Finite Group Databasebeta

← #14539571

Subgroups — #14539571

Order 2176.

424 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 113 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#11763319C212yesmaximal, complemented
2#11763152C212yesmaximal, complemented
3#11763131C212yesmaximal, complemented
4#11763050C211yesmaximal, characteristic, complemented
5#11763036C212yesmaximal, complemented
6#11762972C212yesmaximal, complemented
7#11762951C212yesmaximal, complemented
8C2×#10587491C211yesmaximal, characteristic, complemented
9C17×#466C211yesmaximal, characteristic, complemented
10C2×#63254C2212yes—
11C22×C4×D17C2211yescharacteristic
12C2×D8×C17C2211yescharacteristic, complemented
13C22×C8×C17C2211yescharacteristic
14C17×#121C2212yes—
15#10587624C2212yes—
16#10587623—22no—
17D4×Dic17—22no—
18C2×#63229C2212yescomplemented
19#10587610C2212yes—
20C22×#3622C2211yescharacteristic
21C2×Dic68C2211yescharacteristic, complemented
22#10587583C2212yes—
23#10587578C2212yes—
24#10587569—22no—
25#10587567—22no—
26C17×#107C2211yescharacteristic, complemented
27C17×#102C2212yescomplemented
28#10587503C2212yescomplemented
29#10587491C2212yes—
30#10587491C2212yes—
31#10587489C2211yescharacteristic, complemented
32#10587487C2212yescomplemented
33#10587481C2212yescomplemented
34#10587479C2212yescomplemented
35C23×D17D411yescharacteristic
36C2×D4×C17—22no—
37C2×D4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C2×C22⋊D17—22no—
40C2×C22⋊D17—22no—
41C22×Dic17—22no—
42C22×Dic17D411yescharacteristic
43#63254—42no—
44#63254—42no—
45C2×C4×D17—41no—
46C2×C4×D17—41no—
47C2×C4×D17D412yes—
48C2×C4×D17C2312yes—
49C2×Dic34C2312yes—
50D8×C17—41no—
51C2×C8×C17—41no—
52C2×C8×C17C2311yescharacteristic
53C2×C8×C17C2311yescharacteristic
54C4⋊C4×C17C2312yes—
55C22⋊C4×C17—22no—
56C22⋊Dic17—22no—
57C22⋊Dic17—22no—
58#63229—42no—
59#63227—22no—
60#63227—22no—
61C4⋊Dic17—22no—
62C4⋊Dic17C2312yes—
63#63225—22no—
64C4×Dic17—22no—
65C2×#3622—41no—
66C2×#3622C2312yes—
67Dic68—41no—
68C23×C17—22no—
69C23×C17C2×D411yescharacteristic
70C22×D17—22no—
71C22×D17—41no—
72C22×D17—41no—
73C22×D17C2×D412yes—
74D4×C17—22no—
75D4×C17—22no—
76D4×C17—44no—
77C2×C4×C17—21no—
78C2×C4×C17—21no—
79C2×C4×C17—22no—
80C2×C4×C17C2×D411yescharacteristic
81C2×C4×C17C2411yescharacteristic
82C22⋊D17—44no—
83C22⋊D17—44no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—41no—
87C2×Dic17—44no—
88C2×Dic17C2×D412yes—
89C4×D17—22no—
90C4×D17—22no—
91C4×D17—42no—
92Dic34—22no—
93Dic34—22no—
94C8×C17—21no—
95C8×C17—21no—
96#3622—22no—
97#3191—171nomaximal, Sylow
98C22×C17—21no—
99C22×C17—21no—
100C22×C17—42no—
101C22×C17—42no—
102C22×C17—42no—
103C22×C17#13512yes—
104C22×C17C22×D411yescharacteristic
105C2×D17—22no—
106C2×D17—22no—
107C2×D17—22no—
108C2×D17—44no—
109C4×C17—21no—
110C4×C17—42no—
111C4×C17H(5,𝔽2)11yescharacteristic
112C4×C17H(5,𝔽2)11yescharacteristic
113Dic17—22no—
114Dic17—42no—
115Dic17—42no—
116#525—172no—
117#485—172no—
118#482—172no—
119#467—171no—
120#466D1711yescharacteristic, complemented
121#465—172no—
122#452—172no—
123#449—172no—
124C2×C22⋊C8—171no—
125C2×C17—22no—
126C2×C17—82no—
127C2×C17#53411yescharacteristic
128C2×C17C2×#13511yescharacteristic
129C2×C17#57511yescharacteristic
130D17—42no—
131C2×#41—172no—
132C23×C4—171no—
133C2×Q16—171no—
134C2×SD16—172no—
135C2×D8C2×D1711yescharacteristic, complemented
136C22×C8—171no—
137C22×C8C2×D1711yescharacteristic
138#123—342no—
139C22⋊Q8—172no—
140C22⋊Q8—172no—
141#121—172no—
142#121C2×D1712yes—
143#120—342no—
144C4×D4—342no—
145C4×D4—342no—
146#117—172no—
147#107—171no—
148#107C2×D1711yescharacteristic, complemented
149#106—172no—
150#103—172no—
151#103—172no—
152#102—172no—
153#102C2×D1712yescomplemented
154C22⋊C8—172no—
155C22⋊C8—172no—
156C17#319111yescharacteristic, complemented, Sylow
157C24—171no—
158#41—682no—
159#41—682no—
160C2×Q8—172no—
161C2×D4—22no—
162C2×D4—342no—
163C2×D4—342no—
164C2×D4C22×D1712yes—
165C22×C4—171no—
166C22×C4—172no—
167C22×C4—172no—
168C22×C4—342no—
169C22×C4—681no—
170C22×C4—681no—
171C22×C4C22×D1711yescharacteristic
172Q16—681no—
173SD16—682no—
174D8—41no—
175C2×C8—41no—
176C2×C8—172no—
177C2×C8—681no—
178C2×C8C22×D1711yescharacteristic
179C2×C8C22×D1711yescharacteristic
180C4⋊C4—172no—
181C4⋊C4—342no—
182C4⋊C4—342no—
183C4⋊C4C22×D1712yes—
184C22⋊C4—22no—
185C22⋊C4—342no—
186C22⋊C4—342no—
187C22⋊C4—342no—
188C22⋊C4—342no—
189C42—342no—
190C23—22no—
191C23—172no—
192C23—342no—
193C23—681no—
194C23—681no—
195C23D4×D1711yescharacteristic
196Q8—342no—
197Q8—342no—
198D4—22no—
199D4—22no—
200D4—44no—
201D4—684no—
202D4—684no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—172no—
207C2×C4—342no—
208C2×C4—342no—
209C2×C4—342no—
210C2×C4—342no—
211C2×C4—682no—
212C2×C4—681no—
213C2×C4—684no—
214C2×C4D4×D1711yescharacteristic
215C2×C4C23×D1711yescharacteristic
216C8—21no—
217C8—21no—
218C8—342no—
219C22—21no—
220C22—21no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—342no—
225C22—342no—
226C22—342no—
227C22—684no—
228C22#1058759812yes—
229C22C2×D4×D1711yescharacteristic
230C4—21no—
231C4—42no—
232C4—342no—
233C4—682no—
234C4—682no—
235C4#1058769111yescharacteristic
236C4#1058769611yescharacteristic
237C2—22no—
238C2—82no—
239C2—682no—
240C2#1176336711yescharacteristic
241C2C2×#1058759811yescharacteristic
242C2#1176360611yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#466C2
socleC22×C17C22×D4
hypercenter#466D17
hypocenterC17#3191
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #11763319C2semidirect
2. #11763152C2semidirect
3. #11763131C2semidirect
4. #11763050C2semidirect
5. #11763036C2semidirect
6. #11762972C2semidirect
7. #11762951C2semidirect
8. C2×#10587491C2semidirect
9. C17×#466C2semidirect
12. C2×D8×C17C22semidirect
18. C2×#63229C22semidirect
21. C2×Dic68C22semidirect
26. C17×#107C22semidirect
27. C17×#102C22semidirect
28. #10587503C22semidirect
31. #10587489C22semidirect
32. #10587487C22semidirect
33. #10587481C22semidirect
34. #10587479C22semidirect
120. #466D17semidirect
135. C2×D8C2×D17semidirect
148. #107C2×D17semidirect
153. #102C2×D17semidirect
156. C17#3191semidirect

Maximal subgroups

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

subgroupindex
#117633192
#117631522
#117631312
#117630502
#117630362
#117629722
#117629512
C2×#105874912
C17×#4662
#319117