Finite Group Databasebeta

← #16937454

Subgroups — #16937454

Order 4352.

416 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 101 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#14529765C211yesmaximal, characteristic
2#14527282C212yesmaximal
3#14526687C212yesmaximal
4#14526613C211yesmaximal, characteristic
5C17×#2399C211yesmaximal, characteristic
6#11763366C2211yescharacteristic
7#11762862C2212yes—
8#11762833C2211yescharacteristic
9#11762569—21no—
10#11762569—21no—
11#11762545—22no—
12#11762545—22no—
13C2×C4×Q8×C17—21no—
14C17×#398—22no—
15C17×#395—22no—
16C4×C4⋊C4×C17—21no—
17C4×C4⋊C4×C17C2212yes—
18C17×#376C2211yescharacteristic
19C22×Q8×C17—21no—
20C4×Q8×C17—42no—
21C4×Q8×C17—42no—
22C2×C4⋊C4×C17—22no—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17—22no—
25C2×C4⋊C4×C17—22no—
26C2×C42×C17—22no—
27C2×C42×C17D411yescharacteristic
28C2×C42×C17D411yescharacteristic
29C2×C42×C17C2311yescharacteristic
30#10587612—21no—
31#10587610—22no—
32C2×#63223—21no—
33#10587556—21no—
34#10587555Q812yescomplemented
35C4×Dic34—22no—
36C17×#95—21no—
37C17×#95—21no—
38C17×#95—22no—
39C17×#95D412yes—
40C17×#95D412yes—
41C22⋊#3622—22no—
42C4⋊#3622—22no—
43C4⋊#3622—22no—
44C4×#3622—22no—
45C2×Q8×C17—44no—
46C2×Q8×C17—42no—
47C22×C4×C17—21no—
48C22×C4×C17—22no—
49C22×C4×C17—21no—
50C22×C4×C17—22no—
51C22×C4×C17C2×D412yes—
52C22×C4×C17C2×D411yescharacteristic
53C2×Dic34—22no—
54C4⋊C4×C17—44no—
55C4⋊C4×C17—44no—
56C4⋊C4×C17—44no—
57C4⋊C4×C17—42no—
58C42×C17—21no—
59C42×C17—21no—
60C42×C17—42no—
61C42×C17—42no—
62C42×C17SD1612yes—
63C42×C17Q1612yes—
64C42×C17C2×Q811yescharacteristic
65C42×C17#4112yes—
66C42×C17#4111yescharacteristic
67C22⋊Dic17—41no—
68C4⋊Dic17—22no—
69C4⋊Dic17—22no—
70#63225—42no—
71C4×Dic17—41no—
72#63223—41no—
73#63223—41no—
74C2×#3622—42no—
75C2×#3622—41no—
76#10371—171nomaximal, Sylow
77C23×C17#12011yescharacteristic
78Q8×C17—48no—
79C2×C4×C17—24no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—42no—
83C2×C4×C17—42no—
84C2×C4×C17—42no—
85C2×C4×C17—42no—
86C2×C4×C17—42no—
87C2×C4×C17—42no—
88C2×C4×C17—42no—
89C2×C4×C17—42no—
90C2×C4×C17C22⋊Q812yes—
91C2×C4×C17C22⋊Q811yescharacteristic
92C2×C4×C17#12412yes—
93C2×C4×C17#12411yescharacteristic
94C2×C4×C17C2×SD1611yescharacteristic
95C2×C4×C17C2×Q1611yescharacteristic
96C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
97C2×C4×C17#13711yescharacteristic
98C2×Dic17—42no—
99Dic34—82no—
100#3622—82no—
101#3622—81no—
102#2399D1711yescharacteristic
103#1995—171no—
104#1503—172no—
105#1303—172no—
106#1274—171no—
107C22×C17—22no—
108C22×C17#39311yescharacteristic
109C22×C17#45011yescharacteristic
110C22×C17#45111yescharacteristic
111C4×C17—22no—
112C4×C17—22no—
113C4×C17—42no—
114C4×C17—44no—
115C4×C17—42no—
116C4×C17—82no—
117C4×C17#47511yescharacteristic
118C4×C17#47711yescharacteristic
119C4×C17#48711yescharacteristic
120C4×C17#48911yescharacteristic
121Dic17—82no—
122#533—171no—
123C2×C4×Q8—21no—
124#431—172no—
125#423—171no—
126#398—22no—
127#395—22no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2×D1712yes—
130#376C2×D1711yescharacteristic
131#339—341no—
132#339—341no—
133#328—342no—
134#328—342no—
135C2×C17—21no—
136C2×C17—21no—
137C2×C17#202411yescharacteristic
138C2×C17#203112yes—
139C22×Q8—21no—
140C2×M16—341no—
141C4⋊Q8—172no—
142#125—341no—
143C22⋊Q8—342no—
144C4×Q8—42no—
145C4×Q8—42no—
146C4×Q8—342no—
147#117—341no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C42—22no—
153C2×C42C22⋊D1711yescharacteristic
154C2×C42C22⋊D1711yescharacteristic
155C2×C42C22×D1711yescharacteristic
156C4⋊C8—342no—
157C4⋊C8—342no—
158C22⋊C8—342no—
159C4×C8—342no—
160#95—21no—
161#95—21no—
162#95—22no—
163#95C22⋊D1712yes—
164#95C22⋊D1712yes—
165C17#1037111yescharacteristic, complemented, Sylow
166C2×Q8—42no—
167C2×Q8—44no—
168C2×Q8—342no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4C2×C22⋊D1712yes—
174C22×C4C2×C22⋊D1711yescharacteristic
175M16—681no—
176M16—681no—
177C2×C8—682no—
178C2×C8—681no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—42no—
183C4⋊C4—342no—
184C4⋊C4—342no—
185C4⋊C4—682no—
186C22⋊C4—681no—
187C42—21no—
188C42—21no—
189C42—42no—
190C42—42no—
191C42—681no—
192C42#6322912yes—
193C42#6323112yes—
194C42Q8×D1711yescharacteristic
195C42#6325611yescharacteristic
196C42#6325612yes—
197C23#1058763911yescharacteristic
198Q8—48no—
199Q8—1362no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—682no—
212C2×C4#1058761811yescharacteristic
213C2×C4C2×#6322911yescharacteristic
214C2×C4#1058762811yescharacteristic
215C2×C4C2×#6323111yescharacteristic
216C2×C4#1058763212yes—
217C2×C4#1058763211yescharacteristic
218C2×C4#1058763312yes—
219C2×C4#1058763311yescharacteristic
220C8—1362no—
221C8—1361no—
222C22—22no—
223C22#1176272011yescharacteristic
224C22#1176296011yescharacteristic
225C22#1176296511yescharacteristic
226C4—22no—
227C4—22no—
228C4—44no—
229C4—42no—
230C4—42no—
231C4—82no—
232C4—1362no—
233C4#1176309111yescharacteristic
234C4#1176310111yescharacteristic
235C4#1176316111yescharacteristic
236C4#1176317111yescharacteristic
237C2—21no—
238C2—21no—
239C2#1452992211yescharacteristic
240C2#1452995312yes—
subgroupG/H
centerC2×C4#10587633
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#2399C2
socleC22×C17#393
prime radicalC23×C17#120
hypercenter#2399D17
hypocenterC17#10371
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
34. #10587555Q8semidirect
165. C17#10371semidirect

Maximal subgroups

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

subgroupindex
#145297652
#145272822
#145266872
#145266132
C17×#23992
#1037117