Finite Group Databasebeta

← #16948765

Subgroups — #16948765

Order 4352.

224 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 63 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#14531013C211yesmaximal, characteristic, complemented
2#14530603C211yesmaximal, characteristic
3#14528170C211yesmaximal, characteristic
4#14528150C211yesmaximal, characteristic
5#14526857C211yesmaximal, characteristic, complemented
6#14526834C211yesmaximal, characteristic, complemented
7C17×#1364C211yesmaximal, characteristic, complemented
8C2×C16×D17—21no—
9#11763189C2211yescharacteristic
10#11763090—21no—
11#11763011—21no—
12C8×D68C2211yescharacteristic
13#11762612—21no—
14#11762612—21no—
15#11762588—21no—
16C4×#63214C2211yescharacteristic
17#11762562C2211yescharacteristic
18#11762555—21no—
19#11762554—21no—
20C17×#500C2211yescharacteristic
21C17×C4⋊C16C2211yescharacteristic
22C17×#335C2211yescharacteristic
23C17×C4⋊Q8—21no—
24C2×#63230—41no—
25C2×Dic68—21no—
26C2×Dic68—21no—
27C2×C8×D17—21no—
28C4×D68D411yescharacteristic
29#10587555—21no—
30C2×C16×C17—21no—
31C17×#107—21no—
32C17×#107—21no—
33C17×C4⋊C8—21no—
34C4×C8×C17C2311yescharacteristic
35#10587506—41no—
36#10587492—41no—
37#10587491—21no—
38#10587489—41no—
39C2×#63214D411yescharacteristic
40C2×#63214D411yescharacteristic
41C4⋊#3622—21no—
42C4⋊#3622D411yescharacteristic
43C16×D17—42no—
44C2×Q8×C17—41no—
45C2×D68D811yescharacteristic
46C2×C4×D17—21no—
47C2×Dic34—41no—
48C2×C8×C17—41no—
49C2×C8×C17C2×D411yescharacteristic
50C2×C8×C17#4111yescharacteristic
51C4⋊C4×C17—41no—
52C4⋊C4×C17—41no—
53C42×C17C2×D411yescharacteristic
54#63230—82no—
55#63227—21no—
56C4⋊Dic17—41no—
57C4⋊Dic17—41no—
58C4⋊Dic17D811yescharacteristic
59C2×#3622—21no—
60C2×#3622—41no—
61Dic68—41no—
62Dic68—41no—
63Dic68—81no—
64C8×D17—41no—
65C8×D17—41no—
66C16×C17—41no—
67#63214—21no—
68#63214—21no—
69#12470—171nomaximal, Sylow
70C22×D17—21no—
71Q8×C17—81no—
72Q8×C17—81no—
73C2×C4×C17—81no—
74C2×C4×C17#12111yescharacteristic
75C2×C4×C17C2×D811yescharacteristic
76C2×C4×C17#13511yescharacteristic
77C2×Dic17—21no—
78C2×Dic17—81no—
79D68—21no—
80D68SD3212yes—
81C4×D17—41no—
82C4×D17—41no—
83Dic34—81no—
84Dic34—81no—
85C8×C17—21no—
86C8×C17—81no—
87C8×C17#13511yescharacteristic
88C8×C17AGL(1,ℤ/8)11yescharacteristic
89#3622—41no—
90#3622—81no—
91#2241—171no—
92D4×C16—171no—
93#1673—171no—
94#1669—171no—
95#1372—171no—
96#1364D1711yescharacteristic, complemented
97#1362—171no—
98C22×C17#46611yescharacteristic
99C2×D17—22no—
100C2×D17—41no—
101C2×D17—41no—
102C4×C17—21no—
103C4×C17—161no—
104C4×C17#46311yescharacteristic
105C4×C17#47111yescharacteristic
106C4×C17C2×SD3211yescharacteristic
107C4×C17#50811yescharacteristic
108Dic17—41no—
109Dic17—161no—
110C22×C16—341no—
111#500C2×D1711yescharacteristic
112#494—171no—
113#474—341no—
114#461—341no—
115C8×D4—171no—
116C4⋊C16C2×D1711yescharacteristic
117#358—341no—
118#358—341no—
119C22⋊C16—341no—
120C4×C16—171no—
121#335—171no—
122#335C2×D1711yescharacteristic
123#332—341no—
124#332—341no—
125C2×C17#168111yescharacteristic
126C2×C17#221711yescharacteristic
127C2×C17#222211yescharacteristic
128D17—42no—
129C2×Q16—341no—
130C2×Q16—341no—
131C2×SD16—681no—
132C22×C8—341no—
133C4⋊Q8—21no—
134C4⋊Q8—341no—
135C4×D4—171no—
136C2×C16—21no—
137C2×C16—171no—
138C2×C16—171no—
139C2×C16—682no—
140#107—21no—
141#107—21no—
142#107—681no—
143C4⋊C8—21no—
144C4⋊C8—171no—
145C4⋊C8—341no—
146#103—681no—
147#102—681no—
148C22⋊C8—341no—
149C4×C8C22×D1711yescharacteristic
150C17#1247011yescharacteristic, complemented, Sylow
151C2×Q8—41no—
152C2×Q8—681no—
153C2×D4—171no—
154C22×C4—341no—
155Q16—681no—
156Q16—681no—
157Q16—1361no—
158SD16—1362no—
159C2×C8—41no—
160C2×C8—341no—
161C2×C8—681no—
162C2×C8—681no—
163C2×C8—681no—
164C2×C8#6325211yescharacteristic
165C2×C8D4×D1711yescharacteristic
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—171no—
169C4⋊C4—681no—
170C4⋊C4—681no—
171C22⋊C4—341no—
172C42D4×D1711yescharacteristic
173C16—41no—
174C16—341no—
175C16—341no—
176C23—341no—
177Q8—81no—
178Q8—81no—
179Q8—1361no—
180Q8—1361no—
181D4—172no—
182D4—341no—
183C2×C4—81no—
184C2×C4—341no—
185C2×C4—681no—
186C2×C4—681no—
187C2×C4—1361no—
188C2×C4#1058757011yescharacteristic
189C2×C4D8×D1711yescharacteristic
190C2×C4#1058760211yescharacteristic
191C8—21no—
192C8—81no—
193C8—681no—
194C8—1361no—
195C8#1058760011yescharacteristic
196C8#1058760511yescharacteristic
197C22—342no—
198C22—681no—
199C22—681no—
200C22#1176304411yescharacteristic
201C4—21no—
202C4—161no—
203C4—2721no—
204C4—681no—
205C4#1176302711yescharacteristic
206C4#1176307511yescharacteristic
207C4SD32×D1711yescharacteristic
208C4#1176323811yescharacteristic
209C2—682no—
210C2#1452822411yescharacteristic
211C2#1453087911yescharacteristic
212C2#1453090811yescharacteristic
subgroupG/H
centerC22#11763044
commutator, cosocleC4×C8×C17C23
FrattiniC4×C8C22×D17
FittingC17×#1364C2
socleC22×C17#466
prime radicalC2×D68D8
hypercenter#1364D17
hypocenterC17#12470
LCSG ▷ C4×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
1. #14531013C2semidirect
5. #14526857C2semidirect
6. #14526834C2semidirect
7. C17×#1364C2semidirect
96. #1364D17semidirect
150. C17#12470semidirect

Maximal subgroups

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

subgroupindex
#145310132
#145306032
#145281702
#145281502
#145268572
#145268342
C17×#13642
#1247017