Finite Group Databasebeta

← #14535158

Subgroups — #14535158

Order 2176.

592 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10587582C212yesmaximal, complemented
2C2×#10587632C212yesmaximal, complemented
3#11762733C212yesmaximal, complemented
4#11762731C212yesmaximal, complemented
5#11762729C211yesmaximal, characteristic, complemented
6#11762710C211yesmaximal, characteristic
7#11762706C212yesmaximal, complemented
8#11762687C212yesmaximal
9C17×#395C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×D68C2211yescharacteristic
12C22×C4×D17C2212yes—
13C22×Dic34C2211yescharacteristic
14C2×C4⋊C4×C17C2212yes—
15C2×C4⋊C4×C17C2211yescharacteristic
16#10587632—28no—
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×C4⋊Dic17C2212yes—
23C2×#63225C2212yes—
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26C2×#63225C2212yes—
27C2×C4×Dic17C2211yescharacteristic
28#10587582—28no—
29C17×#95C2212yes—
30C17×#95C2211yescharacteristic
31#10587502C2212yes—
32#10587502C2211yescharacteristic
33C23×D17C2312yes—
34C2×Q8×C17—24no—
35C2×Q8×C17—41no—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2312yes—
39C22×C4×C17C2312yes—
40C22×C4×C17C2311yescharacteristic
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C2×D68—41no—
45C2×D68—41no—
46C2×D68Q814yescomplemented
47C2×C4×D17—24no—
48C2×C4×D17—24no—
49C2×C4×D17—42no—
50C2×Dic34—24no—
51C2×Dic34—41no—
52C4⋊C4×C17—24no—
53C4⋊C4×C17—41no—
54#63227—24no—
55#63227—24no—
56#63227—24no—
57#63227—24no—
58#63227—42no—
59C4⋊Dic17—24no—
60#63225—24no—
61#63225—24no—
62#63225—42no—
63#63225D418yescomplemented
64C4×Dic17—41no—
65C23×C17C2411yescharacteristic
66C22×D17—22no—
67C22×D17—22no—
68C22×D17—22no—
69C22×D17—22no—
70C22×D17—22no—
71C22×D17—22no—
72C22×D17C2×Q814yes—
73Q8×C17—44no—
74C2×C4×C17—22no—
75C2×C4×C17—21no—
76C2×C4×C17—22no—
77C2×C4×C17—21no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—21no—
82C2×C4×C17—21no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17C2×D414yes—
86C2×C4×C17C2×Q812yes—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17C2×D414yes—
95C2×Dic17C2×D414yes—
96D68—44no—
97C4×D17—48no—
98Dic34—44no—
99#2683—171nomaximal, Sylow
100C22×C17C22×D412yes—
101C22×C17C22×Q811yescharacteristic
102C22×C17H(5,𝔽2)11yescharacteristic
103C22×C17H(5,𝔽2)11yescharacteristic
104C22×C172−1+412yes—
105C2×D17—24no—
106C2×D17—24no—
107C2×D17—24no—
108C2×D17—42no—
109C2×D17—42no—
110C2×D17—42no—
111C2×D17—42no—
112C4×C17—22no—
113C4×C17—24no—
114C4×C17—42no—
115C4×C17—41no—
116C4×C17—41no—
117Dic17—24no—
118Dic17—24no—
119Dic17—42no—
120C2×C22⋊Q8—172no—
121C2×C22⋊Q8—172no—
122#396—172no—
123#396—172no—
124#395—171no—
125#395D1711yescharacteristic, complemented
126#390—171no—
127#389—172no—
128#386—172no—
129C2×C17#53712yes—
130C2×C17#54311yescharacteristic
131C2×C17D4211yescharacteristic
132C2×C17D4×Q812yes—
133C2×C17#56311yescharacteristic
134D17—44no—
135C22×Q8—171no—
136C22×Q8C2×D1711yescharacteristic
137C22×D4—171no—
138C23×C4—172no—
139C22⋊Q8—348no—
140C22⋊Q8—348no—
141C2×C4⋊C4—172no—
142C2×C4⋊C4—172no—
143C2×C4⋊C4—172no—
144C2×C4⋊C4—172no—
145C2×C4⋊C4—172no—
146C2×C4⋊C4C2×D1712yes—
147C2×C4⋊C4C2×D1711yescharacteristic
148C2×C22⋊C4—172no—
149C2×C22⋊C4—172no—
150C2×C22⋊C4—172no—
151C2×C22⋊C4—172no—
152C2×C22⋊C4—172no—
153C2×C42—171no—
154#95—172no—
155#95—171no—
156#95C2×D1712yes—
157#95C2×D1711yescharacteristic
158C17#268311yescharacteristic, complemented, Sylow
159C24—172no—
160C2×Q8—24no—
161C2×Q8—41no—
162C2×Q8—344no—
163C2×Q8—681no—
164C2×D4—174no—
165C2×D4—681no—
166C2×D4—681no—
167C22×C4—172no—
168C22×C4—172no—
169C22×C4—172no—
170C22×C4—344no—
171C22×C4—344no—
172C22×C4—682no—
173C22×C4C22×D1711yescharacteristic
174C22×C4C22×D1712yes—
175C22×C4C22×D1711yescharacteristic
176C22×C4C22×D1712yes—
177C22×C4C22×D1711yescharacteristic
178C4⋊C4—24no—
179C4⋊C4—41no—
180C4⋊C4—178no—
181C4⋊C4—344no—
182C4⋊C4—344no—
183C4⋊C4—344no—
184C4⋊C4—682no—
185C22⋊C4—344no—
186C22⋊C4—344no—
187C22⋊C4—344no—
188C22⋊C4—344no—
189C22⋊C4—682no—
190C42—681no—
191C23—174no—
192C23—342no—
193C23—342no—
194C23—342no—
195C23—342no—
196C23—342no—
197C23—342no—
198C23C23×D1711yescharacteristic
199Q8—44no—
200Q8—684no—
201D4—684no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—174no—
214C2×C4—174no—
215C2×C4—342no—
216C2×C4—342no—
217C2×C4—342no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—688no—
223C2×C4D4×D1714yes—
224C2×C4Q8×D1712yes—
225C22—344no—
226C22—344no—
227C22—344no—
228C22—682no—
229C22—682no—
230C22—682no—
231C22—682no—
232C22C2×D4×D1712yes—
233C22#1058769111yescharacteristic
234C22C2×Q8×D1711yescharacteristic
235C22#1058769412yes—
236C22#1058769611yescharacteristic
237C4—22no—
238C4—24no—
239C4—42no—
240C4—41no—
241C4—41no—
242C4—344no—
243C4—344no—
244C4—682no—
245C2—684no—
246C2#1176338912yes—
247C2#1176343411yescharacteristic
248C2#1176344111yescharacteristic
249C2#1176347412yes—
250C2#1176355811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#395C2
prime radicalC22×D68C22
hypercenter#395D17
hypocenterC17#2683
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#10587582C2semidirect
2. C2×#10587632C2semidirect
3. #11762733C2semidirect
4. #11762731C2semidirect
5. #11762729C2semidirect
7. #11762706C2semidirect
9. C17×#395C2semidirect
46. C2×D68Q8semidirect
63. #63225D4semidirect
125. #395D17semidirect
158. C17#2683semidirect

Maximal subgroups

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

subgroupindex
C2×#105875822
C2×#105876322
#117627332
#117627312
#117627292
#117627102
#117627062
#117626872
C17×#3952
#268317