Finite Group Databasebeta

← #31806367

Subgroups — #31806367

Order 17792.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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#22581789C211yesmaximal, characteristic
2#22580969C212yesmaximal, complemented
3#22580950C212yesmaximal, complemented
4#22580927C211yesmaximal, characteristic, complemented
5#408×C139C211yesmaximal, characteristic, complemented
6#17984890C412yescomplemented
7#17984889—22no—
8#17984888—22no—
9#17984887—22no—
10#17984887C412yescomplemented
11C2×#14555785—22no—
12C2×#14555785C2212yes—
13#41×D139—24no—
14#17984801C2212yescomplemented
15#17984768—24no—
16#17984767—22no—
17#17984764—22no—
18#17984757—22no—
19C2×#41×C139C2211yescharacteristic
20C2×M16×C139C2211yescharacteristic, complemented
21C22×C8×C139C2211yescharacteristic, complemented
22C22⋊C8×C139—22no—
23#14555786—24no—
24#14555785—24no—
25#14555785—24no—
26#14555785—24no—
27#14555785—22no—
28#14555785—22no—
29#14555785C2×C414yes—
30#14555785D414yes—
31#14555784—28no—
32#14555784—24no—
33C2×Dic278—22no—
34C2×Dic278D412yes—
35Q8×D139—24no—
36C2×D556—22no—
37C2×D556D412yes—
38C2×C22⋊D139—24no—
39C2×C22⋊D139—22no—
40D4×D139—28no—
41D4×D139—24no—
42C2×C4×D139—24no—
43C2×C4×D139—22no—
44C4⋊Dic139—22no—
45C22⋊Dic139—22no—
46C4×Dic139—22no—
47#41×C139—22no—
48#41×C139—22no—
49C2×Q8×C139C2×C411yescharacteristic
50C2×D4×C139C2×C411yescharacteristic
51C2×D4×C139D412yes—
52C22×C4×C139D412yes—
53C22×C4×C139C2311yescharacteristic
54M16×C139—22no—
55C2×C8×C139—21no—
56C2×C8×C139—21no—
57C2×C8×C139—22no—
58C22×D139—24no—
59C22×D139—22no—
60Dic278—24no—
61Dic278—22no—
62Dic278C22⋊C414yes—
63D556—24no—
64D556—22no—
65D556C22⋊C414yes—
66C22⋊D139—28no—
67C22⋊D139—24no—
68C22⋊D139—24no—
69C2×Dic139—24no—
70C2×Dic139—22no—
71C2×Dic139—42no—
72C4×D139—28no—
73C4×D139—24no—
74C4×D139—24no—
75C23×C139C22⋊C412yes—
76C23×C139C2×D411yescharacteristic
77Q8×C139—22no—
78D4×C139—24no—
79D4×C139—22no—
80C2×C4×C139—22no—
81C2×C4×C139—22no—
82C2×C4×C139C22⋊C412yes—
83C2×C4×C139C22×C411yescharacteristic
84C2×C4×C139C2×D412yes—
85C2×C4×C139C2×D412yes—
86C2×C4×C139C2×D411yescharacteristic
87C8×C139—41no—
88C8×C139—41no—
89C22×C139—22no—
90C22×C139—41no—
91C22×C139—42no—
92C22×C139C2×C22⋊C411yescharacteristic
93C22×C139#12012yes—
94C2×D139—24no—
95C2×D139—82no—
96C2×D139—84no—
97C4×C139—22no—
98C4×C139C2×C22⋊C412yes—
99C4×C139#12011yescharacteristic
100C4×C139#12011yescharacteristic
101Dic139—24no—
102Dic139—82no—
103C2×C139—22no—
104C2×C139—21no—
105C2×C139—42no—
106C2×C139#37911yescharacteristic
107D139—44no—
108C139#179411yescharacteristic, complemented, Sylow
109#1794—1391nomaximal, Sylow
110#585—1391no—
111#421—1392no—
112#416—1392no—
113#410—1391no—
114#408D13911yescharacteristic, complemented
1152−1+4—2782no—
1162−1+4—1392no—
117H(5,𝔽2)—2782no—
118H(5,𝔽2)—2782no—
119H(5,𝔽2)—1392no—
120C2×#41—2784no—
121C2×#41—2782no—
122C2×#41—1392no—
123C2×#41C2×D13911yescharacteristic
124C2×M16C2×D13911yescharacteristic, complemented
125C22×C8C2×D13911yescharacteristic, complemented
126#117—1392no—
127#104—2784no—
128#103—2782no—
129#102—2782no—
130#99—2782no—
131C22⋊C8—22no—
132#41—22no—
133#41—22no—
134#41—2784no—
135#41—2784no—
136#41—2788no—
137#41—2784no—
138#41—2784no—
139#41—2784no—
140#41—2782no—
141#41—2782no—
142#41—1394no—
143#41—1394no—
144C2×Q8—2782no—
145C2×Q8—2784no—
146C2×Q8—1392no—
147C2×Q8C4×D13911yescharacteristic
148C2×D4—2782no—
149C2×D4—2788no—
150C2×D4—2784no—
151C2×D4—2784no—
152C2×D4—2782no—
153C2×D4—1392no—
154C2×D4C4×D13911yescharacteristic
155C2×D4D55612yes—
156C22×C4—2784no—
157C22×C4—2782no—
158C22×C4C22⋊D13912yes—
159C22×C4C22×D13911yescharacteristic
160M16—22no—
161C2×C8—22no—
162C2×C8—21no—
163C2×C8—21no—
164C4⋊C4—2782no—
165C22⋊C4—2782no—
166C42—2782no—
167C23—2784no—
168C23—2782no—
169C23(C2×D139)⋊C412yes—
170C23C2×D55611yescharacteristic
171Q8—22no—
172Q8—2784no—
173Q8—2782no—
174Q8—1394no—
175D4—24no—
176D4—22no—
177D4—2784no—
178D4—2788no—
179D4—2784no—
180D4—2784no—
181D4—2782no—
182D4—1394no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—2788no—
186C2×C4—2784no—
187C2×C4—2782no—
188C2×C4—2784no—
189C2×C4—2784no—
190C2×C4—5562no—
191C2×C4(C2×D139)⋊C412yes—
192C2×C4C2×C4×D13911yescharacteristic
193C2×C4D4×D13912yes—
194C2×C4D4×D13912yes—
195C2×C4C2×C22⋊D13911yescharacteristic
196C8—41no—
197C8—41no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—2784no—
202C22—11122no—
203C22—11124no—
204C22C2×(C2×D139)⋊C411yescharacteristic
205C22#1798481212yes—
206C4—22no—
207C4—2784no—
208C4—11122no—
209C4C22⋊C4×D13912yes—
210C4#1798481011yescharacteristic
211C4#1798481011yescharacteristic
212C2—22no—
213C2—21no—
214C2—42no—
215C2—5564no—
216C2#2258080811yescharacteristic
subgroupG/H
centerC4#17984810
commutatorC2×C4×C139C22×C4
FrattiniC22×C4C22×D139
Fitting#408×C139C2
socleC2×C139#379
prime radical#22581789C2
cosocleC22×C4×C139C23
hypercenter#408D139
hypocenterC139#1794
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #22580969C2semidirect
3. #22580950C2semidirect
4. #22580927C2semidirect
5. #408×C139C2semidirect
6. #17984890C4semidirect
10. #17984887C4semidirect
14. #17984801C22semidirect
20. C2×M16×C139C22semidirect
21. C22×C8×C139C22semidirect
108. C139#1794semidirect
114. #408D139semidirect
124. C2×M16C2×D139semidirect
125. C22×C8C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225817892
#225809692
#225809502
#225809272
#408×C1392
#1794139