Finite Group Databasebeta

← #14544654

Subgroups — #14544654

Order 2176.

356 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 84 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#11763810C211yesmaximal, characteristic, complemented
2C2×#10587547C212yesmaximal
3#11763753C211yesmaximal, characteristic, complemented
4#11763682C212yesmaximal, complemented
5C4×D4×D17C211yesmaximal, characteristic
6C2×D4×D17C2211yescharacteristic
7C22×C4×D17—21no—
8C2×C4⋊#595C2212yes—
9C2×#63242C2212yes—
10C4×D4×C17C411yescharacteristic, complemented
11D4×Dic17C411yescharacteristic, complemented
12C4×C22⋊D17—21no—
13#10587579C411yescharacteristic, complemented
14C4⋊C4×D17C2211yescharacteristic
15#10587567—21no—
16C22⋊C4×D17—21no—
17C4×D68C411yescharacteristic, complemented
18C42×D17C2211yescharacteristic
19#10587547—24no—
20(C2×#595)⋊C4—21no—
21(C2×#595)⋊C4—21no—
22#10587544—21no—
23C4⋊#3624—21no—
24C23×D17—21no—
25C22×#595—22no—
26C2×D4×C17C2×C411yescharacteristic
27C22×C4×C17—21no—
28C2×C22⋊D17—21no—
29C22×Dic17—21no—
30D4×D17—21no—
31D4×D17—21no—
32D4×D17D412yes—
33D4×D17Q812yes—
34C2×D68C2×C411yescharacteristic
35C2×C4×D17—21no—
36C2×C4×D17—21no—
37C2×C4×D17—21no—
38C2×C4×D17—44no—
39C2×C4×D17D411yescharacteristic
40C2×C4×D17D411yescharacteristic
41C2×C4×D17C2311yescharacteristic
42C2×#3624—22no—
43C4⋊#595—22no—
44C4⋊#595—22no—
45#63242—22no—
46#63242—22no—
47C4⋊C4×C17C2×C411yescharacteristic
48C22⋊C4×C17—21no—
49C42×C17C2×C411yescharacteristic
50C22⋊Dic17—21no—
51#63227—21no—
52#63227—21no—
53C4⋊Dic17C2×C411yescharacteristic
54#63225—21no—
55C4×Dic17C2×C411yescharacteristic
56C4×Dic17D411yescharacteristic
57C4×Dic17D411yescharacteristic
58C23×C17—21no—
59C22×D17—21no—
60C22×D17—22no—
61C22×D17—22no—
62C22×D17—42no—
63C22×D17—42no—
64C22×D17C2×D411yescharacteristic
65C2×#595—42no—
66C2×#595—42no—
67C2×#595—42no—
68D4×C17—21no—
69D4×C17C4⋊C412yes—
70C2×C4×C17—21no—
71C2×C4×C17—42no—
72C2×C4×C17C22⋊C411yescharacteristic
73C2×C4×C17C22⋊C411yescharacteristic
74C2×C4×C17C22×C411yescharacteristic
75C22⋊D17—22no—
76C22⋊D17—22no—
77C2×Dic17—21no—
78C2×Dic17—42no—
79C2×Dic17C22⋊C411yescharacteristic
80C2×Dic17C22⋊C411yescharacteristic
81C2×Dic17C2×D411yescharacteristic
82D68—21no—
83D68C4⋊C412yes—
84C4×D17—22no—
85C4×D17—22no—
86C4×D17—42no—
87C4×D17—42no—
88C4×D17—42no—
89C4×D17C2×D411yescharacteristic
90C4×D17C2×Q811yescharacteristic
91C4×D17#4111yescharacteristic
92C4×D17#4111yescharacteristic
93#3624—42no—
94#1804—171nomaximal, Sylow
95C22×C17—22no—
96C22×C17—41no—
97C22×C17—41no—
98C22×C17C2×C22⋊C411yescharacteristic
99C2×D17—22no—
100C2×D17—41no—
101C2×D17—41no—
102C2×D17—44no—
103C2×D17—44no—
104C2×D17#12111yescharacteristic
105C2×D17C22⋊Q811yescharacteristic
106C2×D17#13512yes—
107C2×D17AGL(1,ℤ/8)12yes—
108#595—82no—
109C4×C17—21no—
110C4×C17—21no—
111C4×C17—41no—
112C4×C17C2×C4⋊C411yescharacteristic
113C4×C17#11711yescharacteristic
114Dic17—21no—
115Dic17—21no—
116Dic17—41no—
117Dic17#12111yescharacteristic
118Dic17C22⋊Q811yescharacteristic
119C2×C4×D4—171no—
120C2×C4⋊C8—171no—
121C2×#102—172no—
122#382—171no—
123#340—172no—
124C2×C17—42no—
125C2×C17#38011yescharacteristic
126C2×C17#41611yescharacteristic
127C2×C17#41811yescharacteristic
128D17—42no—
129D17#46312yes—
130D17#47812yes—
131C22×D4—171no—
132C23×C4—341no—
133C22×C8—172no—
134C4×D4—171no—
135C4×D4—171no—
136C4×D4—171no—
137C4×D4—341no—
138C4×D4—341no—
139C4×D4#59511yescharacteristic, complemented
140C2×C4⋊C4—171no—
141C2×C4⋊C4—172no—
142C2×C22⋊C4—341no—
143C2×C42—171no—
144C4⋊C8—341no—
145C4⋊C8—341no—
146#102—344no—
147#95—341no—
148#95—341no—
149C17#180411yescharacteristic, complemented, Sylow
150C24—341no—
151C2×D4—172no—
152C2×D4—172no—
153C2×D4—171no—
154C2×D4—341no—
155C2×D4—341no—
156C2×D4—341no—
157C2×D4C2×#59511yescharacteristic
158C22×C4—21no—
159C22×C4—171no—
160C22×C4—171no—
161C22×C4—171no—
162C22×C4—341no—
163C22×C4—341no—
164C22×C4—341no—
165C22×C4—341no—
166C22×C4—342no—
167C22×C4—684no—
168C2×C8—342no—
169C2×C8—342no—
170C2×C8—342no—
171C4⋊C4—171no—
172C4⋊C4—341no—
173C4⋊C4—342no—
174C4⋊C4—342no—
175C4⋊C4C2×#59511yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—341no—
178C22⋊C4—341no—
179C22⋊C4—341no—
180C42—171no—
181C42—171no—
182C42—171no—
183C42C2×#59511yescharacteristic
184C23—21no—
185C23—171no—
186C23—342no—
187C23—342no—
188C23—341no—
189C23—682no—
190C23—682no—
191D4—21no—
192D4—172no—
193D4—342no—
194D4—342no—
195D4—341no—
196D4C4⋊#59512yes—
197C2×C4—21no—
198C2×C4—42no—
199C2×C4—171no—
200C2×C4—171no—
201C2×C4—171no—
202C2×C4—171no—
203C2×C4—171no—
204C2×C4—171no—
205C2×C4—171no—
206C2×C4—342no—
207C2×C4—342no—
208C2×C4—341no—
209C2×C4—682no—
210C2×C4—682no—
211C2×C4—682no—
212C2×C4—682no—
213C2×C4—682no—
214C2×C4—682no—
215C2×C4—682no—
216C2×C4C22⋊#59511yescharacteristic
217C2×C4C22⋊#59511yescharacteristic
218C2×C4C22×#59511yescharacteristic
219C8—682no—
220C22—22no—
221C22—41no—
222C22—41no—
223C22—171no—
224C22—171no—
225C22—172no—
226C22—172no—
227C22—342no—
228C22—684no—
229C22—684no—
230C22—681no—
231C22—681no—
232C22C2×C22⋊#59511yescharacteristic
233C4—21no—
234C4—21no—
235C4—41no—
236C4—171no—
237C4—171no—
238C4—341no—
239C4—341no—
240C4—681no—
241C4—1362no—
242C4C2×C4⋊#59511yescharacteristic
243C4#1058767711yescharacteristic
244C2—42no—
245C2—172no—
246C2—172no—
247C2—682no—
248C2#1176375111yescharacteristic
249C2#1176379911yescharacteristic
250C2#1176380111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC4×D4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC2×D4×D17C22
cosocleC2×C4×D17C23
hypercenterC4×D4#595
hypocenterC17#1804
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #11763810C2semidirect
3. #11763753C2semidirect
4. #11763682C2semidirect
10. C4×D4×C17C4semidirect
11. D4×Dic17C4semidirect
13. #10587579C4semidirect
17. C4×D68C4semidirect
139. C4×D4#595semidirect
149. C17#1804semidirect

Maximal subgroups

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

subgroupindex
#117638102
C2×#105875472
#117637532
#117636822
C4×D4×D172
#180417