Finite Group Databasebeta

← #14539728

Subgroups — #14539728

Order 2176.

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

normal subgroups only

This group has 384 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#11763364C211yesmaximal, characteristic
2C4⋊Q8×D17C211yesmaximal, characteristic
3#11763188C211yesmaximal, characteristic, complemented
4#11763160C211yesmaximal, characteristic, complemented
5#11763159C211yesmaximal, characteristic, complemented
6#11763017C214yesmaximal, complemented
7#11762963C214yesmaximal
8#11762860C211yesmaximal, characteristic
9C17×#494C211yesmaximal, characteristic, complemented
10C2×Q8×D17C2212yes—
11C2×Q16×C17C2214yes—
12C17×C4⋊Q8C2211yescharacteristic
13C17×C4⋊Q8C2211yescharacteristic
14#10587632C2212yes—
15Q8×Dic17C2212yes—
16#10587630C2212yes—
17C2×#63231C2214yes—
18#10587583—21no—
19#10587578—21no—
20C4⋊C4×D17—21no—
21C4⋊C4×D17—21no—
22#10587576—21no—
23#10587574—21no—
24#10587574—21no—
25#10587573—21no—
26C42×D17C2211yescharacteristic
27#10587555C2211yescharacteristic
28#10587555C2211yescharacteristic
29C4×C8×C17C2211yescharacteristic
30#10587506C2214yes—
31#10587491C2212yes—
32#10587487C2214yes—
33#10587485C2212yes—
34C4×#3622C2211yescharacteristic
35C2×Q8×C17C2312yes—
36C2×Q8×C17C2312yes—
37Q8×D17—44no—
38C2×C4×D17—22no—
39C2×C4×D17D411yescharacteristic
40C2×C4×D17D411yescharacteristic
41C2×C4×D17C2311yescharacteristic
42C2×Dic34—21no—
43C2×Dic34—22no—
44C2×Dic34C2312yes—
45Q16×C17—44no—
46C2×C8×C17C2312yes—
47C4⋊C4×C17—21no—
48C4⋊C4×C17—21no—
49C4⋊C4×C17—21no—
50C4⋊C4×C17—21no—
51C42×C17C2311yescharacteristic
52#63231—44no—
53#63227—22no—
54C4⋊Dic17—21no—
55C4⋊Dic17—21no—
56C4⋊Dic17—21no—
57C4⋊Dic17—22no—
58C4⋊Dic17C2312yes—
59#63225—22no—
60#63225—22no—
61#63225—22no—
62C4×Dic17—22no—
63C4×Dic17D411yescharacteristic
64C4×Dic17D411yescharacteristic
65C4×Dic17C2311yescharacteristic
66C2×#3622C2312yes—
67C22×D17C2×D411yescharacteristic
68Q8×C17—24no—
69Q8×C17—24no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17C2×D411yescharacteristic
73C2×C4×C17C2×D411yescharacteristic
74C2×C4×C17C2411yescharacteristic
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17C2×D411yescharacteristic
78C2×Dic17C2×D411yescharacteristic
79C2×Dic17C2×D411yescharacteristic
80C4×D17—22no—
81C4×D17—22no—
82C4×D17—44no—
83C4×D17Q1614yescomplemented
84Dic34—24no—
85Dic34—44no—
86Dic34—42no—
87C8×C17—22no—
88#3622—22no—
89#3209—171nomaximal, Sylow
90C22×C17C22×D411yescharacteristic
91C2×D17—21no—
92C2×D17—21no—
93C2×D17C2×Q1612yes—
94C4×C17—42no—
95C4×C17—42no—
96C4×C17C2×Q1612yes—
97C4×C17#13712yes—
98C4×C17H(5,𝔽2)12yes—
99Dic17—21no—
100Dic17—21no—
101Dic17—42no—
102Dic17—42no—
103Dic17C2×Q1612yes—
104#533—171no—
105C2×C4⋊Q8—171no—
106#494—171no—
107#494D1711yescharacteristic, complemented
108#487—171no—
109#487—171no—
110#462—174no—
111#451—174no—
112#431—171no—
113C2×C17#53511yescharacteristic
114C2×C17C22×Q1611yescharacteristic
115C2×C17C2×#13711yescharacteristic
116D17—22no—
117C22×Q8—172no—
118C2×Q16—174no—
119C2×Q16C2×D1714yes—
120C4⋊Q8—172no—
121C4⋊Q8—171no—
122C4⋊Q8—171no—
123C4⋊Q8—341no—
124C4⋊Q8—341no—
125C4⋊Q8C2×D1711yescharacteristic
126C4⋊Q8C2×D1711yescharacteristic
127#125—341no—
128C22⋊Q8—172no—
129C22⋊Q8—341no—
130C4×Q8—172no—
131C4×Q8—341no—
132#117—341no—
133C2×C4⋊C4—341no—
134C2×C4⋊C4—341no—
135C2×C42—171no—
136C4⋊C8—172no—
137#103—174no—
138#103—174no—
139C22⋊C8—172no—
140C4×C8—171no—
141C4×C8C2×D1711yescharacteristic
142C17#320911yescharacteristic, complemented, Sylow
143C2×Q8—172no—
144C2×Q8—342no—
145C2×Q8—341no—
146C2×Q8—684no—
147C2×Q8C22×D1712yes—
148C2×Q8C22×D1712yes—
149C22×C4—171no—
150C22×C4—171no—
151C22×C4—171no—
152C22×C4—342no—
153Q16—44no—
154Q16—684no—
155C2×C8—172no—
156C2×C8C22×D1712yes—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—172no—
162C4⋊C4—342no—
163C4⋊C4—342no—
164C4⋊C4—342no—
165C4⋊C4—342no—
166C4⋊C4—341no—
167C4⋊C4—341no—
168C4⋊C4—341no—
169C22⋊C4—342no—
170C42—171no—
171C42—171no—
172C42—171no—
173C42—342no—
174C42C22×D1711yescharacteristic
175C23—171no—
176Q8—24no—
177Q8—24no—
178Q8—344no—
179Q8—684no—
180Q8—682no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—171no—
184C2×C4—174no—
185C2×C4—171no—
186C2×C4—171no—
187C2×C4—342no—
188C2×C4—342no—
189C2×C4—342no—
190C2×C4—342no—
191C2×C4—684no—
192C2×C4D4×D1711yescharacteristic
193C2×C4D4×D1711yescharacteristic
194C2×C4C23×D1711yescharacteristic
195C8—22no—
196C8—342no—
197C22—172no—
198C22—341no—
199C22—341no—
200C22C2×D4×D1711yescharacteristic
201C4—42no—
202C4—42no—
203C4—172no—
204C4—341no—
205C4—341no—
206C4—682no—
207C4—682no—
208C4Q16×D1712yes—
209C4#1058760412yes—
210C4#1058769112yes—
211C2—342no—
212C2#1176337211yescharacteristic
213C2C2×Q16×D1711yescharacteristic
214C2C2×#1058760411yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#494C2
socleC22×C17C22×D4
prime radicalC22×D17C2×D4
hypercenter#494D17
hypocenterC17#3209
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
3. #11763188C2semidirect
4. #11763160C2semidirect
5. #11763159C2semidirect
6. #11763017C2semidirect
9. C17×#494C2semidirect
83. C4×D17Q16semidirect
107. #494D17semidirect
142. C17#3209semidirect

Maximal subgroups

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

subgroupindex
#117633642
C4⋊Q8×D172
#117631882
#117631602
#117631592
#117630172
#117629632
#117628602
C17×#4942
#320917