Finite Group Databasebeta

← #31807436

Subgroups — #31807436

Order 17792.

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

normal subgroups only

This group has 394 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×#17984875C211yesmaximal, characteristic, complemented
2#22581522C211yesmaximal, characteristic
3C2×#17984830C211yesmaximal, characteristic, complemented
4C2×#17984770C211yesmaximal, characteristic, complemented
5#22580961C211yesmaximal, characteristic, complemented
6C2×#17984758C211yesmaximal, characteristic, complemented
7#339×C139C211yesmaximal, characteristic, complemented
8C2×#14555785C2211yescharacteristic
9C22×Dic278C2211yescharacteristic
10C22×D556C2211yescharacteristic
11#17984875—22no—
12#17984875—22no—
13C2×Dic556—21no—
14C2×#14555774—21no—
15C2×#14555769C2211yescharacteristic
16#17984839—21no—
17#17984830—21no—
18#17984830—22no—
19#17984830—22no—
20#17984817—22no—
21#17984812—22no—
22C2×(C2×D139)⋊C4—22no—
23#17984770—22no—
24#17984765—22no—
25#17984758—22no—
26C2×M16×C139C2211yescharacteristic
27#117×C139C2211yescharacteristic
28C2×C42×C139C2211yescharacteristic
29C23×D139D412yes—
30#14555785—21no—
31#14555785—21no—
32#14555785—21no—
33#14555785—21no—
34C2×Dic278—21no—
35C2×Dic278—21no—
36C2×Dic278—21no—
37C2×Dic278—21no—
38C2×Dic278—21no—
39C2×Dic278D411yescharacteristic
40C2×Dic278D412yescomplemented
41C2×D556—24no—
42C2×D556—21no—
43C2×D556—21no—
44C2×D556—21no—
45C2×D556—21no—
46C2×D556—21no—
47C2×D556—21no—
48C2×D556D411yescharacteristic
49C2×C22⋊D139—21no—
50C22×Dic139—21no—
51C2×C4×D139—21no—
52Dic556—24no—
53#14555774—24no—
54#14555769—21no—
55#14555769—21no—
56C2×C139⋊C8—21no—
57(C2×D139)⋊C4—28no—
58(C2×D139)⋊C4—24no—
59C22×C4×C139—21no—
60C22×C4×C139C2311yescharacteristic
61M16×C139—21no—
62M16×C139D412yescomplemented
63C2×C8×C139—21no—
64C4⋊C4×C139D412yes—
65C22⋊C4×C139—21no—
66C42×C139—21no—
67C42×C139—21no—
68C42×C139D412yescomplemented
69C22×D139—22no—
70C22×D139—22no—
71C22×D139—21no—
72C22×D139—22no—
73C22×D139—44no—
74Dic278—22no—
75Dic278—21no—
76Dic278—22no—
77Dic278—21no—
78Dic278—41no—
79Dic278—41no—
80D556—21no—
81D556—21no—
82D556—41no—
83D556—41no—
84D556—41no—
85D556—41no—
86D556—84no—
87C22⋊D139—41no—
88C22⋊D139—41no—
89C2×Dic139—21no—
90C2×Dic139—22no—
91C2×Dic139—41no—
92C2×Dic139—41no—
93C4×D139—41no—
94C4×D139—41no—
95C139⋊C8—41no—
96C23×C139C2×D411yescharacteristic
97C2×C4×C139—22no—
98C2×C4×C139—22no—
99C2×C4×C139—42no—
100C2×C4×C139C2×D411yescharacteristic
101C2×C4×C139C2×D411yescharacteristic
102C2×C4×C139C2×D411yescharacteristic
103C2×C4×C139C2×D411yescharacteristic
104C2×C4×C139C2×D411yescharacteristic
105C2×C4×C139#4111yescharacteristic
106C8×C139—22no—
107C22×C139—22no—
108C22×C139#12011yescharacteristic
109C22×C139#12011yescharacteristic
110C22×C139#12111yescharacteristic
111C2×D139—44no—
112C2×D139—41no—
113C2×D139—42no—
114C2×D139—42no—
115C2×D139—44no—
116C2×D139—82no—
117C4×C139—42no—
118C4×C139—42no—
119C4×C139#12011yescharacteristic
120C4×C139#12111yescharacteristic
121C4×C139#12111yescharacteristic
122C4×C139#12711yescharacteristic
123Dic139—41no—
124Dic139—42no—
125C2×C139—21no—
126C2×C139—21no—
127C2×C139#39211yescharacteristic
128C2×C139#45512yes—
129D139—81no—
130D139—82no—
131C139#200411yescharacteristic, complemented, Sylow
132#2004—1391nomaximal, Sylow
133C2×#137—1391no—
134#535—1391no—
135C2×#124—1391no—
136C2×#104—1391no—
137#419—1391no—
138C2×#100—1391no—
139#339D13911yescharacteristic, complemented
140C2×#41—1391no—
141C22×Q8—1391no—
142C22×D4—1391no—
143#137—2782no—
144#137—2782no—
145C2×Q16—2781no—
146C2×SD16—2781no—
147C2×M16—1391no—
148C2×M16C2×D13911yescharacteristic
149#127—2781no—
150#124—2782no—
151#124—2781no—
152#124—2782no—
153#121—2782no—
154#120—2782no—
155#117C2×D13911yescharacteristic
156C2×C22⋊C4—2782no—
157C2×C42C2×D13911yescharacteristic
158#104—2782no—
159#103—2782no—
160#100—2782no—
161C24—1392no—
162#41—2781no—
163#41—2781no—
164#41—2781no—
165#41—2781no—
166C2×Q8—2781no—
167C2×Q8—2781no—
168C2×Q8—2781no—
169C2×Q8—2781no—
170C2×Q8—2781no—
171C2×Q8—1391no—
172C2×Q8—1392no—
173C2×D4—2784no—
174C2×D4—2781no—
175C2×D4—2781no—
176C2×D4—2781no—
177C2×D4—2781no—
178C2×D4—2781no—
179C2×D4—2781no—
180C2×D4—2781no—
181C2×D4—1391no—
182C22×C4—21no—
183C22×C4—2781no—
184C22×C4—2781no—
185C22×C4C22×D13911yescharacteristic
186Q16—2784no—
187SD16—2784no—
188M16—21no—
189M16—2781no—
190M16—2781no—
191M16D55612yescomplemented
192C2×C8—21no—
193C2×C8—2781no—
194C4⋊C4D55612yes—
195C22⋊C4—21no—
196C22⋊C4—2788no—
197C22⋊C4—2784no—
198C42—21no—
199C42—21no—
200C42D55612yescomplemented
201C23—2782no—
202C23—2782no—
203C23—2781no—
204C23—2782no—
205C23—5564no—
206C23D4×D13911yescharacteristic
207Q8—2782no—
208Q8—2781no—
209Q8—2782no—
210Q8—2781no—
211Q8—5561no—
212Q8—5561no—
213D4—2781no—
214D4—2781no—
215D4—5561no—
216D4—5561no—
217D4—5561no—
218D4—5561no—
219D4—5561no—
220D4—5561no—
221D4—11124no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—2782no—
226C2×C4—2781no—
227C2×C4—5561no—
228C2×C4—5561no—
229C2×C4—5561no—
230C2×C4—5561no—
231C2×C4D4×D13911yescharacteristic
232C2×C4D4×D13911yescharacteristic
233C2×C4C2×D55611yescharacteristic
234C2×C4C2×D55611yescharacteristic
235C2×C4C2×D55611yescharacteristic
236C2×C4#1455578611yescharacteristic
237C8—22no—
238C8—5561no—
239C22—22no—
240C22—5564no—
241C22—5561no—
242C22—5562no—
243C22—5562no—
244C22—5564no—
245C22—11122no—
246C22#1798481211yescharacteristic
247C22#1798481211yescharacteristic
248C22#1798481711yescharacteristic
249C4—42no—
250C4—42no—
251C4—5561no—
252C4—5562no—
253C4#1798481211yescharacteristic
254C4#1798481711yescharacteristic
255C4#1798481711yescharacteristic
256C4#1798483911yescharacteristic
257C2—21no—
258C2—21no—
259C2—11122no—
260C2—11121no—
261C2#2258086311yescharacteristic
262C2#2258113212yes—
subgroupG/H
centerC22#17984812
commutator, cosocleC22×C4×C139C23
FrattiniC22×C4C22×D139
Fitting#339×C139C2
socleC22×C139#120
prime radical#22581522C2
hypercenter#339D139
hypocenterC139#2004
LCSG ▷ C22×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. C2×#17984875C2semidirect
3. C2×#17984830C2semidirect
4. C2×#17984770C2semidirect
5. #22580961C2semidirect
6. C2×#17984758C2semidirect
7. #339×C139C2semidirect
40. C2×Dic278D4semidirect
62. M16×C139D4semidirect
68. C42×C139D4semidirect
131. C139#2004semidirect
139. #339D139semidirect
191. M16D556semidirect
200. C42D556semidirect

Maximal subgroups

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

subgroupindex
C2×#179848752
#225815222
C2×#179848302
C2×#179847702
#225809612
C2×#179847582
#339×C1392
#2004139