Finite Group Databasebeta

← #31814708

Subgroups — #31814708

Order 17792.

696 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 165 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×#17984831C211yesmaximal, characteristic, complemented
2C2×#17984818C211yesmaximal, characteristic, complemented
3C2×#17984817C211yesmaximal, characteristic
4C2×C42×D139C211yesmaximal, characteristic, complemented
5#22580855C211yesmaximal, characteristic, complemented
6#22580842C214yesmaximal, complemented
7#22580831C214yesmaximal, complemented
8#22580818C211yesmaximal, characteristic, complemented
9#391×C139C211yesmaximal, characteristic, complemented
10C22×D556C2212yes—
11C22×D556C2211yescharacteristic
12C22×C4×D139C2212yes—
13C22×C4×D139C2211yescharacteristic
14#17984831—24no—
15#17984818—24no—
16#17984817—24no—
17C2×C4⋊Dic139C2211yescharacteristic
18C2×Dic139⋊C4C2212yes—
19C2×(C2×D139)⋊C4C2212yes—
20C2×(C2×D139)⋊C4C2214yes—
21C2×(C2×D139)⋊C4C2214yes—
22C2×(C2×D139)⋊C4C2212yes—
23C2×C4×Dic139C2212yes—
24C2×C4×Dic139C2211yescharacteristic
25C42×D139—24no—
26C139⋊#95C2214yes—
27C22×Q8×C139C2211yescharacteristic
28C2×C4⋊C4×C139C2211yescharacteristic
29C2×C42×C139C2211yescharacteristic
30#95×C139C2214yes—
31C23×D139C2312yes—
32C23×D139C2311yescharacteristic
33C2×D556—24no—
34C2×D556—28no—
35C2×D556—22no—
36C2×D556—22no—
37C2×D556—22no—
38C22×Dic139C2312yes—
39C22×Dic139C2311yescharacteristic
40C22×Dic139C2312yes—
41C2×C4×D139—24no—
42C2×C4×D139—24no—
43C2×C4×D139—22no—
44C2×C4×D139—22no—
45C2×C4×D139—24no—
46C2×C4×D139D414yes—
47C4⋊Dic139—22no—
48Dic139⋊C4—24no—
49(C2×D139)⋊C4—24no—
50(C2×D139)⋊C4—24no—
51(C2×D139)⋊C4—28no—
52(C2×D139)⋊C4—28no—
53C4×Dic139—24no—
54C4×Dic139D414yescomplemented
55C2×Q8×C139—22no—
56C2×Q8×C139—22no—
57C2×Q8×C139—22no—
58C22×C4×C139C2312yes—
59C22×C4×C139C2312yes—
60C22×C4×C139C2312yes—
61C22×C4×C139C2311yescharacteristic
62C4⋊C4×C139—22no—
63C42×C139—22no—
64C22×D139—21no—
65C22×D139—21no—
66C22×D139—22no—
67C22×D139—22no—
68C22×D139—22no—
69C22×D139—24no—
70C22×D139—24no—
71C22×D139—22no—
72C22×D139C2×D412yes—
73C22×D139#4114yes—
74D556—48no—
75D556—44no—
76C2×Dic139—21no—
77C2×Dic139—22no—
78C2×Dic139—21no—
79C2×Dic139—22no—
80C2×Dic139—22no—
81C2×Dic139—24no—
82C2×Dic139C2×D414yes—
83C2×Dic139C2×D412yes—
84C4×D139—216no—
85C4×D139—28no—
86C23×C139C2411yescharacteristic
87Q8×C139—44no—
88C2×C4×C139—22no—
89C2×C4×C139—22no—
90C2×C4×C139—22no—
91C2×C4×C139—22no—
92C2×C4×C139—22no—
93C2×C4×C139—22no—
94C2×C4×C139—24no—
95C2×C4×C139C2×D412yes—
96C2×C4×C139C2×D412yes—
97C2×C4×C139#4112yes—
98C2×C4×C139#4114yes—
99C22×C139C22×D411yescharacteristic
100C22×C139C22×D411yescharacteristic
101C22×C139C2×#4112yes—
102C22×C139C2×#4111yescharacteristic
103C22×C139C2×#4112yes—
104C2×D139—22no—
105C2×D139—24no—
106C2×D139—24no—
107C2×D139—22no—
108C2×D139—42no—
109C2×D139—44no—
110C2×D139—42no—
111C2×D139—44no—
112C2×D139—42no—
113C2×D139#12414yescomplemented
114C4×C139—24no—
115C4×C139—42no—
116C4×C139—42no—
117C4×C139#12114yes—
118Dic139—22no—
119Dic139—24no—
120Dic139—42no—
121C2×C139C2×#12111yescharacteristic
122C2×C139#52512yes—
123C2×C139C2×#12411yescharacteristic
124C2×C139#52912yes—
125C2×C139#53211yescharacteristic
126D139—24no—
127D139—82no—
128C139#285011yescharacteristic, complemented, Sylow
129#2850—1391nomaximal, Sylow
130C2×#124—1391no—
131C2×C22⋊Q8—1391no—
132C2×#121—1391no—
133C22×C42—1391no—
134#391D13911yescharacteristic, complemented
135#390—1391no—
136#388—1394no—
137#386—1394no—
138#382—1391no—
139C22×Q8C2×D13911yescharacteristic
140C22×D4—1391no—
141C22×D4—1392no—
142C23×C4—1392no—
143C23×C4—1391no—
144#124—2784no—
145C22⋊Q8—2784no—
146#121—2784no—
147C2×C4⋊C4—1392no—
148C2×C4⋊C4—1391no—
149C2×C4⋊C4C2×D13911yescharacteristic
150C2×C22⋊C4—1392no—
151C2×C22⋊C4—1392no—
152C2×C22⋊C4—1394no—
153C2×C22⋊C4—1394no—
154C2×C42—2784no—
155C2×C42—1392no—
156C2×C42—1391no—
157C2×C42C2×D13911yescharacteristic
158#95—1394no—
159#95C2×D13914yes—
160C24—1392no—
161C24—1391no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×D4—2784no—
166C2×D4—2788no—
167C2×D4—2782no—
168C2×D4—2782no—
169C2×D4—2782no—
170C22×C4—2784no—
171C22×C4—2784no—
172C22×C4—2782no—
173C22×C4—2782no—
174C22×C4—2784no—
175C22×C4—1392no—
176C22×C4—1392no—
177C22×C4—1391no—
178C22×C4—1394no—
179C22×C4C22×D13912yes—
180C22×C4C22×D13912yes—
181C22×C4C22×D13912yes—
182C22×C4C22×D13911yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—2784no—
185C4⋊C4—2782no—
186C22⋊C4—2784no—
187C22⋊C4—2784no—
188C22⋊C4—2788no—
189C22⋊C4—2788no—
190C42—22no—
191C42—2784no—
192C42—1394no—
193C23—2781no—
194C23—2781no—
195C23—2782no—
196C23—2782no—
197C23—2782no—
198C23—2784no—
199C23—2782no—
200C23—2784no—
201C23—1392no—
202C23—1394no—
203C23C23×D13911yescharacteristic
204Q8—44no—
205D4—5568no—
206D4—5564no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—27816no—
215C2×C4—2781no—
216C2×C4—2782no—
217C2×C4—2781no—
218C2×C4—2782no—
219C2×C4—2788no—
220C2×C4—2782no—
221C2×C4—2784no—
222C2×C4—1394no—
223C2×C4—1392no—
224C2×C4D4×D13912yes—
225C2×C4D4×D13912yes—
226C2×C4#1455578512yes—
227C2×C4#1455578614yes—
228C22—2782no—
229C22—2782no—
230C22—2784no—
231C22—2784no—
232C22—5562no—
233C22—5562no—
234C22—5564no—
235C22—5562no—
236C22—5564no—
237C22—1394no—
238C22C2×D4×D13911yescharacteristic
239C22C2×D4×D13911yescharacteristic
240C22#41×D13912yes—
241C22C2×#1455578511yescharacteristic
242C22C2×#1455578612yes—
243C4—24no—
244C4—42no—
245C4—42no—
246C4—2782no—
247C4—2784no—
248C4—5562no—
249C4#1798481514yes—
250C2—2784no—
251C2—11122no—
252C2C2×#1798481511yescharacteristic
253C2#2258146612yes—
254C2#124×D13911yescharacteristic
255C2#2258149012yes—
256C2#2258150411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D139
commutator, socle, cosocleC23×C139C24
Fitting#391×C139C2
prime radicalC2×#17984817C2
hypercenter#391D139
hypocenterC139#2850
LCSG ▷ C23×C139 ▷ C139
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#17984831C2semidirect
2. C2×#17984818C2semidirect
4. C2×C42×D139C2semidirect
5. #22580855C2semidirect
6. #22580842C2semidirect
7. #22580831C2semidirect
8. #22580818C2semidirect
9. #391×C139C2semidirect
54. C4×Dic139D4semidirect
113. C2×D139#124semidirect
128. C139#2850semidirect
134. #391D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848312
C2×#179848182
C2×#179848172
C2×C42×D1392
#225808552
#225808422
#225808312
#225808182
#391×C1392
#2850139