Finite Group Databasebeta

← #16948896

Subgroups — #16948896

Order 4352.

226 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 65 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#14530922C211yesmaximal, characteristic
2#14530830C211yesmaximal, characteristic
3#14528381C211yesmaximal, characteristic, complemented
4#14527670C211yesmaximal, characteristic
5#14526838C211yesmaximal, characteristic, complemented
6#14526834C211yesmaximal, characteristic, complemented
7C17×#1370C211yesmaximal, characteristic, complemented
8C2×#10587471—21no—
9#11763214C2211yescharacteristic
10#11763207—21no—
11#11763189C2211yescharacteristic
12#11763160—21no—
13C4×D136C2211yescharacteristic
14#11762888—21no—
15#11762635—21no—
16#11762614—21no—
17#11762611—21no—
18#11762562C2211yescharacteristic
19#11762538—21no—
20C17×C4⋊C16C2211yescharacteristic
21C17×C4⋊C16C2211yescharacteristic
22C82×C17C2211yescharacteristic
23C2×Dic68—21no—
24C2×Dic68—21no—
25C2×D136D411yescharacteristic
26C2×#63218—41no—
27C4×D68—21no—
28#10587555—21no—
29#10587555—21no—
30C2×C16×C17—21no—
31C2×C16×C17—21no—
32C4×C8×C17D411yescharacteristic
33C4×C8×C17D411yescharacteristic
34C4×C8×C17C2311yescharacteristic
35#10587492—21no—
36#10587491—41no—
37#10587489—21no—
38#10587489—21no—
39#10587489D411yescharacteristic
40#10587488—42no—
41#10587487—41no—
42#10587487—41no—
43C4⋊#3622—21no—
44#10587471—42no—
45C2×D68—21no—
46C2×C4×D17—41no—
47C2×Dic34—41no—
48C2×Dic34—41no—
49C2×C8×C17—21no—
50C2×C8×C17D811yescharacteristic
51C2×C8×C17D811yescharacteristic
52C2×C8×C17C2×D411yescharacteristic
53C2×C8×C17#4111yescharacteristic
54C42×C17C2×D411yescharacteristic
55#63227—41no—
56C4⋊Dic17—21no—
57C4⋊Dic17—41no—
58C4⋊Dic17—41no—
59C4⋊Dic17—41no—
60C4⋊Dic17—41no—
61C2×#3622—41no—
62Dic68—41no—
63Dic68—41no—
64Dic68—81no—
65D136—21no—
66D136—21no—
67#63218—82no—
68C16×C17—41no—
69C16×C17—41no—
70#12489—171nomaximal, Sylow
71C22×D17—41no—
72C2×C4×C17#12111yescharacteristic
73C2×C4×C17C2×D811yescharacteristic
74C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
75C2×Dic17—41no—
76C2×Dic17—81no—
77C2×Dic17—81no—
78D68—41no—
79D68—41no—
80C4×D17—81no—
81C4×D17—81no—
82Dic34—81no—
83Dic34—81no—
84Dic34—81no—
85Dic34—81no—
86C8×C17—22no—
87C8×C17—21no—
88C8×C17—21no—
89C8×C17—21no—
90C8×C17AGL(1,ℤ/8)11yescharacteristic
91C8×C17#13711yescharacteristic
92#3622—81no—
93#2224—171no—
94#2208—171no—
95#1707—171no—
96#1589—171no—
97#1370D1711yescharacteristic, complemented
98#1364—171no—
99#1362—171no—
100C22×C17#45911yescharacteristic
101C2×D17—81no—
102C2×D17—81no—
103C2×D17—81no—
104C4×C17—21no—
105C4×C17#46811yescharacteristic
106C4×C17#47111yescharacteristic
107C4×C17#50811yescharacteristic
108C4×C17#50911yescharacteristic
109Dic17—81no—
110Dic17—161no—
111Dic17—161no—
112C2×SD32—341no—
113#500—171no—
114#498—341no—
115#494—171no—
116#487—341no—
117C4×D8—171no—
118#436—341no—
119#366—341no—
120C4⋊C16C2×D1711yescharacteristic
121C4⋊C16C2×D1711yescharacteristic
122#358—341no—
123#357—341no—
124#335—171no—
125D4⋊C8—341no—
126C82C2×D1711yescharacteristic
127C2×C17#168511yescharacteristic
128C2×C17#221011yescharacteristic
129C2×C17#221411yescharacteristic
130D17—161no—
131C2×Q16—341no—
132C2×Q16—341no—
133C2×D8—171no—
134C2×M16—681no—
135C4⋊Q8—341no—
136C4⋊Q8—341no—
137C4×D4—341no—
138SD32—682no—
139C2×C16—21no—
140C2×C16—21no—
141#107—171no—
142#107—341no—
143#107—341no—
144#106—682no—
145C4⋊C8—341no—
146#103—681no—
147#103—681no—
148#102—341no—
149C22⋊C8—681no—
150C4×C8D6811yescharacteristic
151C4×C8D6811yescharacteristic
152C4×C8C22×D1711yescharacteristic
153C17#1248911yescharacteristic, complemented, Sylow
154C2×Q8—681no—
155C2×Q8—681no—
156C2×D4—341no—
157C22×C4—681no—
158Q16—681no—
159Q16—681no—
160Q16—1361no—
161D8—341no—
162D8—341no—
163M16—1362no—
164C2×C8—21no—
165C2×C8—681no—
166C2×C8D13611yescharacteristic
167C2×C8D13611yescharacteristic
168C2×C8D4×D1711yescharacteristic
169C2×C8#6325611yescharacteristic
170C4⋊C4—341no—
171C4⋊C4—681no—
172C4⋊C4—681no—
173C4⋊C4—681no—
174C4⋊C4—681no—
175C22⋊C4—681no—
176C42C2×D6811yescharacteristic
177C16—41no—
178C16—41no—
179C23—681no—
180Q8—1361no—
181Q8—1361no—
182Q8—1361no—
183Q8—1361no—
184D4—681no—
185D4—681no—
186C2×C4—681no—
187C2×C4—1361no—
188C2×C4—1361no—
189C2×C4—1361no—
190C2×C4—1361no—
191C2×C4#1058758111yescharacteristic
192C2×C4C2×D13611yescharacteristic
193C2×C4#1058759411yescharacteristic
194C8—22no—
195C8—21no—
196C8—21no—
197C8—21no—
198C8—1361no—
199C8#1058759711yescharacteristic
200C8#1058760411yescharacteristic
201C22—1361no—
202C22—1361no—
203C22—1361no—
204C22#1176300511yescharacteristic
205C4—21no—
206C4—2721no—
207C4—2721no—
208C4—1361no—
209C4#1176305611yescharacteristic
210C4#1176307311yescharacteristic
211C4#1176324111yescharacteristic
212C4#1176324711yescharacteristic
213C2—2721no—
214C2#1452825011yescharacteristic
215C2#1453084011yescharacteristic
216C2#1453086611yescharacteristic
subgroupG/H
centerC22#11763005
commutator, cosocleC4×C8×C17C23
FrattiniC4×C8C22×D17
FittingC17×#1370C2
socleC22×C17#459
prime radicalC2×D136D4
hypercenter#1370D17
hypocenterC17#12489
LCSG ▷ C4×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ #1370

Complements

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

NH ≅ G/Ntype
3. #14528381C2semidirect
5. #14526838C2semidirect
6. #14526834C2semidirect
7. C17×#1370C2semidirect
97. #1370D17semidirect
153. C17#12489semidirect

Maximal subgroups

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

subgroupindex
#145309222
#145308302
#145283812
#145276702
#145268382
#145268342
C17×#13702
#1248917