Finite Group Databasebeta

← #14544799

Subgroups — #14544799

Order 2176.

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C2×D17⋊M16C211yesmaximal, characteristic
2C2×#10587549C212yesmaximal, complemented
3#11763751C211yesmaximal, characteristic
4#11763682C212yesmaximal, complemented
5C22⋊Q8×D17C211yesmaximal, characteristic
6C2×Q8×D17C2211yescharacteristic
7C22×C4×D17C2211yescharacteristic
8C2×C22⋊#595—21no—
9C2×#63247—21no—
10C2×C4⋊#595C2212yes—
11D17⋊M16—24no—
12C2×#63243—21no—
13C2×#63242C2212yes—
14C17×C22⋊Q8C411yescharacteristic, complemented
15#10587632C411yescharacteristic, complemented
16#10587610C411yescharacteristic, complemented
17#10587583C411yescharacteristic, complemented
18#10587582—21no—
19C4⋊C4×D17—21no—
20C4⋊C4×D17C2211yescharacteristic
21C22⋊C4×D17—21no—
22#10587564—21no—
23#10587549—24no—
24(C2×#595)⋊C4—21no—
25C23×D17D411yescharacteristic
26C22×#595—22no—
27C2×Q8×C17C2×C411yescharacteristic
28C22×C4×C17C2×C411yescharacteristic
29C22×Dic17D411yescharacteristic
30Q8×D17—22no—
31Q8×D17—22no—
32C2×C4×D17—22no—
33C2×C4×D17—22no—
34C2×C4×D17—21no—
35C2×C4×D17—21no—
36C2×C4×D17C2×C411yescharacteristic
37C2×C4×D17D411yescharacteristic
38C2×C4×D17D411yescharacteristic
39C2×C4×D17C2311yescharacteristic
40C2×Dic34—21no—
41C2×Dic34C2×C411yescharacteristic
42C22⋊#595—42no—
43#63247—24no—
44C2×#3624—22no—
45C4⋊#595—22no—
46C4⋊#595—22no—
47#63243—24no—
48#63242—22no—
49#63242D414yes—
50C4⋊C4×C17—21no—
51C4⋊C4×C17C2×C411yescharacteristic
52C22⋊C4×C17—21no—
53C22⋊Dic17—21no—
54#63227—21no—
55#63227—21no—
56C4⋊Dic17—21no—
57C4⋊Dic17C2×C411yescharacteristic
58#63225—21no—
59#63225—21no—
60#63225—21no—
61C23×C17C22⋊C411yescharacteristic
62C22×D17—22no—
63C22×D17—21no—
64C22×D17—21no—
65C22×D17—22no—
66C22×D17C22⋊C411yescharacteristic
67C22×D17C2×D411yescharacteristic
68C2×#595—42no—
69C2×#595—42no—
70C2×#595—42no—
71Q8×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17C22⋊C411yescharacteristic
76C2×C4×C17C22×C411yescharacteristic
77C2×Dic17—21no—
78C2×Dic17—21no—
79C2×Dic17—21no—
80C2×Dic17—21no—
81C2×Dic17C22⋊C411yescharacteristic
82C2×Dic17C2×D411yescharacteristic
83C4×D17—22no—
84C4×D17—22no—
85C4×D17—21no—
86C4×D17—21no—
87C4×D17—42no—
88C4×D17—41no—
89C4×D17—41no—
90C4×D17C2×D412yes—
91C4×D17#4112yes—
92Dic34—22no—
93Dic34—41no—
94Dic34—41no—
95#3624—24no—
96#1894—171nomaximal, Sylow
97C22×C17—21no—
98C22×C17—21no—
99C22×C17—21no—
100C22×C17C2×C22⋊C411yescharacteristic
101C2×D17—21no—
102C2×D17—21no—
103C2×D17—21no—
104C2×D17—42no—
105C2×D17—42no—
106C2×D17#12112yes—
107C2×D17AGL(1,ℤ/8)11yescharacteristic
108C2×D17#13712yes—
109C2×D17#13711yescharacteristic
110#595—82no—
111C4×C17—21no—
112C4×C17—41no—
113C4×C17—41no—
114C4×C17C4×D412yes—
115Dic17—21no—
116Dic17—41no—
117Dic17—41no—
118Dic17#12011yescharacteristic
119Dic17#12311yescharacteristic
120C22×M16—171no—
121C2×C22⋊Q8—171no—
122C2×#103—172no—
123#380—171no—
124#340—172no—
125C2×C17—41no—
126C2×C17#38611yescharacteristic
127C2×C17#41711yescharacteristic
128C2×C17#41911yescharacteristic
129D17—41no—
130D17#46812yes—
131D17#47012yes—
132C22×Q8—171no—
133C23×C4—171no—
134C2×M16—344no—
135C2×M16—341no—
136C2×M16—341no—
137C22×C8—172no—
138C22⋊Q8—171no—
139C22⋊Q8—171no—
140C22⋊Q8—171no—
141C22⋊Q8—341no—
142C22⋊Q8—341no—
143C22⋊Q8#59511yescharacteristic, complemented
144C2×C4⋊C4—171no—
145C2×C4⋊C4—172no—
146C2×C4⋊C4—341no—
147C2×C22⋊C4—341no—
148C2×C22⋊C4—341no—
149#103—344no—
150#95—341no—
151C17#189411yescharacteristic, complemented, Sylow
152C24—171no—
153C2×Q8—171no—
154C2×Q8—341no—
155C2×Q8—342no—
156C2×Q8—342no—
157C2×Q8C2×#59511yescharacteristic
158C22×C4—171no—
159C22×C4—171no—
160C22×C4—171no—
161C22×C4—171no—
162C22×C4—171no—
163C22×C4—342no—
164C22×C4—342no—
165C22×C4—341no—
166C22×C4—341no—
167C22×C4—342no—
168C22×C4C2×#59511yescharacteristic
169M16—344no—
170M16—344no—
171C2×C8—174no—
172C2×C8—342no—
173C2×C8—342no—
174C4⋊C4—21no—
175C4⋊C4—171no—
176C4⋊C4—341no—
177C4⋊C4—341no—
178C4⋊C4—341no—
179C4⋊C4—342no—
180C4⋊C4—342no—
181C4⋊C4—341no—
182C4⋊C4C2×#59511yescharacteristic
183C22⋊C4—21no—
184C22⋊C4—341no—
185C22⋊C4—341no—
186C22⋊C4—341no—
187C22⋊C4—682no—
188C23—171no—
189C23—171no—
190C23—341no—
191C23—342no—
192C23—341no—
193C23—342no—
194C23C22⋊#59511yescharacteristic
195Q8—22no—
196Q8—342no—
197Q8—681no—
198Q8—681no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—171no—
203C2×C4—172no—
204C2×C4—172no—
205C2×C4—171no—
206C2×C4—341no—
207C2×C4—341no—
208C2×C4—342no—
209C2×C4—341no—
210C2×C4—341no—
211C2×C4—342no—
212C2×C4—341no—
213C2×C4—341no—
214C2×C4—682no—
215C2×C4—681no—
216C2×C4—681no—
217C2×C4—682no—
218C2×C4—682no—
219C2×C4—682no—
220C2×C4C22⋊#59511yescharacteristic
221C2×C4C22×#59511yescharacteristic
222C8—344no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—172no—
227C22—171no—
228C22—171no—
229C22—172no—
230C22—341no—
231C22—341no—
232C22—341no—
233C22—682no—
234C22—682no—
235C22C2×C22⋊#59511yescharacteristic
236C4—21no—
237C4—41no—
238C4—41no—
239C4—171no—
240C4—171no—
241C4—341no—
242C4—681no—
243C4—681no—
244C4—1362no—
245C4D4×#59512yes—
246C2—41no—
247C2—172no—
248C2—172no—
249C2—681no—
250C2#1176375711yescharacteristic
251C2#1176380011yescharacteristic
252C2#1176380211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×C22⋊Q8C4
socleC22×C17C2×C22⋊C4
prime radicalC23×D17D4
cosocleC2×C4×D17C23
hypercenterC22⋊Q8#595
hypocenterC17#1894
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#10587549C2semidirect
4. #11763682C2semidirect
14. C17×C22⋊Q8C4semidirect
15. #10587632C4semidirect
16. #10587610C4semidirect
17. #10587583C4semidirect
143. C22⋊Q8#595semidirect
151. C17#1894semidirect

Maximal subgroups

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

subgroupindex
C2×D17⋊M162
C2×#105875492
#117637512
#117636822
C22⋊Q8×D172
#189417