Finite Group Databasebeta

← #14534034

Subgroups — #14534034

Order 2176.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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#11762726C211yesmaximal, characteristic
2#11762722C212yesmaximal, complemented
3#11762714C211yesmaximal, characteristic, complemented
4#11762699C212yesmaximal, complemented
5#11762687C212yesmaximal
6#11762676C212yesmaximal, complemented
7#11762671C212yesmaximal, complemented
8#11762656C212yesmaximal, complemented
9C17×#400C211yesmaximal, characteristic, complemented
10C22×D68C2211yescharacteristic
11C22×C4×D17C2212yes—
12C22×Dic34C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×#63227C2212yes—
16C2×#63227C2212yes—
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×C4×Dic17C2212yes—
21C2×C4×Dic17C2211yescharacteristic
22C17×#95C2212yes—
23C17×#95C2211yescharacteristic
24C17×#95C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2211yescharacteristic
30#10587502C2212yes—
31C23×D17C2312yes—
32C22×C4×C17C2312yes—
33C22×C4×C17C2311yescharacteristic
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×Dic17C2312yes—
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C2×D68—24no—
41C2×D68—41no—
42C2×C4×D17—24no—
43C2×C4×D17—24no—
44C2×C4×D17—42no—
45C2×Dic34—22no—
46C2×Dic34—41no—
47C2×Dic34—41no—
48C4⋊C4×C17—41no—
49C4⋊C4×C17—41no—
50#63227—24no—
51#63227—24no—
52#63227—24no—
53#63227—42no—
54#63227—42no—
55C4×Dic17—24no—
56C4×Dic17—41no—
57C23×C17C2411yescharacteristic
58C22×D17—22no—
59C22×D17—22no—
60C22×D17—22no—
61C22×D17—22no—
62C22×D17—22no—
63C22×D17—22no—
64C22×D17#4114yes—
65C2×C4×C17—21no—
66C2×C4×C17—21no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17#4112yes—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17#4114yes—
87D68—44no—
88C4×D17—48no—
89Dic34—44no—
90#2572—171nomaximal, Sylow
91C22×C17C2×#4111yescharacteristic
92C22×C17C2×#4112yes—
93C22×C17C2×#4112yes—
94C22×C17H(5,𝔽2)11yescharacteristic
95C22×C17H(5,𝔽2)11yescharacteristic
96C2×D17—24no—
97C2×D17—24no—
98C2×D17—24no—
99C2×D17—42no—
100C2×D17—42no—
101C2×D17—42no—
102C2×D17—42no—
103C4×C17—22no—
104C4×C17—42no—
105C4×C17—41no—
106C4×C17—41no—
107C4×C17—42no—
108Dic17—24no—
109Dic17—42no—
110Dic17—42no—
111#400D1711yescharacteristic, complemented
112#394—171no—
113#393—172no—
114#391—171no—
115#388—172no—
116#386—172no—
117#383—172no—
118#381—172no—
119C4×C22⋊C4—172no—
120C2×C17#52912yes—
121C2×C17#53811yescharacteristic
122C2×C17#55112yes—
123C2×C17#55111yescharacteristic
124C2×C17#55511yescharacteristic
125D17—44no—
126C22×Q8—171no—
127C22×D4—171no—
128C23×C4—172no—
129C2×C4⋊C4C2×D1711yescharacteristic
130C2×C4⋊C4C2×D1711yescharacteristic
131C2×C22⋊C4—172no—
132C2×C22⋊C4—172no—
133C2×C22⋊C4—172no—
134C2×C22⋊C4—172no—
135C2×C22⋊C4—172no—
136C2×C42—172no—
137C2×C42—171no—
138#95—172no—
139#95—172no—
140#95—172no—
141#95—171no—
142#95—172no—
143#95—172no—
144#95C2×D1712yes—
145#95C2×D1711yescharacteristic
146#95C2×D1712yes—
147C17#257211yescharacteristic, complemented, Sylow
148C24—172no—
149C2×Q8—342no—
150C2×Q8—681no—
151C2×Q8—681no—
152C2×D4—344no—
153C2×D4—681no—
154C22×C4—172no—
155C22×C4—172no—
156C22×C4—172no—
157C22×C4—344no—
158C22×C4—344no—
159C22×C4—682no—
160C22×C4C22×D1711yescharacteristic
161C22×C4C22×D1712yes—
162C22×C4C22×D1711yescharacteristic
163C22×C4C22×D1712yes—
164C22×C4C22×D1711yescharacteristic
165C4⋊C4—41no—
166C4⋊C4—41no—
167C22⋊C4—344no—
168C22⋊C4—344no—
169C22⋊C4—344no—
170C22⋊C4—682no—
171C22⋊C4—682no—
172C42—344no—
173C42—681no—
174C23—174no—
175C23—342no—
176C23—342no—
177C23—342no—
178C23—342no—
179C23—342no—
180C23—342no—
181C23C23×D1711yescharacteristic
182Q8—684no—
183D4—684no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—174no—
197C2×C4—342no—
198C2×C4—342no—
199C2×C4—342no—
200C2×C4—342no—
201C2×C4—342no—
202C2×C4—342no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—688no—
206C2×C4#6325612yes—
207C22—344no—
208C22—344no—
209C22—344no—
210C22—682no—
211C22—682no—
212C22—682no—
213C22—682no—
214C22#1058769111yescharacteristic
215C22C2×#6325611yescharacteristic
216C22#41×D1712yes—
217C22#41×D1712yes—
218C22#1058769611yescharacteristic
219C4—22no—
220C4—42no—
221C4—41no—
222C4—41no—
223C4—42no—
224C4—344no—
225C4—682no—
226C4—682no—
227C2—684no—
228C2#1176334412yes—
229C2#1176339911yescharacteristic
230C2#1176348412yes—
231C2#1176348711yescharacteristic
232C2#1176352111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#400C2
prime radicalC22×D68C22
hypercenter#400D17
hypocenterC17#2572
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. #11762722C2semidirect
3. #11762714C2semidirect
4. #11762699C2semidirect
6. #11762676C2semidirect
7. #11762671C2semidirect
8. #11762656C2semidirect
9. C17×#400C2semidirect
111. #400D17semidirect
147. C17#2572semidirect

Maximal subgroups

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

subgroupindex
#117627262
#117627222
#117627142
#117626992
#117626872
#117626762
#117626712
#117626562
C17×#4002
#257217