Finite Group Databasebeta

← #31797042

Subgroups — #31797042

Order 17536.

264 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 54 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#22578389C211yesmaximal, characteristic, complemented
2#22578382C211yesmaximal, characteristic, complemented
3#22578306C212yesmaximal, complemented
4#22578304C212yesmaximal, complemented
5#22577915C211yesmaximal, characteristic
6D137⋊M16—21no—
7#17984167—21no—
8#17984149—21no—
9#17984148C2212yes—
10#17984147—21no—
11#17984146C2212yes—
12C2×#14555510C2211yescharacteristic
13C22×C4×D137C411yescharacteristic, complemented
14#17984072C411yescharacteristic, complemented
15#17984072C2211yescharacteristic
16#17984068—21no—
17#17984063—21no—
18#17984058C411yescharacteristic, complemented
19#17984058C2211yescharacteristic
20C4×D548—21no—
21C4×D548—21no—
22#117×C137C411yescharacteristic, complemented
23#14555520—41no—
24#14555519—22no—
25#14555518—41no—
26C4⋊#10587739—41no—
27C22⋊#10587739—22no—
28C4×#10587739—41no—
29C23×D137C2×C411yescharacteristic
30#14555510—41no—
31#14555510—41no—
32C2×Dic274D411yescharacteristic
33C2×D548—21no—
34C2×D548—21no—
35C2×D548D411yescharacteristic
36C2×C22⋊D137C2×C411yescharacteristic
37C2×C22⋊D137C2311yescharacteristic
38C22×Dic137C2×C411yescharacteristic
39C2×C4×D137—21no—
40C2×C4×D137—21no—
41C2×C4×D137—41no—
42C2×C4×D137—41no—
43C2×C4×D137D411yescharacteristic
44C2×C4×D137D411yescharacteristic
45C4⋊Dic137—21no—
46C4⋊Dic137—21no—
47#14555487—21no—
48#14555487—21no—
49#14555487—21no—
50#14555487—21no—
51C22×C4×C137C2×C411yescharacteristic
52C4⋊C4×C137—21no—
53C22⋊C4×C137C2×C411yescharacteristic
54C22⋊C4×C137C2×C411yescharacteristic
55C42×C137—21no—
56C2×#10587739—42no—
57#11764047—42no—
58C22×D137—21no—
59C22×D137—42no—
60C22×D137—41no—
61C22×D137C22⋊C411yescharacteristic
62C22×D137C2×D411yescharacteristic
63Dic274—41no—
64D548—41no—
65D548—42no—
66D548—42no—
67C22⋊D137—41no—
68C22⋊D137—41no—
69C2×Dic137—21no—
70C2×Dic137—41no—
71C2×Dic137C22⋊C411yescharacteristic
72C2×Dic137C2×D411yescharacteristic
73C4×D137—21no—
74C4×D137—21no—
75C4×D137—21no—
76C4×D137—21no—
77C4×D137—42no—
78C4×D137—41no—
79C4×D137—41no—
80C23×C137C22×C411yescharacteristic
81C2×C4×C137—21no—
82C2×C4×C137—21no—
83C2×C4×C137—21no—
84C2×C4×C137—21no—
85C2×C4×C137C22⋊C411yescharacteristic
86C2×C4×C137C22⋊C411yescharacteristic
87C22×C137—21no—
88C22×C137—41no—
89C22×C137C2×C22⋊C411yescharacteristic
90C2×D137—22no—
91C2×D137—21no—
92C2×D137—21no—
93C2×D137—41no—
94C2×D137—42no—
95C2×D137—41no—
96C2×D137—41no—
97C2×D137—41no—
98C2×D137—42no—
99#10587739—82no—
100C4×C137—21no—
101C4×C137—41no—
102C4×C137—41no—
103C4×C137#9911yescharacteristic
104C4×C137#9911yescharacteristic
105Dic137—21no—
106Dic137—21no—
107Dic137—41no—
108C2×C137—21no—
109C2×C137—41no—
110C2×C137C2×#9911yescharacteristic
111D137—41no—
112D137—41no—
113D137—42no—
114C137#212011yescharacteristic, complemented, Sylow
115#2120—1371nomaximal, Sylow
116#525—1371no—
117#413—1371no—
118#410—1371no—
119#352—1372no—
120AΣL(1,𝔽16)—1372no—
121C2×#41—1371no—
122C23×C4—1371no—
123C2×M16—2741no—
124#123—1371no—
125#123—1371no—
126C22⋊Q8—2741no—
127#121—2741no—
128#120—1371no—
129#120—1371no—
130C4×D4—2741no—
131C4×D4—2741no—
132#117—2741no—
133#117#1058773911yescharacteristic, complemented
134#101—2741no—
135#100—1372no—
136#99—2741no—
137#99—1372no—
138C24—1371no—
139#41—5481no—
140#41—5481no—
141C2×Q8—1371no—
142C2×D4—2741no—
143C2×D4—2741no—
144C2×D4—1371no—
145C2×D4—1371no—
146C2×D4—1371no—
147C22×C4—2741no—
148C22×C4—2741no—
149C22×C4—5481no—
150C22×C4—5481no—
151C22×C4—1371no—
152C22×C4—1371no—
153C22×C4—1371no—
154C22×C4C2×#1058773911yescharacteristic
155M16—2742no—
156M16—5481no—
157C2×C8—5481no—
158C4⋊C4—21no—
159C4⋊C4—2741no—
160C4⋊C4—2741no—
161C4⋊C4—5481no—
162C22⋊C4—2741no—
163C22⋊C4—2741no—
164C22⋊C4—2741no—
165C22⋊C4—2741no—
166C22⋊C4—2742no—
167C22⋊C4C2×#1058773911yescharacteristic
168C22⋊C4C2×#1058773911yescharacteristic
169C42—21no—
170C42—5481no—
171C23—2741no—
172C23—5481no—
173C23—5482no—
174C23—1371no—
175C23—1371no—
176C23C22×#1058773911yescharacteristic
177Q8—5481no—
178D4—5481no—
179D4—5482no—
180D4—5482no—
181D4—5481no—
182D4—5481no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—2741no—
188C2×C4—2741no—
189C2×C4—2741no—
190C2×C4—2741no—
191C2×C4—2741no—
192C2×C4—5482no—
193C2×C4—5481no—
194C2×C4—5481no—
195C2×C4—5481no—
196C2×C4—5482no—
197C2×C4—1371no—
198C2×C4—1371no—
199C2×C4C22⋊#1058773911yescharacteristic
200C2×C4C22⋊#1058773911yescharacteristic
201C8—5482no—
202C22—21no—
203C22—41no—
204C22—2742no—
205C22—2741no—
206C22—2741no—
207C22—5481no—
208C22—5481no—
209C22—5481no—
210C22—5482no—
211C22—5482no—
212C22—5481no—
213C22C2×C22⋊#1058773911yescharacteristic
214C4—21no—
215C4—41no—
216C4—41no—
217C4—2741no—
218C4—2741no—
219C4—5481no—
220C4—10962no—
221C4#1798414711yescharacteristic
222C4#1798414711yescharacteristic
223C2—21no—
224C2—41no—
225C2—5481no—
226C2—5482no—
227C2—5481no—
228C2C2×#1798414711yescharacteristic
subgroupG/H
centerC4#17984147
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
Fitting#117×C137C4
socleC2×C137C2×#99
prime radical#22577915C2
cosocleC2×C22⋊D137C23
hypercenter#117#10587739
hypocenterC137#2120
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #22578389C2semidirect
2. #22578382C2semidirect
3. #22578306C2semidirect
4. #22578304C2semidirect
13. C22×C4×D137C4semidirect
14. #17984072C4semidirect
18. #17984058C4semidirect
22. #117×C137C4semidirect
114. C137#2120semidirect
133. #117#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783892
#225783822
#225783062
#225783042
#225779152
#2120137