemented
8#17984829C412yescomplemented
9#17984823—22no—
10#17984821—22no—
11#17984801C2212yes—
12#17984801C2211yescharacteristic
13#17984767—24no—
14#17984757—22no—
15C2×#41×C139C2211yescharacteristic
16#131×C139—24no—
17C2×M16×C139—22no—
18C2×M16×C139C2212yes—
19C2×Dic278—21no—
20C2×Dic278—22no—
21C2×Dic278—22no—
22C2×Dic278C2×C412yes—
23C22×Dic139—21no—
24C4⋊Dic139—21no—
25C4⋊Dic139—22no—
26Dic139⋊C4—24no—
27C22⋊Dic139—24no—
28C22⋊Dic139—21no—
29C22⋊Dic139—22no—
30C4×Dic139—22no—
31C4×Dic139C2×C412yes—
32#41×C139—22no—
33#41×C139—22no—
34C2×Q8×C139C2×C411yescharacteristic
35C2×D4×C139C2×C411yescharacteristic
36C2×D4×C139D412yes—
37C22×C4×C139D412yes—
38C22×C4×C139C2311yescharacteristic
39M16×C139—24no—
40M16×C139—22no—
41M16×C139—22no—
42C2×C8×C139—24no—
43C2×C8×C139D414yes—
44Dic278—24no—
45Dic278—41no—
46Dic278—41no—
47C2×Dic139—22no—
48C2×Dic139—22no—
49C2×Dic139—42no—
50C2×Dic139—42no—
51C23×C139C22⋊C412yes—
52C23×C139C2×D411yescharacteristic
53Q8×C139—22no—
54D4×C139—24no—
55D4×C139—22no—
56C2×C4×C139—22no—
57C2×C4×C139—22no—
58C2×C4×C139C22⋊C412yes—
59C2×C4×C139C22×C411yescharacteristic
60C2×C4×C139C2×D412yes—
61C2×C4×C139C2×D411yescharacteristic
62C2×C4×C139#4112yes—
63C8×C139—24no—
64C22×C139—22no—
65C22×C139—41no—
66C22×C139—42no—
67C22×C139C2×C22⋊C411yescharacteristic
68C22×C139#12112yes—
69C4×C139—22no—
70C4×C139C4×D412yes—
71C4×C139#12011yescharacteristic
72C4×C139#12311yescharacteristic
73Dic139—42no—
74Dic139—42no—
75Dic139—82no—
76C2×C139—21no—
77C2×C139—22no—
78C2×C139—42no—
79C2×C139#38611yescharacteristic
80C139#190911yescharacteristic, complemented, Sylow
81#1909—1391nomaximal, Sylow
82#568D13911yescharacteristic
83#536—1391no—
84#419—1392no—
85#410—1391no—
86H(3,ℤ/4)—1392no—
87C2×#41C2×D13911yescharacteristic
88C22×Q8—1391no—
89#131—24no—
90C2×M16—22no—
91C2×M16C2×D13912yes—
92C4⋊Q8—1392no—
93#124—1392no—
94#123—2782no—
95C22⋊Q8—2782no—
96#117—1392no—
97#117—1391no—
98#103—2784no—
99#99—2782no—
100#41—22no—
101#41—22no—
102C2×Q8—2781no—
103C2×Q8—2782no—
104C2×Q8—2782no—
105C2×Q8—1392no—
106C2×Q8C4×D13911yescharacteristic
107C2×D4C4×D13911yescharacteristic
108C2×D4D55612yes—
109C22×C4—2781no—
110C22×C4C22⋊D13912yes—
111C22×C4C22×D13911yescharacteristic
112M16—24no—
113M16—22no—
114M16—22no—
115C2×C8—24no—
116C2×C8C22⋊D13914yes—
117C4⋊C4—2784no—
118C4⋊C4—2781no—
119C4⋊C4—2782no—
120C22⋊C4—2784no—
121C22⋊C4—2781no—
122C22⋊C4—2782no—
123C42—2782no—
124C42—1392no—
125C23(C2×D139)⋊C412yes—
126C23C2×D55611yescharacteristic
127Q8—22no—
128Q8—2784no—
129Q8—5561no—
130Q8—5561no—
131D4—24no—
132D4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—2782no—
136C2×C4—2782no—
137C2×C4—5562no—
138C2×C4—5562no—
139C2×C4(C2×D139)⋊C412yes—
140C2×C4C2×C4×D13911yescharacteristic
141C2×C4C2×C22⋊D13912yes—
142C2×C4C2×C22⋊D13911yescharacteristic
143C2×C4#1455578512yes—
144C8—24no—
145C22—22no—
146C22—41no—
147C22—42no—
148C22C2×(C2×D139)⋊C411yescharacteristic
149C22#1798481612yes—
150C4—22no—
151C4—5562no—
152C4—5562no—
153C4—11122no—
154C4C4×C22⋊D13912yes—
155C4#1798481111yescharacteristic
156C4#1798482511yescharacteristic
157C2—21no—
158C2—22no—
159C2—42no—
160C2#2258083211yescharacteristic
subgroupG/H
centerC2#22580832
commutatorC2×C4×C139C22×C4
FrattiniC22×C4C22×D139
Fitting#568×C139C2
socleC2×C139#386
prime radicalC2×#41×C139C22
cosocleC22×C4×C139C23
hypercenter#568D139
hypocenterC139#1909
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #22580963C2semidirect
4. #22580710C2semidirect
7. #17984840C4semidirect
8. #17984829C4semidirect
80. C139#1909semidirect

Maximal subgroups

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

subgroupindex
#225815242
#225809632
#225809272
#225807102
#568×C1392
#1909139