Finite Group Databasebeta

← #16948749

Subgroups — #16948749

Order 4352.

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

normal subgroups only

This group has 234 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#14531008C211yesmaximal, characteristic, complemented
2C16×D68C211yesmaximal, characteristic, complemented
3#14528184C211yesmaximal, characteristic, complemented
4#14528140C211yesmaximal, characteristic
5#14526843C211yesmaximal, characteristic, complemented
6#14526829C211yesmaximal, characteristic, complemented
7C17×#1372C211yesmaximal, characteristic, complemented
8C2×C16×D17—21no—
9#11763214C2211yescharacteristic
10#11763186C2211yescharacteristic
11#11763100—21no—
12#11763005—21no—
13C8×D68C2211yescharacteristic
14C4⋊#63214C2211yescharacteristic
15#11762611—21no—
16#11762611—21no—
17#11762588—21no—
18#11762557—21no—
19#11762552—21no—
20C4×C16×C17C2211yescharacteristic
21C17×#335C2211yescharacteristic
22C17×#335C2211yescharacteristic
23C2×D136—21no—
24C2×D136—21no—
25C2×D136—41no—
26C2×C8×D17—21no—
27#10587560—21no—
28C4×D68D411yescharacteristic
29#10587555—21no—
30C2×C16×C17D411yescharacteristic
31C2×C16×C17D411yescharacteristic
32C17×C4⋊C8—21no—
33C17×C4⋊C8—21no—
34C4×C8×C17C2311yescharacteristic
35#10587492—41no—
36#10587492—41no—
37#10587491—21no—
38#10587489—21no—
39#10587489—21no—
40#10587488—41no—
41C2×#63214—21no—
42C4⋊#3622D411yescharacteristic
43C16×D17—42no—
44C2×D68—41no—
45C2×D68—41no—
46C2×D68—41no—
47C2×D68D811yescharacteristic
48C2×C4×D17—21no—
49C2×Dic34—41no—
50C2×C8×C17—41no—
51C2×C8×C17—41no—
52C2×C8×C17C2×D411yescharacteristic
53C2×C8×C17#4111yescharacteristic
54C42×C17C2×D411yescharacteristic
55#63227—21no—
56C4⋊Dic17—41no—
57C4⋊Dic17—41no—
58C4⋊Dic17D811yescharacteristic
59C2×#3622—21no—
60D136—41no—
61D136—41no—
62D136—81no—
63D136—82no—
64C8×D17—41no—
65C8×D17—41no—
66C16×C17—21no—
67C16×C17—21no—
68#63214—41no—
69#12468—171nomaximal, Sylow
70C22×D17—21no—
71C22×D17—81no—
72C2×C4×C17#12111yescharacteristic
73C2×C4×C17C2×D811yescharacteristic
74C2×C4×C17#13511yescharacteristic
75C2×Dic17—21no—
76C2×Dic17—81no—
77D68—21no—
78D68—81no—
79D68—81no—
80D68—81no—
81D68—81no—
82D68—161no—
83D68SD3212yes—
84C4×D17—41no—
85C4×D17—41no—
86Dic34—161no—
87C8×C17—21no—
88C8×C17—81no—
89C8×C17—81no—
90C8×C17#13511yescharacteristic
91C8×C17AGL(1,ℤ/8)11yescharacteristic
92#3622—41no—
93#2240—171no—
94D4×C16—171no—
95#1675—171no—
96#1667—171no—
97#1372D1711yescharacteristic, complemented
98#1365—171no—
99#1361—171no—
100C22×C17#46611yescharacteristic
101C2×D17—22no—
102C2×D17—41no—
103C2×D17—41no—
104C2×D17—161no—
105C2×D17—161no—
106C2×D17—161no—
107C4×C17—21no—
108C4×C17#46311yescharacteristic
109C4×C17#47111yescharacteristic
110C4×C17C2×SD3211yescharacteristic
111C4×C17#50811yescharacteristic
112Dic17—41no—
113Dic17—161no—
114C22×C16—341no—
115#500—171no—
116#493—171no—
117#476—341no—
118#459—341no—
119C8×D4—171no—
120C4⋊C16—171no—
121#357—341no—
122#357—341no—
123C22⋊C16—341no—
124C4×C16C2×D1711yescharacteristic
125#335C2×D1711yescharacteristic
126#335C2×D1711yescharacteristic
127#333—341no—
128#331—341no—
129C2×C17#168111yescharacteristic
130C2×C17#221711yescharacteristic
131C2×C17#222111yescharacteristic
132D17—42no—
133D17—321no—
134C2×D8—341no—
135C2×D8—341no—
136C2×D8—681no—
137C22×C8—341no—
138C4⋊Q8—341no—
139#127—341no—
140C4×D4—171no—
141C2×C16—341no—
142C2×C16—682no—
143C2×C16D6811yescharacteristic
144C2×C16D6811yescharacteristic
145#107—341no—
146#107—341no—
147#106—681no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8—171no—
151#102—681no—
152#102—681no—
153C22⋊C8—341no—
154C4×C8C22×D1711yescharacteristic
155C17#1246811yescharacteristic, complemented, Sylow
156C2×Q8—681no—
157C2×D4—171no—
158C2×D4—681no—
159C2×D4—681no—
160C2×D4—681no—
161C22×C4—341no—
162D8—681no—
163D8—681no—
164D8—1361no—
165D8—1362no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—341no—
169C2×C8—681no—
170C2×C8—681no—
171C2×C8C2×D6811yescharacteristic
172C2×C8#6325611yescharacteristic
173C4⋊C4—171no—
174C4⋊C4—681no—
175C4⋊C4—681no—
176C22⋊C4—341no—
177C42D4×D1711yescharacteristic
178C16—21no—
179C16—21no—
180C16—681no—
181C23—341no—
182C23—1361no—
183Q8—2721no—
184D4—2721no—
185D4—172no—
186D4—341no—
187D4—1361no—
188D4—1361no—
189D4—1361no—
190D4—1361no—
191C2×C4—341no—
192C2×C4—681no—
193C2×C4—681no—
194C2×C4—1361no—
195C2×C4#1058758111yescharacteristic
196C2×C4D8×D1711yescharacteristic
197C2×C4#1058760511yescharacteristic
198C8—21no—
199C8—81no—
200C8—81no—
201C8—681no—
202C8#1058759011yescharacteristic
203C8#1058759411yescharacteristic
204C22—2721no—
205C22—2721no—
206C22—2721no—
207C22—342no—
208C22—681no—
209C22—681no—
210C22#1176304611yescharacteristic
211C4—21no—
212C4—2721no—
213C4—681no—
214C4#1176302511yescharacteristic
215C4#1176307311yescharacteristic
216C4SD32×D1711yescharacteristic
217C4#1176324011yescharacteristic
218C2—5441no—
219C2—682no—
220C2#1452822211yescharacteristic
221C2#1453088111yescharacteristic
222C2#1453090511yescharacteristic
subgroupG/H
centerC22#11763046
commutator, cosocleC4×C8×C17C23
FrattiniC4×C8C22×D17
FittingC17×#1372C2
socleC22×C17#466
prime radical#14528140C2
hypercenter#1372D17
hypocenterC17#12468
LCSG ▷ C4×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1372

Complements

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

NH ≅ G/Ntype
1. #14531008C2semidirect
2. C16×D68C2semidirect
3. #14528184C2semidirect
5. #14526843C2semidirect
6. #14526829C2semidirect
7. C17×#1372C2semidirect
97. #1372D17semidirect
155. C17#12468semidirect

Maximal subgroups

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

subgroupindex
#145310082
C16×D682
#145281842
#145281402
#145268432
#145268292
C17×#13722
#1246817