Finite Group Databasebeta

← #31779151

Subgroups — #31779151

Order 17536.

270 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 65 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×#17984066C211yesmaximal, characteristic
2#22577365C211yesmaximal, characteristic
3C22⋊C8×D137C211yesmaximal, characteristic
4#22577129C211yesmaximal, characteristic, complemented
5#22577128C211yesmaximal, characteristic, complemented
6#22577128C211yesmaximal, characteristic, complemented
7#324×C137C211yesmaximal, characteristic, complemented
8C22×Dic274C411yescharacteristic
9C22×C4×D137C2211yescharacteristic
10C2×#14555493—21no—
11C2×C8×D137—21no—
12#17984066—21no—
13#17984066—21no—
14#17984066—21no—
15#17984066—21no—
16C2×#14555489C411yescharacteristic
17C2×#14555489C2211yescharacteristic
18C2×#14555487C411yescharacteristic
19C2×#14555487C411yescharacteristic
20C22⋊#11764040C2211yescharacteristic
21C22⋊#11764040C2211yescharacteristic
22#17984000—21no—
23#17984000—21no—
24C2×C4⋊C4×C137C2211yescharacteristic
25C22⋊C8×C137C2211yescharacteristic
26C22⋊C8×C137C2211yescharacteristic
27C23×D137C2×C411yescharacteristic
28C2×Dic274—21no—
29C2×Dic274—21no—
30C2×Dic274—42no—
31C22×Dic137C2×C411yescharacteristic
32C22×Dic137C2×C411yescharacteristic
33C22×Dic137C2×C411yescharacteristic
34C2×C4×D137—42no—
35C2×C4×D137D411yescharacteristic
36C2×C4×D137D411yescharacteristic
37C2×C4×D137D411yescharacteristic
38C2×C4×D137D411yescharacteristic
39#14555493—42no—
40C2×#11764040—21no—
41C2×#11764040—21no—
42C8×D137—42no—
43#14555489—21no—
44#14555489—21no—
45#14555489—21no—
46#14555489—21no—
47#14555487—21no—
48#14555487—21no—
49#14555487—21no—
50#14555487—21no—
51C22×C4×C137C2×C411yescharacteristic
52C22×C4×C137C2×C411yescharacteristic
53C22×C4×C137C2311yescharacteristic
54C2×C8×C137—21no—
55C2×C8×C137—21no—
56C4⋊C4×C137—21no—
57C4⋊C4×C137—21no—
58C22×D137—21no—
59C22×D137—21no—
60C22×D137—22no—
61C22×D137—41no—
62C22×D137C22⋊C411yescharacteristic
63C22×D137C22⋊C411yescharacteristic
64Dic274—42no—
65Dic274—42no—
66C2×Dic137—22no—
67C2×Dic137—21no—
68C2×Dic137—21no—
69C2×Dic137—21no—
70C2×Dic137—21no—
71C2×Dic137—41no—
72C2×Dic137C22⋊C411yescharacteristic
73C2×Dic137C22⋊C411yescharacteristic
74C4×D137—22no—
75C4×D137—22no—
76C4×D137—41no—
77C4×D137—41no—
78#11764040—41no—
79#11764040—41no—
80C23×C137C22×C411yescharacteristic
81C2×C4×C137—21no—
82C2×C4×C137—21no—
83C2×C4×C137—21no—
84C2×C4×C137—21no—
85C2×C4×C137—41no—
86C2×C4×C137—41no—
87C2×C4×C137C2×D411yescharacteristic
88C2×C4×C137C2×D411yescharacteristic
89C8×C137—41no—
90C8×C137—41no—
91C22×C137—21no—
92C22×C137—21no—
93C22×C137C2×C22⋊C411yescharacteristic
94C22×C137#13111yescharacteristic
95C22×C137#13111yescharacteristic
96C2×D137—21no—
97C2×D137—21no—
98C2×D137—21no—
99C2×D137—21no—
100C2×D137—21no—
101C2×D137—42no—
102C2×D137—42no—
103C2×D137#10112yes—
104C4×C137—21no—
105C4×C137—21no—
106C4×C137—41no—
107C4×C137—41no—
108Dic137—21no—
109Dic137—41no—
110Dic137—41no—
111Dic137#9912yes—
112C2×C137—21no—
113C2×C137—21no—
114C2×C137#40811yescharacteristic
115C2×C137C2×#9911yescharacteristic
116C2×C137C2×#10111yescharacteristic
117D137—22no—
118D137—41no—
119C137#147211yescharacteristic, complemented, Sylow
120#1472—1371nomaximal, Sylow
121C2×C22⋊Q8—1371no—
122#407—1371no—
123C2×C22⋊C8—1371no—
124#324—1371no—
125#324—1371no—
126#324—1371no—
127#324D13711yescharacteristic, complemented
128C22×Q8—1371no—
129C23×C4—1371no—
130C2×M16—2741no—
131C22×C8—2741no—
132C22⋊Q8—2741no—
133C22⋊Q8—2741no—
134C22⋊Q8—2741no—
135C22⋊Q8—2741no—
136C2×C4⋊C4—1371no—
137C2×C4⋊C4—1371no—
138C2×C4⋊C4C2×D13711yescharacteristic
139C2×C22⋊C4—1371no—
140C2×C22⋊C4—1371no—
141C22⋊C8—2741no—
142C22⋊C8—2741no—
143C22⋊C8—1371no—
144C22⋊C8—1371no—
145C22⋊C8C2×D13711yescharacteristic
146C22⋊C8C2×D13711yescharacteristic
147C24—1371no—
148C2×Q8—2741no—
149C2×Q8—2741no—
150C2×Q8—5482no—
151C22×C4—5482no—
152C22×C4—1371no—
153C22×C4—1371no—
154C22×C4—1371no—
155C22×C4—1371no—
156C22×C4—1371no—
157C22×C4—1371no—
158C22×C4—1371no—
159C22×C4C4×D13711yescharacteristic
160C22×C4C4×D13711yescharacteristic
161C22×C4C22×D13711yescharacteristic
162M16—5482no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—2741no—
166C2×C8—2741no—
167C2×C8—5482no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—2741no—
171C4⋊C4—2741no—
172C4⋊C4—2741no—
173C4⋊C4—2741no—
174C22⋊C4—2741no—
175C22⋊C4—2741no—
176C22⋊C4—2741no—
177C22⋊C4—2741no—
178C23—2742no—
179C23—2741no—
180C23—2741no—
181C23—5481no—
182C23—1371no—
183C23—1371no—
184C23C2×C4×D13711yescharacteristic
185Q8—5482no—
186Q8—5482no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—2742no—
194C2×C4—2742no—
195C2×C4—2742no—
196C2×C4—2741no—
197C2×C4—2741no—
198C2×C4—2741no—
199C2×C4—2741no—
200C2×C4—5481no—
201C2×C4—5481no—
202C2×C4—5481no—
203C2×C4—1371no—
204C2×C4—1371no—
205C2×C4D4×D13711yescharacteristic
206C2×C4D4×D13711yescharacteristic
207C8—41no—
208C8—41no—
209C8—5481no—
210C8—5481no—
211C22—21no—
212C22—21no—
213C22—2741no—
214C22—2741no—
215C22—2741no—
216C22—2741no—
217C22—2741no—
218C22—5482no—
219C22—5482no—
220C22—1372no—
221C22C22⋊C4×D13711yescharacteristic
222C22#1798409411yescharacteristic
223C22#1798409611yescharacteristic
224C4—21no—
225C4—21no—
226C4—41no—
227C4—41no—
228C4—2741no—
229C4—5481no—
230C4—5481no—
231C4—1372no—
232C2—21no—
233C2—21no—
234C2—2742no—
235C2—5481no—
236C2#2257736811yescharacteristic
237C2#99×D13711yescharacteristic
238C2#101×D13711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D137
commutatorC23×C137C22×C4
FrattiniC22×C4C22×D137
Fitting#324×C137C2
socleC22×C137C2×C22⋊C4
prime radicalC23×D137C2×C4
cosocleC22×C4×C137C23
hypercenter#324D137
hypocenterC137#1472
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
4. #22577129C2semidirect
5. #22577128C2semidirect
6. #22577128C2semidirect
7. #324×C137C2semidirect
119. C137#1472semidirect
127. #324D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840662
#225773652
C22⋊C8×D1372
#225771292
#225771282
#324×C1372
#1472137