Finite Group Databasebeta

← #31809645

Subgroups — #31809645

Order 17792.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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×#17984804C212yesmaximal, complemented
2C2×#17984804C211yesmaximal, characteristic, complemented
3C2×#17984800C211yesmaximal, characteristic, complemented
4#22580879C211yesmaximal, characteristic, complemented
5#22580875C212yesmaximal, complemented
6#22580871C212yesmaximal, complemented
7#22580870C211yesmaximal, characteristic, complemented
8#22580863C211yesmaximal, characteristic
9#22580804C211yesmaximal, characteristic, complemented
10#22580800C212yesmaximal, complemented
11#398×C139C211yesmaximal, characteristic, complemented
12C22×D556C2211yescharacteristic
13C22×D556C2212yes—
14C22×C4×D139C2212yes—
15C22×C4×D139C2211yescharacteristic
16#17984804—28no—
17#17984804—24no—
18#17984800—24no—
19C2×C4⋊Dic139C2211yescharacteristic
20C2×Dic139⋊C4C2212yes—
21C2×(C2×D139)⋊C4C2212yes—
22C2×(C2×D139)⋊C4C2212yes—
23C2×(C2×D139)⋊C4C2211yescharacteristic
24C2×(C2×D139)⋊C4C2212yes—
25C2×(C2×D139)⋊C4C2211yescharacteristic
26C2×(C2×D139)⋊C4C2212yes—
27C2×(C2×D139)⋊C4C2212yes—
28C2×C4×Dic139C2211yescharacteristic
29C2×C4×Dic139C2212yes—
30C2×C4×Dic139C2211yescharacteristic
31C139⋊#95C2212yes—
32C139⋊#95C2211yescharacteristic
33C2×C4⋊C4×C139C2211yescharacteristic
34C2×C4⋊C4×C139C2212yes—
35C2×C4⋊C4×C139C2211yescharacteristic
36#95×C139C2212yes—
37#95×C139C2211yescharacteristic
38C23×D139C2311yescharacteristic
39C23×D139C2312yes—
40C2×D556—24no—
41C2×D556—24no—
42C2×D556—24no—
43C2×D556—22no—
44C2×D556—24no—
45C22×Dic139C2312yes—
46C22×Dic139C2311yescharacteristic
47C22×Dic139C2311yescharacteristic
48C22×Dic139C2311yescharacteristic
49C2×C4×D139—22no—
50C2×C4×D139—22no—
51C2×C4×D139—24no—
52C2×C4×D139—24no—
53C2×C4×D139—24no—
54C2×C4×D139D414yescomplemented
55C4⋊Dic139—22no—
56Dic139⋊C4—24no—
57(C2×D139)⋊C4—24no—
58(C2×D139)⋊C4—24no—
59(C2×D139)⋊C4—22no—
60(C2×D139)⋊C4—24no—
61(C2×D139)⋊C4—22no—
62(C2×D139)⋊C4—24no—
63(C2×D139)⋊C4—24no—
64C4×Dic139—22no—
65C4×Dic139—24no—
66C4×Dic139D414yescomplemented
67C22×C4×C139C2311yescharacteristic
68C22×C4×C139C2312yes—
69C22×C4×C139C2311yescharacteristic
70C22×C4×C139C2311yescharacteristic
71C22×C4×C139C2312yes—
72C4⋊C4×C139—22no—
73C4⋊C4×C139—24no—
74C4⋊C4×C139—22no—
75C22×D139—21no—
76C22×D139—22no—
77C22×D139—22no—
78C22×D139—22no—
79C22×D139—21no—
80C22×D139—22no—
81C22×D139—22no—
82C22×D139—22no—
83C22×D139—22no—
84C22×D139—22no—
85C22×D139C2×D412yes—
86C22×D139#4114yes—
87D556—28no—
88D556—216no—
89C2×Dic139—22no—
90C2×Dic139—22no—
91C2×Dic139—21no—
92C2×Dic139—22no—
93C2×Dic139—22no—
94C2×Dic139—22no—
95C2×Dic139C2×D412yes—
96C2×Dic139C2×D412yes—
97C2×Dic139C2×D412yes—
98C2×Dic139#4112yes—
99C4×D139—28no—
100C4×D139—48no—
101C23×C139C2411yescharacteristic
102C2×C4×C139—22no—
103C2×C4×C139—21no—
104C2×C4×C139—21no—
105C2×C4×C139—22no—
106C2×C4×C139—22no—
107C2×C4×C139—21no—
108C2×C4×C139—22no—
109C2×C4×C139—22no—
110C2×C4×C139—22no—
111C2×C4×C139—22no—
112C2×C4×C139C2×D412yes—
113C2×C4×C139C2×D412yes—
114C2×C4×C139#4114yes—
115C22×C139C22×D411yescharacteristic
116C22×C139C22×D411yescharacteristic
117C22×C139C2×#4111yescharacteristic
118C22×C139C2×#4112yes—
119C22×C139C2×#4112yes—
120C2×D139—22no—
121C2×D139—24no—
122C2×D139—24no—
123C2×D139—24no—
124C2×D139—24no—
125C2×D139—42no—
126C2×D139—41no—
127C2×D139—42no—
128C2×D139—42no—
129C2×D139—42no—
130C2×D139—41no—
131C2×D139—42no—
132C4×C139—22no—
133C4×C139—22no—
134C4×C139—24no—
135C4×C139—42no—
136C4×C139—41no—
137Dic139—22no—
138Dic139—42no—
139Dic139—41no—
140Dic139#12114yes—
141C2×C139C2×#12111yescharacteristic
142C2×C139#52511yescharacteristic
143C2×C139#52512yes—
144C2×C139#52912yes—
145C2×C139#53211yescharacteristic
146D139—42no—
147D139—44no—
148C139#239211yescharacteristic, complemented, Sylow
149#2392—1391nomaximal, Sylow
150C2×C4×D4—1392no—
151C2×C4×D4—1391no—
152C2×#117—1391no—
153#398D13911yescharacteristic, complemented
154#397—1391no—
155#396—1392no—
156#394—1391no—
157#394—1392no—
158#392—1391no—
159C4×C4⋊C4—1391no—
160C4×C22⋊C4—1392no—
161C22×D4—1391no—
162C22×D4—1392no—
163C23×C4—1392no—
164C23×C4—1391no—
165C4×D4—2788no—
166C4×D4—2784no—
167#117—2784no—
168C2×C4⋊C4—1391no—
169C2×C4⋊C4—1392no—
170C2×C4⋊C4C2×D13911yescharacteristic
171C2×C4⋊C4C2×D13912yes—
172C2×C4⋊C4C2×D13911yescharacteristic
173C2×C22⋊C4—1391no—
174C2×C22⋊C4—1391no—
175C2×C22⋊C4—1392no—
176C2×C22⋊C4—1392no—
177C2×C22⋊C4—1392no—
178C2×C22⋊C4—1392no—
179C2×C22⋊C4—1392no—
180C2×C42—1391no—
181C2×C42—1392no—
182C2×C42—1391no—
183#95—1392no—
184#95—1391no—
185#95C2×D13912yes—
186#95C2×D13911yescharacteristic
187C24—1391no—
188C24—1392no—
189C2×D4—2784no—
190C2×D4—2784no—
191C2×D4—2784no—
192C2×D4—2782no—
193C2×D4—2784no—
194C22×C4—2782no—
195C22×C4—2782no—
196C22×C4—2784no—
197C22×C4—2784no—
198C22×C4—2784no—
199C22×C4—1392no—
200C22×C4—1391no—
201C22×C4—1391no—
202C22×C4—1394no—
203C22×C4—1391no—
204C22×C4C22×D13912yes—
205C22×C4C22×D13911yescharacteristic
206C22×C4C22×D13911yescharacteristic
207C22×C4C22×D13911yescharacteristic
208C22×C4C22×D13912yes—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—2782no—
213C4⋊C4—2784no—
214C22⋊C4—2784no—
215C22⋊C4—2784no—
216C22⋊C4—2782no—
217C22⋊C4—2784no—
218C22⋊C4—2782no—
219C22⋊C4—2784no—
220C22⋊C4—2784no—
221C42—2782no—
222C42—2784no—
223C42—1394no—
224C23—2781no—
225C23—2782no—
226C23—2782no—
227C23—2781no—
228C23—2782no—
229C23—2782no—
230C23—2782no—
231C23—2782no—
232C23—2782no—
233C23—2782no—
234C23—1392no—
235C23—1394no—
236C23C23×D13911yescharacteristic
237D4—2788no—
238D4—27816no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—2788no—
250C2×C4—2782no—
251C2×C4—2782no—
252C2×C4—2781no—
253C2×C4—2782no—
254C2×C4—2782no—
255C2×C4—2782no—
256C2×C4—5568no—
257C2×C4—1392no—
258C2×C4—1392no—
259C2×C4—1392no—
260C2×C4—1392no—
261C2×C4D4×D13912yes—
262C2×C4D4×D13912yes—
263C2×C4#1455578614yes—
264C22—2782no—
265C22—2784no—
266C22—2784no—
267C22—2784no—
268C22—2784no—
269C22—5561no—
270C22—5562no—
271C22—5562no—
272C22—5561no—
273C22—5562no—
274C22—5562no—
275C22—5562no—
276C22C2×D4×D13911yescharacteristic
277C22C2×D4×D13911yescharacteristic
278C22#41×D13911yescharacteristic
279C22#41×D13912yes—
280C22C2×#1455578612yes—
281C4—22no—
282C4—22no—
283C4—24no—
284C4—42no—
285C4—41no—
286C4—2782no—
287C4—5561no—
288C4—5562no—
289C4—1394no—
290C2—5562no—
291C2—5564no—
292C2#121×D13911yescharacteristic
293C2#2258146111yescharacteristic
294C2#2258146612yes—
295C2#2258148812yes—
296C2#2258150411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D139
commutator, socle, cosocleC23×C139C24
Fitting#398×C139C2
prime radical#22580863C2
hypercenter#398D139
hypocenterC139#2392
LCSG ▷ C23×C139 ▷ C139
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#17984804C2semidirect
2. C2×#17984804C2semidirect
3. C2×#17984800C2semidirect
4. #22580879C2semidirect
5. #22580875C2semidirect
6. #22580871C2semidirect
7. #22580870C2semidirect
9. #22580804C2semidirect
10. #22580800C2semidirect
11. #398×C139C2semidirect
54. C2×C4×D139D4semidirect
66. C4×Dic139D4semidirect
148. C139#2392semidirect
153. #398D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848042
C2×#179848002
#225808792
#225808752
#225808712
#225808702
#225808632
#225808042
#225808002
#398×C1392
#2392139