Finite Group Databasebeta

← #16938823

Subgroups — #16938823

Order 4352.

374 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 97 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#14528953C211yesmaximal, characteristic
2#14528917C212yesmaximal
3#14526665C211yesmaximal, characteristic
4#14526665C212yesmaximal
5C17×#2844C211yesmaximal, characteristic
6C2×#10587478—21no—
7C2×#10587479—21no—
8C2×#10587474C2212yes—
9C2×#10587474C2211yescharacteristic
10#11762679C2211yescharacteristic
11#11762571—22no—
12#11762571—22no—
13C17×#400—22no—
14C17×#400—21no—
15C17×#395—22no—
16C17×#395—21no—
17C17×#382C2212yes—
18C17×#382C2211yescharacteristic
19C22×Q8×C17—21no—
20C2×C4⋊C4×C17—21no—
21C2×C4⋊C4×C17—22no—
22C2×C4⋊C4×C17D412yes—
23C2×C4⋊C4×C17D412yes—
24C2×C4⋊C4×C17D411yescharacteristic
25C2×C4⋊C4×C17D411yescharacteristic
26C2×C42×C17C2311yescharacteristic
27C2×C4⋊Dic17—22no—
28C2×C4⋊Dic17—21no—
29C22×#3622—21no—
30C22×#3622—22no—
31C17×#95—21no—
32C17×#95—22no—
33C17×#95—21no—
34C17×#95—22no—
35C17×#95—22no—
36C17×#95—22no—
37#10587479—42no—
38#10587478—42no—
39#10587474—21no—
40#10587474—22no—
41#10587474Q812yescomplemented
42#10587474Q814yescomplemented
43C2×Q8×C17—81no—
44C2×Q8×C17—82no—
45C22×C4×C17—21no—
46C22×C4×C17—22no—
47C22×C4×C17—21no—
48C22×C4×C17—22no—
49C22×C4×C17C2×D411yescharacteristic
50C22×C4×C17C2×D412yes—
51C22×Dic17—41no—
52C4⋊C4×C17—41no—
53C4⋊C4×C17—41no—
54C4⋊C4×C17—42no—
55C4⋊C4×C17—42no—
56C4⋊C4×C17—81no—
57C4⋊C4×C17—82no—
58C42×C17C2×Q811yescharacteristic
59C42×C17C2×Q812yes—
60C42×C17#4111yescharacteristic
61C4⋊Dic17—41no—
62C4⋊Dic17—41no—
63C4⋊Dic17—44no—
64C2×#3622—22no—
65C2×#3622—22no—
66C2×#3622—22no—
67C2×#3622—24no—
68C2×#3622—42no—
69C2×#3622—42no—
70#10584—171nomaximal, Sylow
71C23×C17#12011yescharacteristic
72Q8×C17—84no—
73C2×C4×C17—21no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17—21no—
77C2×C4×C17—41no—
78C2×C4×C17—41no—
79C2×C4×C17—42no—
80C2×C4×C17—42no—
81C2×C4×C17—42no—
82C2×C4×C17—42no—
83C2×C4×C17—42no—
84C2×C4×C17—42no—
85C2×C4×C17—42no—
86C2×C4×C17—42no—
87C2×C4×C17C22⋊Q811yescharacteristic
88C2×C4×C17C22⋊Q812yes—
89C2×C4×C17C4⋊Q811yescharacteristic
90C2×C4×C17C4⋊Q812yes—
91C2×Dic17—81no—
92C2×Dic17—82no—
93#3622—24no—
94#3622—44no—
95#2844D1711yescharacteristic
96#1847—171no—
97#1836—172no—
98#1295—172no—
99#1295—171no—
100C22×C17#39511yescharacteristic
101C22×C17#45311yescharacteristic
102C22×C17#45412yes—
103C22×C17#45512yes—
104C22×C17#45611yescharacteristic
105C4×C17—21no—
106C4×C17—21no—
107C4×C17—22no—
108C4×C17—22no—
109C4×C17—81no—
110C4×C17—81no—
111C4×C17—82no—
112C4×C17—82no—
113Dic17—161no—
114C2×#107—341no—
115C2×#106—341no—
116C2×#97—172no—
117C2×#97—171no—
118#400—22no—
119#400—21no—
120#395—22no—
121#395—21no—
122#384—171no—
123#382C2×D1712yes—
124#382C2×D1711yescharacteristic
125#340—342no—
126#340—342no—
127C2×C17#165711yescharacteristic
128C2×C17#165911yescharacteristic
129C2×C17#166412yes—
130C2×C17#206011yescharacteristic
131C2×C17#206112yes—
132C22×Q8—21no—
133C22×C8—341no—
134C22×C8—342no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—342no—
138C2×C4⋊C4—341no—
139C2×C4⋊C4C22⋊D1712yes—
140C2×C4⋊C4C22⋊D1712yes—
141C2×C4⋊C4C22⋊D1711yescharacteristic
142C2×C4⋊C4C22⋊D1711yescharacteristic
143C2×C42C22×D1711yescharacteristic
144#107—682no—
145#106—682no—
146#97—172no—
147#97—174no—
148#97—341no—
149#97—342no—
150#95—22no—
151#95—21no—
152#95—21no—
153#95—22no—
154#95—22no—
155#95—22no—
156C17#1058411yescharacteristic, complemented, Sylow
157C2×Q8—81no—
158C2×Q8—82no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—681no—
164C22×C4C2×C22⋊D1711yescharacteristic
165C22×C4C2×C22⋊D1712yes—
166C2×C8—342no—
167C2×C8—342no—
168C2×C8—342no—
169C2×C8—344no—
170C2×C8—682no—
171C2×C8—682no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—81no—
177C4⋊C4—82no—
178C4⋊C4—681no—
179C4⋊C4—681no—
180C4⋊C4—684no—
181C42Q8×D1712yes—
182C42Q8×D1711yescharacteristic
183C42#6325611yescharacteristic
184C23#1058763911yescharacteristic
185Q8—84no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—1361no—
201C2×C4—1362no—
202C2×C4#1058763011yescharacteristic
203C2×C4#1058763012yes—
204C2×C4#1058763211yescharacteristic
205C2×C4#1058763212yes—
206C8—344no—
207C8—684no—
208C22#1176272811yescharacteristic
209C22#1176297711yescharacteristic
210C22#1176298212yes—
211C22#1176298712yes—
212C22#1176299211yescharacteristic
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—81no—
218C4—81no—
219C4—82no—
220C4—82no—
221C4—2721no—
222C2#1452808211yescharacteristic
223C2#1452809011yescharacteristic
224C2#1452811812yes—
225C2#1453012611yescharacteristic
226C2#1453013112yes—
subgroupG/H
centerC23#10587639
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#2844C2
socle, prime radicalC23×C17#120
hypercenter#2844D17
hypocenterC17#10584
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2844

Complements

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

NH ≅ G/Ntype
41. #10587474Q8semidirect
42. #10587474Q8semidirect
156. C17#10584semidirect

Maximal subgroups

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

subgroupindex
#145289532
#145289172
#145266652
C17×#28442
#1058417