Finite Group Databasebeta

← #14544770

Subgroups — #14544770

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×#10587677C211yesmaximal, characteristic
2C2×#10587549C212yesmaximal, complemented
3#11763777C211yesmaximal, characteristic
4#11763682C212yesmaximal, complemented
5C22⋊Q8×D17C211yesmaximal, characteristic
6C2×Q8×D17C2211yescharacteristic
7C22×C4×D17C2211yescharacteristic
8C2×C22⋊#595—21no—
9#10587677—24no—
10C2×C4⋊#595C2212yes—
11C2×C4×#595—21no—
12C2×#63242C2212yes—
13C17×C22⋊Q8C411yescharacteristic, complemented
14#10587632C411yescharacteristic, complemented
15#10587610C411yescharacteristic, complemented
16#10587583C411yescharacteristic, complemented
17#10587582—21no—
18C4⋊C4×D17—21no—
19C4⋊C4×D17C2211yescharacteristic
20C22⋊C4×D17—21no—
21#10587564—21no—
22C22⋊#3624—21no—
23#10587549—24no—
24#10587543—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—24no—
43C2×#3624—22no—
44C4⋊#595—22no—
45C4⋊#595D414yes—
46C4×#595—24no—
47#63242—22no—
48#63242—22no—
49C4⋊C4×C17—21no—
50C4⋊C4×C17C2×C411yescharacteristic
51C22⋊C4×C17—21no—
52C22⋊Dic17—21no—
53#63227—21no—
54#63227—21no—
55C4⋊Dic17—21no—
56C4⋊Dic17C2×C411yescharacteristic
57#63225—21no—
58#63225—21no—
59#63225—21no—
60C23×C17C22⋊C411yescharacteristic
61C22×D17—21no—
62C22×D17—21no—
63C22×D17—22no—
64C22×D17—22no—
65C22×D17C22⋊C411yescharacteristic
66C22×D17C2×D411yescharacteristic
67C2×#595—24no—
68C2×#595—42no—
69C2×#595—42no—
70Q8×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—21no—
73C2×C4×C17—21no—
74C2×C4×C17C22⋊C411yescharacteristic
75C2×C4×C17C22×C411yescharacteristic
76C2×Dic17—22no—
77C2×Dic17—21no—
78C2×Dic17—21no—
79C2×Dic17C22⋊C411yescharacteristic
80C2×Dic17C2×D411yescharacteristic
81C4×D17—21no—
82C4×D17—21no—
83C4×D17—22no—
84C4×D17—22no—
85C4×D17—41no—
86C4×D17—41no—
87C4×D17—42no—
88C4×D17C2×D412yes—
89C4×D17#4112yes—
90Dic34—22no—
91Dic34—42no—
92#3624—42no—
93#1873—171nomaximal, Sylow
94C22×C17—21no—
95C22×C17—21no—
96C22×C17—21no—
97C22×C17C2×C22⋊C411yescharacteristic
98C2×D17—21no—
99C2×D17—21no—
100C2×D17—21no—
101C2×D17—42no—
102C2×D17—42no—
103C2×D17#12011yescharacteristic
104C2×D17#12311yescharacteristic
105C2×D17#13512yes—
106C2×D17#13712yes—
107#595—44no—
108C4×C17—21no—
109C4×C17—41no—
110C4×C17—41no—
111C4×C17C4×D412yes—
112Dic17—21no—
113Dic17—41no—
114Dic17—41no—
115Dic17#12112yes—
116C2×C22⋊Q8—171no—
117C2×#117—171no—
118C2×#103—172no—
119C2×C22⋊C8—171no—
120#340—172no—
121C2×C17—41no—
122C2×C17#38611yescharacteristic
123C2×C17#41611yescharacteristic
124C2×C17#41911yescharacteristic
125D17—41no—
126D17#45212yes—
127D17#48512yes—
128C22×Q8—171no—
129C23×C4—171no—
130C22×C8—172no—
131C22⋊Q8—171no—
132C22⋊Q8—171no—
133C22⋊Q8—171no—
134C22⋊Q8—341no—
135C22⋊Q8—341no—
136C22⋊Q8#59511yescharacteristic, complemented
137#117—344no—
138C2×C4⋊C4—172no—
139C2×C4⋊C4—171no—
140C2×C4⋊C4—341no—
141C2×C22⋊C4—341no—
142C2×C22⋊C4—341no—
143C2×C42—341no—
144#103—344no—
145C22⋊C8—341no—
146C22⋊C8—341no—
147C17#187311yescharacteristic, complemented, Sylow
148C24—171no—
149C2×Q8—171no—
150C2×Q8—341no—
151C2×Q8—342no—
152C2×Q8—342no—
153C2×Q8C2×#59511yescharacteristic
154C22×C4—171no—
155C22×C4—171no—
156C22×C4—171no—
157C22×C4—171no—
158C22×C4—171no—
159C22×C4—342no—
160C22×C4—342no—
161C22×C4—341no—
162C22×C4—341no—
163C22×C4—342no—
164C22×C4C2×#59511yescharacteristic
165C2×C8—342no—
166C2×C8—342no—
167C2×C8—342no—
168C4⋊C4—21no—
169C4⋊C4—174no—
170C4⋊C4—171no—
171C4⋊C4—341no—
172C4⋊C4—341no—
173C4⋊C4—341no—
174C4⋊C4—342no—
175C4⋊C4—341no—
176C4⋊C4C2×#59511yescharacteristic
177C22⋊C4—21no—
178C22⋊C4—341no—
179C22⋊C4—341no—
180C22⋊C4—344no—
181C22⋊C4—341no—
182C42—344no—
183C23—171no—
184C23—171no—
185C23—341no—
186C23—341no—
187C23—342no—
188C23—342no—
189C23C22⋊#59511yescharacteristic
190Q8—22no—
191Q8—342no—
192Q8—682no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—171no—
197C2×C4—172no—
198C2×C4—172no—
199C2×C4—171no—
200C2×C4—342no—
201C2×C4—341no—
202C2×C4—341no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—344no—
206C2×C4—341no—
207C2×C4—341no—
208C2×C4—681no—
209C2×C4—681no—
210C2×C4—682no—
211C2×C4—682no—
212C2×C4—682no—
213C2×C4C22⋊#59511yescharacteristic
214C2×C4C22×#59511yescharacteristic
215C8—682no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—171no—
220C22—171no—
221C22—172no—
222C22—172no—
223C22—341no—
224C22—341no—
225C22—341no—
226C22—682no—
227C22—682no—
228C22C2×C22⋊#59511yescharacteristic
229C4—21no—
230C4—41no—
231C4—41no—
232C4—172no—
233C4—341no—
234C4—684no—
235C4—681no—
236C4—681no—
237C4D4×#59512yes—
238C2—41no—
239C2—172no—
240C2—172no—
241C2—681no—
242C2#1176375711yescharacteristic
243C2#1176379911yescharacteristic
244C2#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#1873
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
13. C17×C22⋊Q8C4semidirect
14. #10587632C4semidirect
15. #10587610C4semidirect
16. #10587583C4semidirect
136. C22⋊Q8#595semidirect
147. C17#1873semidirect

Maximal subgroups

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

subgroupindex
C2×#105876772
C2×#105875492
#117637772
#117636822
C22⋊Q8×D172
#187317