Finite Group Databasebeta

← #31791172

Subgroups — #31791172

Order 17536.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#22577959C211yesmaximal, characteristic, complemented
2#124×D137C211yesmaximal, characteristic
3#22577764C212yesmaximal, complemented
4#22577761C212yesmaximal, complemented
5#22577760C211yesmaximal, characteristic, complemented
6#22577758C211yesmaximal, characteristic, complemented
7#22577757C211yesmaximal, characteristic, complemented
8#22577410C212yesmaximal, complemented
9#22577406C212yesmaximal
10C4×#14555493C211yesmaximal, characteristic, complemented
11#488×C137C211yesmaximal, characteristic, complemented
12C2×Q8×D137C2211yescharacteristic
13C2×D4×D137C2211yescharacteristic
14C2×#14555493C2212yes—
15#17984087C2212yescomplemented
16#17984080C2211yescharacteristic, complemented
17#17984077C2211yescharacteristic
18#17984076C2211yescharacteristic
19#17984075C2211yescharacteristic
20#17984074—22no—
21#17984066—22no—
22#17984053—22no—
23#17984046C2212yes—
24C22⋊C4×D137—22no—
25C42×D137C2211yescharacteristic
26#17984013C2212yescomplemented
27#17984012C2212yescomplemented
28#17984011C2212yescomplemented
29#17984010C2212yes—
30#17984009C2212yes—
31#17984008C2212yes—
32#17983998C2212yescomplemented
33C4×#11764040C2211yescharacteristic, complemented
34#125×C137C2211yescharacteristic, complemented
35#124×C137C2211yescharacteristic
36#103×C137C2212yescomplemented
37#102×C137C2212yes—
38C4×C8×C137C2211yescharacteristic, complemented
39C23×D137—21no—
40C2×Dic274—22no—
41C2×Dic274—21no—
42C2×Dic274C2311yescharacteristic
43Q8×D137—24no—
44C2×D548C2311yescharacteristic
45C2×C22⋊D137—21no—
46D4×D137—24no—
47C2×C4×D137—21no—
48C2×C4×D137D412yes—
49C2×C4×D137C2311yescharacteristic
50#14555493—24no—
51C2×#11764040C2312yes—
52C4⋊Dic137C2312yes—
53#14555489—22no—
54#14555489—22no—
55C22⋊Dic137—22no—
56#14555487—22no—
57#14555487—22no—
58#14555487—22no—
59C4×Dic137—22no—
60C4×Dic137D412yes—
61C4×Dic137C2311yescharacteristic
62C2×Q8×C137C2311yescharacteristic
63C2×D4×C137C2311yescharacteristic
64C2×C8×C137C2312yes—
65C4⋊C4×C137—22no—
66C4⋊C4×C137C2312yes—
67C22⋊C4×C137—22no—
68C42×C137C2311yescharacteristic
69C22×D137—21no—
70C22×D137—44no—
71C22×D137—42no—
72C22×D137C2×D411yescharacteristic
73Dic274—22no—
74Dic274—44no—
75Dic274—42no—
76D548—22no—
77C22⋊D137—42no—
78C2×Dic137—22no—
79C2×Dic137—21no—
80C2×Dic137C2×D412yes—
81C2×Dic137C2×D411yescharacteristic
82C4×D137—42no—
83C4×D137—42no—
84C4×D137#4114yes—
85#11764040—22no—
86C23×C137—21no—
87Q8×C137—22no—
88D4×C137—22no—
89C2×C4×C137—21no—
90C2×C4×C137—22no—
91C2×C4×C137C2×D412yes—
92C2×C4×C137C2411yescharacteristic
93C8×C137—22no—
94C22×C137—41no—
95C22×C137—42no—
96C22×C137C22×D411yescharacteristic
97C2×D137—22no—
98C2×D137—41no—
99C2×D137—42no—
100C2×D137—84no—
101C2×D137#12412yes—
102C4×C137—22no—
103C4×C137—41no—
104C4×C137—42no—
105C4×C137C2×#4112yes—
106Dic137—22no—
107Dic137—41no—
108Dic137—42no—
109Dic137#12412yes—
110C2×C137—81no—
111C2×C137C2×#12411yescharacteristic
112C2×C137#57512yes—
113D137—22no—
114D137—81no—
115C137#314011yescharacteristic, complemented, Sylow
116#3140—1371nomaximal, Sylow
117#533—1371no—
118C2×#124—1371no—
119#489—1372no—
120#489—1372no—
121#488—1371no—
122#488—1371no—
123#488—1371no—
124#488D13711yescharacteristic, complemented
125#419—1372no—
126#418—1372no—
127C4×M16—1371no—
128C22×Q8—1371no—
129C22×D4—1371no—
130C2×M16—1372no—
131C4⋊Q8—1372no—
132#125—1371no—
133#125C2×D13711yescharacteristic, complemented
134#124—2742no—
135#124—1371no—
136#124—1371no—
137#124—1371no—
138#124C2×D13711yescharacteristic
139C22⋊Q8—2742no—
140C4×Q8—2742no—
141#117—1372no—
142C2×C22⋊C4—2742no—
143C2×C42—1371no—
144#103—1372no—
145#103—1372no—
146#103—1372no—
147#103C2×D13712yescomplemented
148#102—1372no—
149#102—1372no—
150#102—1372no—
151#102C2×D13712yes—
152#97—1372no—
153C4×C8—1371no—
154C4×C8C2×D13711yescharacteristic, complemented
155C24—2741no—
156C2×Q8—2742no—
157C2×Q8—2741no—
158C2×Q8—2744no—
159C2×Q8—1371no—
160C2×Q8C22×D13711yescharacteristic
161C2×D4—2744no—
162C2×D4—2741no—
163C2×D4—1371no—
164C2×D4C22×D13711yescharacteristic
165C22×C4—2741no—
166C22×C4—1372no—
167C22×C4—1371no—
168M16—2744no—
169C2×C8—1372no—
170C2×C8C22×D13712yes—
171C4⋊C4—22no—
172C4⋊C4—2742no—
173C4⋊C4—2742no—
174C4⋊C4—1372no—
175C4⋊C4C22×D13712yes—
176C22⋊C4—22no—
177C22⋊C4—2742no—
178C22⋊C4—2742no—
179C22⋊C4—2742no—
180C22⋊C4—2742no—
181C42—2742no—
182C42—1372no—
183C42—1371no—
184C42C22×D13711yescharacteristic
185C23—21no—
186C23—2741no—
187C23—5484no—
188C23—5482no—
189C23—1371no—
190Q8—22no—
191Q8—2742no—
192Q8—5484no—
193Q8—5482no—
194D4—22no—
195D4—2742no—
196D4—5482no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—2742no—
200C2×C4—2741no—
201C2×C4—5482no—
202C2×C4—5482no—
203C2×C4—1372no—
204C2×C4—1371no—
205C2×C4—1374no—
206C2×C4D4×D13712yes—
207C2×C4C23×D13711yescharacteristic
208C8—22no—
209C8—2742no—
210C22—42no—
211C22—41no—
212C22—2742no—
213C22—5482no—
214C22—5481no—
215C22—10964no—
216C22—1372no—
217C22C2×D4×D13711yescharacteristic
218C4—22no—
219C4—42no—
220C4—41no—
221C4—2742no—
222C4—5482no—
223C4—5481no—
224C4—1372no—
225C4#41×D13712yes—
226C2—81no—
227C2—2742no—
228C2—10961no—
229C2#124×D13711yescharacteristic
230C2#2257820012yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#488×C137C2
socleC22×C137C22×D4
prime radicalC2×D4×D137C22
hypercenter#488D137
hypocenterC137#3140
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #22577959C2semidirect
3. #22577764C2semidirect
4. #22577761C2semidirect
5. #22577760C2semidirect
6. #22577758C2semidirect
7. #22577757C2semidirect
8. #22577410C2semidirect
10. C4×#14555493C2semidirect
11. #488×C137C2semidirect
15. #17984087C22semidirect
16. #17984080C22semidirect
26. #17984013C22semidirect
27. #17984012C22semidirect
28. #17984011C22semidirect
32. #17983998C22semidirect
33. C4×#11764040C22semidirect
34. #125×C137C22semidirect
36. #103×C137C22semidirect
38. C4×C8×C137C22semidirect
115. C137#3140semidirect
124. #488D137semidirect
133. #125C2×D137semidirect
147. #103C2×D137semidirect
154. C4×C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225779592
#124×D1372
#225777642
#225777612
#225777602
#225777582
#225777572
#225774102
#225774062
C4×#145554932
#488×C1372
#3140137