Finite Group Databasebeta

← #31780821

Subgroups — #31780821

Order 17536.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#14555493C211yesmaximal, characteristic
2C2×#17984021C211yesmaximal, characteristic, complemented
3C2×#17984019C211yesmaximal, characteristic, complemented
4#22577260C211yesmaximal, characteristic
5#22577260C211yesmaximal, characteristic
6#22577166C211yesmaximal, characteristic, complemented
7#340×C137C211yesmaximal, characteristic, complemented
8C22×C4×D137C2211yescharacteristic
9C2×#14555493—22no—
10C2×#14555493—22no—
11C2×#14555493C414yescomplemented
12C22×#11764040C2211yescharacteristic
13C2×C4⋊Dic137C2211yescharacteristic
14C2×C4⋊Dic137C2211yescharacteristic
15C2×#14555487—21no—
16C2×#14555487—21no—
17#17984021—22no—
18#17984019—22no—
19#17983995—21no—
20#17983995—21no—
21C22×C8×C137C2211yescharacteristic
22C2×C4⋊C4×C137C2211yescharacteristic
23C2×C4⋊C4×C137C2211yescharacteristic
24C23×D137D411yescharacteristic
25C22×Dic137—21no—
26C22×Dic137—21no—
27C22×Dic137D411yescharacteristic
28C2×C4×D137—21no—
29C2×C4×D137—21no—
30C2×C4×D137—21no—
31C2×C4×D137—21no—
32C2×C4×D137C2×C412yes—
33C2×C4×D137Q812yes—
34#14555493—28no—
35C2×#11764040C2×C412yes—
36C2×#11764040D412yes—
37C2×#11764040D412yes—
38C4⋊Dic137—21no—
39C4⋊Dic137—21no—
40C4⋊Dic137—21no—
41C4⋊Dic137—21no—
42#14555487—42no—
43#14555487—42no—
44C22×C4×C137—21no—
45C22×C4×C137—21no—
46C22×C4×C137C2311yescharacteristic
47C2×C8×C137—21no—
48C2×C8×C137—21no—
49C2×C8×C137C2×C412yes—
50C4⋊C4×C137—21no—
51C4⋊C4×C137—21no—
52C4⋊C4×C137—21no—
53C4⋊C4×C137—21no—
54C22×D137—21no—
55C22×D137—21no—
56C22×D137—21no—
57C22×D137—21no—
58C22×D137—21no—
59C22×D137—21no—
60C22×D137C4⋊C412yes—
61C2×Dic137—21no—
62C2×Dic137—21no—
63C2×Dic137—41no—
64C2×Dic137—41no—
65C2×Dic137—41no—
66C2×Dic137—41no—
67C2×Dic137—41no—
68C2×Dic137—41no—
69C2×Dic137C4⋊C412yes—
70C4×D137—22no—
71C4×D137—22no—
72C4×D137—22no—
73C4×D137—22no—
74#11764040C22⋊C414yescomplemented
75C23×C137C2×D411yescharacteristic
76C2×C4×C137—41no—
77C2×C4×C137—41no—
78C2×C4×C137—41no—
79C2×C4×C137—41no—
80C2×C4×C137—41no—
81C2×C4×C137—41no—
82C2×C4×C137C22×C411yescharacteristic
83C2×C4×C137C2×D411yescharacteristic
84C2×C4×C137C2×D411yescharacteristic
85C2×C4×C137C2×Q811yescharacteristic
86C2×C4×C137#4111yescharacteristic
87C2×C4×C137#4111yescharacteristic
88C8×C137—22no—
89C22×C137C2×C4⋊C411yescharacteristic
90C22×C137#12111yescharacteristic
91C22×C137#12111yescharacteristic
92C22×C137AGL(1,ℤ/8)11yescharacteristic
93C22×C137AGL(1,ℤ/8)11yescharacteristic
94C22×C137#13711yescharacteristic
95C22×C137#13711yescharacteristic
96C2×D137—22no—
97C2×D137—22no—
98C2×D137—22no—
99C2×D137—41no—
100C2×D137—41no—
101C2×D137—41no—
102C2×D137—41no—
103C4×C137—81no—
104C4×C137—81no—
105C4×C137C2×C22⋊C411yescharacteristic
106C4×C137#11711yescharacteristic
107C4×C137C22⋊Q811yescharacteristic
108C4×C137C22⋊Q811yescharacteristic
109Dic137—22no—
110Dic137—81no—
111Dic137—81no—
112C2×C137#38011yescharacteristic
113C2×C137#42811yescharacteristic
114C2×C137#42811yescharacteristic
115C2×C137#46811yescharacteristic
116C2×C137#46811yescharacteristic
117C2×C137#46911yescharacteristic
118C2×C137#47011yescharacteristic
119D137—42no—
120C137#180911yescharacteristic, complemented, Sylow
121#1809—1371nomaximal, Sylow
122C22×M16—1371no—
123C2×#107—1371no—
124C2×#106—1371no—
125#380—1371no—
126#380—1371no—
127#340—1371no—
128#340D13711yescharacteristic, complemented
129C23×C4—1371no—
130C2×M16—2742no—
131C2×M16—2742no—
132C2×M16—1374no—
133C22×C8—1371no—
134C22×C8C2×D13711yescharacteristic
135C2×C4⋊C4—1371no—
136C2×C4⋊C4—1371no—
137C2×C4⋊C4C2×D13711yescharacteristic
138C2×C4⋊C4C2×D13711yescharacteristic
139C2×C22⋊C4—2741no—
140C2×C22⋊C4—2741no—
141#107—2742no—
142#106—2742no—
143#95—2741no—
144#95—2741no—
145C24—1371no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—2741no—
149C22×C4—2741no—
150C22×C4—2741no—
151C22×C4—2741no—
152C22×C4—2741no—
153C22×C4—2741no—
154C22×C4—1371no—
155C22×C4—1372no—
156C22×C4—1372no—
157C22×C4C22×D13711yescharacteristic
158M16—2748no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—1372no—
162C2×C8—1372no—
163C2×C8—1372no—
164C2×C8C4×D13712yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—2741no—
170C4⋊C4—2741no—
171C4⋊C4—2741no—
172C4⋊C4—2741no—
173C22⋊C4—5482no—
174C22⋊C4—5482no—
175C23—2741no—
176C23—2741no—
177C23—2741no—
178C23—2741no—
179C23—2741no—
180C23—2741no—
181C23—1372no—
182C23D4×D13711yescharacteristic
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—2741no—
190C2×C4—2741no—
191C2×C4—2742no—
192C2×C4—2742no—
193C2×C4—2742no—
194C2×C4—2742no—
195C2×C4—5481no—
196C2×C4—5481no—
197C2×C4—5481no—
198C2×C4—5481no—
199C2×C4—5481no—
200C2×C4—5481no—
201C2×C4—1372no—
202C2×C4C2×C4×D13711yescharacteristic
203C2×C4D4×D13711yescharacteristic
204C2×C4D4×D13711yescharacteristic
205C2×C4Q8×D13711yescharacteristic
206C2×C4#1455551011yescharacteristic
207C2×C4#1455551011yescharacteristic
208C8—22no—
209C8—1374no—
210C22—2742no—
211C22—2742no—
212C22—2742no—
213C22—5481no—
214C22—5481no—
215C22—5481no—
216C22—5481no—
217C22C4⋊C4×D13711yescharacteristic
218C22#1798406111yescharacteristic
219C22#1798406111yescharacteristic
220C22#1798411211yescharacteristic
221C22#1798411511yescharacteristic
222C22#1798411711yescharacteristic
223C22#1798412011yescharacteristic
224C4—81no—
225C4—81no—
226C4—2742no—
227C4—10961no—
228C4—10961no—
229C4C22⋊C4×D13711yescharacteristic
230C4#1798404511yescharacteristic
231C4#1798406611yescharacteristic
232C4#1798406611yescharacteristic
233C2—5482no—
234C2#2257726211yescharacteristic
235C2#2257744211yescharacteristic
236C2#2257744411yescharacteristic
237C2#2257764811yescharacteristic
238C2#2257765211yescharacteristic
239C2#2257765511yescharacteristic
240C2#2257766011yescharacteristic
subgroupG/H
centerC23D4×D137
commutatorC2×C4×C137C22×C4
FrattiniC22×C4C22×D137
Fitting#340×C137C2
socleC23×C137C2×D4
prime radicalC23×D137D4
cosocleC22×C4×C137C23
hypercenter#340D137
hypocenterC137#1809
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#17984021C2semidirect
3. C2×#17984019C2semidirect
6. #22577166C2semidirect
7. #340×C137C2semidirect
11. C2×#14555493C4semidirect
74. #11764040C22⋊C4semidirect
120. C137#1809semidirect
128. #340D137semidirect

Maximal subgroups

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

subgroupindex
C22×#145554932
C2×#179840212
C2×#179840192
#225772602
#225771662
#340×C1372
#1809137