Finite Group Databasebeta

← #16948745

Subgroups — #16948745

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
1C16×D68C211yesmaximal, characteristic, complemented
2#14528180C211yesmaximal, characteristic, complemented
3#14528136C211yesmaximal, characteristic
4#14526858C211yesmaximal, characteristic, complemented
5#14526842C211yesmaximal, characteristic, complemented
6#14526830C211yesmaximal, characteristic, complemented
7C17×#2240C211yesmaximal, characteristic, complemented
8C2×C16×D17—21no—
9#11763096—21no—
10#11763001—21no—
11C8×D68C2211yescharacteristic
12C4⋊#63214C2211yescharacteristic
13#11762588—21no—
14#11762562C2211yescharacteristic
15#11762562C2211yescharacteristic
16#11762558—21no—
17#11762553—21no—
18C17×#500C2211yescharacteristic
19C17×#493C2211yescharacteristic
20C17×#357—21no—
21C17×#357—21no—
22C4×C16×C17C2211yescharacteristic
23C2×D8×C17—21no—
24C2×D8×C17—21no—
25C17×C4⋊Q8—21no—
26C17×#127—21no—
27C2×#63228—41no—
28C2×C8×D17—21no—
29C4×D68D411yescharacteristic
30C2×C16×C17D411yescharacteristic
31C2×C16×C17D411yescharacteristic
32C17×#107—21no—
33C17×#107—21no—
34C4×C8×C17C2311yescharacteristic
35#10587503—41no—
36#10587491—21no—
37C2×#63214—21no—
38#10587480—41no—
39#10587479—41no—
40C4⋊#3622—21no—
41C4⋊#3622—21no—
42C4⋊#3622D411yescharacteristic
43C16×D17—42no—
44C2×Q8×C17—41no—
45C2×D4×C17—41no—
46C2×D4×C17—41no—
47C2×D4×C17—41no—
48C2×D68D811yescharacteristic
49C2×C4×D17—21no—
50D8×C17—41no—
51D8×C17—41no—
52D8×C17—81no—
53C2×C8×C17C2×D411yescharacteristic
54C2×C8×C17#4111yescharacteristic
55C4⋊C4×C17—41no—
56C4⋊C4×C17—41no—
57C42×C17C2×D411yescharacteristic
58#63228—82no—
59#63227—21no—
60C4⋊Dic17D811yescharacteristic
61C2×#3622—21no—
62C2×#3622—41no—
63C2×#3622—41no—
64C8×D17—41no—
65C8×D17—41no—
66C16×C17—21no—
67C16×C17—21no—
68#63214—41no—
69#12468—171nomaximal, Sylow
70C23×C17—81no—
71C22×D17—21no—
72Q8×C17—161no—
73D4×C17—81no—
74D4×C17—81no—
75D4×C17—81no—
76D4×C17—81no—
77D4×C17—161no—
78C2×C4×C17—81no—
79C2×C4×C17#12111yescharacteristic
80C2×C4×C17C2×D811yescharacteristic
81C2×C4×C17#13511yescharacteristic
82C2×Dic17—21no—
83D68—21no—
84D68SD3212yes—
85C4×D17—41no—
86C4×D17—41no—
87C8×C17—21no—
88C8×C17#13511yescharacteristic
89C8×C17AGL(1,ℤ/8)11yescharacteristic
90#3622—41no—
91#3622—81no—
92#3622—81no—
93#2240D1711yescharacteristic, complemented
94D4×C16—171no—
95#1675—171no—
96#1667—171no—
97#1372—171no—
98#1365—171no—
99#1361—171no—
100C22×C17—161no—
101C22×C17—161no—
102C22×C17—161no—
103C22×C17#46611yescharacteristic
104C2×D17—22no—
105C2×D17—41no—
106C2×D17—41no—
107C4×C17—21no—
108C4×C17—161no—
109C4×C17#46311yescharacteristic
110C4×C17#47111yescharacteristic
111C4×C17C2×SD3211yescharacteristic
112C4×C17#50811yescharacteristic
113Dic17—41no—
114C22×C16—341no—
115#500C2×D1711yescharacteristic
116#493C2×D1711yescharacteristic
117#476—341no—
118#459—341no—
119C8×D4—171no—
120C4⋊C16—171no—
121#357—21no—
122#357—21no—
123C22⋊C16—341no—
124C4×C16C2×D1711yescharacteristic
125#335—171no—
126#335—171no—
127#333—341no—
128#331—341no—
129C2×C17—321no—
130C2×C17#168111yescharacteristic
131C2×C17#221711yescharacteristic
132C2×C17#222111yescharacteristic
133D17—42no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—681no—
137C22×C8—341no—
138C4⋊Q8—21no—
139#127—21no—
140C4×D4—171no—
141C2×C16—341no—
142C2×C16—682no—
143C2×C16C22⋊D1711yescharacteristic
144C2×C16C22⋊D1711yescharacteristic
145#107—21no—
146#107—21no—
147#106—681no—
148C4⋊C8—171no—
149C4⋊C8—341no—
150C4⋊C8—341no—
151#102—681no—
152#102—681no—
153C22⋊C8—341no—
154C4×C8C22×D1711yescharacteristic
155C17#1246811yescharacteristic, complemented, Sylow
156C2×Q8—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—171no—
161C22×C4—341no—
162D8—41no—
163D8—41no—
164D8—81no—
165D8—1362no—
166C2×C8—341no—
167C2×C8—681no—
168C2×C8—681no—
169C2×C8—681no—
170C2×C8—681no—
171C2×C8#6325411yescharacteristic
172C2×C8C2×C22⋊D1711yescharacteristic
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—171no—
176C22⋊C4—341no—
177C42D4×D1711yescharacteristic
178C16—21no—
179C16—21no—
180C16—681no—
181C23—81no—
182C23—341no—
183Q8—161no—
184D4—81no—
185D4—81no—
186D4—81no—
187D4—81no—
188D4—161no—
189D4—172no—
190D4—341no—
191C2×C4—81no—
192C2×C4—341no—
193C2×C4—681no—
194C2×C4—681no—
195C2×C4D8×D1711yescharacteristic
196C2×C4#1058759811yescharacteristic
197C2×C4#1058762411yescharacteristic
198C8—21no—
199C8—681no—
200C8—1361no—
201C8—1361no—
202C8#1058763511yescharacteristic
203C8#1058763611yescharacteristic
204C22—161no—
205C22—161no—
206C22—161no—
207C22—342no—
208C22—681no—
209C22—681no—
210C22#1176304211yescharacteristic
211C4—21no—
212C4—161no—
213C4—681no—
214C4#1176302111yescharacteristic
215C4#1176306911yescharacteristic
216C4SD32×D1711yescharacteristic
217C4#1176324011yescharacteristic
218C2—321no—
219C2—682no—
220C2#1452821811yescharacteristic
221C2#1453087711yescharacteristic
222C2#1453090111yescharacteristic
subgroupG/H
centerC22#11763042
commutator, cosocleC4×C8×C17C23
FrattiniC4×C8C22×D17
FittingC17×#2240C2
socleC22×C17#466
prime radical#14528136C2
hypercenter#2240D17
hypocenterC17#12468
LCSG ▷ C4×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2240

Complements

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

NH ≅ G/Ntype
1. C16×D68C2semidirect
2. #14528180C2semidirect
4. #14526858C2semidirect
5. #14526842C2semidirect
6. #14526830C2semidirect
7. C17×#2240C2semidirect
93. #2240D17semidirect
155. C17#12468semidirect

Maximal subgroups

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

subgroupindex
C16×D682
#145281802
#145281362
#145268582
#145268422
#145268302
C17×#22402
#1246817