Finite Group Databasebeta

← #14533774

Subgroups — #14533774

Order 2176.

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

normal subgroups only

This group has 540 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×#10587555C211yesmaximal, characteristic
2C2×#10587630C211yesmaximal, characteristic
3C2×#10587630C211yesmaximal, characteristic
4C2×#10587573C211yesmaximal, characteristic
5#11762743C214yesmaximal
6#11762714C211yesmaximal, characteristic
7#11762706C214yesmaximal
8C4×#63225C211yesmaximal, characteristic
9C17×#391C211yesmaximal, characteristic
10C22×Q8×C17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C22×Dic34C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C42×C17C2211yescharacteristic
15#10587630—24no—
16#10587630—24no—
17C2×C4⋊Dic17C2214yes—
18C2×#63225C2212yes—
19C2×#63225C2214yes—
20C2×#63225C2212yes—
21C2×#63225C2212yes—
22C2×#63225C2212yes—
23C2×C4×Dic17C2212yes—
24C2×C4×Dic17C2211yescharacteristic
25C2×C4×Dic17C2211yescharacteristic
26#10587573—24no—
27#10587555—24no—
28C17×#95C2214yes—
29#10587502C2214yes—
30#10587502C2212yes—
31C2×Q8×C17—22no—
32C2×Q8×C17—22no—
33C2×Q8×C17—22no—
34C22×C4×C17C2312yes—
35C22×C4×C17C2312yes—
36C22×C4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2311yescharacteristic
41C22×Dic17C2312yes—
42C22×Dic17C2311yescharacteristic
43C2×Dic34—22no—
44C2×Dic34—22no—
45C2×Dic34—22no—
46C2×Dic34—22no—
47C2×Dic34—22no—
48C2×Dic34D414yes—
49C4⋊C4×C17—22no—
50C42×C17—22no—
51C4⋊Dic17—28no—
52#63225—24no—
53#63225—28no—
54#63225—24no—
55#63225—24no—
56#63225Q818yescomplemented
57C4×Dic17—22no—
58C4×Dic17—24no—
59C4×Dic17D414yes—
60C23×C17C2411yescharacteristic
61Q8×C17—44no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17—24no—
69C2×C4×C17C2×D412yes—
70C2×C4×C17C2×D412yes—
71C2×C4×C17C2×Q814yes—
72C2×C4×C17#4112yes—
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×Dic17C2×D414yes—
82C2×Dic17C2×D412yes—
83C2×Dic17C2×D412yes—
84C2×Dic17C2×Q814yes—
85C2×Dic17C2×Q814yes—
86Dic34—28no—
87Dic34—44no—
88#2546—171nomaximal, Sylow
89C22×C17C22×D411yescharacteristic
90C22×C17C22×D411yescharacteristic
91C22×C17C22×Q812yes—
92C22×C17C2×#4111yescharacteristic
93C22×C172−1+412yes—
94C4×C17—24no—
95C4×C17—42no—
96C4×C17—42no—
97C4×C17#12114yes—
98Dic17—24no—
99Dic17—22no—
100Dic17—24no—
101Dic17—42no—
102Dic17C4⋊Q814yescomplemented
103C2×C4⋊Q8—171no—
104C2×C4⋊Q8—171no—
105C2×C4⋊Q8—171no—
106C2×C4×Q8—171no—
107#398—174no—
108#391—171no—
109#391D1711yescharacteristic
110#389—174no—
111C4×C4⋊C4—171no—
112C2×C17C2×#12111yescharacteristic
113C2×C17C2×C4⋊Q811yescharacteristic
114C2×C17#53611yescharacteristic
115C2×C17D4×Q812yes—
116C2×C17#55712yes—
117C22×Q8—171no—
118C22×Q8—171no—
119C22×Q8C2×D1711yescharacteristic
120C4⋊Q8—344no—
121C4⋊Q8—344no—
122C4⋊Q8—344no—
123C4×Q8—344no—
124C2×C4⋊C4—172no—
125C2×C4⋊C4—172no—
126C2×C4⋊C4—172no—
127C2×C4⋊C4—174no—
128C2×C4⋊C4—174no—
129C2×C4⋊C4—172no—
130C2×C4⋊C4C2×D1711yescharacteristic
131C2×C42—171no—
132C2×C42—172no—
133C2×C42—171no—
134C2×C42C2×D1711yescharacteristic
135#95—172no—
136#95—174no—
137#95C2×D1714yes—
138C17#254611yescharacteristic, complemented, Sylow
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—174no—
143C2×Q8—342no—
144C2×Q8—342no—
145C2×Q8—342no—
146C2×Q8—342no—
147C2×Q8—342no—
148C22×C4—171no—
149C22×C4—172no—
150C22×C4—171no—
151C22×C4—172no—
152C22×C4—172no—
153C22×C4C22×D1712yes—
154C22×C4C22×D1712yes—
155C22×C4C22×D1712yes—
156C22×C4C22×D1711yescharacteristic
157C4⋊C4—22no—
158C4⋊C4—178no—
159C4⋊C4—344no—
160C4⋊C4—344no—
161C4⋊C4—344no—
162C4⋊C4—348no—
163C4⋊C4—348no—
164C42—22no—
165C42—174no—
166C42—344no—
167C42—342no—
168C23C23×D1711yescharacteristic
169Q8—44no—
170Q8—348no—
171Q8—684no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—172no—
180C2×C4—174no—
181C2×C4—172no—
182C2×C4—174no—
183C2×C4—174no—
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#6325212yes—
193C2×C4D4×D1712yes—
194C2×C4D4×D1712yes—
195C2×C4Q8×D1714yes—
196C22C2×#6325211yescharacteristic
197C22C2×D4×D1711yescharacteristic
198C22C2×D4×D1711yescharacteristic
199C22C2×Q8×D1712yes—
200C22#1058769712yes—
201C4—24no—
202C4—42no—
203C4—42no—
204C4—174no—
205C4—344no—
206C4—344no—
207C4—342no—
208C4—682no—
209C4#1058757014yes—
210C2C2×#1058757011yescharacteristic
211C2C4⋊Q8×D1711yescharacteristic
212C2#1176337911yescharacteristic
213C2#1176347212yes—
214C2#1176353712yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#391C2
hypercenter#391D17
hypocenterC17#2546
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
56. #63225Q8semidirect
102. Dic17C4⋊Q8semidirect
138. C17#2546semidirect

Maximal subgroups

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

subgroupindex
C2×#105875552
C2×#105876302
C2×#105875732
#117627432
#117627142
#117627062
C4×#632252
C17×#3912
#254617