Finite Group Databasebeta

← #14535790

Subgroups — #14535790

Order 2176.

492 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 141 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×#10587633C211yesmaximal, characteristic
2#11762731C212yesmaximal, complemented
3#11762726C212yesmaximal
4#11762720C212yesmaximal, complemented
5#11762716C211yesmaximal, characteristic, complemented
6#11762700C212yesmaximal, complemented
7#11762699C212yesmaximal, complemented
8#11762690C212yesmaximal, complemented
9C2×C17×C4⋊Q8C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×Q8×C17C2211yescharacteristic
12C22×D68C2211yescharacteristic
13C17×C4⋊Q8—24no—
14C2×C4⋊C4×C17C2212yes—
15C2×C4⋊C4×C17C2212yes—
16C2×C42×C17C2211yescharacteristic
17#10587633—24no—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×C4⋊Dic17C2211yescharacteristic
25C2×#63225C2212yes—
26C2×C4×Dic17C2211yescharacteristic
27C2×C4×Dic17C2211yescharacteristic
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2212yes—
32#10587502C2212yes—
33C23×D17C2312yes—
34C2×Q8×C17—22no—
35C2×Q8×C17—41no—
36C2×Q8×C17—41no—
37C2×Q8×C17—41no—
38C2×Q8×C17—41no—
39C2×Q8×C17D414yes—
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2312yes—
47C22×Dic17C2312yes—
48C2×D68—22no—
49C2×D68—41no—
50C2×D68—41no—
51C4⋊C4×C17—24no—
52C4⋊C4×C17—24no—
53C42×C17—22no—
54#63227—24no—
55#63227—24no—
56#63227—24no—
57#63227—24no—
58#63227—42no—
59#63227—42no—
60C4⋊Dic17—41no—
61#63225—24no—
62C4×Dic17—22no—
63C4×Dic17—41no—
64C23×C17C2411yescharacteristic
65C22×D17—22no—
66C22×D17—22no—
67C22×D17—22no—
68C22×D17—22no—
69C22×D17—22no—
70C22×D17—22no—
71C22×D17—22no—
72Q8×C17—44no—
73Q8×C17—44no—
74C2×C4×C17—22no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—21no—
78C2×C4×C17—21no—
79C2×C4×C17—21no—
80C2×C4×C17—21no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17C2×D412yes—
86C2×C4×C17C2×D412yes—
87C2×C4×C17C2×D412yes—
88C2×C4×C17#4114yes—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98D68—44no—
99#2731—171nomaximal, Sylow
100C22×C17C22×D411yescharacteristic
101C22×C17C2×#4112yes—
102C22×C17H(5,𝔽2)11yescharacteristic
103C22×C17H(5,𝔽2)11yescharacteristic
104C22×C172−1+412yes—
105C2×D17—42no—
106C2×D17—42no—
107C2×D17—42no—
108C2×D17—42no—
109C2×D17—42no—
110C2×D17—42no—
111C2×D17—42no—
112C4×C17—22no—
113C4×C17—22no—
114C4×C17—22no—
115C4×C17—24no—
116C4×C17—42no—
117Dic17—42no—
118Dic17—42no—
119Dic17—42no—
120C2×C4⋊Q8D1711yescharacteristic, complemented
121C2×#124—171no—
122#396—172no—
123#394—172no—
124#393—172no—
125#391—171no—
126#388—172no—
127#388—172no—
128#387—172no—
129C2×C17#53411yescharacteristic
130C2×C17#54212yes—
131C2×C17#54812yes—
132C2×C17#55511yescharacteristic
133C2×C17#56311yescharacteristic
134D17—82no—
135C22×Q8C2×D1711yescharacteristic
136C22×Q8C2×D1711yescharacteristic
137C22×D4—171no—
138C4⋊Q8—24no—
139#124—344no—
140C2×C4⋊C4—172no—
141C2×C4⋊C4—171no—
142C2×C4⋊C4C2×D1712yes—
143C2×C4⋊C4C2×D1712yes—
144C2×C22⋊C4—172no—
145C2×C22⋊C4—172no—
146C2×C22⋊C4—172no—
147C2×C22⋊C4—172no—
148C2×C22⋊C4—172no—
149C2×C22⋊C4—172no—
150C2×C42—171no—
151C2×C42—171no—
152C2×C42C2×D1711yescharacteristic
153#95—172no—
154#95—172no—
155#95—172no—
156#95—172no—
157#95—172no—
158C17#273111yescharacteristic, complemented, Sylow
159C24—172no—
160C2×Q8—22no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8C22⋊D1714yes—
166C2×D4—342no—
167C2×D4—681no—
168C2×D4—681no—
169C22×C4—172no—
170C22×C4—172no—
171C22×C4—172no—
172C22×C4C22×D1712yes—
173C22×C4C22×D1712yes—
174C22×C4C22×D1711yescharacteristic
175C22×C4C22×D1711yescharacteristic
176C22×C4C22×D1711yescharacteristic
177C4⋊C4—24no—
178C4⋊C4—24no—
179C4⋊C4—344no—
180C4⋊C4—681no—
181C22⋊C4—344no—
182C22⋊C4—344no—
183C22⋊C4—344no—
184C22⋊C4—344no—
185C22⋊C4—682no—
186C22⋊C4—682no—
187C42—22no—
188C42—342no—
189C42—681no—
190C23—342no—
191C23—342no—
192C23—342no—
193C23—342no—
194C23—342no—
195C23—342no—
196C23—342no—
197C23C23×D1711yescharacteristic
198Q8—44no—
199Q8—44no—
200D4—684no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—342no—
213C2×C4—342no—
214C2×C4—342no—
215C2×C4—342no—
216C2×C4—342no—
217C2×C4—342no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—342no—
221C2×C4#6325614yes—
222C2×C4C2×C22⋊D1712yes—
223C2×C4C2×C22⋊D1712yes—
224C2×C4C2×C22⋊D1712yes—
225C22—682no—
226C22—682no—
227C22—682no—
228C22—682no—
229C22—682no—
230C22—682no—
231C22—682no—
232C22#1058769111yescharacteristic
233C22#1058769111yescharacteristic
234C22C2×#6325612yes—
235C22#1058769412yes—
236C22C22×C22⋊D1711yescharacteristic
237C4—24no—
238C4—22no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—682no—
243C4—682no—
244C4—682no—
245C2—1362no—
246C2#1176336811yescharacteristic
247C2#1176343312yes—
248C2#1176347012yes—
249C2#1176352511yescharacteristic
250C2#1176355811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×C4⋊Q8C2
prime radicalC22×D68C22
hypercenterC2×C4⋊Q8D17
hypocenterC17#2731
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #11762731C2semidirect
4. #11762720C2semidirect
5. #11762716C2semidirect
6. #11762700C2semidirect
7. #11762699C2semidirect
8. #11762690C2semidirect
9. C2×C17×C4⋊Q8C2semidirect
120. C2×C4⋊Q8D17semidirect
158. C17#2731semidirect

Maximal subgroups

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

subgroupindex
C2×#105876332
#117627312
#117627262
#117627202
#117627162
#117627002
#117626992
#117626902
C2×C17×C4⋊Q82
#273117