Finite Group Databasebeta

← #17949799

Subgroups — #17949799

Order 4352.

440 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 118 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#14544711C211yesmaximal, characteristic
2#14544711C211yesmaximal, characteristic
3#14543602C214yesmaximal
4C4⋊Q8×Dic17C211yesmaximal, characteristic
5#11763675—21no—
6#11763675—21no—
7#11763675—21no—
8#11763675—21no—
9#11763675C2212yes—
10#11763675C2212yes—
11C2×Q8×Dic17—21no—
12#11762716—21no—
13#11762708—21no—
14#11762708C2212yes—
15#11762679C411yescharacteristic
16C4⋊C4×Dic17—21no—
17C4⋊C4×Dic17C412yes—
18C42×Dic17C2211yescharacteristic
19C2×C17×C4⋊Q8C411yescharacteristic
20C22×Q8×C17—21no—
21C22×#3624—22no—
22C22×#3624—22no—
23C17×C4⋊Q8—22no—
24C17×C4⋊Q8C814yescomplemented
25C2×C4⋊C4×C17—21no—
26C2×C4⋊C4×C17C2×C412yes—
27C2×C42×C17C2×C411yescharacteristic
28Q8×Dic17—44no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—22no—
32C2×C4⋊Dic17C2×C412yes—
33C2×C4×Dic17—22no—
34C2×C4×Dic17C2×C411yescharacteristic
35C2×C4×Dic17D411yescharacteristic
36C2×C4×Dic17D411yescharacteristic
37C2×C4×Dic17D411yescharacteristic
38C2×C4×Dic17D411yescharacteristic
39C2×C4×Dic17C2311yescharacteristic
40#10587502—22no—
41#10587502—22no—
42C2×Q8×C17—24no—
43C2×Q8×C17—42no—
44C2×Q8×C17—42no—
45C22×C4×C17—22no—
46C22×C4×C17C22⋊C411yescharacteristic
47C22×C4×C17C22⋊C411yescharacteristic
48C22×C4×C17C22×C411yescharacteristic
49C22×Dic17—22no—
50C22×Dic17C22⋊C411yescharacteristic
51C22×Dic17C22⋊C411yescharacteristic
52C22×Dic17C2×D411yescharacteristic
53C22×Dic17C2×D411yescharacteristic
54C2×#3624—42no—
55C2×#3624—42no—
56C2×#3624—42no—
57C2×#3624—42no—
58C2×#3624—42no—
59C2×#3624—42no—
60C4⋊C4×C17—24no—
61C4⋊C4×C17—22no—
62C4⋊C4×C17C2×C814yes—
63C42×C17C2×C812yes—
64C42×C17M1612yes—
65C4⋊Dic17—42no—
66C4⋊Dic17—42no—
67C4⋊Dic17—44no—
68C4⋊Dic17—42no—
69C4×Dic17—22no—
70C4×Dic17—22no—
71C4×Dic17—22no—
72C4×Dic17—41no—
73C4×Dic17—41no—
74C4×Dic17—41no—
75C4×Dic17—44no—
76#7604—171nomaximal, Sylow
77C23×C17C2×C22⋊C411yescharacteristic
78Q8×C17—48no—
79C2×C4×C17—21no—
80C2×C4×C17—21no—
81C2×C4×C17—21no—
82C2×C4×C17—21no—
83C2×C4×C17—21no—
84C2×C4×C17—21no—
85C2×C4×C17—24no—
86C2×C4×C17—42no—
87C2×C4×C17—42no—
88C2×C4×C17C22⋊C812yes—
89C2×C4×C17C22⋊C812yes—
90C2×C4×C17C22×C811yescharacteristic
91C2×C4×C17C2×M1611yescharacteristic
92C2×Dic17—21no—
93C2×Dic17—21no—
94C2×Dic17—21no—
95C2×Dic17—21no—
96C2×Dic17—21no—
97C2×Dic17—21no—
98C2×Dic17—21no—
99C2×Dic17—21no—
100C2×Dic17—21no—
101C2×Dic17—42no—
102C2×Dic17—42no—
103C2×Dic17—42no—
104C2×Dic17#10112yes—
105C2×Dic17AGL(1,ℤ/8)12yes—
106C2×Dic17#13712yes—
107#3624—82no—
108#3624—82no—
109C4×C4⋊Q8—171no—
110#1839—171no—
111#1839—171no—
112#1271—174no—
113C22×C17C2×C22⋊C811yescharacteristic
114C22×C17#41011yescharacteristic
115C22×C17C2×#10111yescharacteristic
116C22×C17#41811yescharacteristic
117C22×C17#41911yescharacteristic
118C22×C17#42111yescharacteristic
119C22×C17#42111yescharacteristic
120C4×C17—21no—
121C4×C17—21no—
122C4×C17—22no—
123C4×C17—22no—
124C4×C17—44no—
125Dic17—22no—
126Dic17—22no—
127Dic17—22no—
128Dic17—41no—
129Dic17—82no—
130C2×C4⋊Q8#59511yescharacteristic
131C2×C4×Q8—341no—
132#391—341no—
133#389—172no—
134#389—341no—
135#384—171no—
136C4×C4⋊C4—172no—
137C4×C4⋊C4—341no—
138C43—171no—
139#336—172no—
140#336—172no—
141#336—341no—
142#336—341no—
143#336—341no—
144#336—341no—
145C2×C17#146111yescharacteristic
146C2×C17#147811yescharacteristic
147C2×C17#147911yescharacteristic
148C2×C17#151411yescharacteristic
149C2×C17#151511yescharacteristic
150C2×C17#153711yescharacteristic
151C2×C17#155611yescharacteristic
152C22×Q8—21no—
153C22×C8—342no—
154C22×C8—342no—
155C4⋊Q8—22no—
156C4⋊Q8#362414yescomplemented
157C4×Q8—684no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—172no—
160C2×C4⋊C4—342no—
161C2×C4⋊C4—341no—
162C2×C4⋊C4—341no—
163C2×C4⋊C4C2×#59512yes—
164C2×C42—171no—
165C2×C42—171no—
166C2×C42—171no—
167C2×C42—171no—
168C2×C42—171no—
169C2×C42—171no—
170C2×C42—342no—
171C2×C42C2×#59511yescharacteristic
172#95—342no—
173#95—342no—
174C17#760411yescharacteristic, complemented, Sylow
175C2×Q8—24no—
176C2×Q8—42no—
177C2×Q8—42no—
178C22×C4—22no—
179C22×C4—171no—
180C22×C4—171no—
181C22×C4—171no—
182C22×C4—171no—
183C22×C4—342no—
184C22×C4C22⋊#59511yescharacteristic
185C22×C4C22⋊#59511yescharacteristic
186C22×C4C22×#59511yescharacteristic
187C2×C8—682no—
188C2×C8—682no—
189C2×C8—682no—
190C2×C8—682no—
191C2×C8—682no—
192C2×C8—682no—
193C4⋊C4—22no—
194C4⋊C4—24no—
195C4⋊C4—682no—
196C4⋊C4—682no—
197C4⋊C4—682no—
198C4⋊C4—684no—
199C4⋊C4C2×#362414yes—
200C42—342no—
201C42—342no—
202C42—342no—
203C42—681no—
204C42—681no—
205C42—681no—
206C42—684no—
207C42C2×#362412yes—
208C42#6324712yes—
209C23C2×C22⋊#59511yescharacteristic
210Q8—48no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—24no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—172no—
221C2×C4—172no—
222C2×C4—172no—
223C2×C4—341no—
224C2×C4—341no—
225C2×C4—341no—
226C2×C4—341no—
227C2×C4—341no—
228C2×C4—341no—
229C2×C4—341no—
230C2×C4—341no—
231C2×C4—341no—
232C2×C4—682no—
233C2×C4—682no—
234C2×C4—682no—
235C2×C4C22⋊#362412yes—
236C2×C4C22⋊#362412yes—
237C2×C4C22×#362411yescharacteristic
238C2×C4C2×#6324711yescharacteristic
239C8—1362no—
240C8—1362no—
241C22C2×C22⋊#362411yescharacteristic
242C22#1176379011yescharacteristic
243C22D17⋊#10111yescharacteristic
244C22#1176380111yescharacteristic
245C22#1176380211yescharacteristic
246C22#1176380711yescharacteristic
247C22#1176380811yescharacteristic
248C4—21no—
249C4—21no—
250C4—22no—
251C4—22no—
252C4—44no—
253C4—342no—
254C4—342no—
255C4—342no—
256C4—681no—
257C4—1362no—
258C2#1454402711yescharacteristic
259C2#1454408311yescharacteristic
260C2#1454408611yescharacteristic
261C2#1454420511yescharacteristic
262C2#1454420711yescharacteristic
263C2#1454427611yescharacteristic
264C2#1454431411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socle, prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#7604
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
24. C17×C4⋊Q8C8semidirect
156. C4⋊Q8#3624semidirect
174. C17#7604semidirect

Maximal subgroups

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

subgroupindex
#145447112
#145436022
C4⋊Q8×Dic172
#760417