Finite Group Databasebeta

← #31796219

Subgroups — #31796219

Order 17536.

264 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 66 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#22578422C211yesmaximal, characteristic, complemented
2#22578347C211yesmaximal, characteristic, complemented
3#22578255C212yesmaximal, complemented
4#22578254C212yesmaximal, complemented
5#22577950C211yesmaximal, characteristic
6C2×C4⋊#10587739—21no—
7C4⋊#11764047—21no—
8#17984145C2212yes—
9C4×#11764047—21no—
10(C2×#10587739)⋊C4—21no—
11(C2×#10587739)⋊C4C2212yes—
12C2×#14555510—21no—
13#17984087C411yescharacteristic, complemented
14#17984084C411yescharacteristic, complemented
15#17984074—21no—
16#17984063C2211yescharacteristic
17#17984061—21no—
18#17984059C2211yescharacteristic
19C4×D548C411yescharacteristic, complemented
20C4×C22⋊D137—21no—
21C42×D137C2211yescharacteristic
22C4×D4×C137C411yescharacteristic, complemented
23C22×#10587739—22no—
24C2×#11764047—22no—
25C4⋊#10587739—42no—
26#14555510—42no—
27#14555510—42no—
28C2×Dic274—21no—
29C2×D548—21no—
30C2×D548C2×C411yescharacteristic
31C2×C22⋊D137—21no—
32C2×C22⋊D137—21no—
33C2×C22⋊D137—21no—
34C2×C4×D137—21no—
35C2×C4×D137D411yescharacteristic
36C2×C4×D137D411yescharacteristic
37C2×C4×D137C2311yescharacteristic
38C4⋊Dic137C2×C411yescharacteristic
39#14555489—21no—
40C22⋊Dic137—21no—
41#14555487—21no—
42#14555487—21no—
43C4×Dic137C2×C411yescharacteristic
44C4×Dic137D411yescharacteristic
45C4×Dic137D411yescharacteristic
46C2×D4×C137C2×C411yescharacteristic
47C22×C4×C137—21no—
48C4⋊C4×C137C2×C411yescharacteristic
49C22⋊C4×C137—21no—
50C42×C137C2×C411yescharacteristic
51C2×#10587739—42no—
52C2×#10587739—42no—
53C2×#10587739—42no—
54#11764047—42no—
55C22×D137—21no—
56C22×D137C2×D411yescharacteristic
57Dic274—42no—
58D548—41no—
59D548—41no—
60D548—41no—
61C22⋊D137—42no—
62C22⋊D137—41no—
63C22⋊D137—41no—
64C22⋊D137—42no—
65C2×Dic137—21no—
66C2×Dic137C22⋊C411yescharacteristic
67C2×Dic137C22⋊C411yescharacteristic
68C2×Dic137C2×D411yescharacteristic
69C4×D137—41no—
70C4×D137—41no—
71C4×D137—42no—
72C4×D137D812yes—
73C4×D137SD1612yes—
74C23×C137—21no—
75D4×C137—41no—
76C2×C4×C137—21no—
77C2×C4×C137—42no—
78C2×C4×C137C22⋊C411yescharacteristic
79C2×C4×C137C22⋊C411yescharacteristic
80C2×C4×C137C22×C411yescharacteristic
81C22×C137—41no—
82C22×C137—41no—
83C22×C137—41no—
84C22×C137C2×C22⋊C411yescharacteristic
85C2×D137—21no—
86C2×D137—21no—
87C2×D137—41no—
88C2×D137—41no—
89C2×D137—41no—
90C2×D137C2×D811yescharacteristic
91C2×D137C2×SD1611yescharacteristic
92#10587739—82no—
93C4×C137—21no—
94C4×C137—21no—
95C4×C137—41no—
96C4×C137#10212yes—
97Dic137—21no—
98Dic137—21no—
99Dic137—41no—
100Dic137#10212yes—
101C2×C137—81no—
102C2×C137#41011yescharacteristic
103C2×C137C2×#10211yescharacteristic
104C2×C137#42111yescharacteristic
105D137—21no—
106D137—21no—
107D137—81no—
108C137#151711yescharacteristic, complemented, Sylow
109#1517—1371nomaximal, Sylow
110#532—1371no—
111#431—1371no—
112#382—1371no—
113#327—1372no—
114#327—1372no—
115C2×#41—2741no—
116C4⋊Q8—1371no—
117#127—1371no—
118#124—2741no—
119#121—2741no—
120#121—1371no—
121#121—1371no—
122C4×D4—2741no—
123C4×D4—1371no—
124C4×D4#1058773911yescharacteristic, complemented
125C2×C4⋊C4—2741no—
126C2×C42—1371no—
127C4⋊C8—2741no—
128C22⋊C8—1372no—
129C4×C8—2741no—
130#95—2741no—
131#95—1372no—
132#41—5482no—
133#41—5482no—
134C2×Q8—2741no—
135C2×D4—2741no—
136C2×D4—2741no—
137C2×D4—2741no—
138C2×D4—2741no—
139C2×D4—1371no—
140C2×D4C2×#1058773911yescharacteristic
141C22×C4—21no—
142C22×C4—2741no—
143C22×C4—2742no—
144C22×C4—1371no—
145C22×C4—1371no—
146C22×C4—1371no—
147C2×C8—2742no—
148C4⋊C4—2741no—
149C4⋊C4—5482no—
150C4⋊C4—1371no—
151C4⋊C4C2×#1058773911yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—2741no—
154C22⋊C4—2741no—
155C22⋊C4—2741no—
156C42—1371no—
157C42—1371no—
158C42—1371no—
159C42C2×#1058773911yescharacteristic
160C23—21no—
161C23—2741no—
162C23—1371no—
163Q8—5482no—
164D4—41no—
165D4—5481no—
166D4—5481no—
167D4—5481no—
168D4—5482no—
169D4—5481no—
170D4—5481no—
171D4—5482no—
172C2×C4—21no—
173C2×C4—42no—
174C2×C4—2741no—
175C2×C4—5481no—
176C2×C4—5481no—
177C2×C4—5482no—
178C2×C4—5482no—
179C2×C4—5482no—
180C2×C4—5482no—
181C2×C4—1371no—
182C2×C4—1371no—
183C2×C4—1371no—
184C2×C4—1372no—
185C2×C4—1372no—
186C2×C4C22⋊#1058773911yescharacteristic
187C2×C4C22⋊#1058773911yescharacteristic
188C2×C4C22×#1058773911yescharacteristic
189C8—5482no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22—2741no—
194C22—2741no—
195C22—5481no—
196C22—5481no—
197C22—5481no—
198C22—1371no—
199C22—1371no—
200C22C2×C22⋊#1058773911yescharacteristic
201C4—21no—
202C4—21no—
203C4—41no—
204C4—2741no—
205C4—2741no—
206C4—5481no—
207C4—10962no—
208C4—1372no—
209C4#1798415012yes—
210C2—81no—
211C2—2741no—
212C2—2741no—
213C2—10961no—
214C2#2257838211yescharacteristic
215C2C2×#1798415011yescharacteristic
216C2#2257840011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4×D4×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577950C2
cosocleC2×C4×D137C23
hypercenterC4×D4#10587739
hypocenterC137#1517
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #22578422C2semidirect
2. #22578347C2semidirect
3. #22578255C2semidirect
4. #22578254C2semidirect
13. #17984087C4semidirect
14. #17984084C4semidirect
19. C4×D548C4semidirect
22. C4×D4×C137C4semidirect
108. C137#1517semidirect
124. C4×D4#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784222
#225783472
#225782552
#225782542
#225779502
#1517137