Finite Group Databasebeta

← #14537041

Subgroups — #14537041

Order 2176.

396 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 133 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#11762751C212yesmaximal, complemented
2#11762745C211yesmaximal, characteristic
3#11762743C212yesmaximal, complemented
4#11762741C212yesmaximal, complemented
5#11762732C212yesmaximal
6#11762730C212yesmaximal, complemented
7#11762721C211yesmaximal, characteristic
8#11762669C212yesmaximal
9C17×#381C211yesmaximal, characteristic
10C23×C4×C17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C2×C22⋊C4×C17C2211yescharacteristic
13C2×C22⋊C4×C17C2211yescharacteristic
14C2×C22⋊Dic17C2212yes—
15C2×C22⋊Dic17C2212yes—
16C2×C4⋊Dic17C2212yes—
17C2×C4⋊Dic17C2212yes—
18C2×C4⋊Dic17C2211yescharacteristic
19C2×C4⋊Dic17C2212yes—
20C2×#63225C2212yes—
21C2×#63225C2212yes—
22C2×#63225C2212yes—
23C17×#95C2212yes—
24C17×#95C2212yes—
25#10587502C2212yes—
26#10587502C2211yescharacteristic
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2211yescharacteristic
31C24×C17C2311yescharacteristic
32C22×C4×C17—41no—
33C22×C4×C17—41no—
34C22×C4×C17Q814yescomplemented
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2312yes—
38C22×C4×C17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C2×Dic34—42no—
44C2×Dic34—41no—
45C22⋊C4×C17—41no—
46C22⋊C4×C17—41no—
47C22⋊Dic17—42no—
48C22⋊Dic17—42no—
49C4⋊Dic17—42no—
50C4⋊Dic17—42no—
51C4⋊Dic17—42no—
52C4⋊Dic17—41no—
53#63225—42no—
54#63225—42no—
55#63225—42no—
56C23×C17—22no—
57C23×C17—21no—
58C23×C17—21no—
59C23×C17—22no—
60C23×C17C2×Q812yes—
61C23×C17C2411yescharacteristic
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—21no—
65C2×C4×C17—22no—
66C2×C4×C17—21no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—44no—
72C2×C4×C17C2×Q814yes—
73C2×Dic17—22no—
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85Dic34—44no—
86#2841—171nomaximal, Sylow
87C22×C17—22no—
88C22×C17—24no—
89C22×C17—41no—
90C22×C17—42no—
91C22×C17—41no—
92C22×C17C22×Q811yescharacteristic
93C22×C17H(5,𝔽2)11yescharacteristic
94C22×C17H(5,𝔽2)11yescharacteristic
95C22×C17H(5,𝔽2)12yes—
96C22×C172−1+412yes—
97C4×C17—24no—
98C4×C17—41no—
99C4×C17—42no—
100C4×C17—41no—
101Dic17—42no—
102Dic17—42no—
103Dic17—42no—
104Dic17—42no—
105#400—172no—
106#399—171no—
107#398—172no—
108#398—172no—
109#396—172no—
110#395—172no—
111#393—171no—
112#381D1711yescharacteristic
113#380—172no—
114C2×C17—42no—
115C2×C17#54311yescharacteristic
116C2×C17#54412yes—
117C2×C17#56011yescharacteristic
118C2×C17#56212yes—
119C2×C17#56311yescharacteristic
120C22×Q8—171no—
121C23×C4C2×D1711yescharacteristic
122C2×C4⋊C4—172no—
123C2×C4⋊C4—172no—
124C2×C4⋊C4—172no—
125C2×C4⋊C4—172no—
126C2×C4⋊C4—172no—
127C2×C4⋊C4—172no—
128C2×C4⋊C4—171no—
129C2×C22⋊C4—172no—
130C2×C22⋊C4—172no—
131C2×C22⋊C4C2×D1711yescharacteristic
132C2×C22⋊C4C2×D1711yescharacteristic
133#95—172no—
134#95—172no—
135#95—171no—
136#95—172no—
137#95—172no—
138#95—171no—
139#95C2×D1712yes—
140#95C2×D1712yes—
141C17#284111yescharacteristic, complemented, Sylow
142C24C22×D1711yescharacteristic
143C2×Q8—681no—
144C2×Q8—682no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—172no—
148C22×C4—172no—
149C22×C4—172no—
150C22×C4—172no—
151C22×C4Dic3414yescomplemented
152C22×C4C22×D1711yescharacteristic
153C22×C4C22×D1712yes—
154C22×C4C22×D1711yescharacteristic
155C22×C4C22×D1712yes—
156C4⋊C4—682no—
157C4⋊C4—682no—
158C4⋊C4—682no—
159C4⋊C4—682no—
160C4⋊C4—682no—
161C4⋊C4—681no—
162C4⋊C4—682no—
163C22⋊C4—41no—
164C22⋊C4—41no—
165C22⋊C4—682no—
166C22⋊C4—682no—
167C23—21no—
168C23—21no—
169C23—22no—
170C23—22no—
171C23C2×Dic3412yes—
172C23C23×D1711yescharacteristic
173Q8—684no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—44no—
184C2×C4—342no—
185C2×C4—342no—
186C2×C4—342no—
187C2×C4—342no—
188C2×C4—342no—
189C2×C4—342no—
190C2×C4—342no—
191C2×C4—342no—
192C2×C4—342no—
193C2×C4—342no—
194C2×C4—342no—
195C2×C4—342no—
196C2×C4C2×Dic3414yes—
197C22—22no—
198C22—24no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C22×Dic3411yescharacteristic
203C22#1058769111yescharacteristic
204C22#1058769111yescharacteristic
205C22#1058769612yes—
206C22#1058769712yes—
207C4—24no—
208C4—41no—
209C4—42no—
210C4—41no—
211C4—682no—
212C4—682no—
213C4—682no—
214C4—682no—
215C2—42no—
216C2#1176343511yescharacteristic
217C2#1176343712yes—
218C2#1176354911yescharacteristic
219C2#1176355612yes—
220C2#1176356211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#381C2
prime radicalC24×C17C23
hypercenter#381D17
hypocenterC17#2841
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. #11762751C2semidirect
3. #11762743C2semidirect
4. #11762741C2semidirect
6. #11762730C2semidirect
34. C22×C4×C17Q8semidirect
141. C17#2841semidirect
151. C22×C4Dic34semidirect

Maximal subgroups

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

subgroupindex
#117627512
#117627452
#117627432
#117627412
#117627322
#117627302
#117627212
#117626692
C17×#3812
#284117