Finite Group Databasebeta

← #14540159

Subgroups — #14540159

Order 2176.

348 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 113 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#11763364C211yesmaximal, characteristic, complemented
2#11763341C211yesmaximal, characteristic
3#11763119C212yesmaximal, complemented
4#11763101C211yesmaximal, characteristic, complemented
5#11763068C212yesmaximal, complemented
6#11763054C212yesmaximal
7#11763030C212yesmaximal, complemented
8#11763020C211yesmaximal, characteristic, complemented
9#11763014C211yesmaximal, characteristic, complemented
10#11762831C211yesmaximal, characteristic, complemented
11C17×#476C211yesmaximal, characteristic, complemented
12C17×C4⋊Q8C2211yescharacteristic
13C4×D4×C17C2211yescharacteristic
14#10587632—21no—
15Q8×Dic17—21no—
16#10587624C2211yescharacteristic
17#10587622C2211yescharacteristic
18C2×#63229C2212yes—
19C4×C22⋊D17—21no—
20#10587612—21no—
21C2×Dic68C2212yes—
22C2×#63218C2212yes—
23#10587583C2212yes—
24#10587583C2211yescharacteristic
25#10587578—21no—
26#10587576—21no—
27#10587576C2211yescharacteristic
28#10587573C2212yes—
29#10587569—21no—
30#10587565—21no—
31C42×D17C2211yescharacteristic
32#10587555C2211yescharacteristic
33C4×Dic34C2211yescharacteristic
34C17×#106C2212yes—
35C17×C4⋊C8C2211yescharacteristic
36C17×#102C2212yes—
37#10587487C2212yes—
38#10587485C2212yes—
39#10587481C2212yes—
40#10587481C2212yes—
41#10587478C2212yes—
42C4⋊#3622C2211yescharacteristic
43C2×Q8×C17—21no—
44C2×D4×C17C2311yescharacteristic
45C22×C4×C17—21no—
46C2×C22⋊D17—21no—
47C2×C4×D17D411yescharacteristic
48C2×C4×D17D411yescharacteristic
49C2×C4×D17C2311yescharacteristic
50C2×Dic34C2311yescharacteristic
51C2×Dic34C2312yes—
52C2×C8×C17C2312yes—
53C4⋊C4×C17—21no—
54C4⋊C4×C17C2312yes—
55C4⋊C4×C17C2311yescharacteristic
56C22⋊C4×C17—21no—
57C42×C17C2311yescharacteristic
58C22⋊Dic17—21no—
59C22⋊Dic17—21no—
60C22⋊Dic17—21no—
61#63229—42no—
62#63227—21no—
63#63227—22no—
64C4⋊Dic17—21no—
65C4⋊Dic17—21no—
66C4⋊Dic17—21no—
67C4⋊Dic17—21no—
68C4⋊Dic17—22no—
69C4⋊Dic17C2311yescharacteristic
70#63225—21no—
71#63225—21no—
72#63225—22no—
73C4×Dic17—21no—
74C4×Dic17—22no—
75C4×Dic17D411yescharacteristic
76C4×Dic17D411yescharacteristic
77C4×Dic17C2311yescharacteristic
78C2×#3622C2312yes—
79Dic68—42no—
80#63218—42no—
81C23×C17—21no—
82C22×D17C2×D411yescharacteristic
83Q8×C17—42no—
84D4×C17—21no—
85D4×C17—21no—
86C2×C4×C17—21no—
87C2×C4×C17—22no—
88C2×C4×C17—42no—
89C2×C4×C17C2×D411yescharacteristic
90C2×C4×C17C2×D411yescharacteristic
91C2×C4×C17C2411yescharacteristic
92C22⋊D17—42no—
93C2×Dic17—21no—
94C2×Dic17—21no—
95C2×Dic17—22no—
96C2×Dic17C2×D411yescharacteristic
97C2×Dic17C2×D411yescharacteristic
98C2×Dic17C2×D411yescharacteristic
99C4×D17—22no—
100C4×D17—21no—
101C4×D17—21no—
102C4×D17—41no—
103Dic34—21no—
104Dic34—22no—
105Dic34—21no—
106Dic34—22no—
107C8×C17—22no—
108#3622—22no—
109#3252—171nomaximal, Sylow
110C22×C17—41no—
111C22×C17—41no—
112C22×C17—41no—
113C22×C17C22×D411yescharacteristic
114C2×D17—21no—
115C2×D17—21no—
116C2×D17—21no—
117C4×C17—21no—
118C4×C17—41no—
119C4×C17—42no—
120C4×C17#13712yes—
121C4×C17H(5,𝔽2)11yescharacteristic
122C4×C172−1+411yescharacteristic
123Dic17—21no—
124Dic17—21no—
125Dic17—21no—
126Dic17—41no—
127Dic17—42no—
128Dic17—41no—
129#533—171no—
130#529—171no—
131#479—172no—
132#477—171no—
133#476D1711yescharacteristic, complemented
134#470—172no—
135#468—172no—
136#464—172no—
137#462—171no—
138#461—171no—
139#423—171no—
140C2×C17—81no—
141C2×C17#53711yescharacteristic
142C2×C17C2×#13711yescharacteristic
143C2×C17#57511yescharacteristic
144D17—41no—
145C2×Q16—172no—
146C2×SD16—172no—
147C2×M16—172no—
148C4⋊Q8—171no—
149C4⋊Q8C2×D1711yescharacteristic
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—172no—
160C4×Q8—171no—
161C4×Q8—341no—
162C4×D4—341no—
163C4×D4C2×D1711yescharacteristic
164#117—341no—
165#117—341no—
166C2×C42—171no—
167#107—172no—
168#106C2×D1712yes—
169C4⋊C8—172no—
170C4⋊C8—171no—
171C4⋊C8C2×D1711yescharacteristic
172#103—172no—
173#103—172no—
174#103—172no—
175#102C2×D1712yes—
176C17#325211yescharacteristic, complemented, Sylow
177C2×Q8—21no—
178C2×Q8—171no—
179C2×Q8—172no—
180C2×D4—341no—
181C2×D4C22×D1711yescharacteristic
182C22×C4—21no—
183C22×C4—171no—
184C22×C4—171no—
185C22×C4—171no—
186Q16—682no—
187SD16—682no—
188M16—682no—
189C2×C8—172no—
190C2×C8C22×D1712yes—
191C4⋊C4—21no—
192C4⋊C4—171no—
193C4⋊C4—341no—
194C4⋊C4—341no—
195C4⋊C4—341no—
196C4⋊C4—342no—
197C4⋊C4—342no—
198C4⋊C4—341no—
199C4⋊C4—341no—
200C4⋊C4—341no—
201C4⋊C4C22×D1712yes—
202C4⋊C4C22×D1711yescharacteristic
203C22⋊C4—21no—
204C22⋊C4—342no—
205C22⋊C4—341no—
206C22⋊C4—341no—
207C22⋊C4—341no—
208C22⋊C4—341no—
209C42—171no—
210C42—171no—
211C42—171no—
212C42—342no—
213C42—341no—
214C42C22×D1711yescharacteristic
215C23—21no—
216C23—171no—
217Q8—42no—
218Q8—341no—
219Q8—342no—
220Q8—342no—
221Q8—341no—
222D4—21no—
223D4—21no—
224D4—682no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—171no—
229C2×C4—171no—
230C2×C4—171no—
231C2×C4—341no—
232C2×C4—341no—
233C2×C4—342no—
234C2×C4—341no—
235C2×C4—341no—
236C2×C4—342no—
237C2×C4—681no—
238C2×C4D4×D1711yescharacteristic
239C2×C4D4×D1711yescharacteristic
240C2×C4C23×D1711yescharacteristic
241C8—22no—
242C8—342no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—341no—
247C22—341no—
248C22—341no—
249C22C2×D4×D1711yescharacteristic
250C4—21no—
251C4—41no—
252C4—42no—
253C4—341no—
254C4—341no—
255C4—341no—
256C4—681no—
257C4—681no—
258C4—682no—
259C4#1058760112yes—
260C4#1058769411yescharacteristic
261C4#1058769611yescharacteristic
262C2—81no—
263C2—681no—
264C2#1176338911yescharacteristic
265C2C2×#1058760111yescharacteristic
266C2#1176360511yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#476C2
socleC22×C17C22×D4
prime radical#10587624C22
hypercenter#476D17
hypocenterC17#3252
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #11763364C2semidirect
3. #11763119C2semidirect
4. #11763101C2semidirect
5. #11763068C2semidirect
7. #11763030C2semidirect
8. #11763020C2semidirect
9. #11763014C2semidirect
10. #11762831C2semidirect
11. C17×#476C2semidirect
133. #476D17semidirect
176. C17#3252semidirect

Maximal subgroups

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

subgroupindex
#117633642
#117633412
#117631192
#117631012
#117630682
#117630542
#117630302
#117630202
#117630142
#117628312
C17×#4762
#325217