Finite Group Databasebeta

← #31780776

Subgroups — #31780776

Order 17536.

380 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#17984048C211yesmaximal, characteristic
2C2×#17984002C212yesmaximal
3#22577261C212yesmaximal, complemented
4#22577170C211yesmaximal, characteristic, complemented
5#342×C137C211yesmaximal, characteristic, complemented
6C22×C22⋊D137C2211yescharacteristic
7C23×Dic137C2211yescharacteristic
8C22×C4×D137C411yescharacteristic, complemented
9#17984048—22no—
10#17984048—22no—
11C2×#14555489—22no—
12C2×#14555489C411yescharacteristic, complemented
13C2×C22⋊Dic137C2212yes—
14C2×#14555487C411yescharacteristic, complemented
15C2×C4×Dic137C411yescharacteristic, complemented
16#17984002—24no—
17#17983995—22no—
18C2×C22⋊C4×C137C2212yes—
19C2×C22⋊C4×C137C2211yescharacteristic
20C23×D137C2×C411yescharacteristic
21C2×C22⋊D137—22no—
22C2×C22⋊D137—22no—
23C2×C22⋊D137D412yes—
24C2×C22⋊D137Q812yes—
25C22×Dic137—22no—
26C22×Dic137—42no—
27C22×Dic137C2×C411yescharacteristic
28C22×Dic137C2×C411yescharacteristic
29C22×Dic137C2×C411yescharacteristic
30C22×Dic137D412yes—
31C22×Dic137D412yes—
32C2×C4×D137—21no—
33C2×C4×D137—21no—
34C2×C4×D137—42no—
35#14555489—22no—
36#14555489—44no—
37C22⋊Dic137—22no—
38C22⋊Dic137—22no—
39#14555487—22no—
40C4×Dic137—21no—
41C4×Dic137—21no—
42C24×C137C2311yescharacteristic
43C22×C4×C137—22no—
44C22×C4×C137C2×C411yescharacteristic
45C22×C4×C137C2×C411yescharacteristic
46C22⋊C4×C137—22no—
47C22⋊C4×C137—22no—
48C22⋊C4×C137—22no—
49C22×D137—21no—
50C22×D137—21no—
51C22×D137—41no—
52C22×D137—41no—
53C22×D137C4⋊C412yescomplemented
54C22⋊D137—24no—
55C22⋊D137—24no—
56C2×Dic137—21no—
57C2×Dic137—21no—
58C2×Dic137—24no—
59C2×Dic137—22no—
60C2×Dic137—41no—
61C2×Dic137—42no—
62C2×Dic137—42no—
63C2×Dic137—42no—
64C2×Dic137—42no—
65C2×Dic137C22⋊C411yescharacteristic
66C2×Dic137C22⋊C412yes—
67C2×Dic137C22⋊C411yescharacteristic
68C2×Dic137C4⋊C412yescomplemented
69C4×D137—42no—
70C4×D137—42no—
71C23×C137—41no—
72C23×C137—41no—
73C23×C137C22×C411yescharacteristic
74C23×C137C2×D412yes—
75C23×C137C2×D411yescharacteristic
76C23×C137C2×Q811yescharacteristic
77C23×C137#4112yes—
78C2×C4×C137—21no—
79C2×C4×C137—21no—
80C2×C4×C137—41no—
81C2×C4×C137—41no—
82C2×C4×C137—42no—
83C2×C4×C137—42no—
84C2×C4×C137—42no—
85C22×C137—21no—
86C22×C137—22no—
87C22×C137—21no—
88C22×C137—22no—
89C22×C137—41no—
90C22×C137—41no—
91C22×C137—42no—
92C22×C137C2×C22⋊C411yescharacteristic
93C22×C137C2×C4⋊C411yescharacteristic
94C22×C137#11711yescharacteristic
95C22×C137#12112yes—
96C22×C137C22⋊Q812yes—
97C2×D137—22no—
98C2×D137—42no—
99C2×D137—41no—
100C2×D137—41no—
101C2×D137—42no—
102C4×C137—41no—
103C4×C137—41no—
104C4×C137—82no—
105Dic137—22no—
106Dic137—21no—
107Dic137—22no—
108Dic137—82no—
109Dic137#9912yescomplemented
110C2×C137—22no—
111C2×C137—21no—
112C2×C137—21no—
113C2×C137—22no—
114C2×C137#38011yescharacteristic
115C2×C137C2×#9911yescharacteristic
116C2×C137#41011yescharacteristic
117D137—42no—
118C137#179911yescharacteristic, complemented, Sylow
119#1799—1371nomaximal, Sylow
120C2×C4×D4—1371no—
121C2×#99—1372no—
122#380—1372no—
123#342—1371no—
124#342D13711yescharacteristic, complemented
125C22×D4—1371no—
126C23×C4—1371no—
127C23×C4—1371no—
128C4×D4—2742no—
129C4×D4—2742no—
130C2×C4⋊C4—2742no—
131C2×C4⋊C4—1371no—
132C2×C22⋊C4—1372no—
133C2×C22⋊C4—1371no—
134C2×C22⋊C4C2×D13712yes—
135C2×C22⋊C4C2×D13711yescharacteristic
136C2×C42—1371no—
137#99—2744no—
138#95—2742no—
139C24—1371no—
140C24C22×D13711yescharacteristic
141C2×D4—2742no—
142C2×D4—2742no—
143C2×D4—1372no—
144C2×D4—1372no—
145C22×C4—22no—
146C22×C4—2741no—
147C22×C4—2741no—
148C22×C4—2742no—
149C22×C4—5482no—
150C22×C4—5482no—
151C22×C4—1371no—
152C22×C4—1371no—
153C22×C4—1372no—
154C22×C4—1371no—
155C22×C4—1372no—
156C22×C4C4×D13711yescharacteristic
157C22×C4C4×D13711yescharacteristic
158C4⋊C4—2742no—
159C4⋊C4—5484no—
160C22⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—2742no—
164C22⋊C4—2742no—
165C22⋊C4—2742no—
166C42—2741no—
167C42—2741no—
168C23—41no—
169C23—41no—
170C23—2741no—
171C23—2741no—
172C23—5481no—
173C23—5481no—
174C23—1372no—
175C23C2×C4×D13711yescharacteristic
176C23D4×D13712yes—
177C23D4×D13711yescharacteristic
178C23Q8×D13711yescharacteristic
179C23#1455551012yes—
180D4—2744no—
181D4—2744no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—2741no—
190C2×C4—2741no—
191C2×C4—2744no—
192C2×C4—2742no—
193C2×C4—5482no—
194C2×C4—5482no—
195C2×C4—5482no—
196C2×C4—5481no—
197C2×C4—5482no—
198C2×C4—5482no—
199C2×C4—5482no—
200C2×C4—1371no—
201C2×C4—1372no—
202C2×C4—1372no—
203C2×C4—1371no—
204C22—22no—
205C22—21no—
206C22—22no—
207C22—21no—
208C22—41no—
209C22—41no—
210C22—42no—
211C22—2742no—
212C22—5482no—
213C22—5482no—
214C22—5481no—
215C22—5481no—
216C22C22⋊C4×D13711yescharacteristic
217C22C4⋊C4×D13711yescharacteristic
218C22#1798404511yescharacteristic
219C22#1798406112yes—
220C22#1798406612yes—
221C4—41no—
222C4—41no—
223C4—82no—
224C4—2742no—
225C4—2741no—
226C4—2742no—
227C4—10962no—
228C4—1372no—
229C2—22no—
230C2—21no—
231C2—21no—
232C2—22no—
233C2—5482no—
234C2#2257726211yescharacteristic
235C2#99×D13711yescharacteristic
236C2#2257737511yescharacteristic
subgroupG/H
centerC22C22⋊C4×D137
commutatorC23×C137C22×C4
FrattiniC24C22×D137
Fitting#342×C137C2
socleC22×C137C2×C22⋊C4
prime radicalC22×C22⋊D137C22
cosocleC24×C137C23
hypercenter#342D137
hypocenterC137#1799
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
3. #22577261C2semidirect
4. #22577170C2semidirect
5. #342×C137C2semidirect
8. C22×C4×D137C4semidirect
12. C2×#14555489C4semidirect
14. C2×#14555487C4semidirect
15. C2×C4×Dic137C4semidirect
53. C22×D137C4⋊C4semidirect
68. C2×Dic137C4⋊C4semidirect
109. Dic137#99semidirect
118. C137#1799semidirect
124. #342D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840482
C2×#179840022
#225772612
#225771702
#342×C1372
#1799137