Finite Group Databasebeta

← #16947167

Subgroups — #16947167

Order 4352.

380 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 63 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#14529945C211yesmaximal, characteristic
2#14527757C212yesmaximal
3#14526814C212yesmaximal
4#14526718C211yesmaximal, characteristic
5C17×#2811C211yesmaximal, characteristic
6#11763440C2211yescharacteristic
7#11762965—22no—
8#11762818—21no—
9#11762578C2211yescharacteristic
10#11762569—21no—
11#11762545—22no—
12#11762535C2212yes—
13C17×#395—22no—
14C17×#395C2212yes—
15C17×#391—21no—
16C17×#391—22no—
17C17×#391—21no—
18C17×#391C2211yescharacteristic
19C22×Q8×C17—21no—
20C22×Q8×C17D411yescharacteristic
21C22×Q8×C17D412yes—
22C2×C4⋊C4×C17—21no—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17—22no—
25C2×C4⋊C4×C17C2311yescharacteristic
26C2×C42×C17—21no—
27C2×C42×C17D411yescharacteristic
28C2×#63231—42no—
29#10587612—21no—
30#10587610—22no—
31C2×#63223—21no—
32#10587576—41no—
33#10587574—41no—
34C17×#95—22no—
35C17×#95—22no—
36C17×#95—22no—
37C17×#95—22no—
38C17×#95D412yes—
39#10587506—42no—
40#10587506—41no—
41#10587506—41no—
42C22⋊#3622—22no—
43C2×Q8×C17—24no—
44C2×Q8×C17—21no—
45C2×Q8×C17—41no—
46C2×Q8×C17—41no—
47C2×Q8×C17—42no—
48C2×Q8×C17—82no—
49C2×Q8×C17—81no—
50C2×Q8×C17—81no—
51C2×Q8×C17Q1612yescomplemented
52C22×C4×C17—21no—
53C22×C4×C17—22no—
54C22×C4×C17—21no—
55C22×C4×C17—22no—
56C22×C4×C17C2×D411yescharacteristic
57C22×C4×C17C2×D412yes—
58C2×Dic34—22no—
59C4⋊C4×C17—41no—
60C4⋊C4×C17—82no—
61C4⋊C4×C17—81no—
62C4⋊C4×C17—81no—
63C42×C17—41no—
64C42×C17—81no—
65C22⋊Dic17—41no—
66#63231—84no—
67C4⋊Dic17—21no—
68C4⋊Dic17—22no—
69C4⋊Dic17—21no—
70#63225—42no—
71C4×Dic17—41no—
72#63223—41no—
73#63223—41no—
74C2×#3622—42no—
75C2×#3622—41no—
76#12224—171nomaximal, Sylow
77C23×C17#12011yescharacteristic
78Q8×C17—42no—
79Q8×C17—44no—
80Q8×C17—81no—
81Q8×C17—84no—
82Q8×C17—82no—
83C2×C4×C17—21no—
84C2×C4×C17—21no—
85C2×C4×C17—22no—
86C2×C4×C17—24no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—41no—
90C2×C4×C17—42no—
91C2×C4×C17—41no—
92C2×C4×C17—41no—
93C2×C4×C17—42no—
94C2×C4×C17—41no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17—42no—
98C2×C4×C17—42no—
99C2×C4×C17C2×Q1611yescharacteristic
100C2×C4×C17#13711yescharacteristic
101C2×Dic17—41no—
102C2×Dic17—41no—
103Dic34—82no—
104#3622—81no—
105#3622—82no—
106#2811D1711yescharacteristic
107#2028—171no—
108#1605—172no—
109#1354—172no—
110#1319—171no—
111C22×C17—21no—
112C22×C17—21no—
113C22×C17#45111yescharacteristic
114C22×C17#45511yescharacteristic
115C22×C17B(4,𝔽2)11yescharacteristic
116C4×C17—21no—
117C4×C17—21no—
118C4×C17—42no—
119C4×C17—44no—
120C4×C17—42no—
121C4×C17—42no—
122C4×C17—82no—
123Dic17—81no—
124Dic17—81no—
125#544—171no—
126#451—342no—
127#419—341no—
128#395—22no—
129#395C2×D1712yes—
130#391—21no—
131#391—21no—
132#391—22no—
133#391C2×D1711yescharacteristic
134#343—171no—
135#339—341no—
136#328—342no—
137#324—172no—
138C2×C17—21no—
139C2×C17—21no—
140C2×C17#160211yescharacteristic
141C2×C17#219911yescharacteristic
142C2×C17#220511yescharacteristic
143C22×Q8—21no—
144C22×Q8C22⋊D1711yescharacteristic
145C22×Q8C22⋊D1712yes—
146C2×Q16—682no—
147C2×M16—341no—
148C4⋊Q8—681no—
149#125—681no—
150C22⋊Q8—342no—
151#117—341no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C22×D1711yescharacteristic
156C2×C42—21no—
157C2×C42C22⋊D1711yescharacteristic
158#103—681no—
159#103—681no—
160#103—682no—
161C22⋊C8—342no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95C22⋊D1712yes—
167C17#1222411yescharacteristic, complemented, Sylow
168C2×Q8—24no—
169C2×Q8—21no—
170C2×Q8—41no—
171C2×Q8—42no—
172C2×Q8—41no—
173C2×Q8—82no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8—342no—
177C2×Q8#6323112yescomplemented
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4C2×C22⋊D1711yescharacteristic
183C22×C4C2×C22⋊D1712yes—
184Q16—1364no—
185M16—681no—
186M16—681no—
187C2×C8—682no—
188C2×C8—681no—
189C4⋊C4—41no—
190C4⋊C4—82no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—341no—
194C4⋊C4—341no—
195C4⋊C4—342no—
196C4⋊C4—682no—
197C22⋊C4—681no—
198C42—41no—
199C42—81no—
200C42—681no—
201C23#1058763911yescharacteristic
202Q8—42no—
203Q8—44no—
204Q8—81no—
205Q8—84no—
206Q8—82no—
207Q8—1362no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—681no—
225C2×C4—681no—
226C2×C4#1058762811yescharacteristic
227C2×C4C2×#6323111yescharacteristic
228C8—1361no—
229C8—1362no—
230C22—21no—
231C22—21no—
232C22#1176296511yescharacteristic
233C22#1176298711yescharacteristic
234C22#1176299611yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—82no—
242C4—1361no—
243C4—1361no—
244C2—21no—
245C2—21no—
246C2#1452774111yescharacteristic
247C2#1453078511yescharacteristic
248C2#1453081411yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2811C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#2811D17
hypocenterC17#12224
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2811

Complements

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

NH ≅ G/Ntype
51. C2×Q8×C17Q16semidirect
167. C17#12224semidirect
177. C2×Q8#63231semidirect

Maximal subgroups

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

subgroupindex
#145299452
#145277572
#145268142
#145267182
C17×#28112
#1222417