Finite Group Databasebeta

← #16947148

Subgroups — #16947148

Order 4352.

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

normal subgroups only

This group has 392 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#14529940C211yesmaximal, characteristic
2#14527825C212yesmaximal, complemented
3#14526811C212yesmaximal, complemented
4#14526716C211yesmaximal, characteristic
5C17×#2804C211yesmaximal, characteristic
6#11763440C2211yescharacteristic
7#11763144—22no—
8#11762814—21no—
9#11762578C2211yescharacteristic
10#11762569—21no—
11#11762545—22no—
12#11762535C2212yes—
13C17×#400C2212yes—
14C17×#393—22no—
15C17×#391—21no—
16C17×#390C2211yescharacteristic
17C17×#386—21no—
18C17×#385—22no—
19C22×Q8×C17—21no—
20C22×D4×C17D411yescharacteristic
21C23×C4×C17—21no—
22C2×C4⋊C4×C17—21no—
23C2×C4⋊C4×C17D412yes—
24C2×C4⋊C4×C17C2311yescharacteristic
25C2×C22⋊C4×C17—22no—
26C2×C22⋊C4×C17—22no—
27C2×C22⋊C4×C17—21no—
28C2×C42×C17D411yescharacteristic
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×#95D412yes—
38#10587503—41no—
39#10587503—41no—
40C22⋊#3622—22no—
41#10587481—42no—
42#10587478—42no—
43C24×C17—21no—
44C2×Q8×C17—42no—
45C2×Q8×C17—81no—
46C2×Q8×C17—81no—
47C2×D4×C17—21no—
48C2×D4×C17—41no—
49C2×D4×C17—41no—
50C2×D4×C17Q1612yes—
51C22×C4×C17—21no—
52C22×C4×C17—22no—
53C22×C4×C17—22no—
54C22×C4×C17—82no—
55C22×C4×C17—81no—
56C22×C4×C17C2×D411yescharacteristic
57C22×C4×C17C2×D412yes—
58C2×Dic34—22no—
59C4⋊C4×C17—41no—
60C4⋊C4×C17—42no—
61C4⋊C4×C17—81no—
62C22⋊C4×C17—44no—
63C22⋊C4×C17—82no—
64C22⋊C4×C17—81no—
65C42×C17—41no—
66C22⋊Dic17—41no—
67C4⋊Dic17—22no—
68C4⋊Dic17—21no—
69C4⋊Dic17—21no—
70#63225—42no—
71C4×Dic17—41no—
72#63223—41no—
73#63223—41no—
74C2×#3622—42no—
75C2×#3622—41no—
76#12221—171nomaximal, Sylow
77C23×C17—22no—
78C23×C17—42no—
79C23×C17—42no—
80C23×C17—41no—
81C23×C17—41no—
82C23×C17#12011yescharacteristic
83Q8×C17—84no—
84D4×C17—42no—
85D4×C17—81no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—41no—
91C2×C4×C17—41no—
92C2×C4×C17—42no—
93C2×C4×C17—42no—
94C2×C4×C17—42no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17—42no—
98C2×C4×C17—42no—
99C2×C4×C17—84no—
100C2×C4×C17C2×Q1611yescharacteristic
101C2×C4×C17#13711yescharacteristic
102C2×Dic17—41no—
103C2×Dic17—41no—
104Dic34—82no—
105#3622—81no—
106#3622—82no—
107#2804D1711yescharacteristic
108#2027—171no—
109#1617—172no—
110#1353—172no—
111#1318—171no—
112C22×C17—21no—
113C22×C17—21no—
114C22×C17—44no—
115C22×C17—42no—
116C22×C17—82no—
117C22×C17—81no—
118C22×C17—81no—
119C22×C17#45111yescharacteristic
120C22×C17#45511yescharacteristic
121C22×C17B(4,𝔽2)11yescharacteristic
122C4×C17—21no—
123C4×C17—21no—
124C4×C17—42no—
125C4×C17—42no—
126C4×C17—82no—
127C4×C17—82no—
128Dic17—81no—
129Dic17—81no—
130#544—171no—
131#484—342no—
132#418—341no—
133#400C2×D1712yes—
134#393—22no—
135#391—21no—
136#390C2×D1711yescharacteristic
137#386—21no—
138#385—22no—
139#343—171no—
140#339—341no—
141#328—342no—
142#324—172no—
143C2×C17—21no—
144C2×C17—21no—
145C2×C17—82no—
146C2×C17#160211yescharacteristic
147C2×C17#219811yescharacteristic
148C2×C17#220211yescharacteristic
149C22×Q8—21no—
150C22×D4C22⋊D1711yescharacteristic
151C23×C4—21no—
152C2×M16—341no—
153C4⋊Q8—681no—
154#125—681no—
155C22⋊Q8—342no—
156#117—341no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4C22⋊D1712yes—
159C2×C4⋊C4C22×D1711yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C42C22⋊D1711yescharacteristic
164#107—682no—
165#103—682no—
166#102—681no—
167#102—681no—
168C22⋊C8—342no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95C22⋊D1712yes—
173C17#1222111yescharacteristic, complemented, Sylow
174C24—21no—
175C2×Q8—42no—
176C2×Q8—81no—
177C2×Q8—81no—
178C2×Q8—342no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4#6323112yes—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—82no—
187C22×C4—81no—
188C22×C4C2×C22⋊D1711yescharacteristic
189C22×C4C2×C22⋊D1712yes—
190M16—681no—
191M16—681no—
192C2×C8—682no—
193C2×C8—681no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4—341no—
198C4⋊C4—341no—
199C4⋊C4—342no—
200C4⋊C4—682no—
201C22⋊C4—44no—
202C22⋊C4—82no—
203C22⋊C4—81no—
204C22⋊C4—681no—
205C42—41no—
206C42—681no—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23#1058763911yescharacteristic
213Q8—84no—
214Q8—1362no—
215D4—42no—
216D4—81no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—84no—
231C2×C4—681no—
232C2×C4—681no—
233C2×C4#1058762811yescharacteristic
234C2×C4C2×#6323111yescharacteristic
235C8—1361no—
236C8—1362no—
237C22—21no—
238C22—21no—
239C22—44no—
240C22—42no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22#1176296511yescharacteristic
245C22#1176298711yescharacteristic
246C22#1176299611yescharacteristic
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—82no—
252C4—82no—
253C4—1361no—
254C4—1361no—
255C2—21no—
256C2—21no—
257C2—82no—
258C2#1452774111yescharacteristic
259C2#1453078011yescharacteristic
260C2#1453080111yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2804C2
socleC22×C17B(4,𝔽2)
prime radicalC22×D4×C17D4
hypercenter#2804D17
hypocenterC17#12221
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2804

Complements

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

NH ≅ G/Ntype
2. #14527825C2semidirect
3. #14526811C2semidirect
173. C17#12221semidirect

Maximal subgroups

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

subgroupindex
#145299402
#145278252
#145268112
#145267162
C17×#28042
#1222117