Finite Group Databasebeta

← #17958312

Subgroups — #17958312

Order 4352.

428 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 100 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#14544027C212yesmaximal
2#14543726C212yesmaximal
3#14543714C212yesmaximal
4C4⋊Q8×Dic17C211yesmaximal, characteristic
5#11763825—22no—
6#11763654—22no—
7#11763654C2214yes—
8C2×Q8×Dic17C411yescharacteristic
9C2×Q8×Dic17C2211yescharacteristic
10#11762716C411yescharacteristic
11#11762716C2211yescharacteristic
12#11762708—22no—
13#11762679C2211yescharacteristic
14C4⋊C4×Dic17—22no—
15C42×Dic17C411yescharacteristic
16C2×C17×C4⋊Q8C411yescharacteristic
17C22×Q8×C17C2×C411yescharacteristic
18C22×Q8×C17C2×C411yescharacteristic
19C17×C4⋊Q8—22no—
20C17×C4⋊Q8C814yescomplemented
21C2×C4⋊C4×C17—22no—
22C2×C42×C17C2×C411yescharacteristic
23Q8×Dic17—41no—
24Q8×Dic17—41no—
25Q8×Dic17—41no—
26Q8×Dic17—41no—
27C2×C4⋊Dic17—21no—
28C2×C4⋊Dic17—22no—
29C2×C4⋊Dic17C2×C411yescharacteristic
30C2×C4⋊Dic17D412yes—
31C2×C4⋊Dic17C2311yescharacteristic
32C2×C4×Dic17—21no—
33C2×C4×Dic17—21no—
34C2×C4×Dic17—21no—
35C2×C4×Dic17C2×C411yescharacteristic
36C2×C4×Dic17C2×C411yescharacteristic
37C2×C4×Dic17D412yes—
38C22⋊#3624—24no—
39C4⋊#3624—42no—
40C4×#3624—42no—
41#10587502—22no—
42#10587502—22no—
43C2×Q8×C17—21no—
44C2×Q8×C17—41no—
45C2×Q8×C17—41no—
46C2×Q8×C17—41no—
47C2×Q8×C17—41no—
48C2×Q8×C17C2×C812yes—
49C2×Q8×C17C2×C812yes—
50C2×Q8×C17M1612yes—
51C22×C4×C17—21no—
52C22×C4×C17—21no—
53C22×C4×C17C22⋊C412yes—
54C22×C4×C17C22×C411yescharacteristic
55C22×Dic17—21no—
56C22×Dic17—21no—
57C22×Dic17C22⋊C411yescharacteristic
58C22×Dic17C22⋊C411yescharacteristic
59C22×Dic17C2×D412yes—
60C2×#3624—44no—
61C4⋊C4×C17—28no—
62C42×C17—21no—
63C42×C17C2×C812yes—
64C4⋊Dic17—42no—
65C4⋊Dic17—41no—
66C4⋊Dic17—42no—
67C4⋊Dic17—44no—
68C4⋊Dic17—41no—
69C4×Dic17—24no—
70C4×Dic17—22no—
71C4×Dic17—22no—
72C4×Dic17—21no—
73C4×Dic17—21no—
74C4×Dic17—42no—
75C4×Dic17—41no—
76C4×Dic17—41no—
77C4×Dic17—41no—
78#11478—171nomaximal, Sylow
79C23×C17C2×C22⋊C411yescharacteristic
80Q8×C17—42no—
81Q8×C17—42no—
82Q8×C17—42no—
83Q8×C17—42no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—21no—
89C2×C4×C17—22no—
90C2×C4×C17—21no—
91C2×C4×C17—22no—
92C2×C4×C17—41no—
93C2×C4×C17—41no—
94C2×C4×C17C22⋊C814yes—
95C2×C4×C17C22×C811yescharacteristic
96C2×C4×C17C2×M1611yescharacteristic
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—21no—
101C2×Dic17—41no—
102C2×Dic17—41no—
103C2×Dic17—41no—
104C2×Dic17—42no—
105C2×Dic17—42no—
106C2×Dic17—41no—
107C2×Dic17#9911yescharacteristic
108C2×Dic17#9911yescharacteristic
109#3624—84no—
110C4×C4⋊Q8—171no—
111#1461—172no—
112#1325—172no—
113#1319—172no—
114C22×C17—21no—
115C22×C17—21no—
116C22×C17C2×C22⋊C811yescharacteristic
117C22×C17C2×#9911yescharacteristic
118C22×C17#41311yescharacteristic
119C4×C17—24no—
120C4×C17—24no—
121C4×C17—21no—
122C4×C17—21no—
123C4×C17—42no—
124Dic17—21no—
125Dic17—42no—
126Dic17—42no—
127Dic17—41no—
128Dic17—81no—
129Dic17#35612yes—
130C2×C4⋊Q8#59511yescharacteristic
131C2×C4×Q8—171no—
132C2×C4×Q8—171no—
133#431—342no—
134#391—171no—
135#391—171no—
136#389—342no—
137#384—171no—
138C4×C4⋊C4—342no—
139C43—171no—
140#324—174no—
141#324—342no—
142C2×C17—21no—
143C2×C17—21no—
144C2×C17#145911yescharacteristic
145C2×C17#212611yescharacteristic
146C2×C17C2×#35611yescharacteristic
147C22×Q8C2×#59511yescharacteristic
148C22×Q8C2×#59511yescharacteristic
149C4⋊Q8—22no—
150C4⋊Q8#362414yescomplemented
151C4×Q8—681no—
152C4×Q8—681no—
153C4×Q8—681no—
154C4×Q8—681no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—172no—
157C2×C4⋊C4—171no—
158C2×C4⋊C4—171no—
159C2×C4⋊C4—341no—
160C2×C4⋊C4—342no—
161C2×C42—172no—
162C2×C42—171no—
163C2×C42—171no—
164C2×C42—341no—
165C2×C42—341no—
166C2×C42—341no—
167C2×C42C2×#59511yescharacteristic
168C4⋊C8—682no—
169C22⋊C8—344no—
170C4×C8—682no—
171#95—342no—
172#95—342no—
173C17#1147811yescharacteristic, complemented, Sylow
174C2×Q8—21no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8C2×#362412yes—
180C2×Q8C2×#362412yes—
181C2×Q8#6324712yes—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—171no—
185C22×C4—171no—
186C22×C4—172no—
187C22×C4—341no—
188C22×C4—341no—
189C22×C4C22⋊#59512yes—
190C22×C4C22×#59511yescharacteristic
191C2×C8—684no—
192C4⋊C4—28no—
193C4⋊C4—681no—
194C4⋊C4—682no—
195C4⋊C4—681no—
196C4⋊C4—684no—
197C4⋊C4—682no—
198C42—21no—
199C42—342no—
200C42—342no—
201C42—341no—
202C42—341no—
203C42—344no—
204C42—681no—
205C42—682no—
206C42—681no—
207C42—681no—
208C42C2×#362412yes—
209C23C2×C22⋊#59511yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—42no—
213Q8—42no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—171no—
225C2×C4—171no—
226C2×C4—342no—
227C2×C4—342no—
228C2×C4—341no—
229C2×C4—342no—
230C2×C4—681no—
231C2×C4—681no—
232C2×C4—681no—
233C2×C4—682no—
234C2×C4—682no—
235C2×C4—681no—
236C2×C4C22⋊#362414yes—
237C2×C4C22×#362411yescharacteristic
238C2×C4C2×#6324711yescharacteristic
239C8—1364no—
240C22—21no—
241C22—21no—
242C22C2×C22⋊#362411yescharacteristic
243C22D17⋊#9911yescharacteristic
244C22#1176379611yescharacteristic
245C4—24no—
246C4—21no—
247C4—21no—
248C4—24no—
249C4—42no—
250C4—172no—
251C4—341no—
252C4—682no—
253C4—682no—
254C4—681no—
255C4—1361no—
256C2—21no—
257C2—21no—
258C2#1454402011yescharacteristic
259C2#1454505811yescharacteristic
260C2D17⋊#35611yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socleC22×C17C2×#99
prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4⋊Dic17C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#11478
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
20. C17×C4⋊Q8C8semidirect
150. C4⋊Q8#3624semidirect
173. C17#11478semidirect

Maximal subgroups

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

subgroupindex
#145440272
#145437262
#145437142
C4⋊Q8×Dic172
#1147817