Finite Group Databasebeta

← #31808391

Subgroups — #31808391

Order 17792.

258 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 41 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#22581631C211yesmaximal, characteristic, complemented
2#22581140C211yesmaximal, characteristic
3#22581131C211yesmaximal, characteristic, complemented
4#22580747C211yesmaximal, characteristic, complemented
5#22580743C211yesmaximal, characteristic, complemented
6#22580739C211yesmaximal, characteristic, complemented
7#458×C139C211yesmaximal, characteristic, complemented
8#17984872—21no—
9#17984835—21no—
10#17984829C2211yescharacteristic
11#17984824—21no—
12#17984820—21no—
13#17984810—21no—
14#17984810C2211yescharacteristic
15C4×C22⋊D139—21no—
16#17984800—21no—
17C22⋊C4×D139—21no—
18#17984769—21no—
19#17984759C2211yescharacteristic
20#17984757—21no—
21#17984757C2211yescharacteristic
22H(5,𝔽2)×C139C2211yescharacteristic
23#123×C139—21no—
24#123×C139C2211yescharacteristic
25#99×C139—21no—
26#99×C139C2211yescharacteristic
27C23×D139D411yescharacteristic
28C2×Dic278D411yescharacteristic, complemented
29C2×C22⋊D139—21no—
30C2×C22⋊D139—41no—
31C2×C4×D139—21no—
32C2×C4×D139—21no—
33#14555775—41no—
34#14555773—41no—
35#14555769—21no—
36C4⋊Dic139—21no—
37Dic139⋊C4—21no—
38Dic139⋊C4—21no—
39C22⋊Dic139—21no—
40C22⋊Dic139—21no—
41C22⋊Dic139—21no—
42C22⋊Dic139D411yescharacteristic
43(C2×D139)⋊C4—21no—
44(C2×D139)⋊C4—21no—
45(C2×D139)⋊C4—21no—
46(C2×D139)⋊C4—41no—
47C4×Dic139—21no—
48C4×Dic139D411yescharacteristic, complemented
49#41×C139—21no—
50#41×C139—41no—
51C2×D4×C139—21no—
52C2×D4×C139—21no—
53C2×D4×C139—41no—
54C2×D4×C139C2311yescharacteristic
55C22×C4×C139D411yescharacteristic, complemented
56C4⋊C4×C139—21no—
57C4⋊C4×C139—41no—
58C22⋊C4×C139—21no—
59C22⋊C4×C139—21no—
60C22⋊C4×C139—41no—
61C22⋊C4×C139D411yescharacteristic, complemented
62C22×D139—21no—
63C22×D139—42no—
64C22×D139—41no—
65Dic278—41no—
66C22⋊D139—42no—
67C22⋊D139—82no—
68C2×Dic139—21no—
69C2×Dic139—21no—
70C2×Dic139—21no—
71C2×Dic139—41no—
72C4×D139—42no—
73C4×D139—81no—
74C139⋊C8—41no—
75C23×C139—21no—
76C23×C139—21no—
77C23×C139C2×D411yescharacteristic
78C23×C139C2×D411yescharacteristic
79Q8×C139—21no—
80D4×C139—21no—
81D4×C139—41no—
82D4×C139—41no—
83D4×C139—41no—
84D4×C139—41no—
85C2×C4×C139—21no—
86C2×C4×C139—21no—
87C2×C4×C139—21no—
88C2×C4×C139—21no—
89C2×C4×C139—41no—
90C2×C4×C139—41no—
91C2×C4×C139—41no—
92C2×C4×C139C2×D411yescharacteristic
93C22×C139—21no—
94C22×C139—21no—
95C22×C139—41no—
96C22×C139—41no—
97C22×C139—41no—
98C22×C139—81no—
99C22×C139—81no—
100C22×C139#12011yescharacteristic
101C2×D139—22no—
102C2×D139—41no—
103C2×D139—41no—
104C2×D139—42no—
105C2×D139—81no—
106C4×C139—21no—
107C4×C139—41no—
108C4×C139—41no—
109C4×C139—41no—
110C4×C139—81no—
111Dic139—41no—
112Dic139—41no—
113Dic139—41no—
114Dic139—81no—
115C2×C139—21no—
116C2×C139—41no—
117C2×C139—41no—
118C2×C139—81no—
119C2×C139B(4,𝔽2)11yescharacteristic
120D139—42no—
121C139#219711yescharacteristic, complemented, Sylow
122#2197—1391nomaximal, Sylow
123#551—1391no—
124#458D13911yescharacteristic, complemented
125B(4,𝔽2)—1391no—
126#455—1391no—
127#354—1391no—
128#352—1391no—
129AΣL(1,𝔽16)—1391no—
130H(5,𝔽2)C2×D13911yescharacteristic
131#137—2781no—
132#126—2781no—
133#124—1391no—
134#123—21no—
135#123—2781no—
136#123C2×D13911yescharacteristic
137C22⋊Q8—2781no—
138#120—2781no—
139#120—1391no—
140C4×D4—2781no—
141#117—2781no—
142C2×C22⋊C4—2781no—
143#104—2781no—
144#100—1391no—
145#99—21no—
146#99—2781no—
147#99—1391no—
148#99C2×D13911yescharacteristic
149C24—1391no—
150#41—21no—
151#41—41no—
152C2×Q8—1391no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—2781no—
157C2×D4—5561no—
158C2×D4C22×D13911yescharacteristic
159C22×C4—2781no—
160C22×C4—2781no—
161C22×C4C22⋊D13911yescharacteristic, complemented
162Q16—5561no—
163SD16—5561no—
164M16—2781no—
165C4⋊C4—21no—
166C4⋊C4—41no—
167C4⋊C4—2781no—
168C4⋊C4—2781no—
169C4⋊C4—2781no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—41no—
173C22⋊C4—2781no—
174C22⋊C4—2781no—
175C22⋊C4—2781no—
176C22⋊C4—2781no—
177C22⋊C4—2781no—
178C22⋊C4—2781no—
179C22⋊C4—5561no—
180C22⋊C4—1391no—
181C22⋊C4C22⋊D13911yescharacteristic, complemented
182C42—2781no—
183C42—1391no—
184C23—21no—
185C23—21no—
186C23—2781no—
187C23—5561no—
188C23—5562no—
189C23D4×D13911yescharacteristic
190C23C2×C22⋊D13911yescharacteristic
191Q8—21no—
192Q8—5561no—
193D4—21no—
194D4—41no—
195D4—41no—
196D4—41no—
197D4—41no—
198D4—5562no—
199D4—11122no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—2781no—
208C2×C4—2781no—
209C2×C4—2781no—
210C2×C4—5562no—
211C2×C4—5561no—
212C2×C4—11121no—
213C2×C4D4×D13911yescharacteristic
214C8—5561no—
215C22—21no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—81no—
222C22—2782no—
223C22—5561no—
224C22—5562no—
225C22—5561no—
226C22—11121no—
227C22#1798481011yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—81no—
233C4—5561no—
234C4—5561no—
235C4—5561no—
236C4—11121no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—81no—
241C2—5562no—
242C2#2258113911yescharacteristic
subgroupG/H
centerC2#22581139
commutator, cosocleC2×D4×C139C23
FrattiniC2×D4C22×D139
Fitting#458×C139C2
socleC2×C139B(4,𝔽2)
prime radical#22581140C2
hypercenter#458D139
hypocenterC139#2197
LCSG ▷ C2×D4×C139 ▷ C22×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #458

Complements

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

NH ≅ G/Ntype
1. #22581631C2semidirect
3. #22581131C2semidirect
4. #22580747C2semidirect
5. #22580743C2semidirect
6. #22580739C2semidirect
7. #458×C139C2semidirect
28. C2×Dic278D4semidirect
48. C4×Dic139D4semidirect
55. C22×C4×C139D4semidirect
61. C22⋊C4×C139D4semidirect
121. C139#2197semidirect
124. #458D139semidirect
161. C22×C4C22⋊D139semidirect
181. C22⋊C4C22⋊D139semidirect

Maximal subgroups

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

subgroupindex
#225816312
#225811402
#225811312
#225807472
#225807432
#225807392
#458×C1392
#2197139