Finite Group Databasebeta

← #14539627

Subgroups — #14539627

Order 2176.

418 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 109 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#11763322C212yesmaximal, complemented
2#11763148C212yesmaximal, complemented
3#11763135C212yesmaximal, complemented
4#11763062C211yesmaximal, characteristic, complemented
5#11763057C212yesmaximal, complemented
6#11762970C212yesmaximal, complemented
7#11762957C212yesmaximal, complemented
8#11762772C211yesmaximal, characteristic, complemented
9C17×#470C211yesmaximal, characteristic, complemented
10C2×#63256C2212yes—
11C22×C4×D17C2211yescharacteristic
12C2×Q16×C17C2211yescharacteristic, complemented
13C2×M16×C17C2211yescharacteristic
14C17×C22⋊Q8C2212yes—
15#10587632C2212yes—
16C2×#63230C2212yescomplemented
17#10587616C2212yes—
18C2×#63223C2211yescharacteristic
19C2×D136C2211yescharacteristic, complemented
20#10587581C2212yes—
21#10587580—22no—
22#10587579—22no—
23#10587578C2212yes—
24#10587568—22no—
25#10587567—22no—
26C17×#106C2211yescharacteristic, complemented
27C17×#103C2212yescomplemented
28#10587506C2212yescomplemented
29#10587492C2212yescomplemented
30#10587491C2212yes—
31#10587491C2212yes—
32#10587488C2211yescharacteristic, complemented
33#10587480C2212yescomplemented
34#10587478C2212yescomplemented
35C23×D17D411yescharacteristic
36C2×Q8×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C2×C22⋊D17—22no—
39C22×Dic17D411yescharacteristic
40#63256—42no—
41#63256—42no—
42C2×D68—22no—
43C2×D68—22no—
44C2×D68C2312yes—
45C2×C4×D17—22no—
46C2×C4×D17—41no—
47C2×C4×D17—41no—
48C2×C4×D17D412yes—
49C2×C4×D17C2312yes—
50Q16×C17—41no—
51M16×C17—41no—
52C2×C8×C17C2311yescharacteristic
53C2×C8×C17C2311yescharacteristic
54C4⋊C4×C17—22no—
55C4⋊C4×C17C2312yes—
56C22⋊C4×C17—22no—
57#63230—42no—
58#63227—22no—
59#63227—22no—
60#63227—22no—
61#63227—22no—
62C4⋊Dic17C2312yes—
63#63225—22no—
64C4×Dic17—22no—
65#63223—41no—
66C2×#3622C2312yes—
67D136—41no—
68C23×C17C2×D411yescharacteristic
69C22×D17—22no—
70C22×D17—41no—
71C22×D17—42no—
72C22×D17C2×D412yes—
73Q8×C17—22no—
74Q8×C17—22no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17C2×D411yescharacteristic
80C2×C4×C17C2411yescharacteristic
81C22⋊D17—44no—
82C2×Dic17—22no—
83C2×Dic17—41no—
84C2×Dic17C2×D412yes—
85D68—22no—
86D68—22no—
87D68—44no—
88D68—44no—
89C4×D17—22no—
90C4×D17—22no—
91C4×D17—42no—
92C4×D17—44no—
93C8×C17—21no—
94C8×C17—21no—
95#3622—22no—
96#3197—171nomaximal, Sylow
97C22×C17—21no—
98C22×C17—22no—
99C22×C17C22×D411yescharacteristic
100C2×D17—22no—
101C2×D17—22no—
102C2×D17—22no—
103C2×D17—44no—
104C2×D17—42no—
105C2×D17—42no—
106C2×D17—42no—
107C4×C17—21no—
108C4×C17—42no—
109C4×C17—42no—
110C4×C17H(5,𝔽2)11yescharacteristic
111C4×C17H(5,𝔽2)11yescharacteristic
112Dic17—22no—
113Dic17—42no—
114#525—172no—
115#485—172no—
116#482—172no—
117#470D1711yescharacteristic, complemented
118#469—171no—
119#468—172no—
120#452—172no—
121#449—172no—
122#407—171no—
123C2×C17—41no—
124C2×C17#53411yescharacteristic
125C2×C17#57512yes—
126D17—42no—
127D17—82no—
128C2×#41—172no—
129C23×C4—171no—
130C2×Q16C2×D1711yescharacteristic, complemented
131C2×SD16—172no—
132C2×D8—171no—
133C2×M16—171no—
134C2×M16C2×D1711yescharacteristic
135#123—342no—
136C22⋊Q8—172no—
137C22⋊Q8C2×D1712yes—
138#121—172no—
139#121—172no—
140#120—342no—
141C4×D4—342no—
142C4×D4—342no—
143#117—172no—
144#107—172no—
145#106—171no—
146#106C2×D1711yescharacteristic, complemented
147#103—172no—
148#103C2×D1712yescomplemented
149#102—172no—
150#102—172no—
151C22⋊C8—172no—
152C22⋊C8—172no—
153C17#319711yescharacteristic, complemented, Sylow
154C24—171no—
155#41—682no—
156#41—682no—
157C2×Q8C22×D1712yes—
158C2×D4—172no—
159C2×D4—342no—
160C2×D4—342no—
161C2×D4—342no—
162C22×C4—171no—
163C22×C4—172no—
164C22×C4—172no—
165C22×C4—342no—
166C22×C4—681no—
167C22×C4—681no—
168C22×C4C22×D1711yescharacteristic
169Q16—41no—
170SD16—682no—
171D8—681no—
172M16—41no—
173M16—681no—
174C2×C8—172no—
175C2×C8C22×D1711yescharacteristic
176C2×C8C22×D1711yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—172no—
179C4⋊C4—342no—
180C4⋊C4C22×D1712yes—
181C22⋊C4—22no—
182C22⋊C4—342no—
183C22⋊C4—342no—
184C22⋊C4—342no—
185C22⋊C4—342no—
186C42—342no—
187C23—172no—
188C23—342no—
189C23—681no—
190C23—682no—
191C23D4×D1711yescharacteristic
192Q8—22no—
193Q8—22no—
194D4—342no—
195D4—342no—
196D4—684no—
197D4—684no—
198D4—684no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—172no—
204C2×C4—342no—
205C2×C4—342no—
206C2×C4—342no—
207C2×C4—682no—
208C2×C4—684no—
209C2×C4—681no—
210C2×C4D4×D1711yescharacteristic
211C2×C4C23×D1711yescharacteristic
212C8—21no—
213C8—21no—
214C8—342no—
215C22—22no—
216C22—21no—
217C22—342no—
218C22—342no—
219C22—342no—
220C22—684no—
221C22—682no—
222C22—682no—
223C22—682no—
224C22C2×D4×D1711yescharacteristic
225C4—21no—
226C4—42no—
227C4—42no—
228C4—342no—
229C4—682no—
230C4#1058769111yescharacteristic
231C4#1058769611yescharacteristic
232C2—41no—
233C2—682no—
234C2—1362no—
235C2#1176336711yescharacteristic
236C2#1176360812yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#470C2
socleC22×C17C22×D4
hypercenter#470D17
hypocenterC17#3197
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #11763322C2semidirect
2. #11763148C2semidirect
3. #11763135C2semidirect
4. #11763062C2semidirect
5. #11763057C2semidirect
6. #11762970C2semidirect
7. #11762957C2semidirect
8. #11762772C2semidirect
9. C17×#470C2semidirect
12. C2×Q16×C17C22semidirect
16. C2×#63230C22semidirect
19. C2×D136C22semidirect
26. C17×#106C22semidirect
27. C17×#103C22semidirect
28. #10587506C22semidirect
29. #10587492C22semidirect
32. #10587488C22semidirect
33. #10587480C22semidirect
34. #10587478C22semidirect
117. #470D17semidirect
130. C2×Q16C2×D17semidirect
146. #106C2×D17semidirect
148. #103C2×D17semidirect
153. C17#3197semidirect

Maximal subgroups

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

subgroupindex
#117633222
#117631482
#117631352
#117630622
#117630572
#117629702
#117629572
#117627722
C17×#4702
#319717