Finite Group Databasebeta

← #14533101

Subgroups — #14533101

Order 2176.

584 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C22×C4⋊Dic17C211yesmaximal, characteristic, complemented
2#11762748C212yesmaximal, complemented
3#11762745C212yesmaximal, complemented
4#11762707C212yesmaximal, complemented
5#11762689C212yesmaximal
6#11762681C212yesmaximal, complemented
7#11762669C212yesmaximal, complemented
8#11762669C211yesmaximal, characteristic, complemented
9C2×C4×D4×C17C211yesmaximal, characteristic
10C22×D4×C17C2211yescharacteristic
11C23×C4×C17C2211yescharacteristic
12C23×C4×C17C2211yescharacteristic
13C23×Dic17C2212yes—
14C4×D4×C17—24no—
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C22⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C42×C17C2211yescharacteristic
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2214yes—
21C2×C4⋊Dic17—24no—
22C2×C4⋊Dic17C2212yes—
23C2×C4⋊Dic17C2212yes—
24C2×C4⋊Dic17C2212yes—
25C2×#63225C2214yes—
26#10587502C2214yes—
27#10587502C2212yes—
28#10587502C2214yes—
29C24×C17C2311yescharacteristic
30C24×C17C2311yescharacteristic
31C2×D4×C17—22no—
32C2×D4×C17—22no—
33C2×D4×C17Q814yes—
34C22×C4×C17—24no—
35C22×C4×C17—22no—
36C22×C4×C17—24no—
37C22×C4×C17D414yes—
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×Dic17—24no—
44C22×Dic17—44no—
45C22×Dic17C2314yes—
46C22×Dic17C2314yes—
47C4⋊C4×C17—22no—
48C22⋊C4×C17—22no—
49C22⋊C4×C17—22no—
50C42×C17—22no—
51C22⋊Dic17—24no—
52C22⋊Dic17—44no—
53C4⋊Dic17—24no—
54C4⋊Dic17—24no—
55C4⋊Dic17—24no—
56#63225—44no—
57C23×C17—22no—
58C23×C17—22no—
59C23×C17—21no—
60C23×C17—22no—
61C23×C17—21no—
62C23×C17—22no—
63C23×C17C2×D412yes—
64C23×C17C2×Q812yes—
65C23×C17C2×Q812yes—
66C23×C17#4112yes—
67C23×C17C2411yescharacteristic
68D4×C17—28no—
69C2×C4×C17—21no—
70C2×C4×C17—22no—
71C2×C4×C17—21no—
72C2×C4×C17—28no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—44no—
78C2×C4×C17C2×D412yes—
79C2×C4×C17C2×D412yes—
80C2×C4×C17C2×Q812yes—
81C2×C4×C17#4114yes—
82C2×Dic17—24no—
83C2×Dic17—24no—
84C2×Dic17—24no—
85C2×Dic17—24no—
86C2×Dic17—24no—
87C2×Dic17—24no—
88C2×Dic17—48no—
89#2487—171nomaximal, Sylow
90C22×C17—22no—
91C22×C17—22no—
92C22×C17—24no—
93C22×C17—24no—
94C22×C17—22no—
95C22×C17—24no—
96C22×C17—41no—
97C22×C17—41no—
98C22×C17—42no—
99C22×C17C22⋊Q814yes—
100C22×C17C22×D411yescharacteristic
101C22×C17C22×Q811yescharacteristic
102C22×C17C2×#4111yescharacteristic
103C22×C17C2×#4112yes—
104C22×C17H(5,𝔽2)12yes—
105C4×C17—22no—
106C4×C17—22no—
107C4×C17—41no—
108C4×C17—41no—
109C4×C17#12314yes—
110Dic17—44no—
111Dic17—44no—
112C2×C4×D4D1711yescharacteristic
113C22×C4⋊C4—171no—
114#400—172no—
115#399—172no—
116#389—172no—
117#386—172no—
118#385—172no—
119#380—172no—
120#380—171no—
121C2×C17—24no—
122C2×C17—42no—
123C2×C17C2×C22⋊Q811yescharacteristic
124C2×C17C2×#12311yescharacteristic
125C2×C17#53511yescharacteristic
126C2×C17#55312yes—
127C2×C17#55412yes—
128C22×D4C2×D1711yescharacteristic
129C23×C4—172no—
130C23×C4C2×D1711yescharacteristic
131C23×C4C2×D1711yescharacteristic
132C4×D4—24no—
133C2×C4⋊C4—172no—
134C2×C4⋊C4—172no—
135C2×C4⋊C4—172no—
136C2×C4⋊C4—174no—
137C2×C4⋊C4—344no—
138C2×C4⋊C4C2×D1711yescharacteristic
139C2×C22⋊C4—172no—
140C2×C22⋊C4—174no—
141C2×C22⋊C4C2×D1711yescharacteristic
142C2×C22⋊C4C2×D1711yescharacteristic
143C2×C42C2×D1711yescharacteristic
144#95—174no—
145#95—172no—
146#95—174no—
147C17#248711yescharacteristic, complemented, Sylow
148C24C22×D1711yescharacteristic
149C24C22×D1711yescharacteristic
150C2×D4—22no—
151C2×D4—22no—
152C2×D4Dic3414yes—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—174no—
157C22×C4—174no—
158C22×C4—344no—
159C22×C4—684no—
160C22×C4C22⋊D1714yes—
161C22×C4C22×D1711yescharacteristic
162C22×C4C22×D1711yescharacteristic
163C22×C4C22×D1711yescharacteristic
164C22×C4C22×D1711yescharacteristic
165C22×C4C22×D1711yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—344no—
168C4⋊C4—344no—
169C4⋊C4—344no—
170C4⋊C4—684no—
171C22⋊C4—22no—
172C22⋊C4—22no—
173C22⋊C4—344no—
174C22⋊C4—684no—
175C42—22no—
176C23—22no—
177C23—22no—
178C23—21no—
179C23—22no—
180C23—22no—
181C23—21no—
182C23C2×Dic3412yes—
183C23C2×Dic3412yes—
184C23#6325212yes—
185C23C2×C22⋊D1712yes—
186C23C23×D1711yescharacteristic
187D4—28no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—28no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—44no—
197C2×C4—344no—
198C2×C4—344no—
199C2×C4—344no—
200C2×C4—344no—
201C2×C4—344no—
202C2×C4—344no—
203C2×C4—688no—
204C2×C4C2×Dic3412yes—
205C2×C4#6325414yes—
206C2×C4C2×C22⋊D1712yes—
207C2×C4C2×C22⋊D1712yes—
208C22—24no—
209C22—24no—
210C22—22no—
211C22—22no—
212C22—24no—
213C22—22no—
214C22—42no—
215C22—41no—
216C22—41no—
217C22#1058761014yes—
218C22C22×Dic3411yescharacteristic
219C22C2×#6325211yescharacteristic
220C22C2×#6325412yes—
221C22#1058769612yes—
222C22C22×C22⋊D1711yescharacteristic
223C4—22no—
224C4—22no—
225C4—41no—
226C4—41no—
227C4—684no—
228C4—684no—
229C4#1058762114yes—
230C2—24no—
231C2—42no—
232C2C2×#1058761011yescharacteristic
233C2C2×#1058762111yescharacteristic
234C2#1176337311yescharacteristic
235C2#1176350512yes—
236C2#1176351612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C4×D4×C17C2
prime radicalC22×D4×C17C22
hypercenterC2×C4×D4D17
hypocenterC17#2487
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊Dic17C2semidirect
2. #11762748C2semidirect
3. #11762745C2semidirect
4. #11762707C2semidirect
6. #11762681C2semidirect
7. #11762669C2semidirect
8. #11762669C2semidirect
147. C17#2487semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic172
#117627482
#117627452
#117627072
#117626892
#117626812
#117626692
C2×C4×D4×C172
#248717