Finite Group Databasebeta

← #14533777

Subgroups — #14533777

Order 2176.

604 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 165 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×#10587630C211yesmaximal, characteristic, complemented
2C2×#10587622C211yesmaximal, characteristic
3C2×D4×Dic17C211yesmaximal, characteristic
4#11762737C214yesmaximal, complemented
5#11762716C211yesmaximal, characteristic, complemented
6#11762715C211yesmaximal, characteristic, complemented
7#11762667C214yesmaximal, complemented
8C4×#63225C211yesmaximal, characteristic, complemented
9C2×C17×#124C211yesmaximal, characteristic
10C22×Q8×C17C2211yescharacteristic
11C22×D4×C17C2211yescharacteristic
12C23×Dic17C2212yes—
13C22×Dic34C2211yescharacteristic
14C17×#124—24no—
15C2×C22⋊C4×C17C2214yes—
16C2×C42×C17C2211yescharacteristic
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2214yes—
19#10587630—24no—
20#10587622—24no—
21D4×Dic17—24no—
22C2×C4⋊Dic17C2211yescharacteristic
23C2×#63225C2212yes—
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26C2×C4×Dic17C2211yescharacteristic
27C2×C4×Dic17C2211yescharacteristic
28#10587502C2214yes—
29#10587502C2214yes—
30#10587502C2212yes—
31C24×C17C2312yes—
32C2×Q8×C17—22no—
33C2×Q8×C17—22no—
34C2×Q8×C17—22no—
35C2×D4×C17—22no—
36C2×D4×C17—22no—
37C2×D4×C17D414yes—
38C22×C4×C17C2312yes—
39C22×C4×C17C2312yes—
40C22×C4×C17C2311yescharacteristic
41C22×Dic17—24no—
42C22×Dic17—28no—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C2×Dic34—22no—
49C2×Dic34—22no—
50C2×Dic34—22no—
51C22⋊C4×C17—28no—
52C42×C17—22no—
53C22⋊Dic17—28no—
54C22⋊Dic17—24no—
55C4⋊Dic17—22no—
56#63225—24no—
57#63225—24no—
58#63225—24no—
59C4×Dic17—22no—
60C4×Dic17D414yes—
61C23×C17—22no—
62C23×C17—24no—
63C23×C17—22no—
64C23×C17—24no—
65C23×C17C2×D414yes—
66C23×C17C2411yescharacteristic
67Q8×C17—44no—
68D4×C17—28no—
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×C17C2×D412yes—
76C2×C4×C17C2×D412yes—
77C2×C4×C17#4112yes—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—24no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—48no—
88C2×Dic17C2×D412yes—
89C2×Dic17C2×D412yes—
90C2×Dic17#4114yes—
91Dic34—44no—
92#2547—171nomaximal, Sylow
93C22×C17—24no—
94C22×C17—28no—
95C22×C17—42no—
96C22×C17—42no—
97C22×C17—44no—
98C22×C17C22×D411yescharacteristic
99C22×C17C22×D411yescharacteristic
100C22×C17C2×#4111yescharacteristic
101C22×C17C2×#4112yes—
102C22×C172−1+412yes—
103C4×C17—42no—
104C4×C17—42no—
105C4×C17#12114yes—
106Dic17—22no—
107Dic17—42no—
108Dic17—42no—
109Dic17—42no—
110Dic17#12414yescomplemented
111C2×C4⋊Q8—171no—
112C2×#124—171no—
113C2×#124D1711yescharacteristic
114C2×C4×D4—171no—
115#397—174no—
116#391—171no—
117#391—171no—
118#380—174no—
119C4×C4⋊C4—171no—
120C2×C17—44no—
121C2×C17C2×#12111yescharacteristic
122C2×C17C2×#12411yescharacteristic
123C2×C17#53611yescharacteristic
124C2×C17#54712yes—
125C2×C17#55612yes—
126C22×Q8—171no—
127C22×Q8C2×D1711yescharacteristic
128C22×D4C2×D1711yescharacteristic
129C23×C4—172no—
130C4⋊Q8—344no—
131#124—24no—
132#124—344no—
133C4×D4—344no—
134C2×C4⋊C4—171no—
135C2×C4⋊C4—172no—
136C2×C4⋊C4—172no—
137C2×C4⋊C4—172no—
138C2×C22⋊C4—172no—
139C2×C22⋊C4—174no—
140C2×C22⋊C4C2×D1714yes—
141C2×C42—171no—
142C2×C42—171no—
143C2×C42C2×D1711yescharacteristic
144#95—172no—
145#95—174no—
146#95—174no—
147C17#254711yescharacteristic, complemented, Sylow
148C24C22×D1712yes—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—342no—
153C2×Q8—342no—
154C2×Q8—342no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4C22⋊D1714yes—
158C22×C4—171no—
159C22×C4—171no—
160C22×C4—172no—
161C22×C4—172no—
162C22×C4—172no—
163C22×C4—344no—
164C22×C4—348no—
165C22×C4C22×D1712yes—
166C22×C4C22×D1712yes—
167C22×C4C22×D1711yescharacteristic
168C4⋊C4—344no—
169C4⋊C4—344no—
170C4⋊C4—344no—
171C4⋊C4—342no—
172C22⋊C4—28no—
173C22⋊C4—344no—
174C22⋊C4—348no—
175C42—22no—
176C42—174no—
177C42—342no—
178C23—22no—
179C23—22no—
180C23—24no—
181C23—24no—
182C23C2×C22⋊D1714yes—
183C23C23×D1711yescharacteristic
184Q8—44no—
185Q8—684no—
186D4—28no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—172no—
194C2×C4—174no—
195C2×C4—172no—
196C2×C4—342no—
197C2×C4—342no—
198C2×C4—342no—
199C2×C4—342no—
200C2×C4—342no—
201C2×C4—344no—
202C2×C4—342no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—688no—
206C2×C4D4×D1712yes—
207C2×C4#6325412yes—
208C2×C4C2×C22⋊D1712yes—
209C22—28no—
210C22—24no—
211C22—44no—
212C22—42no—
213C22—42no—
214C22C2×D4×D1711yescharacteristic
215C22C2×#6325411yescharacteristic
216C22#41×D1712yes—
217C22#1058769712yes—
218C22C22×C22⋊D1711yescharacteristic
219C4—42no—
220C4—42no—
221C4—174no—
222C4—342no—
223C4—682no—
224C4—682no—
225C4—682no—
226C4#1058762514yes—
227C2—44no—
228C2C2×#1058762511yescharacteristic
229C2#124×D1711yescharacteristic
230C2#1176337911yescharacteristic
231C2#1176345812yes—
232C2#1176352812yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#124C2
prime radicalC22×D4×C17C22
hypercenterC2×#124D17
hypocenterC17#2547
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#10587630C2semidirect
4. #11762737C2semidirect
5. #11762716C2semidirect
6. #11762715C2semidirect
7. #11762667C2semidirect
8. C4×#63225C2semidirect
110. Dic17#124semidirect
147. C17#2547semidirect

Maximal subgroups

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

subgroupindex
C2×#105876302
C2×#105876222
C2×D4×Dic172
#117627372
#117627162
#117627152
#117626672
C4×#632252
C2×C17×#1242
#254717