Finite Group Databasebeta

← #14540158

Subgroups — #14540158

Order 2176.

348 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 121 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#11763366C211yesmaximal, characteristic, complemented
2#11763343C211yesmaximal, characteristic
3#11763117C212yesmaximal, complemented
4#11763105C211yesmaximal, characteristic, complemented
5#11763100C211yesmaximal, characteristic, complemented
6#11763067C212yesmaximal, complemented
7#11763055C212yesmaximal
8#11763032C212yesmaximal, complemented
9#11763020C211yesmaximal, characteristic, complemented
10#11763015C211yesmaximal, characteristic, complemented
11C17×#423C211yesmaximal, characteristic, complemented
12C2×M16×C17C2212yes—
13C2×C42×C17C2211yescharacteristic
14#10587616C2211yescharacteristic
15#10587615—21no—
16C4×C22⋊D17—21no—
17#10587612—21no—
18#10587610—21no—
19#10587610C2212yes—
20#10587610C2211yescharacteristic
21C2×Dic68C2212yes—
22C2×#63219C2212yes—
23#10587562—21no—
24#10587561C2211yescharacteristic
25C4×D68C2211yescharacteristic
26#10587558—21no—
27#10587556—21no—
28#10587556C2211yescharacteristic
29#10587555C2211yescharacteristic
30#10587555C2211yescharacteristic
31C4×Dic34—21no—
32C4×Dic34C2211yescharacteristic
33C4×Dic34C2212yes—
34C17×C4⋊C8C2211yescharacteristic
35C17×C4⋊C8C2211yescharacteristic
36C17×C4⋊C8C2212yes—
37#10587492C2212yes—
38#10587489C2212yes—
39#10587488C2212yes—
40#10587487C2212yes—
41#10587487C2212yes—
42#10587487C2212yes—
43C22×C4×C17D411yescharacteristic
44C22×C4×C17D411yescharacteristic
45C22×C4×C17C2311yescharacteristic
46C2×C22⋊D17—21no—
47C2×D68C2311yescharacteristic
48C2×C4×D17—21no—
49C2×Dic34—21no—
50C2×Dic34C2312yes—
51C2×Dic34C2311yescharacteristic
52M16×C17—42no—
53C2×C8×C17C2312yes—
54C2×C8×C17C2312yes—
55C42×C17D411yescharacteristic
56C42×C17D411yescharacteristic
57C42×C17C2311yescharacteristic
58C42×C17C2311yescharacteristic
59C22⋊Dic17—22no—
60C22⋊Dic17—21no—
61#63227—21no—
62#63227—21no—
63#63227—21no—
64#63227—21no—
65C4⋊Dic17—21no—
66C4⋊Dic17—21no—
67C4⋊Dic17—21no—
68C4⋊Dic17C2311yescharacteristic
69C4⋊Dic17C2311yescharacteristic
70C4⋊Dic17C2312yes—
71#63225—22no—
72#63225—22no—
73#63225—21no—
74#63225—21no—
75#63225—21no—
76#63225—21no—
77C4×Dic17—22no—
78C4×Dic17—21no—
79Dic68—42no—
80#63219—42no—
81C23×C17C2×D411yescharacteristic
82C22×D17—21no—
83C2×C4×C17—22no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—41no—
87C2×C4×C17C2×D411yescharacteristic
88C2×C4×C17C2×D411yescharacteristic
89C2×C4×C17C2×D411yescharacteristic
90C2×C4×C17C2×D411yescharacteristic
91C2×C4×C17C2×D411yescharacteristic
92C2×C4×C17C2411yescharacteristic
93C22⋊D17—42no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—21no—
97C2×Dic17—21no—
98C2×Dic17—21no—
99D68—21no—
100D68—21no—
101C4×D17—42no—
102Dic34—22no—
103Dic34—21no—
104Dic34—22no—
105Dic34—21no—
106Dic34—42no—
107C8×C17—22no—
108C8×C17—22no—
109#3252—171nomaximal, Sylow
110C22×C17—21no—
111C22×C17—21no—
112C22×C17—21no—
113C22×C17C22×D411yescharacteristic
114C2×D17—41no—
115C2×D17—41no—
116C2×D17—41no—
117C4×C17—21no—
118C4×C17—21no—
119C4×C17—21no—
120C4×C17—21no—
121C4×C17#13712yes—
122C4×C17H(5,𝔽2)11yescharacteristic
123C4×C172−1+411yescharacteristic
124Dic17—42no—
125Dic17—41no—
126Dic17—41no—
127Dic17—42no—
128Dic17—41no—
129#533—171no—
130#529—171no—
131#479—172no—
132#477—171no—
133#476—171no—
134#470—172no—
135#468—172no—
136#464—172no—
137#462—171no—
138#461—171no—
139#423D1711yescharacteristic, complemented
140C2×C17—41no—
141C2×C17#53711yescharacteristic
142C2×C17C2×#13711yescharacteristic
143C2×C17#57511yescharacteristic
144D17—81no—
145C2×Q16—172no—
146C2×SD16—172no—
147C2×M16C2×D1712yes—
148C4⋊Q8—171no—
149C4⋊Q8—171no—
150#126—341no—
151#125—171no—
152#125—341no—
153#124—171no—
154#123—341no—
155C22⋊Q8—171no—
156C22⋊Q8—172no—
157C22⋊Q8—341no—
158#121—171no—
159C4×Q8—171no—
160C4×Q8—172no—
161C4×Q8—341no—
162C4×D4—171no—
163C4×D4—341no—
164#117—341no—
165#117—341no—
166C2×C42C2×D1711yescharacteristic
167#107—172no—
168#106—172no—
169C4⋊C8C2×D1711yescharacteristic
170C4⋊C8C2×D1711yescharacteristic
171C4⋊C8C2×D1712yes—
172#103—172no—
173#103—172no—
174#103—172no—
175#102—172no—
176C17#325211yescharacteristic, complemented, Sylow
177C2×Q8—171no—
178C2×Q8—172no—
179C2×Q8—341no—
180C2×D4—171no—
181C2×D4—341no—
182C22×C4—341no—
183C22×C4D6811yescharacteristic
184C22×C4D6811yescharacteristic
185C22×C4C22×D1711yescharacteristic
186Q16—682no—
187SD16—682no—
188M16—42no—
189C2×C8C22×D1712yes—
190C2×C8C22×D1712yes—
191C4⋊C4—172no—
192C4⋊C4—171no—
193C4⋊C4—171no—
194C4⋊C4—341no—
195C4⋊C4—341no—
196C4⋊C4—341no—
197C4⋊C4—341no—
198C4⋊C4—341no—
199C4⋊C4—342no—
200C4⋊C4—341no—
201C4⋊C4—342no—
202C4⋊C4—341no—
203C22⋊C4—342no—
204C22⋊C4—341no—
205C22⋊C4—341no—
206C22⋊C4—341no—
207C22⋊C4—341no—
208C22⋊C4—341no—
209C42—341no—
210C42—342no—
211C42D6811yescharacteristic
212C42D6811yescharacteristic
213C42C22×D1711yescharacteristic
214C42C22×D1711yescharacteristic
215C23—341no—
216C23C2×D6811yescharacteristic
217Q8—342no—
218Q8—341no—
219Q8—341no—
220Q8—342no—
221Q8—682no—
222D4—341no—
223D4—341no—
224D4—682no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—341no—
230C2×C4—341no—
231C2×C4—342no—
232C2×C4—342no—
233C2×C4—341no—
234C2×C4—682no—
235C2×C4C2×D6811yescharacteristic
236C2×C4C2×D6811yescharacteristic
237C2×C4C2×D6811yescharacteristic
238C2×C4C2×D6811yescharacteristic
239C2×C4C2×D6811yescharacteristic
240C2×C4C23×D1711yescharacteristic
241C8—22no—
242C8—22no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—681no—
247C22—681no—
248C22—681no—
249C22C22×D6811yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—681no—
255C4—681no—
256C4—682no—
257C4—681no—
258C4—682no—
259C4#1058759512yes—
260C4#1058769111yescharacteristic
261C4#1058769411yescharacteristic
262C2—41no—
263C2—1361no—
264C2#1176339011yescharacteristic
265C2C2×#1058759511yescharacteristic
266C2#1176360911yescharacteristic
subgroupG/H
centerC22C22×D68
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#423C2
socleC22×C17C22×D4
prime radical#10587616C22
hypercenter#423D17
hypocenterC17#3252
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #11763366C2semidirect
3. #11763117C2semidirect
4. #11763105C2semidirect
5. #11763100C2semidirect
6. #11763067C2semidirect
8. #11763032C2semidirect
9. #11763020C2semidirect
10. #11763015C2semidirect
11. C17×#423C2semidirect
139. #423D17semidirect
176. C17#3252semidirect

Maximal subgroups

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

subgroupindex
#117633662
#117633432
#117631172
#117631052
#117631002
#117630672
#117630552
#117630322
#117630202
#117630152
C17×#4232
#325217