Finite Group Databasebeta

← #31796933

Subgroups — #31796933

Order 17536.

312 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 70 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#22578406C211yesmaximal, characteristic, complemented
2C2×#17984149C212yesmaximal, complemented
3C2×#17984148C212yesmaximal
4C4×#14555519C211yesmaximal, characteristic, complemented
5C2×#17984074C211yesmaximal, characteristic
6C2×#14555519C2212yes—
7C2×#14555519C2212yes—
8#17984155—21no—
9C4⋊#11764047—21no—
10#17984149—24no—
11#17984148—24no—
12#17984143—21no—
13C4×#11764047—21no—
14C22×Dic274C2211yescharacteristic
15C22×C22⋊D137C2211yescharacteristic
16#17984074—21no—
17#17984074—21no—
18#17984074—21no—
19#17984074—21no—
20C2×C22⋊Dic137C411yescharacteristic
21C2×#14555487C411yescharacteristic
22C2×#14555487C411yescharacteristic
23C2×C4×Dic137C2211yescharacteristic
24C2×C22⋊C4×C137C411yescharacteristic
25#14555519—22no—
26#14555519—22no—
27#14555519—22no—
28#14555519D414yes—
29C2×#11764047—22no—
30C2×#11764047—22no—
31C23×D137C2×C411yescharacteristic
32C2×Dic274—21no—
33C2×Dic274—21no—
34C2×Dic274—22no—
35C2×Dic274—22no—
36C2×C22⋊D137—21no—
37C2×C22⋊D137—21no—
38C2×C22⋊D137—22no—
39C2×C22⋊D137—22no—
40C22×Dic137C2×C411yescharacteristic
41C22×Dic137C2×C411yescharacteristic
42C22×Dic137C2311yescharacteristic
43C22⋊Dic137—21no—
44C22⋊Dic137—21no—
45#14555487—21no—
46#14555487—21no—
47#14555487—21no—
48#14555487—21no—
49C4×Dic137D411yescharacteristic
50C4×Dic137D411yescharacteristic
51C4×Dic137D411yescharacteristic
52C4×Dic137D411yescharacteristic
53C24×C137C2×C411yescharacteristic
54C22×C4×C137C2×C411yescharacteristic
55C22×C4×C137C2×C411yescharacteristic
56C22⋊C4×C137—21no—
57C22⋊C4×C137—21no—
58#11764047—24no—
59#11764047—42no—
60C22×D137—21no—
61C22×D137—22no—
62C22×D137—41no—
63C22×D137—41no—
64Dic274—42no—
65Dic274—41no—
66Dic274—41no—
67C22⋊D137—42no—
68C22⋊D137—41no—
69C22⋊D137—41no—
70C2×Dic137—21no—
71C2×Dic137—22no—
72C2×Dic137—41no—
73C2×Dic137C22⋊C411yescharacteristic
74C2×Dic137C22⋊C411yescharacteristic
75C2×Dic137C2×D411yescharacteristic
76C2×Dic137C2×D412yes—
77C2×Dic137C2×D411yescharacteristic
78C2×Dic137#4112yes—
79C23×C137—21no—
80C23×C137—22no—
81C23×C137—41no—
82C23×C137—41no—
83C23×C137C22×C411yescharacteristic
84C2×C4×C137—21no—
85C2×C4×C137—22no—
86C2×C4×C137—41no—
87C2×C4×C137C22⋊C411yescharacteristic
88C2×C4×C137C22⋊C411yescharacteristic
89C22×C137—21no—
90C22×C137—21no—
91C22×C137—41no—
92C22×C137—41no—
93C22×C137—41no—
94C22×C137—82no—
95C22×C137C2×C22⋊C411yescharacteristic
96C22×C137C4×D412yes—
97C2×D137—41no—
98C2×D137—41no—
99C2×D137—41no—
100C2×D137—82no—
101C4×C137—21no—
102C4×C137—21no—
103C4×C137—41no—
104Dic137—21no—
105Dic137—21no—
106Dic137—41no—
107Dic137#12112yes—
108Dic137#12411yescharacteristic
109Dic137#12711yescharacteristic
110C2×C137—22no—
111C2×C137—81no—
112C2×C137#39011yescharacteristic
113C2×C137#41311yescharacteristic
114C2×C137#41311yescharacteristic
115D137—81no—
116C137#196711yescharacteristic, complemented, Sylow
117#1967—1371nomaximal, Sylow
118C2×#124—1371no—
119#423—1371no—
120C2×#101—1372no—
121C2×#100—1372no—
122C4×M16—1371no—
123C22×Q8—1371no—
124C22×D4—1371no—
125C2×M16—1372no—
126C2×M16—1372no—
127#124—2741no—
128#124—2741no—
129#124—2741no—
130#124—2741no—
131C2×C22⋊C4—1371no—
132C2×C22⋊C4—1371no—
133C2×C22⋊C4—1371no—
134C2×C22⋊C4#1058773911yescharacteristic
135C2×C42—1371no—
136C4⋊C8—2741no—
137C4⋊C8—2741no—
138#101—2744no—
139#100—2744no—
140#97—2741no—
141C4×C8—2741no—
142C24—1371no—
143C24C2×#1058773911yescharacteristic
144C2×Q8—2742no—
145C2×Q8—2741no—
146C2×Q8—2741no—
147C2×Q8—2742no—
148C2×D4—2742no—
149C2×D4—2741no—
150C2×D4—2741no—
151C2×D4—2742no—
152C22×C4—1371no—
153C22×C4—1371no—
154C22×C4—1371no—
155C22×C4C2×#1058773911yescharacteristic
156C22×C4C2×#1058773911yescharacteristic
157M16—2742no—
158M16—2742no—
159M16—2742no—
160M16—1374no—
161C2×C8—2742no—
162C2×C8—2742no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—2741no—
166C22⋊C4—2741no—
167C22⋊C4—2741no—
168C22⋊C4—2741no—
169C22⋊C4—2741no—
170C22⋊C4—2741no—
171C42—1371no—
172C42—1371no—
173C42—1371no—
174C42—1371no—
175C23—21no—
176C23—22no—
177C23—41no—
178C23—41no—
179C23—2741no—
180C23—2742no—
181C23—5481no—
182C23—5481no—
183C23C22×#1058773911yescharacteristic
184Q8—5482no—
185Q8—5481no—
186Q8—5481no—
187D4—5482no—
188D4—5481no—
189D4—5481no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—2741no—
194C2×C4—2742no—
195C2×C4—5481no—
196C2×C4—1371no—
197C2×C4—1371no—
198C2×C4—1371no—
199C2×C4—1371no—
200C2×C4—1372no—
201C2×C4—1372no—
202C2×C4C22⋊#1058773911yescharacteristic
203C2×C4C22⋊#1058773911yescharacteristic
204C8—2744no—
205C8—5482no—
206C22—21no—
207C22—21no—
208C22—41no—
209C22—41no—
210C22—41no—
211C22—82no—
212C22—5481no—
213C22—5481no—
214C22—5481no—
215C22—10962no—
216C22C2×C22⋊#1058773911yescharacteristic
217C22D4×#1058773912yes—
218C4—21no—
219C4—21no—
220C4—41no—
221C4—2741no—
222C4—2741no—
223C4—5481no—
224C4—1372no—
225C4—1371no—
226C4—1371no—
227C2—22no—
228C2—81no—
229C2—10961no—
230C2#2257835511yescharacteristic
231C2#2257838911yescharacteristic
232C2#2257839011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C22⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×C22⋊D137C22
cosocleC22×Dic137C23
hypercenterC2×C22⋊C4#10587739
hypocenterC137#1967
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #22578406C2semidirect
2. C2×#17984149C2semidirect
4. C4×#14555519C2semidirect
116. C137#1967semidirect

Maximal subgroups

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

subgroupindex
#225784062
C2×#179841492
C2×#179841482
C4×#145555192
C2×#179840742
#1967137