Finite Group Databasebeta

← #14531537

Subgroups — #14531537

Order 2176.

484 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 177 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×#10587610C211yesmaximal, characteristic
2#11762713C212yesmaximal, complemented
3#11762703C212yesmaximal
4#11762697C212yesmaximal
5#11762692C212yesmaximal, complemented
6#11762691C212yesmaximal, complemented
7#11762660C212yesmaximal, complemented
8C4×C22⋊Dic17C211yesmaximal, characteristic
9C17×#381C211yesmaximal, characteristic
10C23×C4×C17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C2×C22⋊C4×C17C2211yescharacteristic
13C2×C22⋊C4×C17C2211yescharacteristic
14C2×C22⋊Dic17C2212yes—
15C2×C22⋊Dic17C2212yes—
16C2×C4⋊Dic17C2211yescharacteristic
17#10587610C418yescomplemented
18C2×#63225C2212yes—
19C2×#63225C2212yes—
20C2×#63225C2212yes—
21C2×C4×Dic17C2212yes—
22C2×C4×Dic17C2211yescharacteristic
23C2×C4×Dic17C2212yes—
24C2×C4×Dic17C2211yescharacteristic
25C17×#95C2212yes—
26C17×#95C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2212yes—
32C24×C17C2311yescharacteristic
33C22×C4×C17—41no—
34C22×C4×C17—41no—
35C22×C4×C17C2×C414yes—
36C22×C4×C17C2312yes—
37C22×C4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C2×Dic34—41no—
45C2×Dic34—41no—
46C2×Dic34C2×C414yes—
47C22⋊C4×C17—41no—
48C22⋊C4×C17—41no—
49C22⋊Dic17—24no—
50C22⋊Dic17C2×C418yes—
51C4⋊Dic17C2×C414yes—
52#63225—24no—
53#63225—42no—
54#63225C2×C418yes—
55C4×Dic17—24no—
56C4×Dic17—42no—
57C4×Dic17—41no—
58C4×Dic17—41no—
59C23×C17—22no—
60C23×C17—21no—
61C23×C17—21no—
62C23×C17—22no—
63C23×C17C22×C412yes—
64C23×C17C2411yescharacteristic
65C2×C4×C17—21no—
66C2×C4×C17—21no—
67C2×C4×C17—21no—
68C2×C4×C17—21no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—44no—
76C2×C4×C17C22×C412yes—
77C2×C4×C17C22×C412yes—
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—22no—
87C2×Dic17C22×C414yes—
88C2×Dic17C22×C414yes—
89C2×Dic17#4114yes—
90Dic34—44no—
91#2332—171nomaximal, Sylow
92C22×C17—22no—
93C22×C17—22no—
94C22×C17—22no—
95C22×C17—42no—
96C22×C17—42no—
97C22×C17C23×C411yescharacteristic
98C22×C17C2×#4112yes—
99C22×C17H(5,𝔽2)11yescharacteristic
100C22×C17H(5,𝔽2)11yescharacteristic
101C22×C172−1+412yes—
102C4×C17—22no—
103C4×C17—22no—
104C4×C17—42no—
105C4×C17—42no—
106Dic17—24no—
107Dic17—24no—
108Dic17—24no—
109Dic17—42no—
110C2×C22⋊Q8—171no—
111#391—172no—
112#388—172no—
113#388—172no—
114#387—172no—
115#387—172no—
116#381D1711yescharacteristic
117C4×C4⋊C4—172no—
118C4×C22⋊C4—171no—
119C2×C17—42no—
120C2×C17C4×#4111yescharacteristic
121C2×C17#51811yescharacteristic
122C2×C17#51911yescharacteristic
123C2×C17#53912yes—
124C2×C17#54212yes—
125C22×Q8—171no—
126C23×C4C2×D1711yescharacteristic
127C22⋊Q8—178no—
128C2×C4⋊C4—172no—
129C2×C4⋊C4—171no—
130C2×C4⋊C4—172no—
131C2×C4⋊C4—172no—
132C2×C22⋊C4—172no—
133C2×C22⋊C4—172no—
134C2×C22⋊C4C2×D1711yescharacteristic
135C2×C22⋊C4C2×D1711yescharacteristic
136C2×C42—172no—
137C2×C42—171no—
138C2×C42—171no—
139C2×C42—172no—
140#95—172no—
141#95—172no—
142#95—172no—
143#95—172no—
144#95—172no—
145#95C2×D1712yes—
146#95C2×D1712yes—
147C17#233211yescharacteristic, complemented, Sylow
148C24C22×D1711yescharacteristic
149C2×Q8—174no—
150C2×Q8—681no—
151C2×Q8—681no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—172no—
155C22×C4—172no—
156C22×C4—172no—
157C22×C4—172no—
158C22×C4C4×D1714yes—
159C22×C4C22×D1712yes—
160C22×C4C22×D1712yes—
161C22×C4C22×D1711yescharacteristic
162C22×C4C22×D1711yescharacteristic
163C4⋊C4—178no—
164C4⋊C4—174no—
165C4⋊C4—344no—
166C4⋊C4—682no—
167C22⋊C4—41no—
168C22⋊C4—41no—
169C22⋊C4—178no—
170C22⋊C4—344no—
171C42—344no—
172C42—681no—
173C42—681no—
174C42—682no—
175C23—21no—
176C23—21no—
177C23—22no—
178C23—22no—
179C23C2×C4×D1712yes—
180C23C23×D1711yescharacteristic
181Q8—684no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—44no—
193C2×C4—174no—
194C2×C4—174no—
195C2×C4—174no—
196C2×C4—342no—
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×C4C2×C4×D1712yes—
206C2×C4C2×C4×D1712yes—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22C22×C4×D1711yescharacteristic
213C22#1058769111yescharacteristic
214C22#1058769111yescharacteristic
215C22#41×D1712yes—
216C22#1058769712yes—
217C4—22no—
218C4—22no—
219C4—42no—
220C4—42no—
221C4—344no—
222C4—344no—
223C4—344no—
224C4—682no—
225C2—42no—
226C2#1176327711yescharacteristic
227C2#1176328211yescharacteristic
228C2#1176328611yescharacteristic
229C2#1176340212yes—
230C2#1176342712yes—
subgroupG/H
center, FrattiniC23C23×D17
commutatorC22×C17C23×C4
FittingC17×#381C2
socle, cosocleC23×C17C24
prime radicalC24×C17C23
hypercenter#381D17
hypocenterC17#2332
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. #11762713C2semidirect
5. #11762692C2semidirect
6. #11762691C2semidirect
7. #11762660C2semidirect
17. #10587610C4semidirect
147. C17#2332semidirect

Maximal subgroups

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

subgroupindex
C2×#105876102
#117627132
#117627032
#117626972
#117626922
#117626912
#117626602
C4×C22⋊Dic172
C17×#3812
#233217