Finite Group Databasebeta

← #14535100

Subgroups — #14535100

Order 2176.

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

normal subgroups only

This group has 544 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×#10587574C212yesmaximal, complemented
2C2×#10587564C212yesmaximal
3#11762740C212yesmaximal, complemented
4#11762729C212yesmaximal, complemented
5#11762721C211yesmaximal, characteristic
6#11762697C212yesmaximal
7#11762694C212yesmaximal, complemented
8#11762681C211yesmaximal, characteristic
9C17×#399C211yesmaximal, characteristic
10C23×Dic17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C22×Dic34C2212yes—
13C2×C4⋊C4×C17C2212yes—
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊C4×C17C2211yescharacteristic
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2211yescharacteristic
19C2×C4⋊Dic17C2212yes—
20C2×#63225C2212yes—
21C2×#63225C2212yes—
22C2×#63225C2212yes—
23C2×#63225C2212yes—
24C2×C4×Dic17C2212yes—
25#10587574—28no—
26#10587564—28no—
27C17×#95C2211yescharacteristic
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2212yes—
32#10587502C2211yescharacteristic
33C24×C17C2311yescharacteristic
34C22×C4×C17C2312yes—
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2312yes—
38C22×Dic17—24no—
39C22×Dic17—41no—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C2×Dic34—24no—
45C2×Dic34—24no—
46C2×Dic34—24no—
47C2×Dic34—41no—
48C2×Dic34—42no—
49C4⋊C4×C17—24no—
50C4⋊C4×C17—41no—
51C22⋊C4×C17—24no—
52C22⋊C4×C17—41no—
53C22⋊Dic17—24no—
54C22⋊Dic17Q814yescomplemented
55C4⋊Dic17D418yes—
56#63225—24no—
57#63225—24no—
58#63225—24no—
59#63225—24no—
60C4×Dic17—24no—
61C23×C17—22no—
62C23×C17—21no—
63C23×C17—21no—
64C23×C17—22no—
65C23×C17C2×Q812yes—
66C23×C17C2411yescharacteristic
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—21no—
73C2×C4×C17—21no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17C2×D414yes—
77C2×Dic17—22no—
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—44no—
87C2×Dic17C2×D414yes—
88C2×Dic17C2×D414yes—
89C2×Dic17C2×Q814yes—
90Dic34—44no—
91Dic34—48no—
92#2677—171nomaximal, Sylow
93C22×C17—22no—
94C22×C17—24no—
95C22×C17—42no—
96C22×C17—41no—
97C22×C17—41no—
98C22×C17C22×D412yes—
99C22×C17C22×Q811yescharacteristic
100C22×C17H(5,𝔽2)11yescharacteristic
101C22×C172−1+411yescharacteristic
102C22×C172−1+412yes—
103C4×C17—24no—
104C4×C17—42no—
105C4×C17—41no—
106C4×C17—41no—
107Dic17—24no—
108Dic17—24no—
109Dic17—24no—
110Dic17—42no—
111C2×C4⋊Q8—172no—
112C2×C22⋊Q8—172no—
113#399D1711yescharacteristic
114#398—172no—
115#395—172no—
116#393—171no—
117#388—172no—
118#387—172no—
119#385—171no—
120C2×C17—42no—
121C2×C17#53612yes—
122C2×C17#54411yescharacteristic
123C2×C17D4211yescharacteristic
124C2×C17D4×Q812yes—
125C2×C17#56311yescharacteristic
126C22×Q8—172no—
127C22×Q8—171no—
128C23×C4—171no—
129C4⋊Q8—348no—
130C22⋊Q8—348no—
131C2×C4⋊C4—172no—
132C2×C4⋊C4—172no—
133C2×C4⋊C4—172no—
134C2×C4⋊C4—172no—
135C2×C4⋊C4—172no—
136C2×C4⋊C4C2×D1712yes—
137C2×C4⋊C4C2×D1711yescharacteristic
138C2×C22⋊C4—171no—
139C2×C22⋊C4—172no—
140C2×C22⋊C4C2×D1712yes—
141C2×C22⋊C4C2×D1711yescharacteristic
142C2×C42—172no—
143#95—172no—
144#95—172no—
145#95—172no—
146#95—172no—
147#95—171no—
148#95C2×D1711yescharacteristic
149C17#267711yescharacteristic, complemented, Sylow
150C24C22×D1711yescharacteristic
151C2×Q8—344no—
152C2×Q8—344no—
153C2×Q8—344no—
154C2×Q8—681no—
155C2×Q8—682no—
156C22×C4—172no—
157C22×C4—172no—
158C22×C4—172no—
159C22×C4—172no—
160C22×C4—344no—
161C22×C4—681no—
162C22×C4C22×D1711yescharacteristic
163C22×C4C22×D1712yes—
164C22×C4C22×D1711yescharacteristic
165C22×C4C22×D1712yes—
166C4⋊C4—24no—
167C4⋊C4—41no—
168C4⋊C4—178no—
169C4⋊C4—344no—
170C4⋊C4—344no—
171C4⋊C4—344no—
172C4⋊C4—344no—
173C22⋊C4—24no—
174C22⋊C4—41no—
175C22⋊C4—174no—
176C22⋊C4—344no—
177C42—344no—
178C23—22no—
179C23—21no—
180C23—22no—
181C23—21no—
182C23Q8×D1712yes—
183C23C23×D1711yescharacteristic
184Q8—688no—
185Q8—684no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—174no—
196C2×C4—174no—
197C2×C4—174no—
198C2×C4—342no—
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×C4—684no—
208C2×C4D4×D1714yes—
209C22—24no—
210C22—22no—
211C22—41no—
212C22—42no—
213C22—41no—
214C22C2×D4×D1712yes—
215C22#1058769111yescharacteristic
216C22C2×Q8×D1711yescharacteristic
217C22#1058769712yes—
218C22#1058769711yescharacteristic
219C4—24no—
220C4—41no—
221C4—42no—
222C4—41no—
223C4—344no—
224C4—344no—
225C4—344no—
226C4—682no—
227C2—42no—
228C2#1176338112yes—
229C2#1176343811yescharacteristic
230C2#1176344411yescharacteristic
231C2#1176347212yes—
232C2#1176356211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#399C2
prime radicalC24×C17C23
hypercenter#399D17
hypocenterC17#2677
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#10587574C2semidirect
3. #11762740C2semidirect
4. #11762729C2semidirect
7. #11762694C2semidirect
54. C22⋊Dic17Q8semidirect
149. C17#2677semidirect

Maximal subgroups

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

subgroupindex
C2×#105875742
C2×#105875642
#117627402
#117627292
#117627212
#117626972
#117626942
#117626812
C17×#3992
#267717