Finite Group Databasebeta

← #14534461

Subgroups — #14534461

Order 2176.

540 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 141 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×#10587558C212yesmaximal, complemented
2#11762739C211yesmaximal, characteristic, complemented
3#11762731C212yesmaximal, complemented
4#11762726C211yesmaximal, characteristic
5#11762701C212yesmaximal, complemented
6#11762694C212yesmaximal, complemented
7#11762687C212yesmaximal
8#11762683C212yesmaximal, complemented
9C17×#382C211yesmaximal, characteristic, complemented
10C22×D68C2211yescharacteristic
11C22×C4×D17C2212yes—
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C42×C17C2211yescharacteristic
15C2×#63227C2212yes—
16C2×#63227C2212yes—
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×C4⋊Dic17C2212yes—
21C2×C4⋊Dic17C2211yescharacteristic
22C2×#63225C2212yes—
23C2×#63225C2212yes—
24C2×C4×Dic17C2212yes—
25#10587558—28no—
26C17×#95C2212yes—
27C17×#95C2212yes—
28#10587502C2211yescharacteristic
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2211yescharacteristic
32C23×D17C2312yes—
33C22×C4×C17C2311yescharacteristic
34C22×C4×C17C2312yes—
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C2×D68—24no—
42C2×D68—41no—
43C2×C4×D17—24no—
44C2×C4×D17—24no—
45C2×C4×D17—24no—
46C4⋊C4×C17—41no—
47C4⋊C4×C17—41no—
48C42×C17D414yescomplemented
49#63227—24no—
50#63227—24no—
51#63227—24no—
52#63227—24no—
53#63227—42no—
54C4⋊Dic17—24no—
55C4⋊Dic17—41no—
56#63225—24no—
57#63225—24no—
58C4×Dic17—24no—
59C23×C17C2411yescharacteristic
60C22×D17—22no—
61C22×D17—22no—
62C22×D17—22no—
63C22×D17—22no—
64C22×D17—22no—
65C22×D17—22no—
66C22×D17#4114yes—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—21no—
70C2×C4×C17—21no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—22no—
78C2×C4×C17C2×D414yes—
79C2×C4×C17C2×D412yes—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17#4114yes—
89D68—44no—
90C4×D17—216no—
91#2609—171nomaximal, Sylow
92C22×C17C22×D411yescharacteristic
93C22×C17C2×#4112yes—
94C22×C17C2×#4112yes—
95C22×C17H(5,𝔽2)11yescharacteristic
96C22×C172−1+411yescharacteristic
97C2×D17—24no—
98C2×D17—24no—
99C2×D17—24no—
100C2×D17—42no—
101C2×D17—42no—
102C2×D17—42no—
103C2×D17—42no—
104C4×C17—24no—
105C4×C17—22no—
106C4×C17—41no—
107C4×C17—41no—
108C4×C17—42no—
109Dic17—24no—
110Dic17—42no—
111Dic17—42no—
112C2×#117—172no—
113#398—171no—
114#396—172no—
115#394—171no—
116#388—172no—
117#387—172no—
118#386—172no—
119#385—172no—
120#382D1711yescharacteristic, complemented
121C2×C17#52512yes—
122C2×C17#53711yescharacteristic
123C2×C17#55111yescharacteristic
124C2×C17#55212yes—
125C2×C17#55511yescharacteristic
126D17—44no—
127C22×D4—171no—
128C23×C4—172no—
129#117—348no—
130C2×C4⋊C4—172no—
131C2×C4⋊C4—172no—
132C2×C4⋊C4—172no—
133C2×C4⋊C4—171no—
134C2×C4⋊C4C2×D1711yescharacteristic
135C2×C4⋊C4C2×D1711yescharacteristic
136C2×C22⋊C4—172no—
137C2×C22⋊C4—172no—
138C2×C22⋊C4—172no—
139C2×C22⋊C4—172no—
140C2×C22⋊C4—172no—
141C2×C42—172no—
142C2×C42C2×D1711yescharacteristic
143#95—171no—
144#95—172no—
145#95—171no—
146#95—172no—
147#95C2×D1712yes—
148#95C2×D1712yes—
149C17#260911yescharacteristic, complemented, Sylow
150C24—172no—
151C2×D4—344no—
152C2×D4—681no—
153C22×C4—172no—
154C22×C4—172no—
155C22×C4—172no—
156C22×C4—344no—
157C22×C4—344no—
158C22×C4—344no—
159C22×C4C22×D1711yescharacteristic
160C22×C4C22×D1712yes—
161C22×C4C22×D1711yescharacteristic
162C22×C4C22×D1712yes—
163C22×C4C22×D1711yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—344no—
167C4⋊C4—344no—
168C4⋊C4—344no—
169C4⋊C4—681no—
170C22⋊C4—344no—
171C22⋊C4—344no—
172C22⋊C4—344no—
173C22⋊C4—344no—
174C22⋊C4—682no—
175C42—344no—
176C42D6814yescomplemented
177C23—174no—
178C23—342no—
179C23—342no—
180C23—342no—
181C23—342no—
182C23—342no—
183C23—342no—
184C23C23×D1711yescharacteristic
185D4—684no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—174no—
198C2×C4—3416no—
199C2×C4—342no—
200C2×C4—342no—
201C2×C4—342no—
202C2×C4—342no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—342no—
206C2×C4—342no—
207C2×C4C2×D6814yes—
208C2×C4C2×D6812yes—
209C22—344no—
210C22—344no—
211C22—344no—
212C22—682no—
213C22—682no—
214C22—682no—
215C22—682no—
216C22C22×D6811yescharacteristic
217C22#1058769111yescharacteristic
218C22#1058769411yescharacteristic
219C22#41×D1712yes—
220C22#41×D1712yes—
221C4—24no—
222C4—22no—
223C4—41no—
224C4—41no—
225C4—42no—
226C4—344no—
227C4—682no—
228C4—682no—
229C2—684no—
230C2#1176332112yes—
231C2#1176339011yescharacteristic
232C2#1176348911yescharacteristic
233C2#1176349112yes—
234C2#1176352111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#382C2
prime radicalC22×D68C22
hypercenter#382D17
hypocenterC17#2609
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. C2×#10587558C2semidirect
2. #11762739C2semidirect
3. #11762731C2semidirect
5. #11762701C2semidirect
6. #11762694C2semidirect
8. #11762683C2semidirect
9. C17×#382C2semidirect
48. C42×C17D4semidirect
120. #382D17semidirect
149. C17#2609semidirect
176. C42D68semidirect

Maximal subgroups

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

subgroupindex
C2×#105875582
#117627392
#117627312
#117627262
#117627012
#117626942
#117626872
#117626832
C17×#3822
#260917