Finite Group Databasebeta

← #31806822

Subgroups — #31806822

Order 17792.

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

normal subgroups only

This group has 342 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#22581460C211yesmaximal, characteristic
2#22580937C211yesmaximal, characteristic, complemented
3#22580916C212yesmaximal, complemented
4#22580724C212yesmaximal, complemented
5#568×C139C211yesmaximal, characteristic, complemented
6C22×C4×D139C2211yescharacteristic
7C2×#14555769C2212yes—
8#17984823C412yes—
9#17984818—21no—
10#17984813—21no—
11#17984810C412yes—
12C4×C22⋊D139—22no—
13#17984801C2211yescharacteristic
14#17984761—21no—
15#17984759—21no—
16(C2×D139)⋊C8—24no—
17C2×#41×C139C2211yescharacteristic
18#131×C139—24no—
19C2×M16×C139—22no—
20C2×M16×C139C2212yes—
21C23×D139C2×C411yescharacteristic
22C2×C22⋊D139—22no—
23C22×Dic139C2×C411yescharacteristic
24C2×C4×D139—21no—
25C2×C4×D139—21no—
26C2×C4×D139—22no—
27C2×C4×D139—22no—
28#14555769—22no—
29#14555769—22no—
30C2×C139⋊C8—22no—
31C4⋊Dic139—21no—
32Dic139⋊C4—22no—
33C22⋊Dic139—22no—
34C22⋊Dic139C2×C412yes—
35(C2×D139)⋊C4—22no—
36C4×Dic139—21no—
37#41×C139—22no—
38#41×C139—22no—
39C2×Q8×C139D411yescharacteristic
40C2×D4×C139C2×C412yes—
41C2×D4×C139D411yescharacteristic
42C22×C4×C139D412yes—
43C22×C4×C139C2311yescharacteristic
44M16×C139—24no—
45M16×C139—22no—
46M16×C139—22no—
47C2×C8×C139—24no—
48C2×C8×C139D414yescomplemented
49C22×D139—21no—
50C22×D139—22no—
51C22×D139—42no—
52C22⋊D139—44no—
53C2×Dic139—22no—
54C2×Dic139—21no—
55C2×Dic139—22no—
56C4×D139—41no—
57C4×D139—42no—
58C4×D139—41no—
59C139⋊C8—42no—
60C23×C139C22⋊C412yes—
61C23×C139C22×C411yescharacteristic
62Q8×C139—22no—
63D4×C139—24no—
64D4×C139—22no—
65C2×C4×C139—22no—
66C2×C4×C139—22no—
67C2×C4×C139C22⋊C412yes—
68C2×C4×C139C2×D412yes—
69C2×C4×C139C2×D411yescharacteristic
70C2×C4×C139C2×D411yescharacteristic
71C2×C4×C139#4112yes—
72C8×C139—24no—
73C22×C139—22no—
74C22×C139—41no—
75C22×C139—42no—
76C22×C139C2×C22⋊C411yescharacteristic
77C22×C139C4×D412yes—
78C2×D139—22no—
79C2×D139—24no—
80C2×D139—41no—
81C2×D139—42no—
82C2×D139—41no—
83C4×C139—22no—
84C4×C139#12011yescharacteristic
85C4×C139#12112yes—
86C4×C139#12311yescharacteristic
87Dic139—42no—
88Dic139—41no—
89C2×C139—21no—
90C2×C139—22no—
91C2×C139—42no—
92C2×C139#38611yescharacteristic
93D139—42no—
94C139#189111yescharacteristic, complemented, Sylow
95#1891—1391nomaximal, Sylow
96#568D13911yescharacteristic, complemented
97#525—1391no—
98#413—1391no—
99#407—1392no—
100#344—1392no—
101C2×#41C2×D13911yescharacteristic
102C23×C4—1391no—
103#131—24no—
104C2×M16—22no—
105C2×M16—1392no—
106C2×M16C2×D13912yes—
107#123—1392no—
108C22⋊Q8—2781no—
109#121—2781no—
110#120—1392no—
111C4×D4—2782no—
112#117—1391no—
113#101—2781no—
114#100—2781no—
115C22⋊C8—2784no—
116C24—1391no—
117#41—22no—
118#41—22no—
119C2×Q8D55611yescharacteristic
120C2×D4—2782no—
121C2×D4C4×D13912yes—
122C2×D4D55611yescharacteristic
123C22×C4—2782no—
124C22×C4—2781no—
125C22×C4—2781no—
126C22×C4—2782no—
127C22×C4—1391no—
128C22×C4C22⋊D13912yes—
129C22×C4C22×D13911yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—2782no—
134M16—2782no—
135C2×C8—24no—
136C2×C8—2782no—
137C2×C8C22⋊D13914yescomplemented
138C4⋊C4—2782no—
139C4⋊C4—2781no—
140C22⋊C4—2782no—
141C22⋊C4—2782no—
142C22⋊C4—1392no—
143C42—2781no—
144C23—2781no—
145C23—2782no—
146C23—5562no—
147C23(C2×D139)⋊C412yes—
148C23C2×C4×D13911yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—5564no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—2782no—
156C2×C4—2781no—
157C2×C4—2782no—
158C2×C4—5561no—
159C2×C4—5562no—
160C2×C4—5561no—
161C2×C4(C2×D139)⋊C412yes—
162C2×C4C2×C22⋊D13912yes—
163C2×C4C2×C22⋊D13911yescharacteristic
164C2×C4C2×D55611yescharacteristic
165C2×C4#1455578512yes—
166C8—24no—
167C8—5562no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—2782no—
172C22—2784no—
173C22—5561no—
174C22—5562no—
175C22—5561no—
176C22C2×(C2×D139)⋊C411yescharacteristic
177C22C4×C22⋊D13912yes—
178C4—22no—
179C4—5562no—
180C4—5561no—
181C4#1798481111yescharacteristic
182C4#1798481612yes—
183C4#1798482511yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—5562no—
188C2#2258083211yescharacteristic
subgroupG/H
centerC4#17984811
commutatorC23×C139C22×C4
FrattiniC22×C4C22×D139
Fitting#568×C139C2
socleC2×C139#386
prime radical#22581460C2
cosocleC22×C4×C139C23
hypercenter#568D139
hypocenterC139#1891
LCSG ▷ C23×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #22580937C2semidirect
3. #22580916C2semidirect
4. #22580724C2semidirect
5. #568×C139C2semidirect
48. C2×C8×C139D4semidirect
94. C139#1891semidirect
96. #568D139semidirect
137. C2×C8C22⋊D139semidirect

Maximal subgroups

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

subgroupindex
#225814602
#225809372
#225809162
#225807242
#568×C1392
#1891139