Finite Group Databasebeta

← #14529013

Subgroups — #14529013

Order 2176.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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#11763290C211yesmaximal, characteristic
2#11762851C211yesmaximal, characteristic, complemented
3C2×#10587496C211yesmaximal, characteristic
4#11762811C211yesmaximal, characteristic
5#11762678C211yesmaximal, characteristic, complemented
6#11762570C211yesmaximal, characteristic, complemented
7C17×#343C211yesmaximal, characteristic, complemented
8C2×#63252C2211yescharacteristic
9C2×M16×C17C2211yescharacteristic
10C2×M16×C17C2211yescharacteristic
11C2×C4⋊C4×C17C2211yescharacteristic
12C4×C22⋊D17—22no—
13#10587612C2211yescharacteristic
14C2×C4⋊Dic17C2211yescharacteristic
15C2×#63225—22no—
16C2×C4×Dic17C2211yescharacteristic
17#10587579—21no—
18#10587578—21no—
19C4⋊C4×D17—21no—
20#10587573—21no—
21#10587496C414yescomplemented
22#10587492—21no—
23#10587489—21no—
24#10587488—21no—
25#10587487—21no—
26C22×C4×C17D412yes—
27C22×C4×C17C2311yescharacteristic
28C2×C22⋊D17—21no—
29C22×Dic17—21no—
30C22×Dic17—21no—
31#63252—21no—
32#63252—21no—
33#63252C2×C412yes—
34#63252Q812yes—
35C2×D68D411yescharacteristic
36C2×C4×D17—21no—
37C2×C4×D17—22no—
38C2×Dic34D411yescharacteristic
39M16×C17—21no—
40M16×C17—21no—
41M16×C17—21no—
42M16×C17C2×C412yescomplemented
43C2×C8×C17—21no—
44C2×C8×C17—21no—
45C4⋊C4×C17—21no—
46C4⋊C4×C17—21no—
47C22⋊Dic17—21no—
48#63227—22no—
49C4⋊Dic17—21no—
50C4⋊Dic17—21no—
51C4⋊Dic17D411yescharacteristic
52C4⋊Dic17D411yescharacteristic
53#63225—22no—
54#63225—44no—
55C4×Dic17—22no—
56C4×Dic17—21no—
57C4×Dic17—21no—
58C4×Dic17C2×C412yescomplemented
59C23×C17C2×D411yescharacteristic
60C22×D17—21no—
61C2×C4×C17—22no—
62C2×C4×C17—42no—
63C2×C4×C17C22×C411yescharacteristic
64C2×C4×C17C2×D411yescharacteristic
65C2×C4×C17C2×D411yescharacteristic
66C2×C4×C17C2×Q811yescharacteristic
67C2×C4×C17#4111yescharacteristic
68C2×C4×C17#4111yescharacteristic
69C22⋊D17—22no—
70C22⋊D17—22no—
71C2×Dic17—22no—
72C2×Dic17—22no—
73C2×Dic17—22no—
74C2×Dic17—21no—
75C2×Dic17—21no—
76C2×Dic17—21no—
77C2×Dic17—42no—
78C2×Dic17—41no—
79D68—21no—
80D68C4⋊C412yescomplemented
81C4×D17—22no—
82C4×D17—22no—
83C4×D17—82no—
84Dic34—21no—
85Dic34C4⋊C412yescomplemented
86C8×C17—22no—
87C8×C17—41no—
88#1859—171nomaximal, Sylow
89C22×C17—21no—
90C22×C17—21no—
91C22×C17C4×D411yescharacteristic
92C22×C17#12011yescharacteristic
93C22×C17C22⋊Q811yescharacteristic
94C2×D17—22no—
95C2×D17—41no—
96C2×D17—41no—
97C4×C17—42no—
98C4×C17C2×C4⋊C411yescharacteristic
99C4×C17C4×D411yescharacteristic
100C4×C17C22⋊Q811yescharacteristic
101C4×C17#12311yescharacteristic
102Dic17—24no—
103Dic17—22no—
104Dic17—41no—
105Dic17—41no—
106Dic17—81no—
107#520—171no—
108#428—171no—
109C2×#104—171no—
110#417—171no—
111#384—171no—
112#343D1711yescharacteristic, complemented
113#339—171no—
114C2×C17—21no—
115C2×C17—21no—
116C2×C17#38511yescharacteristic
117C2×C17#45311yescharacteristic
118C2×C17#45611yescharacteristic
119D17—42no—
120C2×#41—171no—
121C2×M16C2×D1711yescharacteristic
122C2×M16C2×D1711yescharacteristic
123C4×Q8—341no—
124C4×D4—341no—
125C4×D4—342no—
126#117—171no—
127#117—341no—
128C2×C4⋊C4—171no—
129C2×C4⋊C4—341no—
130C2×C4⋊C4—342no—
131C2×C4⋊C4C2×D1711yescharacteristic
132C2×C42—171no—
133#107—341no—
134#106—341no—
135#104—174no—
136#103—341no—
137#102—341no—
138C17#185911yescharacteristic, complemented, Sylow
139#41—172no—
140#41—172no—
141#41—341no—
142#41—341no—
143C2×Q8—171no—
144C2×D4—171no—
145C2×D4—341no—
146C22×C4—342no—
147C22×C4—341no—
148C22×C4—341no—
149C22×C4—341no—
150C22×C4D6812yes—
151C22×C4C22×D1711yescharacteristic
152M16—21no—
153M16—21no—
154M16—21no—
155M16C4×D1712yescomplemented
156C2×C8—21no—
157C2×C8—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—171no—
161C4⋊C4—171no—
162C4⋊C4—342no—
163C4⋊C4—341no—
164C4⋊C4—341no—
165C4⋊C4—684no—
166C22⋊C4—342no—
167C22⋊C4—341no—
168C42—172no—
169C42—342no—
170C42—341no—
171C42—341no—
172C23—341no—
173C23C2×D6811yescharacteristic
174Q8—172no—
175Q8—341no—
176D4—172no—
177D4—342no—
178D4—342no—
179D4—341no—
180C2×C4—22no—
181C2×C4—42no—
182C2×C4—342no—
183C2×C4—342no—
184C2×C4—342no—
185C2×C4—341no—
186C2×C4—342no—
187C2×C4—342no—
188C2×C4—341no—
189C2×C4—341no—
190C2×C4—682no—
191C2×C4—681no—
192C2×C4—1362no—
193C2×C4C2×C4×D1711yescharacteristic
194C2×C4#6325211yescharacteristic
195C2×C4D4×D1711yescharacteristic
196C2×C4D4×D1711yescharacteristic
197C2×C4#6325411yescharacteristic
198C2×C4Q8×D1711yescharacteristic
199C8—22no—
200C8—41no—
201C22—21no—
202C22—21no—
203C22—342no—
204C22—681no—
205C22—681no—
206C22C4×D6811yescharacteristic
207C22#1058756811yescharacteristic
208C22#1058758311yescharacteristic
209C4—42no—
210C4—344no—
211C4—342no—
212C4—681no—
213C4—681no—
214C4—1361no—
215C4#1058756711yescharacteristic
216C4#1058756911yescharacteristic
217C4C4⋊C4×D1711yescharacteristic
218C4#1058758211yescharacteristic
219C2—21no—
220C2—21no—
221C2—682no—
222C2#1176268311yescharacteristic
223C2#1176297911yescharacteristic
224C2#1176299411yescharacteristic
subgroupG/H
centerC22#10587568
commutatorC2×C4×C17C22×C4
FrattiniC22×C4C22×D17
FittingC17×#343C2
socleC22×C17#120
prime radicalC2×#63252C22
cosocleC22×C4×C17C23
hypercenter#343D17
hypocenterC17#1859
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
2. #11762851C2semidirect
5. #11762678C2semidirect
6. #11762570C2semidirect
7. C17×#343C2semidirect
21. #10587496C4semidirect
42. M16×C17C2×C4semidirect
58. C4×Dic17C2×C4semidirect
80. D68C4⋊C4semidirect
85. Dic34C4⋊C4semidirect
112. #343D17semidirect
138. C17#1859semidirect
155. M16C4×D17semidirect

Maximal subgroups

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

subgroupindex
#117632902
#117628512
C2×#105874962
#117628112
#117626782
#117625702
C17×#3432
#185917