Finite Group Databasebeta

← #14528803

Subgroups — #14528803

Order 2176.

312 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#11763277C211yesmaximal, characteristic
2#11762832C211yesmaximal, characteristic, complemented
3C2×#10587476C211yesmaximal, characteristic
4C2×#10587496C211yesmaximal, characteristic
5C4×#63225C211yesmaximal, characteristic, complemented
6#11762568C211yesmaximal, characteristic, complemented
7C17×#339C211yesmaximal, characteristic, complemented
8C2×#63252C2211yescharacteristic
9C2×M16×C17C2211yescharacteristic
10C17×#117C2211yescharacteristic
11C2×C42×C17C2211yescharacteristic
12C2×#63225—21no—
13C2×C4×Dic17C2211yescharacteristic
14C2×C4×Dic17C2211yescharacteristic
15C2×#63223C2211yescharacteristic
16#10587579C411yescharacteristic, complemented
17#10587579C411yescharacteristic, complemented
18#10587573C411yescharacteristic, complemented
19#10587573C411yescharacteristic, complemented
20#10587567—21no—
21#10587567—21no—
22#10587558—21no—
23C42×D17—21no—
24#10587502—21no—
25#10587496—22no—
26#10587485—21no—
27#10587485—21no—
28#10587476—22no—
29C22×C4×C17—21no—
30C22×C4×C17C2311yescharacteristic
31C2×C22⋊D17—21no—
32C22×Dic17—21no—
33C22×Dic17D411yescharacteristic
34C22×Dic17D411yescharacteristic
35#63252—21no—
36#63252—21no—
37#63252D412yes—
38#63252Q812yes—
39C2×D68C2×C411yescharacteristic
40C2×C4×D17—21no—
41C2×C4×D17—21no—
42C2×C4×D17—21no—
43C2×Dic34C2×C411yescharacteristic
44M16×C17—21no—
45M16×C17—21no—
46C2×C8×C17—21no—
47C4⋊C4×C17C2×C411yescharacteristic
48C4⋊C4×C17C2×C411yescharacteristic
49C22⋊C4×C17—21no—
50C42×C17—21no—
51C42×C17—21no—
52C42×C17—21no—
53#63227—21no—
54#63227—21no—
55#63225—24no—
56#63225—21no—
57#63225—21no—
58C4×Dic17—21no—
59C4×Dic17—21no—
60C4×Dic17—21no—
61C4×Dic17—21no—
62C4×Dic17C2×C411yescharacteristic
63C4×Dic17C2×C411yescharacteristic
64C4×Dic17D411yescharacteristic
65C4×Dic17D411yescharacteristic
66#63223—21no—
67#63223—21no—
68C2×#3622—21no—
69C23×C17C2×D411yescharacteristic
70C22×D17—21no—
71C2×C4×C17—22no—
72C2×C4×C17—21no—
73C2×C4×C17—21no—
74C2×C4×C17—41no—
75C2×C4×C17—41no—
76C2×C4×C17C22×C411yescharacteristic
77C2×C4×C17C2×D411yescharacteristic
78C2×C4×C17C2×D411yescharacteristic
79C2×C4×C17C2×Q811yescharacteristic
80C2×C4×C17#4111yescharacteristic
81C2×C4×C17#4111yescharacteristic
82C22⋊D17—22no—
83C22⋊D17—22no—
84C2×Dic17—21no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—41no—
88C2×Dic17—41no—
89C2×Dic17—41no—
90C2×Dic17C22⋊C411yescharacteristic
91C2×Dic17C22⋊C411yescharacteristic
92C2×Dic17C22⋊C411yescharacteristic
93C2×Dic17C22⋊C411yescharacteristic
94D68—21no—
95D68C4⋊C412yescomplemented
96C4×D17—22no—
97C4×D17—22no—
98C4×D17—42no—
99C4×D17—42no—
100Dic34—21no—
101Dic34C4⋊C412yescomplemented
102C8×C17—41no—
103#3622—41no—
104#1806—171nomaximal, Sylow
105C22×C17—21no—
106C22×C17—21no—
107C22×C17C2×C22⋊C411yescharacteristic
108C22×C17#12111yescharacteristic
109C22×C17C22⋊Q811yescharacteristic
110C2×D17—22no—
111C2×D17—41no—
112C2×D17—41no—
113C4×C17—42no—
114C4×C17—41no—
115C4×C17—41no—
116C4×C17C2×C4⋊C411yescharacteristic
117C4×C17#11711yescharacteristic
118C4×C17#12111yescharacteristic
119C4×C17C22⋊Q811yescharacteristic
120Dic17—22no—
121Dic17—21no—
122Dic17—21no—
123Dic17—21no—
124Dic17—42no—
125Dic17#10412yescomplemented
126C4×#41—171no—
127#423—171no—
128C2×#104—171no—
129C2×#104—171no—
130C4×C4⋊C4—171no—
131#339—171no—
132#339D1711yescharacteristic, complemented
133C2×C17—21no—
134C2×C17—21no—
135C2×C17#38011yescharacteristic
136C2×C17C2×#10411yescharacteristic
137C2×C17#42111yescharacteristic
138D17—42no—
139C2×#41—171no—
140C2×M16—171no—
141C2×M16C2×D1711yescharacteristic
142C4×Q8—171no—
143C4×Q8—171no—
144C4×D4—171no—
145C4×D4—171no—
146C4×D4—341no—
147C4×D4—341no—
148#117—341no—
149#117C2×D1711yescharacteristic
150C2×C4⋊C4—341no—
151C2×C42—171no—
152C2×C42—171no—
153C2×C42—341no—
154C2×C42C2×D1711yescharacteristic
155C4⋊C8—341no—
156C4⋊C8—341no—
157#104—342no—
158#104—342no—
159#95—341no—
160C17#180611yescharacteristic, complemented, Sylow
161#41—172no—
162#41—172no—
163#41—341no—
164#41—341no—
165C2×Q8—171no—
166C2×D4—171no—
167C2×D4—341no—
168C22×C4—21no—
169C22×C4—171no—
170C22×C4—171no—
171C22×C4—341no—
172C22×C4—341no—
173C22×C4—341no—
174C22×C4—341no—
175C22×C4C22×D1711yescharacteristic
176M16—21no—
177M16—21no—
178M16—341no—
179M16—341no—
180C2×C8—21no—
181C2×C8—341no—
182C4⋊C4—341no—
183C4⋊C4—341no—
184C4⋊C4—344no—
185C4⋊C4C4×D1711yescharacteristic
186C4⋊C4C4×D1711yescharacteristic
187C22⋊C4—21no—
188C22⋊C4—341no—
189C22⋊C4—341no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—171no—
194C42—171no—
195C42—171no—
196C42—171no—
197C42—341no—
198C42—341no—
199C42—341no—
200C42—341no—
201C23—341no—
202C23D4×D1711yescharacteristic
203Q8—172no—
204Q8—341no—
205D4—172no—
206D4—342no—
207D4—342no—
208D4—341no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—171no—
215C2×C4—171no—
216C2×C4—171no—
217C2×C4—171no—
218C2×C4—342no—
219C2×C4—341no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—682no—
224C2×C4—682no—
225C2×C4—681no—
226C2×C4—681no—
227C2×C4—681no—
228C2×C4C2×C4×D1711yescharacteristic
229C2×C4#6325211yescharacteristic
230C2×C4#6325211yescharacteristic
231C2×C4D4×D1711yescharacteristic
232C2×C4D4×D1711yescharacteristic
233C2×C4Q8×D1711yescharacteristic
234C8—41no—
235C8—681no—
236C22—21no—
237C22—21no—
238C22—342no—
239C22—681no—
240C22—681no—
241C22C22⋊C4×D1711yescharacteristic
242C22#1058757011yescharacteristic
243C22#1058758211yescharacteristic
244C4—42no—
245C4—41no—
246C4—41no—
247C4—172no—
248C4—341no—
249C4—341no—
250C4—341no—
251C4—342no—
252C4—682no—
253C4#1058755811yescharacteristic
254C4#1058757011yescharacteristic
255C4C4⋊C4×D1711yescharacteristic
256C4#1058758211yescharacteristic
257C2—21no—
258C2—21no—
259C2—682no—
260C2#1176266811yescharacteristic
261C2#104×D1711yescharacteristic
262C2#1176282411yescharacteristic
subgroupG/H
centerC2×C4D4×D17
commutatorC2×C4×C17C22×C4
FrattiniC22×C4C22×D17
FittingC17×#339C2
socleC22×C17C2×C22⋊C4
prime radicalC2×#63252C22
cosocleC22×C4×C17C23
hypercenter#339D17
hypocenterC17#1806
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #11762832C2semidirect
5. C4×#63225C2semidirect
6. #11762568C2semidirect
7. C17×#339C2semidirect
16. #10587579C4semidirect
17. #10587579C4semidirect
18. #10587573C4semidirect
19. #10587573C4semidirect
95. D68C4⋊C4semidirect
101. Dic34C4⋊C4semidirect
125. Dic17#104semidirect
132. #339D17semidirect
160. C17#1806semidirect

Maximal subgroups

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

subgroupindex
#117632772
#117628322
C2×#105874762
C2×#105874962
C4×#632252
#117625682
C17×#3392
#180617