Finite Group Databasebeta

← #31807448

Subgroups — #31807448

Order 17792.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#17984870C212yesmaximal, complemented
2#22581534C211yesmaximal, characteristic, complemented
3C2×#17984768C212yesmaximal, complemented
4#22580854C211yesmaximal, characteristic, complemented
5#343×C139C211yesmaximal, characteristic, complemented
6C2×#14555785C2212yes—
7C22×D556C2211yescharacteristic
8#17984870—24no—
9#17984870—24no—
10C2×#14555774—22no—
11C2×D1112—22no—
12#17984822—22no—
13#17984817—22no—
14#17984816—22no—
15C2×(C2×D139)⋊C4—22no—
16C2×C4×Dic139C2211yescharacteristic
17#17984768—24no—
18C2×M16×C139C2212yes—
19C2×C4⋊C4×C139C2211yescharacteristic
20C23×D139—21no—
21#14555785—22no—
22#14555785—22no—
23#14555785—22no—
24#14555785—22no—
25C2×Dic278D412yes—
26C2×D556—22no—
27C2×D556—22no—
28C2×D556—22no—
29C2×D556—21no—
30C2×D556D412yes—
31C2×D556D412yescomplemented
32C2×C22⋊D139—22no—
33C22×Dic139—21no—
34C2×C4×D139—22no—
35#14555774—28no—
36D1112—28no—
37C4⋊Dic139—22no—
38(C2×D139)⋊C4—24no—
39(C2×D139)⋊C4—44no—
40C4×Dic139—22no—
41C22×C4×C139D412yes—
42C22×C4×C139C2311yescharacteristic
43M16×C139—22no—
44M16×C139D414yescomplemented
45C2×C8×C139—22no—
46C4⋊C4×C139—22no—
47C22×D139—22no—
48C22×D139—22no—
49C22×D139—42no—
50C22×D139—42no—
51C22×D139—41no—
52C22×D139—41no—
53Dic278—22no—
54Dic278—22no—
55D556—22no—
56D556—24no—
57D556—22no—
58D556—42no—
59D556—82no—
60C22⋊D139—42no—
61C22⋊D139—42no—
62C2×Dic139—22no—
63C2×Dic139—22no—
64C2×Dic139—41no—
65C2×Dic139—41no—
66C4×D139—42no—
67C4×D139—42no—
68C23×C139C2×D411yescharacteristic
69C2×C4×C139—22no—
70C2×C4×C139—42no—
71C2×C4×C139C2×D411yescharacteristic
72C2×C4×C139C2×D412yes—
73C2×C4×C139C2×D412yes—
74C2×C4×C139#4111yescharacteristic
75C8×C139—24no—
76C22×C139—21no—
77C22×C139—21no—
78C22×C139#12011yescharacteristic
79C22×C139#12111yescharacteristic
80C22×C139#12711yescharacteristic
81C2×D139—44no—
82C2×D139—42no—
83C2×D139—42no—
84C2×D139—42no—
85C2×D139—42no—
86C2×D139—82no—
87C2×D139—81no—
88C2×D139—81no—
89C4×C139—42no—
90C4×C139#12012yes—
91C4×C139#12112yes—
92Dic139—42no—
93Dic139—42no—
94C2×C139—21no—
95C2×C139—21no—
96C2×C139#39211yescharacteristic
97C2×C139#45311yescharacteristic
98C2×C139#45411yescharacteristic
99D139—82no—
100D139—82no—
101C139#200611yescharacteristic, complemented, Sylow
102#2006—1391nomaximal, Sylow
103C2×AGL(1,ℤ/8)—1392no—
104#537—1391no—
105C2×#104—1392no—
106#390—1391no—
107#343D13911yescharacteristic, complemented
108C2×#41—1392no—
109C22×D4—1391no—
110AGL(1,ℤ/8)—2784no—
111AGL(1,ℤ/8)—2784no—
112C2×SD16—2782no—
113C2×D8—2782no—
114C2×M16C2×D13912yes—
115C22⋊Q8—2782no—
116#121—2782no—
117#121—2782no—
118C2×C4⋊C4C2×D13911yescharacteristic
119C2×C22⋊C4—2782no—
120C2×C42—1391no—
121#104—2784no—
122C24—2781no—
123#41—2782no—
124#41—2782no—
125#41—2782no—
126#41—2782no—
127C2×Q8—1392no—
128C2×D4—2782no—
129C2×D4—2782no—
130C2×D4—2782no—
131C2×D4—2782no—
132C2×D4—2781no—
133C2×D4—1392no—
134C2×D4—1392no—
135C22×C4—2781no—
136C22×C4—2782no—
137C22×C4D55612yes—
138C22×C4C22×D13911yescharacteristic
139SD16—2788no—
140D8—2788no—
141M16—22no—
142M16D55614yescomplemented
143C2×C8—22no—
144C4⋊C4—22no—
145C4⋊C4—2782no—
146C22⋊C4—2784no—
147C22⋊C4—5564no—
148C42—2782no—
149C23—2782no—
150C23—2782no—
151C23—5562no—
152C23—5562no—
153C23—5561no—
154C23—5561no—
155C23C2×D55611yescharacteristic
156Q8—2782no—
157Q8—2782no—
158D4—2782no—
159D4—2784no—
160D4—2782no—
161D4—5562no—
162D4—5562no—
163D4—5562no—
164D4—11122no—
165C2×C4—22no—
166C2×C4—42no—
167C2×C4—2782no—
168C2×C4—2782no—
169C2×C4—5561no—
170C2×C4—5561no—
171C2×C4—5562no—
172C2×C4—5562no—
173C2×C4D4×D13911yescharacteristic
174C2×C4D4×D13912yes—
175C2×C4C2×D55612yes—
176C2×C4#1455578611yescharacteristic
177C8—24no—
178C22—21no—
179C22—21no—
180C22—5564no—
181C22—5562no—
182C22—5562no—
183C22—5562no—
184C22—5562no—
185C22—11122no—
186C22—11121no—
187C22—11121no—
188C22#1798481211yescharacteristic
189C22#1798481711yescharacteristic
190C22#1798483911yescharacteristic
191C4—42no—
192C4—5562no—
193C4—5562no—
194C4#1798481212yes—
195C4#1798481712yes—
196C2—21no—
197C2—21no—
198C2—11122no—
199C2—11122no—
200C2#2258086311yescharacteristic
201C2#2258112311yescharacteristic
202C2#2258112811yescharacteristic
subgroupG/H
centerC22#17984812
commutator, cosocleC22×C4×C139C23
FrattiniC22×C4C22×D139
Fitting#343×C139C2
socleC22×C139#120
hypercenter#343D139
hypocenterC139#2006
LCSG ▷ C22×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#17984870C2semidirect
2. #22581534C2semidirect
3. C2×#17984768C2semidirect
4. #22580854C2semidirect
5. #343×C139C2semidirect
31. C2×D556D4semidirect
44. M16×C139D4semidirect
101. C139#2006semidirect
107. #343D139semidirect
142. M16D556semidirect

Maximal subgroups

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

subgroupindex
C2×#179848702
#225815342
C2×#179847682
#225808542
#343×C1392
#2006139