Finite Group Databasebeta

← #16952651

Subgroups — #16952651

Order 4352.

472 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#11762608C214yesmaximal
2#14528780C211yesmaximal, characteristic
3C4×#10587507C211yesmaximal, characteristic
4C17×#2546C211yesmaximal, characteristic
5#11762833—21no—
6C2×#10587507C2212yes—
7C2×#10587507C2212yes—
8C4×#63223—21no—
9#11762608—28no—
10#11762578—21no—
11#11762578—21no—
12C2×C17×C4⋊Q8C411yescharacteristic
13C2×C17×C4⋊Q8C2212yes—
14C2×C4×Q8×C17C2211yescharacteristic
15C17×#398—21no—
16C17×#398—21no—
17C17×#391C412yes—
18C17×#389—21no—
19C17×#389—21no—
20C4×C4⋊C4×C17C411yescharacteristic
21C22×Q8×C17C2×C412yes—
22C22×Q8×C17C2311yescharacteristic
23C17×C4⋊Q8—24no—
24C17×C4⋊Q8—24no—
25C17×C4⋊Q8—41no—
26C17×C4⋊Q8—41no—
27C4×Q8×C17—41no—
28C4×Q8×C17—41no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—22no—
33C2×C4⋊C4×C17—22no—
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C4⋊C4×C17D411yescharacteristic
37C2×C42×C17C2×C412yes—
38C2×C42×C17C2×C411yescharacteristic
39C2×C42×C17D411yescharacteristic
40C2×C42×C17D411yescharacteristic
41C2×#63223—22no—
42C2×#63223—22no—
43C17×#95—21no—
44C17×#95—22no—
45C17×#95—22no—
46#10587507—24no—
47#10587507—22no—
48#10587507D414yes—
49C4⋊#3622—41no—
50C4⋊#3622—41no—
51#10587474—41no—
52C4×#3622—41no—
53C2×Q8×C17—24no—
54C2×Q8×C17—41no—
55C2×Q8×C17—41no—
56C2×Q8×C17—42no—
57C2×Q8×C17—42no—
58C2×Q8×C17C2×D412yes—
59C2×Q8×C17#4112yes—
60C22×C4×C17—22no—
61C22×C4×C17—22no—
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22×C411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C4⋊C4×C17—24no—
70C4⋊C4×C17—41no—
71C4⋊C4×C17—41no—
72C4⋊C4×C17—41no—
73C4⋊C4×C17—42no—
74C4⋊C4×C17—41no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—44no—
77C4⋊C4×C17—42no—
78C4⋊C4×C17—42no—
79C42×C17—21no—
80C42×C17—21no—
81C42×C17—21no—
82C42×C17—21no—
83C42×C17—21no—
84C42×C17—21no—
85C42×C17—24no—
86#63223—24no—
87#63223—42no—
88#63223—42no—
89#63223—42no—
90C2×#3622—42no—
91C2×#3622—42no—
92#13227—171nomaximal, Sylow
93C23×C17C2×C22⋊C411yescharacteristic
94Q8×C17—42no—
95Q8×C17—42no—
96Q8×C17—84no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—22no—
104C2×C4×C17—24no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17#9911yescharacteristic
117C2×C4×C17#9911yescharacteristic
118C2×C4×C17C4×D412yes—
119C2×C4×C17#12112yes—
120C2×C4×C17#12411yescharacteristic
121C2×C4×C17#12711yescharacteristic
122#3622—44no—
123#3622—82no—
124#2546D1711yescharacteristic
125C2×#356—174no—
126#1801—171no—
127C4×#101—171no—
128C22×C17—21no—
129C22×C17—21no—
130C22×C17#39011yescharacteristic
131C22×C17C2×#9911yescharacteristic
132C22×C17#41011yescharacteristic
133C4×C17—21no—
134C4×C17—22no—
135C4×C17—21no—
136C4×C17—22no—
137C4×C17—21no—
138C4×C17—44no—
139C4×C17—41no—
140C4×C17—41no—
141C4×C17—41no—
142C4×C17—82no—
143C4×C17#35612yescomplemented
144C2×C4⋊Q8Dic1711yescharacteristic
145C2×C4⋊Q8C2×D1712yes—
146C2×C4×Q8C2×D1711yescharacteristic
147#423—341no—
148C2×#101—172no—
149C2×#101—172no—
150C4×M16—341no—
151#398—21no—
152#398—21no—
153#391Dic1712yes—
154#389—21no—
155#389—21no—
156C4×C4⋊C4Dic1711yescharacteristic
157#356—348no—
158#343—341no—
159#343—341no—
160C2×C17—21no—
161C2×C17—21no—
162C2×C17#196411yescharacteristic
163C2×C17C2×#35611yescharacteristic
164C2×C17#213511yescharacteristic
165C22×Q8C2×Dic1712yes—
166C22×Q8C22×D1711yescharacteristic
167C2×M16—342no—
168C2×M16—342no—
169C4⋊Q8—24no—
170C4⋊Q8—24no—
171C4⋊Q8—41no—
172C4⋊Q8—41no—
173C4×Q8—41no—
174C4×Q8—41no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4C2×Dic1711yescharacteristic
181C2×C4⋊C4C22⋊D1711yescharacteristic
182C2×C4⋊C4C22⋊D1711yescharacteristic
183C2×C42C2×Dic1712yes—
184C2×C42C2×Dic1711yescharacteristic
185C2×C42C22⋊D1711yescharacteristic
186C2×C42C22⋊D1711yescharacteristic
187C4⋊C8—681no—
188C4⋊C8—681no—
189#101—174no—
190#101—342no—
191#101—344no—
192#97—681no—
193C4×C8—681no—
194#95—21no—
195#95—22no—
196#95—22no—
197C17#1322711yescharacteristic, complemented, Sylow
198C2×Q8—24no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8D4×D1712yes—
204C2×Q8#6325412yes—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4C22⋊Dic1711yescharacteristic
208C22×C4C22⋊Dic1711yescharacteristic
209C22×C4C22⋊Dic1711yescharacteristic
210C22×C4C22⋊Dic1711yescharacteristic
211C22×C4C22×Dic1711yescharacteristic
212C22×C4C2×C22⋊D1711yescharacteristic
213C22×C4C2×C22⋊D1711yescharacteristic
214M16—344no—
215M16—682no—
216M16—682no—
217M16—682no—
218C2×C8—682no—
219C2×C8—682no—
220C4⋊C4—24no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—44no—
229C4⋊C4—42no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—24no—
237C23C2×C22⋊Dic1711yescharacteristic
238Q8—42no—
239Q8—42no—
240Q8—84no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4#1058750511yescharacteristic
261C2×C4#1058750511yescharacteristic
262C2×C4D4×Dic1712yes—
263C2×C4#1058762211yescharacteristic
264C2×C4#1058762412yes—
265C2×C4#1058762611yescharacteristic
266C8—684no—
267C8—1362no—
268C22—21no—
269C22—21no—
270C22#1176271211yescharacteristic
271C22C2×#1058750511yescharacteristic
272C22#1176278311yescharacteristic
273C4—22no—
274C4—21no—
275C4—22no—
276C4—21no—
277C4—21no—
278C4—44no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—82no—
283C4#1176260812yescomplemented
284C2—21no—
285C2—21no—
286C2#1452962111yescharacteristic
287C2C2×#1176260811yescharacteristic
288C2#1453047911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC17×#2546C2
socleC22×C17C2×#99
prime radicalC23×C17C2×C22⋊C4
cosocleC22×Q8×C17C23
hypercenter#2546D17
hypocenterC17#13227
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
143. C4×C17#356semidirect
197. C17#13227semidirect
283. C4#11762608semidirect

Maximal subgroups

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

subgroupindex
C2×#117626082
#145287802
C4×#105875072
C17×#25462
#1322717