Finite Group Databasebeta

← #14539399

Subgroups — #14539399

Order 2176.

362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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#11763424C212yesmaximal, complemented
2#11763206C211yesmaximal, characteristic, complemented
3#11763192C211yesmaximal, characteristic, complemented
4#11763167C212yesmaximal, complemented
5#11763068C211yesmaximal, characteristic, complemented
6#11763059C211yesmaximal, characteristic
7#11763037C212yesmaximal, complemented
8#11762917C211yesmaximal, characteristic, complemented
9C4×#63229C212yesmaximal, complemented
10#11762766C211yesmaximal, characteristic, complemented
11C17×#442C211yesmaximal, characteristic, complemented
12C2×D8×C17C2211yescharacteristic
13C4×D4×C17C2212yes—
14#10587624C2212yes—
15#10587622C2212yes—
16C2×#63229C2212yes—
17#10587615—22no—
18#10587610—22no—
19C2×Dic68C2211yescharacteristic
20C2×#63218C2211yescharacteristic
21C2×C8×D17C2211yescharacteristic
22#10587583C2212yes—
23#10587574C2212yes—
24#10587571—22no—
25#10587565—22no—
26#10587558C2211yescharacteristic
27C4×Dic34C2212yes—
28C17×#106C2211yescharacteristic
29C17×#102C2212yes—
30C17×#97C2211yescharacteristic
31#10587503C2212yes—
32#10587488C2211yescharacteristic
33#10587487C2212yes—
34#10587486C2211yescharacteristic
35C8×Dic17C2211yescharacteristic
36#10587481C2212yes—
37#10587479C2212yes—
38C4×#3622C2211yescharacteristic
39C2×D4×C17C2312yes—
40C22×C4×C17—22no—
41C2×C22⋊D17—22no—
42C2×C4×D17C2311yescharacteristic
43C2×Dic34—22no—
44C2×Dic34C2312yes—
45D8×C17—22no—
46C2×C8×C17C2311yescharacteristic
47C2×C8×C17C2311yescharacteristic
48C4⋊C4×C17C2312yes—
49C22⋊C4×C17—22no—
50C42×C17C2311yescharacteristic
51C22⋊Dic17—22no—
52C22⋊Dic17—22no—
53C22⋊Dic17—22no—
54#63229—24no—
55#63227—22no—
56#63227D412yes—
57C4⋊Dic17—22no—
58C4⋊Dic17C2312yes—
59#63225—22no—
60#63225—22no—
61#63225D412yes—
62C4×Dic17—22no—
63C4×Dic17C2311yescharacteristic
64C2×#3622C2311yescharacteristic
65C2×#3622C2311yescharacteristic
66Dic68—22no—
67#63218—22no—
68C8×D17—22no—
69C23×C17—22no—
70C22×D17C2×D411yescharacteristic
71D4×C17—22no—
72D4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—82no—
75C2×C4×C17C2×D412yes—
76C2×C4×C17C2411yescharacteristic
77C22⋊D17—82no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17C2×D412yes—
81C2×Dic17C2×D411yescharacteristic
82C4×D17—21no—
83C4×D17—21no—
84Dic34—22no—
85Dic34—22no—
86Dic34—82no—
87C8×C17—21no—
88C8×C17#4112yes—
89#3622—21no—
90#3622#4112yes—
91#3173—171nomaximal, Sylow
92C22×C17—42no—
93C22×C17—42no—
94C22×C17—42no—
95C22×C17C22×D411yescharacteristic
96C2×D17—21no—
97C2×D17—22no—
98C4×C17—22no—
99C4×C17—42no—
100C4×C17C2×#4111yescharacteristic
101C4×C17C2×#4111yescharacteristic
102Dic17—21no—
103Dic17—22no—
104Dic17—42no—
105Dic17—42no—
106#542—172no—
107#498—171no—
108#495—171no—
109#489—172no—
110#470—171no—
111#469—171no—
112#465—172no—
113#442D1711yescharacteristic, complemented
114#441—171no—
115C4×SD16—172no—
116#405—171no—
117C2×C17—82no—
118C2×C17#53211yescharacteristic
119C2×C17#57712yes—
120D17—41no—
121C2×Q16—171no—
122C2×SD16—172no—
123C2×D8C2×D1711yescharacteristic
124C2×M16—171no—
125C22×C8—171no—
126C4⋊Q8—172no—
127#126—342no—
128#124—172no—
129#124—342no—
130#123—342no—
131C22⋊Q8—172no—
132C22⋊Q8—342no—
133#121—172no—
134C4×Q8—172no—
135C4×D4C2×D1712yes—
136#117—171no—
137#106—171no—
138#106—172no—
139#106C2×D1711yescharacteristic
140#103—172no—
141#103—172no—
142#102—172no—
143#102C2×D1712yes—
144#97—171no—
145#97C2×D1711yescharacteristic
146C4×C8—171no—
147C4×C8—171no—
148C17#317311yescharacteristic, complemented, Sylow
149C2×Q8—172no—
150C2×Q8—342no—
151C2×D4—342no—
152C2×D4C22×D1712yes—
153C22×C4—22no—
154C22×C4—171no—
155Q16—342no—
156SD16—344no—
157D8—22no—
158M16—342no—
159C2×C8—171no—
160C2×C8—171no—
161C2×C8—342no—
162C2×C8C22×D1711yescharacteristic
163C2×C8C22×D1711yescharacteristic
164C4⋊C4—172no—
165C4⋊C4—172no—
166C4⋊C4—342no—
167C4⋊C4—342no—
168C4⋊C4—342no—
169C4⋊C4C22×D1712yes—
170C22⋊C4—22no—
171C22⋊C4—172no—
172C22⋊C4—342no—
173C22⋊C4—342no—
174C22⋊C4—342no—
175C22⋊C4—342no—
176C42—171no—
177C42—342no—
178C42C22×D1711yescharacteristic
179C23—22no—
180C23—171no—
181Q8—342no—
182Q8—342no—
183Q8—1362no—
184D4—22no—
185D4—22no—
186D4—1362no—
187C2×C4—22no—
188C2×C4—82no—
189C2×C4—172no—
190C2×C4—171no—
191C2×C4—341no—
192C2×C4—341no—
193C2×C4—342no—
194C2×C4—342no—
195C2×C4D4×D1712yes—
196C2×C4C23×D1711yescharacteristic
197C8—21no—
198C8—172no—
199C8—341no—
200C8#6325212yes—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—341no—
205C22—342no—
206C22C2×D4×D1711yescharacteristic
207C4—22no—
208C4—42no—
209C4—341no—
210C4—342no—
211C4—682no—
212C4—682no—
213C4C2×#6325211yescharacteristic
214C4#41×D1711yescharacteristic
215C2—82no—
216C2—681no—
217C2#1176335611yescharacteristic
218C2#1176361912yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#442C2
socleC22×C17C22×D4
prime radical#11763059C2
hypercenter#442D17
hypocenterC17#3173
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #11763424C2semidirect
2. #11763206C2semidirect
3. #11763192C2semidirect
4. #11763167C2semidirect
5. #11763068C2semidirect
7. #11763037C2semidirect
8. #11762917C2semidirect
9. C4×#63229C2semidirect
10. #11762766C2semidirect
11. C17×#442C2semidirect
113. #442D17semidirect
148. C17#3173semidirect

Maximal subgroups

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

subgroupindex
#117634242
#117632062
#117631922
#117631672
#117630682
#117630592
#117630372
#117629172
C4×#632292
#117627662
C17×#4422
#317317