Finite Group Databasebeta

← #16947671

Subgroups — #16947671

Order 4352.

310 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 59 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#14531001C212yesmaximal, complemented
2#14530337C211yesmaximal, characteristic
3#14526772C212yesmaximal, complemented
4#14526722C211yesmaximal, characteristic
5C17×#3202C211yesmaximal, characteristic
6#11762858C2211yescharacteristic
7#11762835—21no—
8#11762826—21no—
9#11762631—22no—
10#11762625—22no—
11#11762591C2212yes—
12#11762566C2211yescharacteristic
13C17×#536—21no—
14C17×#485—22no—
15C17×#470C2212yes—
16C17×#467—21no—
17C17×#451—22no—
18C17×#408C2211yescharacteristic
19C2×#41×C17D411yescharacteristic
20C22×Q8×C17—21no—
21C2×Q16×C17—21no—
22C2×Q16×C17D412yes—
23C2×M16×C17C2311yescharacteristic
24C22×C8×C17D411yescharacteristic
25C17×C4⋊Q8—41no—
26C17×#124—41no—
27C17×#123—41no—
28C17×#123—41no—
29C17×C22⋊Q8—22no—
30C17×C22⋊Q8—22no—
31C17×#117—21no—
32#10587612—21no—
33C2×#63223—21no—
34C17×#107—21no—
35C17×#106D412yes—
36C17×#103—22no—
37C17×#103—22no—
38C17×C22⋊C8—22no—
39#10587508—42no—
40#10587490—22no—
41#10587485—42no—
42#10587483—22no—
43#41×C17—41no—
44#41×C17—41no—
45C2×Q8×C17—22no—
46C2×Q8×C17—41no—
47C2×Q8×C17—81no—
48C2×Q8×C17—81no—
49C2×Q8×C17SD1611yescharacteristic
50C2×D4×C17—21no—
51C2×D4×C17SD1611yescharacteristic
52C22×C4×C17—21no—
53C22×C4×C17—41no—
54C22×C4×C17C2×D411yescharacteristic
55Q16×C17—42no—
56Q16×C17—81no—
57M16×C17—41no—
58C2×C8×C17—21no—
59C2×C8×C17—41no—
60C2×C8×C17C2×D412yes—
61C4⋊C4×C17—22no—
62C4⋊C4×C17—42no—
63C4⋊C4×C17—42no—
64C22⋊C4×C17—42no—
65C22⋊C4×C17—41no—
66C22⋊C4×C17—42no—
67C42×C17—41no—
68C22⋊Dic17—41no—
69C4⋊Dic17—41no—
70C4×Dic17—22no—
71#63223—41no—
72#63223—41no—
73C2×#3622—41no—
74#63214—42no—
75#12302—171nomaximal, Sylow
76C23×C17—21no—
77C23×C17AGL(1,ℤ/8)11yescharacteristic
78Q8×C17—41no—
79Q8×C17—42no—
80Q8×C17—42no—
81Q8×C17—82no—
82D4×C17—41no—
83D4×C17—81no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—21no—
87C2×C4×C17—41no—
88C2×C4×C17—41no—
89C2×C4×C17—42no—
90C2×C4×C17—42no—
91C2×C4×C17—82no—
92C2×C4×C17#12011yescharacteristic
93C2×C4×C17C2×SD1611yescharacteristic
94C2×Dic17—42no—
95C8×C17—22no—
96C8×C17—41no—
97#3622—81no—
98#3202D1711yescharacteristic
99#2239—172no—
100#2098—171no—
101#1340—172no—
102#1321—171no—
103C22×C17—21no—
104C22×C17—41no—
105C22×C17—42no—
106C22×C17—41no—
107C22×C17#45011yescharacteristic
108C4×C17—21no—
109C4×C17—41no—
110C4×C17—82no—
111C4×C17—82no—
112C4×C17#45311yescharacteristic
113C4×C17#45811yescharacteristic
114Dic17—82no—
115#536—21no—
116#485—22no—
117#470C2×D1712yes—
118#467—21no—
119#451—22no—
120#430—171no—
121#424—341no—
122#421—341no—
123#408C2×D1711yescharacteristic
124#365—342no—
125#362—342no—
126#349—172no—
127H(3,ℤ/4)—171no—
128C2×C17—21no—
129C2×C17—41no—
130C2×C17—81no—
131C2×C17#161311yescharacteristic
132C2×#41C22⋊D1711yescharacteristic
133C22×Q8—21no—
134C2×Q16—21no—
135C2×Q16C22⋊D1712yes—
136C2×M16—341no—
137C2×M16C22×D1711yescharacteristic
138C22×C8C22⋊D1711yescharacteristic
139C4⋊Q8—41no—
140#124—41no—
141#123—41no—
142#123—41no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145#117—21no—
146#117—341no—
147M32—342no—
148#108—342no—
149#107—21no—
150#106C22⋊D1712yes—
151C4⋊C8—682no—
152#104—682no—
153#103—22no—
154#103—22no—
155C22⋊C8—22no—
156C17#1230211yescharacteristic, complemented, Sylow
157#41—41no—
158#41—41no—
159C2×Q8—22no—
160C2×Q8—41no—
161C2×Q8—81no—
162C2×Q8—81no—
163C2×Q8#6322911yescharacteristic
164C2×D4—21no—
165C2×D4#6322911yescharacteristic
166C22×C4—21no—
167C22×C4—41no—
168C22×C4C2×C22⋊D1711yescharacteristic
169Q16—42no—
170Q16—81no—
171M16—41no—
172M16—681no—
173M16—681no—
174C2×C8—21no—
175C2×C8—41no—
176C2×C8—681no—
177C2×C8C2×C22⋊D1712yes—
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—681no—
182C22⋊C4—41no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—681no—
186C42—41no—
187C42—342no—
188C16—682no—
189C23—21no—
190C23#1058761811yescharacteristic
191Q8—41no—
192Q8—42no—
193Q8—42no—
194Q8—82no—
195D4—41no—
196D4—81no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—82no—
205C2×C4—682no—
206C2×C4C2×#6322911yescharacteristic
207C2×C4#1058763911yescharacteristic
208C8—22no—
209C8—41no—
210C8—1361no—
211C22—21no—
212C22—41no—
213C22—42no—
214C22—41no—
215C22#1176296011yescharacteristic
216C4—21no—
217C4—41no—
218C4—82no—
219C4—82no—
220C4—1362no—
221C4#1176297711yescharacteristic
222C4#1176299911yescharacteristic
223C2—21no—
224C2—41no—
225C2—81no—
226C2#1452780411yescharacteristic
subgroupG/H
centerC2#14527804
commutator, cosocleC2×M16×C17C23
FrattiniC2×M16C22×D17
FittingC17×#3202C2
socleC2×C17#1613
prime radicalC2×#41×C17D4
hypercenter#3202D17
hypocenterC17#12302
LCSG ▷ C2×M16×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3202

Complements

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

NH ≅ G/Ntype
1. #14531001C2semidirect
3. #14526772C2semidirect
156. C17#12302semidirect

Maximal subgroups

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

subgroupindex
#145310012
#145303372
#145267722
#145267222
C17×#32022
#1230217