Finite Group Databasebeta

← #31791033

Subgroups — #31791033

Order 17536.

418 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 121 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#22578008C211yesmaximal, characteristic, complemented
2#22577877C211yesmaximal, characteristic, complemented
3#22577728C212yesmaximal, complemented
4#22577704C212yesmaximal, complemented
5#22577599C212yesmaximal, complemented
6#22577539C212yesmaximal, complemented
7#22577516C212yesmaximal, complemented
8C4×D1096C212yesmaximal, complemented
9#433×C137C211yesmaximal, characteristic, complemented
10C22×D548C2211yescharacteristic
11C2×D1096C2212yes—
12C2×D1096C2212yes—
13#17984085C2211yescharacteristic, complemented
14#17984080C2211yescharacteristic, complemented
15#17984073—22no—
16#17984062C2212yes—
17#17984061—22no—
18C4×D548C2212yes—
19C4×D548C2212yes—
20C4×D548C2212yescomplemented
21#17984050—22no—
22#17984047C2211yescharacteristic
23C22⋊C4×D137—22no—
24#17984022C2212yescomplemented
25#17984020C2212yescomplemented
26#17984010C2212yes—
27#17984010C2212yes—
28#17984010C2212yescomplemented
29#17984010C2212yescomplemented
30#117×C137C2211yescharacteristic
31C4⋊C8×C137C2212yescomplemented
32C22⋊C8×C137C2212yes—
33#97×C137C2211yescharacteristic, complemented
34C4×C8×C137C2211yescharacteristic, complemented
35C23×D137—21no—
36C2×D548—24no—
37C2×D548—21no—
38C2×D548—22no—
39C2×D548C2312yes—
40C2×D548C2312yes—
41C2×C22⋊D137—22no—
42C2×C4×D137—22no—
43C2×C4×D137—22no—
44C2×C4×D137—22no—
45D1096—24no—
46D1096—24no—
47C4⋊Dic137C2312yes—
48C4⋊Dic137C2312yes—
49#14555489—22no—
50C22⋊Dic137—21no—
51#14555487—22no—
52#14555487—22no—
53#14555487—22no—
54#14555487—22no—
55C4×Dic137—21no—
56C22×C4×C137C2311yescharacteristic
57C2×C8×C137C2312yes—
58C2×C8×C137C2312yes—
59C4⋊C4×C137D412yes—
60C22⋊C4×C137D412yes—
61C42×C137C2311yescharacteristic
62C42×C137C2311yescharacteristic
63C22×D137—22no—
64C22×D137—24no—
65C22×D137—22no—
66C22×D137—82no—
67D548—24no—
68D548—22no—
69D548—22no—
70D548—22no—
71D548—82no—
72D548#4114yes—
73C22⋊D137—82no—
74C2×Dic137—22no—
75C2×Dic137—22no—
76C4×D137—82no—
77C4×D137—82no—
78C4×D137—82no—
79C23×C137C2×D411yescharacteristic
80C2×C4×C137—22no—
81C2×C4×C137C2×D412yes—
82C2×C4×C137C2×D412yes—
83C2×C4×C137C2×D411yescharacteristic
84C2×C4×C137C2411yescharacteristic
85C8×C137—22no—
86C8×C137—22no—
87C22×C137—21no—
88C22×C137—22no—
89C22×C137C22×D411yescharacteristic
90C2×D137—24no—
91C2×D137—42no—
92C2×D137—42no—
93C2×D137—42no—
94C2×D137—42no—
95C2×D137—42no—
96C2×D137—84no—
97C4×C137—21no—
98C4×C137—22no—
99C4×C137—22no—
100C4×C137C2×#4112yes—
101Dic137—42no—
102Dic137—42no—
103C2×C137—41no—
104C2×C137#52511yescharacteristic
105C2×C137#57612yes—
106D137—44no—
107D137—82no—
108C137#312211yescharacteristic, complemented, Sylow
109#3122—1371nomaximal, Sylow
110#540—1371no—
111#518—1371no—
112#482—1372no—
113#478—1372no—
114#459—1372no—
115#447—1372no—
116#442—1372no—
117C4×D8—1372no—
118#433D13711yescharacteristic, complemented
119C22×D4—1371no—
120C2×D8—1372no—
121C2×D8—1372no—
122#127—1371no—
123#125—1371no—
124#123—2742no—
125#121—2742no—
126#121—1372no—
127C4×D4—2742no—
128C4×D4—1372no—
129C4×D4—1372no—
130C4×D4—1372no—
131#117—1371no—
132#117C2×D13711yescharacteristic
133C2×C22⋊C4—2742no—
134#107—1372no—
135#106—1372no—
136C4⋊C8C2×D13712yescomplemented
137#102—1372no—
138#102—1372no—
139#102—1372no—
140#102—1372no—
141C22⋊C8C2×D13712yes—
142#97C2×D13711yescharacteristic, complemented
143C4×C8C2×D13711yescharacteristic, complemented
144C24—2741no—
145C2×D4—2744no—
146C2×D4—2742no—
147C2×D4—2741no—
148C2×D4—2742no—
149C2×D4—1372no—
150C2×D4—1372no—
151C22×C4—2742no—
152C22×C4—2742no—
153C22×C4—2742no—
154C22×C4C22×D13711yescharacteristic
155D8—2744no—
156D8—2744no—
157C2×C8C22×D13712yes—
158C2×C8C22×D13712yes—
159C4⋊C4—2742no—
160C4⋊C4—1372no—
161C4⋊C4—1372no—
162C4⋊C4D54812yes—
163C22⋊C4—2741no—
164C22⋊C4—2742no—
165C22⋊C4—2742no—
166C22⋊C4—2742no—
167C22⋊C4—2742no—
168C22⋊C4D54812yes—
169C42—2741no—
170C42C22×D13711yescharacteristic
171C42C22×D13711yescharacteristic
172C23—2744no—
173C23—2742no—
174C23—2742no—
175C23—10962no—
176C23C2×D54811yescharacteristic
177D4—2744no—
178D4—2742no—
179D4—2742no—
180D4—2742no—
181D4—10962no—
182D4—10962no—
183D4—1374no—
184C2×C4—22no—
185C2×C4—2742no—
186C2×C4—2742no—
187C2×C4—10962no—
188C2×C4—10962no—
189C2×C4—10962no—
190C2×C4C2×D54811yescharacteristic
191C2×C4C2×D54812yes—
192C2×C4C2×D54812yes—
193C2×C4C23×D13711yescharacteristic
194C8—22no—
195C8—22no—
196C22—22no—
197C22—21no—
198C22—2744no—
199C22—5482no—
200C22—5482no—
201C22—5482no—
202C22—5482no—
203C22—5482no—
204C22—10964no—
205C22C22×D54811yescharacteristic
206C4—22no—
207C4—22no—
208C4—21no—
209C4—5482no—
210C4—5482no—
211C4#41×D13712yes—
212C2—41no—
213C2—5484no—
214C2—10962no—
215C2#2257791511yescharacteristic
216C2#2257820812yes—
subgroupG/H
centerC22C22×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#433×C137C2
socleC22×C137C22×D4
hypercenter#433D137
hypocenterC137#3122
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #22578008C2semidirect
2. #22577877C2semidirect
3. #22577728C2semidirect
4. #22577704C2semidirect
5. #22577599C2semidirect
6. #22577539C2semidirect
7. #22577516C2semidirect
8. C4×D1096C2semidirect
9. #433×C137C2semidirect
13. #17984085C22semidirect
14. #17984080C22semidirect
20. C4×D548C22semidirect
24. #17984022C22semidirect
25. #17984020C22semidirect
28. #17984010C22semidirect
29. #17984010C22semidirect
31. C4⋊C8×C137C22semidirect
33. #97×C137C22semidirect
34. C4×C8×C137C22semidirect
108. C137#3122semidirect
118. #433D137semidirect
136. C4⋊C8C2×D137semidirect
142. #97C2×D137semidirect
143. C4×C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780082
#225778772
#225777282
#225777042
#225775992
#225775392
#225775162
C4×D10962
#433×C1372
#3122137