Finite Group Databasebeta

← #31793484

Subgroups — #31793484

Order 17536.

348 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 115 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#22578138C211yesmaximal, characteristic
2#22578128C211yesmaximal, characteristic, complemented
3#22577759C212yesmaximal, complemented
4#22577751C211yesmaximal, characteristic
5#22577702C212yesmaximal, complemented
6#22577628C212yesmaximal, complemented
7#22577604C211yesmaximal, characteristic
8#22577527C211yesmaximal, characteristic, complemented
9C4×#14555500C211yesmaximal, characteristic, complemented
10#22577490C212yesmaximal, complemented
11#475×C137C211yesmaximal, characteristic, complemented
12C2×#14555500C2211yescharacteristic
13C2×#14555499C2212yes—
14#17984086—21no—
15#17984085C2211yescharacteristic
16#17984079C2212yes—
17#17984078—22no—
18#17984077C2212yes—
19#17984076—21no—
20#17984073—22no—
21Q8×Dic137—21no—
22C4×Dic274C2211yescharacteristic
23C4×Dic274C2211yescharacteristic
24#17984053C2212yes—
25C4×D548—21no—
26#17984050C2212yes—
27#17984021C2211yescharacteristic
28#17984017C2212yes—
29#17984012C2211yescharacteristic
30#17984011C2211yescharacteristic
31#17984010C2212yes—
32#17984008C2212yes—
33#17984008C2212yes—
34C4×#11764040C2211yescharacteristic
35C8×Dic137C2212yes—
36C4⋊Q8×C137C2211yescharacteristic
37C4×Q8×C137C2211yescharacteristic
38#106×C137C2212yes—
39C4⋊C8×C137C2211yescharacteristic
40#103×C137C2212yes—
41C2×Dic274D411yescharacteristic
42C2×Dic274C2311yescharacteristic
43C2×D548—21no—
44C2×D548—21no—
45C2×D548C2312yes—
46C2×C4×D137—22no—
47#14555500—22no—
48#14555499—24no—
49C2×#11764040C2311yescharacteristic
50C2×#11764040C2311yescharacteristic
51C4⋊Dic137—22no—
52C4⋊Dic137D411yescharacteristic
53C4⋊Dic137C2311yescharacteristic
54#14555489—22no—
55#14555489—22no—
56#14555489—21no—
57#14555489D412yes—
58#14555487—22no—
59#14555487—22no—
60#14555487—22no—
61C4×Dic137—22no—
62C4×Dic137—21no—
63C4×Dic137C2312yes—
64C2×Q8×C137—21no—
65C2×Q8×C137C2311yescharacteristic
66C2×C8×C137C2312yes—
67C4⋊C4×C137—21no—
68C4⋊C4×C137—21no—
69C4⋊C4×C137C2312yes—
70C4⋊C4×C137C2311yescharacteristic
71C42×C137—21no—
72C42×C137C2311yescharacteristic
73C22×D137—22no—
74Dic274—22no—
75Dic274—21no—
76Dic274#4112yes—
77D548—22no—
78D548—22no—
79D548—42no—
80D548—81no—
81C2×Dic137—21no—
82C2×Dic137—21no—
83C2×Dic137C2×D412yes—
84C2×Dic137C2×D412yes—
85C4×D137—82no—
86#11764040—21no—
87#11764040—21no—
88Q8×C137—21no—
89Q8×C137—21no—
90Q8×C137—42no—
91C2×C4×C137—21no—
92C2×C4×C137—21no—
93C2×C4×C137—22no—
94C2×C4×C137C2×D411yescharacteristic
95C2×C4×C137C2×D411yescharacteristic
96C2×C4×C137C2411yescharacteristic
97C8×C137—22no—
98C22×C137C22×D411yescharacteristic
99C2×D137—42no—
100C2×D137—42no—
101C2×D137—42no—
102C4×C137—21no—
103C4×C137—41no—
104C4×C137—42no—
105C4×C137—41no—
106C4×C137#13512yes—
107C4×C137C2×#4111yescharacteristic
108C4×C1372−1+411yescharacteristic
109Dic137—22no—
110Dic137—22no—
111Dic137—22no—
112Dic137—41no—
113C2×C137#54811yescharacteristic
114C2×C137C2×#13511yescharacteristic
115C2×C137#57711yescharacteristic
116D137—82no—
117C137#336511yescharacteristic, complemented, Sylow
118#3365—1371nomaximal, Sylow
119#559—1371no—
120#557—1371no—
121#488—1372no—
122#486—1371no—
123#478—1372no—
124#475D13711yescharacteristic, complemented
125#464—1372no—
126#460—1371no—
127C8×Q8—1371no—
128C4×Q16—1371no—
129C4×SD16—1372no—
130C2×Q16—1371no—
131C2×SD16—1372no—
132C4⋊Q8—2741no—
133C4⋊Q8C2×D13711yescharacteristic
134#127—1371no—
135#125—2742no—
136#125—1372no—
137#124—2741no—
138#124—1372no—
139#123—2742no—
140C4×Q8—2741no—
141C4×Q8—1372no—
142C4×Q8—1371no—
143C4×Q8—1371no—
144C4×Q8C2×D13711yescharacteristic
145C4×D4—2741no—
146C4×D4—1372no—
147#107—1371no—
148#106C2×D13712yes—
149C4⋊C8—1372no—
150C4⋊C8C2×D13711yescharacteristic
151#103—1371no—
152#103—1371no—
153#103C2×D13712yes—
154#102—1372no—
155#102—1372no—
156#102—1372no—
157C4×C8—1372no—
158C4×C8—1371no—
159C2×Q8—21no—
160C2×Q8—1371no—
161C2×Q8—1371no—
162C2×Q8C22×D13711yescharacteristic
163C2×D4—2741no—
164C2×D4—2741no—
165C2×D4—1372no—
166C22×C4—2742no—
167Q16—2742no—
168SD16—2744no—
169C2×C8—1371no—
170C2×C8—1371no—
171C2×C8C22×D13712yes—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—2742no—
175C4⋊C4—2742no—
176C4⋊C4—2742no—
177C4⋊C4—2741no—
178C4⋊C4—1371no—
179C4⋊C4—1372no—
180C4⋊C4—1371no—
181C4⋊C4C22×D13712yes—
182C4⋊C4C22×D13711yescharacteristic
183C22⋊C4—2742no—
184C22⋊C4—2742no—
185C22⋊C4—2742no—
186C42—21no—
187C42—2742no—
188C42—2741no—
189C42—1372no—
190C42C22×D13711yescharacteristic
191C23—2742no—
192Q8—21no—
193Q8—21no—
194Q8—42no—
195Q8—2742no—
196Q8—2741no—
197Q8—1372no—
198D4—2742no—
199D4—2742no—
200D4—5482no—
201D4—10961no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—2741no—
206C2×C4—2741no—
207C2×C4—10962no—
208C2×C4—1372no—
209C2×C4—1372no—
210C2×C4D4×D13711yescharacteristic
211C2×C4D4×D13711yescharacteristic
212C2×C4C23×D13711yescharacteristic
213C8—22no—
214C8—2741no—
215C8—2741no—
216C22—5482no—
217C22—5482no—
218C22—5482no—
219C22C2×D4×D13711yescharacteristic
220C4—21no—
221C4—42no—
222C4—41no—
223C4—41no—
224C4—2742no—
225C4—2742no—
226C4—2742no—
227C4—5481no—
228C4#1798410912yes—
229C4#41×D13711yescharacteristic
230C4#1798413611yescharacteristic
231C2—10962no—
232C2#2257806311yescharacteristic
233C2C2×#1798410911yescharacteristic
234C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#475×C137C2
socleC22×C137C22×D4
prime radical#17984085C22
hypercenter#475D137
hypocenterC137#3365
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #22578128C2semidirect
3. #22577759C2semidirect
5. #22577702C2semidirect
6. #22577628C2semidirect
8. #22577527C2semidirect
9. C4×#14555500C2semidirect
10. #22577490C2semidirect
11. #475×C137C2semidirect
117. C137#3365semidirect
124. #475D137semidirect

Maximal subgroups

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

subgroupindex
#225781382
#225781282
#225777592
#225777512
#225777022
#225776282
#225776042
#225775272
C4×#145555002
#225774902
#475×C1372
#3365137