Finite Group Databasebeta

← #14539906

Subgroups — #14539906

Order 2176.

404 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 125 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#11763360C211yesmaximal, characteristic
2C4⋊Q8×D17C211yesmaximal, characteristic, complemented
3#11763212C211yesmaximal, characteristic, complemented
4#11763157C211yesmaximal, characteristic, complemented
5#11763156C211yesmaximal, characteristic, complemented
6#11763124C214yesmaximal, complemented
7#11763088C214yesmaximal, complemented
8#11762860C211yesmaximal, characteristic, complemented
9C17×#500C211yesmaximal, characteristic, complemented
10C2×#63256—21no—
11C2×Q8×D17—21no—
12C17×C4⋊Q8C2211yescharacteristic
13C17×C4⋊Q8C2211yescharacteristic
14#10587633—21no—
15#10587630—21no—
16#10587581—21no—
17#10587581C2212yes—
18#10587579—21no—
19#10587579C2212yes—
20C4⋊C4×D17—21no—
21C4⋊C4×D17C2212yes—
22#10587574—21no—
23#10587574C2212yes—
24#10587560C2211yescharacteristic
25C42×D17C2211yescharacteristic
26#10587555C2211yescharacteristic
27C17×#107C2214yes—
28C4×C8×C17C2211yescharacteristic
29#10587492C2214yes—
30#10587491C2212yes—
31#10587485C2212yes—
32#10587480C2214yes—
33#10587478C2214yes—
34C4×#3622C2211yescharacteristic
35C2×Q8×C17—21no—
36C2×Q8×C17—21no—
37#63256—44no—
38Q8×D17—44no—
39C2×D68—22no—
40C2×D68—21no—
41C2×D68—21no—
42C2×D68C2312yes—
43C2×C4×D17—22no—
44C2×C4×D17—22no—
45C2×C4×D17D411yescharacteristic
46C2×C4×D17D411yescharacteristic
47C2×C4×D17C2311yescharacteristic
48C2×Dic34—21no—
49C2×Dic34—22no—
50C2×C8×C17C2312yes—
51C4⋊C4×C17—21no—
52C4⋊C4×C17—21no—
53C4⋊C4×C17C2312yes—
54C4⋊C4×C17C2312yes—
55C42×C17C2311yescharacteristic
56#63227—22no—
57#63227—22no—
58C4⋊Dic17—21no—
59C4⋊Dic17C2312yes—
60#63225—22no—
61#63225—22no—
62C4×Dic17D411yescharacteristic
63C4×Dic17D411yescharacteristic
64C4×Dic17C2311yescharacteristic
65C2×#3622C2312yes—
66C22×D17—22no—
67C22×D17C2×D411yescharacteristic
68Q8×C17—42no—
69Q8×C17—42no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17C2×D411yescharacteristic
73C2×C4×C17C2×D411yescharacteristic
74C2×C4×C17C2411yescharacteristic
75C2×Dic17—22no—
76C2×Dic17C2×D411yescharacteristic
77C2×Dic17C2×D411yescharacteristic
78C2×Dic17C2×D411yescharacteristic
79D68—24no—
80D68—44no—
81D68—42no—
82D68—42no—
83C4×D17—22no—
84C4×D17—22no—
85C4×D17—44no—
86C4×D17—44no—
87C4×D17D814yes—
88Dic34—42no—
89Dic34—44no—
90C8×C17—22no—
91#3622—22no—
92#3226—171nomaximal, Sylow
93C22×C17C22×D411yescharacteristic
94C2×D17—21no—
95C2×D17—21no—
96C2×D17—42no—
97C2×D17—42no—
98C2×D17—42no—
99C2×D17C2×D812yes—
100C4×C17—42no—
101C4×C17—42no—
102C4×C17C2×D812yes—
103C4×C17#13712yes—
104C4×C172−1+412yes—
105Dic17—21no—
106Dic17—21no—
107Dic17—42no—
108Dic17C2×D812yes—
109#532—171no—
110C2×C4⋊Q8—171no—
111#500—171no—
112#500D1711yescharacteristic, complemented
113#486—171no—
114#486—171no—
115#480—174no—
116#474—174no—
117#431—171no—
118C2×C17#53611yescharacteristic
119C2×C17C22×D811yescharacteristic
120C2×C17C2×#13711yescharacteristic
121D17—22no—
122D17—82no—
123C2×#41—341no—
124C22×Q8—341no—
125C4⋊Q8—172no—
126C4⋊Q8—171no—
127C4⋊Q8—341no—
128C4⋊Q8—341no—
129C4⋊Q8C2×D1711yescharacteristic
130C4⋊Q8C2×D1711yescharacteristic
131#127—171no—
132#124—341no—
133#121—172no—
134#121—341no—
135C4×D4—172no—
136C4×D4—341no—
137C2×C4⋊C4—172no—
138C2×C4⋊C4—341no—
139C2×C42—171no—
140#107—174no—
141#107C2×D1714yes—
142C4⋊C8—172no—
143#102—174no—
144#102—174no—
145C22⋊C8—172no—
146C4×C8—171no—
147C4×C8C2×D1711yescharacteristic
148C17#322611yescharacteristic, complemented, Sylow
149#41—684no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—342no—
153C2×Q8—341no—
154C2×Q8—684no—
155C2×D4—172no—
156C2×D4—342no—
157C2×D4—341no—
158C2×D4—341no—
159C22×C4—171no—
160C22×C4—171no—
161C22×C4—171no—
162C22×C4—342no—
163C22×C4—342no—
164C2×C8—172no—
165C2×C8C22×D1712yes—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—172no—
169C4⋊C4—342no—
170C4⋊C4—342no—
171C4⋊C4—341no—
172C4⋊C4C22×D1712yes—
173C4⋊C4C22×D1712yes—
174C22⋊C4—342no—
175C22⋊C4—342no—
176C42—171no—
177C42—171no—
178C42—171no—
179C42C22×D1711yescharacteristic
180C23—171no—
181C23—342no—
182Q8—42no—
183Q8—42no—
184Q8—684no—
185Q8—682no—
186D4—344no—
187D4—684no—
188D4—682no—
189D4—682no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—171no—
193C2×C4—174no—
194C2×C4—171no—
195C2×C4—171no—
196C2×C4—342no—
197C2×C4—342no—
198C2×C4—342no—
199C2×C4—684no—
200C2×C4—684no—
201C2×C4D4×D1711yescharacteristic
202C2×C4D4×D1711yescharacteristic
203C2×C4C23×D1711yescharacteristic
204C8—22no—
205C8—342no—
206C22—172no—
207C22—341no—
208C22—341no—
209C22—682no—
210C22—682no—
211C22—682no—
212C22C2×D4×D1711yescharacteristic
213C4—42no—
214C4—42no—
215C4—172no—
216C4—341no—
217C4—341no—
218C4—682no—
219C4D8×D1712yes—
220C4#1058760412yes—
221C4#1058769412yes—
222C2—342no—
223C2—1362no—
224C2#1176338411yescharacteristic
225C2C2×D8×D1711yescharacteristic
226C2C2×#1058760411yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#500C2
socleC22×C17C22×D4
prime radical#11763360C2
hypercenter#500D17
hypocenterC17#3226
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D17C2semidirect
3. #11763212C2semidirect
4. #11763157C2semidirect
5. #11763156C2semidirect
6. #11763124C2semidirect
7. #11763088C2semidirect
8. #11762860C2semidirect
9. C17×#500C2semidirect
112. #500D17semidirect
148. C17#3226semidirect

Maximal subgroups

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

subgroupindex
#117633602
C4⋊Q8×D172
#117632122
#117631572
#117631562
#117631242
#117630882
#117628602
C17×#5002
#322617