Finite Group Databasebeta

← #14531495

Subgroups — #14531495

Order 2176.

612 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 205 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×#10587561C211yesmaximal, characteristic
2C2×C4×Dic34C211yesmaximal, characteristic, complemented
3C2×C4×D68C211yesmaximal, characteristic
4#11762715C211yesmaximal, characteristic, complemented
5#11762711C211yesmaximal, characteristic
6#11762701C214yesmaximal, complemented
7#11762683C214yesmaximal, complemented
8#11762679C211yesmaximal, characteristic, complemented
9C17×#376C211yesmaximal, characteristic, complemented
10C22×D68C2211yescharacteristic
11C22×C4×D17C2212yes—
12C22×Dic34C2211yescharacteristic
13C2×C42×C17C2211yescharacteristic
14C2×C42×C17C2211yescharacteristic
15C2×C42×C17C2211yescharacteristic
16C2×#63227C2212yes—
17C2×#63227C2214yes—
18C2×#63227C2214yes—
19C2×C4⋊Dic17C2211yescharacteristic
20C2×C4⋊Dic17C2211yescharacteristic
21C2×C4⋊Dic17C2214yes—
22C2×#63225C2212yes—
23C2×C4×Dic17C2212yes—
24#10587561C418yescomplemented
25C4×D68—24no—
26C4×Dic34—24no—
27C17×#95C2212yes—
28C17×#95C2212yes—
29#10587502C2214yes—
30C23×D17C2312yes—
31C22×C4×C17C2312yes—
32C22×C4×C17C2312yes—
33C22×C4×C17C2311yescharacteristic
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2311yescharacteristic
36C22×Dic17C2312yes—
37C22×Dic17C2312yes—
38C22×Dic17C2312yes—
39C2×D68—22no—
40C2×D68—22no—
41C2×D68C2×C414yes—
42C2×C4×D17—24no—
43C2×C4×D17—44no—
44C2×Dic34—22no—
45C2×Dic34—22no—
46C2×Dic34C2×C414yescomplemented
47C42×C17—22no—
48C42×C17C2×C414yescomplemented
49C42×C17D414yes—
50#63227—24no—
51#63227—44no—
52#63227C2×C4116yescomplemented
53C4⋊Dic17—22no—
54C4⋊Dic17—22no—
55C4⋊Dic17—44no—
56#63225—24no—
57C4×Dic17—24no—
58C23×C17C2411yescharacteristic
59C22×D17—22no—
60C22×D17—22no—
61C22×D17—24no—
62C22×D17—24no—
63C22×D17C22×C414yes—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17C22×C414yes—
72C2×C4×C17C22×C412yes—
73C2×C4×C17C2×D412yes—
74C2×C4×C17C2×D412yes—
75C2×C4×C17C2×D412yes—
76C2×C4×C17#4112yes—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—24no—
82C2×Dic17—24no—
83C2×Dic17C22×C414yes—
84D68—28no—
85C4×D17—48no—
86Dic34—28no—
87#2326—171nomaximal, Sylow
88C22×C17C23×C411yescharacteristic
89C22×C17C22×D411yescharacteristic
90C22×C17C2×#4111yescharacteristic
91C22×C17H(5,𝔽2)12yes—
92C22×C172−1+412yes—
93C2×D17—24no—
94C2×D17—28no—
95C2×D17—42no—
96C2×D17—42no—
97C2×D17—44no—
98C4×C17—24no—
99C4×C17—22no—
100C4×C17—22no—
101C4×C17—42no—
102C4×C17C4×D414yes—
103Dic17—24no—
104Dic17—42no—
105Dic17—42no—
106C2×#124—171no—
107C2×C4×Q8—171no—
108C2×C4×D4—171no—
109#391—171no—
110#390—171no—
111#388—174no—
112#385—174no—
113#384—171no—
114#376D1711yescharacteristic, complemented
115C2×C17C2×C4×D411yescharacteristic
116C2×C17#52012yes—
117C2×C17#53511yescharacteristic
118C2×C17#53611yescharacteristic
119C2×C17#54212yes—
120D17—44no—
121C22×Q8—171no—
122C22×D4—171no—
123C23×C4—172no—
124#124—178no—
125C4×Q8—344no—
126C4×D4—344no—
127C2×C4⋊C4—171no—
128C2×C4⋊C4—171no—
129C2×C4⋊C4—172no—
130C2×C4⋊C4—174no—
131C2×C22⋊C4—174no—
132C2×C22⋊C4—172no—
133C2×C22⋊C4—174no—
134C2×C42—172no—
135C2×C42C2×D1711yescharacteristic
136C2×C42C2×D1711yescharacteristic
137C2×C42C2×D1711yescharacteristic
138#95—174no—
139#95C2×D1712yes—
140#95C2×D1712yes—
141C17#232611yescharacteristic, complemented, Sylow
142C24—172no—
143C2×Q8—174no—
144C2×Q8—342no—
145C2×Q8—342no—
146C2×D4—174no—
147C2×D4—342no—
148C2×D4—342no—
149C22×C4—172no—
150C22×C4—172no—
151C22×C4—172no—
152C22×C4—344no—
153C22×C4—684no—
154C22×C4C22×D1712yes—
155C22×C4C22×D1712yes—
156C22×C4C22×D1711yescharacteristic
157C22×C4C22×D1711yescharacteristic
158C22×C4C22×D1711yescharacteristic
159C4⋊C4—342no—
160C4⋊C4—344no—
161C4⋊C4—342no—
162C4⋊C4—684no—
163C22⋊C4—1716no—
164C22⋊C4—344no—
165C22⋊C4—684no—
166C42—22no—
167C42—344no—
168C42C4×D1714yescomplemented
169C42D6814yes—
170C23—174no—
171C23—342no—
172C23—342no—
173C23—344no—
174C23—344no—
175C23C23×D1711yescharacteristic
176Q8—348no—
177D4—348no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—174no—
186C2×C4—342no—
187C2×C4—342no—
188C2×C4—342no—
189C2×C4—342no—
190C2×C4—344no—
191C2×C4—344no—
192C2×C4—688no—
193C2×C4C2×C4×D1714yes—
194C2×C4C2×C4×D1712yes—
195C2×C4C2×D6812yes—
196C2×C4C2×D6812yes—
197C2×C4C2×D6812yes—
198C2×C4#6325212yes—
199C22—344no—
200C22—348no—
201C22—682no—
202C22—682no—
203C22—684no—
204C22C22×C4×D1711yescharacteristic
205C22C22×D6811yescharacteristic
206C22C2×#6325211yescharacteristic
207C22#1058769612yes—
208C22#1058769712yes—
209C4—24no—
210C4—22no—
211C4—22no—
212C4—42no—
213C4—344no—
214C4—682no—
215C4—682no—
216C4C4×D6814yes—
217C2—684no—
218C2C2×C4×D6811yescharacteristic
219C2#1176328912yes—
220C2#1176337811yescharacteristic
221C2#1176338511yescharacteristic
222C2#1176343012yes—
subgroupG/H
center, FrattiniC23C23×D17
commutatorC22×C17C23×C4
FittingC17×#376C2
socle, cosocleC23×C17C24
prime radicalC22×D68C22
hypercenter#376D17
hypocenterC17#2326
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #376

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic34C2semidirect
4. #11762715C2semidirect
6. #11762701C2semidirect
7. #11762683C2semidirect
8. #11762679C2semidirect
9. C17×#376C2semidirect
24. #10587561C4semidirect
46. C2×Dic34C2×C4semidirect
48. C42×C17C2×C4semidirect
52. #63227C2×C4semidirect
114. #376D17semidirect
141. C17#2326semidirect
168. C42C4×D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105875612
C2×C4×Dic342
C2×C4×D682
#117627152
#117627112
#117627012
#117626832
#117626792
C17×#3762
#232617