Finite Group Databasebeta

← #31790961

Subgroups — #31790961

Order 17536.

386 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 121 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#22577939C211yesmaximal, characteristic
2C2×Q8×Dic137C211yesmaximal, characteristic, complemented
3#22577736C211yesmaximal, characteristic, complemented
4#22577722C211yesmaximal, characteristic
5#22577708C212yesmaximal, complemented
6#22577627C212yesmaximal, complemented
7#22577542C211yesmaximal, characteristic
8#22577510C212yesmaximal, complemented
9#22577506C212yesmaximal, complemented
10#22577457C211yesmaximal, characteristic, complemented
11#451×C137C211yesmaximal, characteristic, complemented
12C2×#14555497C2212yes—
13#17984083—21no—
14#17984079C2211yescharacteristic
15#17984077C2211yescharacteristic
16#17984072—21no—
17#17984062C2211yescharacteristic
18Q8×Dic137—21no—
19Q8×Dic137—21no—
20Q8×Dic137C2211yescharacteristic
21Q8×Dic137C2212yes—
22Q8×Dic137C2212yes—
23#17984050C2211yescharacteristic
24#17984048—21no—
25#17984046—21no—
26C2×C4⋊Dic137—21no—
27C2×C4⋊Dic137C2211yescharacteristic
28C2×C4×Dic137—21no—
29C2×C4×Dic137C2211yescharacteristic
30#17984021C2212yes—
31#17984020C2212yes—
32#17984017C2212yes—
33#17984012C2212yes—
34#17984012C2212yes—
35#17984010C2212yes—
36C22⋊#11764040C2211yescharacteristic
37#17983998C2212yes—
38C22×Q8×C137C2211yescharacteristic
39C2×Q16×C137C2212yes—
40C22⋊Q8×C137C2211yescharacteristic
41#103×C137C2212yes—
42C22⋊C8×C137C2211yescharacteristic
43C2×D548C2311yescharacteristic
44C2×C22⋊D137—21no—
45C22×Dic137—21no—
46C22×Dic137D411yescharacteristic
47C22×Dic137D411yescharacteristic
48C2×C4×D137—21no—
49#14555497—24no—
50C2×#11764040C2312yes—
51C4⋊Dic137—22no—
52C4⋊Dic137—22no—
53C4⋊Dic137—21no—
54C4⋊Dic137—21no—
55C4⋊Dic137—21no—
56C4⋊Dic137C2312yes—
57C4⋊Dic137C2311yescharacteristic
58#14555489—21no—
59#14555487—21no—
60#14555487—21no—
61#14555487—21no—
62#14555487—21no—
63C4×Dic137—22no—
64C4×Dic137—22no—
65C4×Dic137—21no—
66C4×Dic137D411yescharacteristic
67C4×Dic137D411yescharacteristic
68C4×Dic137C2311yescharacteristic
69C4×Dic137C2311yescharacteristic
70C2×Q8×C137—24no—
71C2×Q8×C137—21no—
72C2×Q8×C137C2311yescharacteristic
73C2×Q8×C137C2312yes—
74C22×C4×C137—21no—
75C22×C4×C137C2311yescharacteristic
76Q16×C137—24no—
77C2×C8×C137C2312yes—
78C4⋊C4×C137—21no—
79C4⋊C4×C137C2311yescharacteristic
80C22⋊C4×C137—21no—
81C22×D137—21no—
82D548—22no—
83C22⋊D137—42no—
84C2×Dic137—22no—
85C2×Dic137—22no—
86C2×Dic137—21no—
87C2×Dic137—41no—
88C2×Dic137—42no—
89C2×Dic137C2×D411yescharacteristic
90C2×Dic137C2×D411yescharacteristic
91C2×Dic137C2×D411yescharacteristic
92C2×Dic137C2×D411yescharacteristic
93C4×D137—42no—
94#11764040—22no—
95C23×C137C2×D411yescharacteristic
96Q8×C137—24no—
97Q8×C137—22no—
98Q8×C137—22no—
99Q8×C137#4114yes—
100C2×C4×C137—24no—
101C2×C4×C137—22no—
102C2×C4×C137—22no—
103C2×C4×C137—21no—
104C2×C4×C137—21no—
105C2×C4×C137C2×D411yescharacteristic
106C2×C4×C137C2411yescharacteristic
107C8×C137—22no—
108C22×C137—21no—
109C22×C137—21no—
110C22×C137#13712yes—
111C22×C137C22×D411yescharacteristic
112C2×D137—41no—
113C2×D137—41no—
114C2×D137—41no—
115C4×C137—24no—
116C4×C137—21no—
117C4×C137—41no—
118C4×C137—41no—
119C4×C137C2×#4112yes—
120Dic137—21no—
121Dic137—21no—
122Dic137—21no—
123Dic137—21no—
124Dic137—41no—
125Dic137—42no—
126C2×C137—22no—
127C2×C137#52511yescharacteristic
128C2×C137C2×#13711yescharacteristic
129C2×C137#57511yescharacteristic
130D137—81no—
131C137#311711yescharacteristic, complemented, Sylow
132#3117—1371nomaximal, Sylow
133#529—1371no—
134C2×C4×Q8—1371no—
135#484—1371no—
136#481—1371no—
137#479—1372no—
138#464—1372no—
139#451D13711yescharacteristic, complemented
140#448—1371no—
141#441—1372no—
142#440—1372no—
143#432—1371no—
144C22×Q8C2×D13711yescharacteristic
145C2×Q16C2×D13712yes—
146C2×SD16—1372no—
147#126—2741no—
148#125—1371no—
149#124—1371no—
150#123—2741no—
151C22⋊Q8C2×D13711yescharacteristic
152#121—1371no—
153C4×Q8—2741no—
154C4×Q8—2741no—
155C4×Q8—1372no—
156C4×Q8—1372no—
157C4×Q8—1371no—
158C4×D4—2741no—
159C4×D4—1371no—
160#117—2741no—
161C2×C4⋊C4—2741no—
162C2×C4⋊C4—1371no—
163C2×C42—2741no—
164C2×C42—1371no—
165#107—1372no—
166#106—1372no—
167C4⋊C8—1372no—
168#103—1372no—
169#103—1372no—
170#103C2×D13712yes—
171#102—1372no—
172C22⋊C8—1371no—
173C22⋊C8C2×D13711yescharacteristic
174#97—1372no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8C22×D13711yescharacteristic
178C2×Q8C22×D13712yes—
179C2×D4—2741no—
180C2×D4—1371no—
181C22×C4—21no—
182C22×C4—2741no—
183C22×C4—2741no—
184C22×C4—1371no—
185C22×C4—1371no—
186C22×C4C22×D13711yescharacteristic
187Q16—24no—
188SD16—2744no—
189C2×C8—1372no—
190C2×C8C22×D13712yes—
191C4⋊C4—21no—
192C4⋊C4—2741no—
193C4⋊C4—2742no—
194C4⋊C4—2742no—
195C4⋊C4—2741no—
196C4⋊C4—2741no—
197C4⋊C4—2741no—
198C4⋊C4—1372no—
199C4⋊C4—1371no—
200C4⋊C4C22×D13711yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—2741no—
203C22⋊C4—2741no—
204C22⋊C4—2741no—
205C22⋊C4—2741no—
206C42—2742no—
207C42—2742no—
208C42—2741no—
209C42—1371no—
210C42—1371no—
211C42—1371no—
212C42—1371no—
213C23—2741no—
214C23D4×D13711yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—22no—
218Q8#1455550914yes—
219D4—2742no—
220D4—5482no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—2742no—
227C2×C4—2742no—
228C2×C4—2741no—
229C2×C4—5482no—
230C2×C4—5481no—
231C2×C4—5482no—
232C2×C4—1371no—
233C2×C4—1371no—
234C2×C4—1371no—
235C2×C4—1371no—
236C2×C4D4×D13711yescharacteristic
237C2×C4C23×D13711yescharacteristic
238C8—22no—
239C8—2742no—
240C22—21no—
241C22—21no—
242C22—5481no—
243C22—5481no—
244C22—5481no—
245C22#1798412012yes—
246C22C2×D4×D13711yescharacteristic
247C4—24no—
248C4—21no—
249C4—41no—
250C4—41no—
251C4—2741no—
252C4—2741no—
253C4—2741no—
254C4—2741no—
255C4—5481no—
256C4—5482no—
257C4C2×#1455550912yes—
258C2—22no—
259C2—10961no—
260C2#2257790911yescharacteristic
261C2C2×#1798412011yescharacteristic
262C2#2257820211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#451×C137C2
socleC22×C137C22×D4
prime radical#17984062C22
hypercenter#451D137
hypocenterC137#3117
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic137C2semidirect
3. #22577736C2semidirect
5. #22577708C2semidirect
6. #22577627C2semidirect
8. #22577510C2semidirect
9. #22577506C2semidirect
10. #22577457C2semidirect
11. #451×C137C2semidirect
131. C137#3117semidirect
139. #451D137semidirect

Maximal subgroups

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

subgroupindex
#225779392
C2×Q8×Dic1372
#225777362
#225777222
#225777082
#225776272
#225775422
#225775102
#225775062
#225774572
#451×C1372
#3117137