Finite Group Databasebeta

← #14533273

Subgroups — #14533273

Order 2176.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10587581C211yesmaximal, characteristic
2C2×C4⋊C4×D17C211yesmaximal, characteristic, complemented
3#11762738C214yesmaximal, complemented
4#11762711C211yesmaximal, characteristic, complemented
5#11762710C211yesmaximal, characteristic, complemented
6#11762687C214yesmaximal, complemented
7#11762674C211yesmaximal, characteristic, complemented
8C4×#63227C211yesmaximal, characteristic, complemented
9C17×#382C211yesmaximal, characteristic, complemented
10C22×D68C2211yescharacteristic
11C22×D68C2212yes—
12C22×C4×D17C2211yescharacteristic
13C22×C4×D17C2212yes—
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C42×C17C2211yescharacteristic
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2214yes—
20C2×#63227C2214yes—
21C2×C4⋊Dic17C2211yescharacteristic
22C2×#63225C2212yes—
23C2×C4×Dic17C2211yescharacteristic
24#10587581—24no—
25C4⋊C4×D17—24no—
26C17×#95C2214yes—
27#10587502C2212yes—
28#10587502C2214yes—
29C23×D17C2312yes—
30C23×D17C2311yescharacteristic
31C22×C4×C17C2312yes—
32C22×C4×C17C2312yes—
33C22×C4×C17C2312yes—
34C22×C4×C17C2311yescharacteristic
35C22×Dic17C2312yes—
36C22×Dic17C2312yes—
37C22×Dic17C2311yescharacteristic
38C2×D68—24no—
39C2×D68—28no—
40C2×D68—22no—
41C2×D68—22no—
42C2×D68—22no—
43C2×C4×D17—22no—
44C2×C4×D17—22no—
45C2×C4×D17—24no—
46C2×C4×D17—44no—
47C2×C4×D17D414yes—
48C4⋊C4×C17—22no—
49C4⋊C4×C17—22no—
50C42×C17—22no—
51#63227—24no—
52#63227—24no—
53#63227—44no—
54#63227—44no—
55C4⋊Dic17—22no—
56#63225—24no—
57C4×Dic17—22no—
58C23×C17C2411yescharacteristic
59C22×D17—21no—
60C22×D17—21no—
61C22×D17—22no—
62C22×D17—22no—
63C22×D17—22no—
64C22×D17—22no—
65C22×D17—24no—
66C22×D17—24no—
67C22×D17C2×D412yes—
68C22×D17#4114yes—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—24no—
73C2×C4×C17—22no—
74C2×C4×C17—24no—
75C2×C4×C17—22no—
76C2×C4×C17C2×D412yes—
77C2×C4×C17#4112yes—
78C2×C4×C17#4112yes—
79C2×Dic17—21no—
80C2×Dic17—21no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—24no—
85C2×Dic17—22no—
86C2×Dic17C2×D412yes—
87D68—48no—
88D68—44no—
89C4×D17—28no—
90C4×D17—48no—
91#2504—171nomaximal, Sylow
92C22×C17C22×D411yescharacteristic
93C22×C17C2×#4111yescharacteristic
94C22×C17C2×#4111yescharacteristic
95C22×C17C2×#4112yes—
96C22×C17H(5,𝔽2)12yes—
97C2×D17—22no—
98C2×D17—22no—
99C2×D17—24no—
100C2×D17—24no—
101C2×D17—42no—
102C2×D17—42no—
103C2×D17—42no—
104C2×D17—44no—
105C2×D17—44no—
106C2×D17#12414yes—
107C4×C17—42no—
108C4×C17—42no—
109C4×C17—42no—
110C4×C17#12314yes—
111Dic17—22no—
112Dic17—42no—
113Dic17—42no—
114C2×#121—171no—
115C22×C4⋊C4—171no—
116#397—174no—
117#390—171no—
118#390—171no—
119#386—174no—
120#382—171no—
121#382D1711yescharacteristic, complemented
122C4×C22⋊C4—171no—
123C2×C17C2×#12311yescharacteristic
124C2×C17C2×#12411yescharacteristic
125C2×C17#53511yescharacteristic
126C2×C17#55312yes—
127C2×C17#55312yes—
128D17—24no—
129D17—82no—
130C22×D4—172no—
131C22×D4—171no—
132C23×C4—171no—
133C23×C4—172no—
134#121—344no—
135C2×C4⋊C4—171no—
136C2×C4⋊C4—172no—
137C2×C4⋊C4—344no—
138C2×C4⋊C4C2×D1711yescharacteristic
139C2×C4⋊C4C2×D1711yescharacteristic
140C2×C22⋊C4—172no—
141C2×C22⋊C4—172no—
142C2×C22⋊C4—174no—
143C2×C22⋊C4—174no—
144C2×C42—171no—
145C2×C42C2×D1711yescharacteristic
146#95—172no—
147#95—174no—
148#95C2×D1714yes—
149C17#250411yescharacteristic, complemented, Sylow
150C24—171no—
151C24—172no—
152C2×D4—344no—
153C2×D4—348no—
154C2×D4—342no—
155C2×D4—342no—
156C2×D4—342no—
157C22×C4—171no—
158C22×C4—174no—
159C22×C4—172no—
160C22×C4—172no—
161C22×C4—344no—
162C22×C4—342no—
163C22×C4—342no—
164C22×C4—684no—
165C22×C4C22×D1712yes—
166C22×C4C22×D1712yes—
167C22×C4C22×D1712yes—
168C22×C4C22×D1711yescharacteristic
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—344no—
172C4⋊C4—342no—
173C22⋊C4—344no—
174C22⋊C4—344no—
175C22⋊C4—684no—
176C22⋊C4—684no—
177C42—22no—
178C42—342no—
179C23—172no—
180C23—174no—
181C23—341no—
182C23—341no—
183C23—342no—
184C23—342no—
185C23—342no—
186C23—342no—
187C23—344no—
188C23—344no—
189C23C23×D1711yescharacteristic
190D4—688no—
191D4—684no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—172no—
200C2×C4—341no—
201C2×C4—341no—
202C2×C4—348no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—342no—
206C2×C4—342no—
207C2×C4—344no—
208C2×C4—688no—
209C2×C4#6325212yes—
210C2×C4D4×D1712yes—
211C2×C4#6325612yes—
212C22—174no—
213C22—344no—
214C22—344no—
215C22—342no—
216C22—342no—
217C22—684no—
218C22—684no—
219C22—682no—
220C22—682no—
221C22—682no—
222C22C2×#6325211yescharacteristic
223C22C2×D4×D1711yescharacteristic
224C22C2×#6325611yescharacteristic
225C22#41×D1712yes—
226C22#1058769612yes—
227C4—42no—
228C4—42no—
229C4—42no—
230C4—342no—
231C4—682no—
232C4—682no—
233C4#1058758014yes—
234C2—344no—
235C2—1362no—
236C2C2×#1058758011yescharacteristic
237C2#124×D1711yescharacteristic
238C2#1176337711yescharacteristic
239C2#1176350312yes—
240C2#1176351212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#382C2
prime radicalC2×#10587581C2
hypercenter#382D17
hypocenterC17#2504
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D17C2semidirect
3. #11762738C2semidirect
4. #11762711C2semidirect
5. #11762710C2semidirect
6. #11762687C2semidirect
7. #11762674C2semidirect
8. C4×#63227C2semidirect
9. C17×#382C2semidirect
121. #382D17semidirect
149. C17#2504semidirect

Maximal subgroups

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

subgroupindex
C2×#105875812
C2×C4⋊C4×D172
#117627382
#117627112
#117627102
#117626872
#117626742
C4×#632272
C17×#3822
#250417