Finite Group Databasebeta

← #16938449

Subgroups — #16938449

Order 4352.

402 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 99 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#14529762C212yesmaximal
2#14528935C211yesmaximal, characteristic
3#14527474C212yesmaximal
4#14526659C211yesmaximal, characteristic
5C17×#3249C211yesmaximal, characteristic
6#11762833C2212yes—
7C2×C4⋊#3622C2211yescharacteristic
8C2×#10587485—22no—
9C42×Dic17C2211yescharacteristic
10#11762568—22no—
11#11762568—22no—
12#11762558—22no—
13#11762555—22no—
14C17×#533C2212yes—
15C17×#477—22no—
16C17×#467—22no—
17C17×#462—22no—
18C2×C17×C4⋊C8C2211yescharacteristic
19C2×Q16×C17—22no—
20C22×C8×C17D412yes—
21C17×C4⋊Q8D412yes—
22C17×C4⋊Q8D412yes—
23C17×#125—22no—
24C17×C22⋊Q8—22no—
25C17×C22⋊Q8—22no—
26C4×Q8×C17—22no—
27C4×Q8×C17—22no—
28C17×#117—22no—
29C2×C42×C17C2311yescharacteristic
30C2×C4×Dic17—21no—
31C2×C4×Dic17—21no—
32C2×C4×Dic17D411yescharacteristic
33C2×C4×Dic17D411yescharacteristic
34C2×#63223—22no—
35C22×#3622—21no—
36C17×#107—22no—
37C17×C4⋊C8—22no—
38C17×#103—24no—
39#10587485—44no—
40C4⋊#3622—22no—
41C4⋊#3622—22no—
42C4⋊#3622—22no—
43C2×Q8×C17—22no—
44C2×Q8×C17—22no—
45C22×C4×C17C2×D411yescharacteristic
46C22×C4×C17C2×D411yescharacteristic
47C22×C4×C17#4111yescharacteristic
48C22×Dic17—21no—
49C22×Dic17—21no—
50Q16×C17—82no—
51C2×C8×C17—22no—
52C2×C8×C17—42no—
53C4⋊C4×C17—22no—
54C4⋊C4×C17—22no—
55C4⋊C4×C17—22no—
56C4⋊C4×C17—22no—
57C4⋊C4×C17—42no—
58C4⋊C4×C17—42no—
59C22⋊C4×C17—42no—
60C42×C17—42no—
61C42×C17C2×D412yes—
62C42×C17#4112yes—
63C4×Dic17—22no—
64C4×Dic17—22no—
65C4×Dic17—21no—
66C4×Dic17—22no—
67C4×Dic17—42no—
68C4×Dic17Q1612yescomplemented
69#63223—42no—
70#63223—42no—
71C2×#3622—42no—
72C2×#3622—41no—
73C2×#3622—42no—
74#10522—171nomaximal, Sylow
75C23×C17#12111yescharacteristic
76Q8×C17—42no—
77Q8×C17—42no—
78Q8×C17—82no—
79C2×C4×C17—21no—
80C2×C4×C17—21no—
81C2×C4×C17—41no—
82C2×C4×C17—42no—
83C2×C4×C17—42no—
84C2×C4×C17#12011yescharacteristic
85C2×C4×C17#12112yes—
86C2×C4×C17#12312yes—
87C2×C4×C17#12411yescharacteristic
88C2×C4×C17C2×Q1611yescharacteristic
89C2×C4×C17#13511yescharacteristic
90C2×C4×C17#13712yes—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—41no—
96C2×Dic17—41no—
97C8×C17—42no—
98#3622—81no—
99#3622—82no—
100#3249D1711yescharacteristic
101#1994—172no—
102#1842—171no—
103#1550—172no—
104#1292—171no—
105C22×C17—21no—
106C22×C17—21no—
107C22×C17#39411yescharacteristic
108C22×C17#46711yescharacteristic
109C22×C17#47011yescharacteristic
110C4×C17—22no—
111C4×C17—21no—
112C4×C17—82no—
113C4×C17—82no—
114C4×C17#45311yescharacteristic
115C4×C17#45611yescharacteristic
116C4×C17#46212yes—
117C4×C17#46412yes—
118Dic17—24no—
119Dic17—42no—
120#533C2×D1712yes—
121#477—22no—
122#467—22no—
123#462—22no—
124#423—172no—
125C2×C4⋊C8—171no—
126C2×C4⋊C8—342no—
127C2×C4⋊C8C2×D1711yescharacteristic
128C43—171no—
129#339—342no—
130#339—342no—
131#333—342no—
132#332—342no—
133C2×C17—21no—
134C2×C17—21no—
135C2×C17#203911yescharacteristic
136C2×C17#205011yescharacteristic
137C2×C17#205611yescharacteristic
138C2×Q16—22no—
139C2×M16—342no—
140C22×C8—341no—
141C22×C8C22⋊D1712yes—
142C4⋊Q8C22⋊D1712yes—
143C4⋊Q8C22⋊D1712yes—
144#125—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149#117—22no—
150C2×C42—171no—
151C2×C42—171no—
152C2×C42—341no—
153C2×C42—341no—
154C2×C42C22×D1711yescharacteristic
155#107—22no—
156C4⋊C8—22no—
157C4⋊C8—342no—
158C4⋊C8—342no—
159C4⋊C8—342no—
160C4⋊C8—684no—
161#103—24no—
162C17#1052211yescharacteristic, complemented, Sylow
163C2×Q8—22no—
164C2×Q8—22no—
165C22×C4—341no—
166C22×C4—341no—
167C22×C4D4×D1711yescharacteristic
168C22×C4#6325411yescharacteristic
169C22×C4C2×C22⋊D1711yescharacteristic
170Q16—82no—
171M16—682no—
172M16—682no—
173C2×C8—22no—
174C2×C8—42no—
175C2×C8—682no—
176C2×C8—682no—
177C2×C8—681no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C22⋊C4—42no—
185C42—42no—
186C42—172no—
187C42—342no—
188C42—342no—
189C42—341no—
190C42—342no—
191C42—682no—
192C42#6325412yes—
193C42C2×C22⋊D1712yes—
194C23#1058762411yescharacteristic
195Q8—42no—
196Q8—42no—
197Q8—82no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—342no—
206C2×C4—342no—
207C2×C4—681no—
208C2×C4—681no—
209C2×C4Q16×D1711yescharacteristic
210C2×C4#1058760412yes—
211C2×C4#1058760511yescharacteristic
212C2×C4#1058762112yes—
213C2×C4#1058762211yescharacteristic
214C2×C4#1058762512yes—
215C2×C4#1058763911yescharacteristic
216C8—42no—
217C8—1361no—
218C8—1362no—
219C22—21no—
220C22—21no—
221C22#1176272311yescharacteristic
222C22#1176304711yescharacteristic
223C22#1176306411yescharacteristic
224C4—22no—
225C4—21no—
226C4—82no—
227C4—82no—
228C4—344no—
229C4—682no—
230C4#1176297711yescharacteristic
231C4#1176299211yescharacteristic
232C4#1176301712yes—
233C4#1176303312yes—
234C2—21no—
235C2—21no—
236C2#1453000511yescharacteristic
237C2#1453006811yescharacteristic
238C2#1453010411yescharacteristic
subgroupG/H
centerC22#11762723
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#3249C2
socleC22×C17#394
prime radicalC23×C17#121
hypercenter#3249D17
hypocenterC17#10522
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
68. C4×Dic17Q16semidirect
162. C17#10522semidirect

Maximal subgroups

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

subgroupindex
#145297622
#145289352
#145274742
#145266592
C17×#32492
#1052217