Finite Group Databasebeta

← #14535619

Subgroups — #14535619

Order 2176.

512 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 137 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×#10587621C211yesmaximal, characteristic
2#11762737C212yesmaximal, complemented
3#11762737C212yesmaximal, complemented
4#11762728C212yesmaximal, complemented
5#11762693C212yesmaximal, complemented
6#11762689C211yesmaximal, characteristic
7#11762681C212yesmaximal, complemented
8#11762667C212yesmaximal, complemented
9C2×C17×#124C211yesmaximal, characteristic
10C22×Q8×C17C2211yescharacteristic
11C22×D4×C17C2211yescharacteristic
12C23×Dic17C2211yescharacteristic
13C23×Dic17C2211yescharacteristic
14C17×#124—24no—
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊C4×C17C2212yes—
17C2×C42×C17C2211yescharacteristic
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2211yescharacteristic
20C2×C22⋊Dic17C2212yes—
21C2×C22⋊Dic17C2211yescharacteristic
22#10587621—24no—
23C2×#63225C2212yes—
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26C2×#63225C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2212yes—
32#10587502C2212yes—
33C24×C17C2311yescharacteristic
34C24×C17C2311yescharacteristic
35C2×Q8×C17—22no—
36C2×Q8×C17—41no—
37C2×Q8×C17—41no—
38C2×D4×C17—41no—
39C2×D4×C17—41no—
40C2×D4×C17D414yes—
41C22×C4×C17C2312yes—
42C22×C4×C17C2312yes—
43C22×C4×C17C2311yescharacteristic
44C22×Dic17—22no—
45C22×Dic17—24no—
46C22×Dic17—41no—
47C22×Dic17—41no—
48C22×Dic17—41no—
49C22×Dic17C2312yes—
50C22×Dic17C2312yes—
51C22×Dic17C2312yes—
52C22×Dic17C2312yes—
53C22⋊C4×C17—24no—
54C22⋊C4×C17—24no—
55C42×C17—22no—
56C22⋊Dic17—22no—
57C22⋊Dic17—24no—
58C22⋊Dic17—24no—
59C22⋊Dic17—41no—
60#63225—24no—
61#63225—24no—
62#63225—42no—
63#63225—42no—
64C23×C17—22no—
65C23×C17—22no—
66C23×C17—21no—
67C23×C17—21no—
68C23×C17—21no—
69C23×C17—21no—
70C23×C17—22no—
71C23×C17—22no—
72C23×C17C2×D412yes—
73C23×C17C2×D412yes—
74C23×C17C2411yescharacteristic
75Q8×C17—44no—
76D4×C17—44no—
77C2×C4×C17—21no—
78C2×C4×C17—21no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17C2×D412yes—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—22no—
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—44no—
98C2×Dic17—44no—
99C2×Dic17#4114yes—
100#2717—171nomaximal, Sylow
101C22×C17—22no—
102C22×C17—22no—
103C22×C17—24no—
104C22×C17—24no—
105C22×C17—42no—
106C22×C17—42no—
107C22×C17—41no—
108C22×C17—41no—
109C22×C17—41no—
110C22×C17—41no—
111C22×C17C22×D411yescharacteristic
112C22×C17C2×#4112yes—
113C22×C17H(5,𝔽2)11yescharacteristic
114C22×C17H(5,𝔽2)11yescharacteristic
115C22×C172−1+412yes—
116C4×C17—22no—
117C4×C17—42no—
118C4×C17—42no—
119Dic17—24no—
120Dic17—42no—
121Dic17—42no—
122Dic17—42no—
123C2×#124D1711yescharacteristic
124C2×#123—171no—
125#397—172no—
126#397—172no—
127#395—172no—
128#387—172no—
129#386—171no—
130#385—172no—
131#380—172no—
132C2×C17—42no—
133C2×C17—42no—
134C2×C17#53411yescharacteristic
135C2×C17#54212yes—
136C2×C17#54712yes—
137C2×C17#55111yescharacteristic
138C2×C17#56311yescharacteristic
139C22×Q8C2×D1711yescharacteristic
140C22×D4C2×D1711yescharacteristic
141C23×C4—171no—
142C23×C4—171no—
143#124—24no—
144#123—344no—
145C2×C4⋊C4—172no—
146C2×C4⋊C4—172no—
147C2×C4⋊C4—172no—
148C2×C4⋊C4—172no—
149C2×C22⋊C4—172no—
150C2×C22⋊C4—171no—
151C2×C22⋊C4—172no—
152C2×C22⋊C4—171no—
153C2×C22⋊C4C2×D1712yes—
154C2×C22⋊C4C2×D1712yes—
155C2×C42C2×D1711yescharacteristic
156#95—172no—
157#95—172no—
158#95—172no—
159#95—172no—
160#95—172no—
161#95—172no—
162C17#271711yescharacteristic, complemented, Sylow
163C24C22×D1711yescharacteristic
164C24C22×D1711yescharacteristic
165C2×Q8—22no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4C22⋊D1714yes—
171C22×C4—172no—
172C22×C4—172no—
173C22×C4—172no—
174C22×C4—172no—
175C22×C4—344no—
176C22×C4—342no—
177C22×C4—681no—
178C22×C4—681no—
179C22×C4—681no—
180C22×C4C22×D1712yes—
181C22×C4C22×D1712yes—
182C22×C4C22×D1711yescharacteristic
183C4⋊C4—344no—
184C4⋊C4—344no—
185C4⋊C4—682no—
186C4⋊C4—682no—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—344no—
190C22⋊C4—342no—
191C22⋊C4—344no—
192C22⋊C4—681no—
193C42—22no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23C2×C22⋊D1712yes—
203C23C2×C22⋊D1712yes—
204C23C23×D1711yescharacteristic
205Q8—44no—
206D4—44no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—174no—
216C2×C4—342no—
217C2×C4—342no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—342no—
224C2×C4—342no—
225C2×C4—342no—
226C2×C4—342no—
227C2×C4—684no—
228C2×C4—684no—
229C2×C4C2×C22⋊D1712yes—
230C22—24no—
231C22—24no—
232C22—22no—
233C22—22no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—42no—
239C22—42no—
240C22#1058769111yescharacteristic
241C22#1058769111yescharacteristic
242C22#41×D1712yes—
243C22#1058769712yes—
244C22C22×C22⋊D1711yescharacteristic
245C4—22no—
246C4—42no—
247C4—42no—
248C4—344no—
249C4—682no—
250C4—682no—
251C4—682no—
252C2—42no—
253C2—42no—
254C2#1176336811yescharacteristic
255C2#1176342312yes—
256C2#1176345812yes—
257C2#1176348911yescharacteristic
258C2#1176355911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#124C2
prime radicalC22×D4×C17C22
hypercenterC2×#124D17
hypocenterC17#2717
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #11762737C2semidirect
3. #11762737C2semidirect
4. #11762728C2semidirect
5. #11762693C2semidirect
7. #11762681C2semidirect
8. #11762667C2semidirect
162. C17#2717semidirect

Maximal subgroups

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

subgroupindex
C2×#105876212
#117627372
#117627282
#117626932
#117626892
#117626812
#117626672
C2×C17×#1242
#271717