Finite Group Databasebeta

← #14527713

Subgroups — #14527713

Order 2176.

272 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 53 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#11762959C211yesmaximal, characteristic, complemented
2#11762942C211yesmaximal, characteristic
3#11762726C211yesmaximal, characteristic, complemented
4#11762546C211yesmaximal, characteristic, complemented
5#11762543C211yesmaximal, characteristic, complemented
6#11762535C211yesmaximal, characteristic, complemented
7C17×#539C211yesmaximal, characteristic, complemented
8C22×D68C2211yescharacteristic
9C17×#126—21no—
10C17×#125—21no—
11C17×#123—21no—
12C17×#123—21no—
13C17×C22⋊Q8—21no—
14C17×C22⋊Q8C2211yescharacteristic
15C17×#121C2211yescharacteristic
16C4×D4×C17—21no—
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×#63230—21no—
19C2×#63228—21no—
20C2×#63227—21no—
21C2×#63227—21no—
22#10587502C2211yescharacteristic
23C22⋊#3622C2211yescharacteristic
24C22⋊#3622C2211yescharacteristic
25#10587480—21no—
26#10587480—21no—
27C23×D17—21no—
28C2×Q8×C17D411yescharacteristic
29C2×D4×C17—21no—
30C2×D4×C17D411yescharacteristic
31C22×C4×C17—21no—
32C22×C4×C17D411yescharacteristic
33C22×C4×C17D411yescharacteristic
34C22×C4×C17C2311yescharacteristic
35C22×Dic17—21no—
36C2×D68—21no—
37C2×D68—21no—
38C2×D68—42no—
39C4⋊C4×C17—21no—
40C4⋊C4×C17—21no—
41C4⋊C4×C17—21no—
42C4⋊C4×C17—21no—
43C4⋊C4×C17—21no—
44C4⋊C4×C17—21no—
45C4⋊C4×C17D411yescharacteristic
46C4⋊C4×C17D411yescharacteristic
47C22⋊C4×C17—21no—
48C22⋊C4×C17—21no—
49C22⋊C4×C17—21no—
50C22⋊C4×C17—21no—
51C22⋊C4×C17—21no—
52C42×C17—21no—
53#63230—42no—
54#63228—42no—
55#63227—42no—
56#63227—42no—
57C2×#3622—21no—
58C2×#3622—21no—
59C23×C17—21no—
60C23×C17C2×D411yescharacteristic
61C22×D17—22no—
62C22×D17—41no—
63C22×D17—41no—
64C22×D17—41no—
65C22×D17—41no—
66C22×D17—41no—
67C22×D17—41no—
68Q8×C17—41no—
69D4×C17—41no—
70D4×C17—41no—
71D4×C17—41no—
72C2×C4×C17—21no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—21no—
78C2×C4×C17—21no—
79C2×C4×C17—41no—
80C2×C4×C17—41no—
81C2×C4×C17—81no—
82C2×C4×C17C2×D411yescharacteristic
83C2×C4×C17C2×D411yescharacteristic
84C2×Dic17—41no—
85C2×Dic17—41no—
86C2×Dic17—41no—
87D68—22no—
88D68—22no—
89D68—41no—
90D68—41no—
91#3622—41no—
92#3622—41no—
93#1597—171nomaximal, Sylow
94C22×C17—21no—
95C22×C17—21no—
96C22×C17—41no—
97C22×C17—41no—
98C22×C17—41no—
99C22×C17#12011yescharacteristic
100C22×C17#13511yescharacteristic
101C22×C17AGL(1,ℤ/8)11yescharacteristic
102C2×D17—42no—
103C2×D17—42no—
104C2×D17—42no—
105C2×D17—81no—
106C2×D17—81no—
107C2×D17—81no—
108C2×D17—81no—
109C4×C17—21no—
110C4×C17—21no—
111C4×C17—41no—
112C4×C17—41no—
113C4×C17—41no—
114C4×C17—41no—
115C4×C17—41no—
116Dic17—81no—
117#539D1711yescharacteristic, complemented
118#450—171no—
119#447—171no—
120#394—171no—
121#328—171no—
122#327—171no—
123#324—171no—
124C2×C17—21no—
125C2×C17—21no—
126C2×C17—81no—
127C2×C17#44911yescharacteristic
128C2×C17#45411yescharacteristic
129C2×C17B(4,𝔽2)11yescharacteristic
130D17—82no—
131C22×D4—171no—
132C2×SD16—341no—
133C2×D8—341no—
134#126—21no—
135#125—21no—
136#123—21no—
137#123—21no—
138C22⋊Q8—21no—
139C22⋊Q8C2×D1711yescharacteristic
140#121C2×D1711yescharacteristic
141C4×D4—21no—
142C2×C4⋊C4C2×D1711yescharacteristic
143C2×C22⋊C4—341no—
144C2×C22⋊C4—341no—
145#102—341no—
146#102—341no—
147C22⋊C8—171no—
148C22⋊C8—171no—
149#95—171no—
150C17#159711yescharacteristic, complemented, Sylow
151C24—341no—
152C2×Q8C22⋊D1711yescharacteristic
153C2×D4—21no—
154C2×D4—341no—
155C2×D4—341no—
156C2×D4—682no—
157C2×D4C22⋊D1711yescharacteristic
158C22×C4—21no—
159C22×C4—341no—
160C22×C4C22⋊D1711yescharacteristic
161C22×C4C22⋊D1711yescharacteristic
162C22×C4C22×D1711yescharacteristic
163SD16—682no—
164D8—682no—
165C2×C8—341no—
166C2×C8—341no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4C22⋊D1711yescharacteristic
174C4⋊C4C22⋊D1711yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—682no—
181C22⋊C4—682no—
182C42—21no—
183C23—21no—
184C23—342no—
185C23—681no—
186C23—681no—
187C23—681no—
188C23—681no—
189C23—681no—
190C23—681no—
191C23C2×C22⋊D1711yescharacteristic
192Q8—41no—
193D4—41no—
194D4—41no—
195D4—41no—
196D4—342no—
197D4—342no—
198D4—681no—
199D4—681no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—81no—
210C2×C4—681no—
211C2×C4—681no—
212C2×C4—681no—
213C2×C4C2×C22⋊D1711yescharacteristic
214C2×C4C2×C22⋊D1711yescharacteristic
215C8—681no—
216C8—681no—
217C22—21no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—682no—
223C22—682no—
224C22—682no—
225C22—1361no—
226C22—1361no—
227C22—1361no—
228C22—1361no—
229C22#1058763511yescharacteristic
230C22#1058763611yescharacteristic
231C22#1058763911yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—41no—
239C4—1361no—
240C2—21no—
241C2—21no—
242C2—81no—
243C2—1362no—
244C2#1176295411yescharacteristic
245C2#1176298211yescharacteristic
246C2#1176299611yescharacteristic
subgroupG/H
centerC22#10587639
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#539C2
socleC22×C17#120
prime radical#11762942C2
hypercenter#539D17
hypocenterC17#1597
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
1. #11762959C2semidirect
3. #11762726C2semidirect
4. #11762546C2semidirect
5. #11762543C2semidirect
6. #11762535C2semidirect
7. C17×#539C2semidirect
117. #539D17semidirect
150. C17#1597semidirect

Maximal subgroups

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

subgroupindex
#117629592
#117629422
#117627262
#117625462
#117625432
#117625352
C17×#5392
#159717