Finite Group Databasebeta

← #30766941

Subgroups — #30766941

Order 17152.

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

normal subgroups only

This group has 442 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
1C2×#16863632C211yesmaximal, characteristic
2#22561820C211yesmaximal, characteristic, complemented
3#22558223C214yesmaximal, complemented
4C67×#1810C211yesmaximal, characteristic, complemented
5C22×D536C411yescharacteristic
6C2×#14524017—21no—
7C2×#14523983—21no—
8#16863632—21no—
9#16863632—21no—
10#16863632C412yes—
11#16863632C2212yes—
12C2×#14523977C411yescharacteristic
13C2×#14523965—21no—
14C22⋊(C67⋊C16)—21no—
15C22⋊(C67⋊C16)C2212yescomplemented
16C23×C8×C67C2211yescharacteristic
17C2×#108×C67—21no—
18#407×C67—21no—
19#341×C67C2212yescomplemented
20C22×D268—21no—
21C22×C22⋊D67—41no—
22C2×D536—21no—
23C2×D536—42no—
24C2×D536C2×C412yes—
25#14524017—22no—
26#14524017—22no—
27#14524017—41no—
28#14524017—41no—
29C2×C4⋊Dic67—21no—
30C2×(C2×D67)⋊C4—41no—
31#14523983—42no—
32C2×C67⋊C16—22no—
33#14523977—21no—
34#14523977C2×C412yes—
35#14523965—22no—
36#14523965—22no—
37C23×C4×C67D411yescharacteristic
38C2×M16×C67—22no—
39C22×C8×C67—42no—
40C22×C8×C67C2×C411yescharacteristic
41C22×C8×C67C2×C411yescharacteristic
42C22×C8×C67D411yescharacteristic
43C22×C8×C67D411yescharacteristic
44C22×C8×C67D411yescharacteristic
45C22×C8×C67C2311yescharacteristic
46#108×C67—42no—
47C22⋊C8×C67—41no—
48C22⋊C8×C67—41no—
49C23×D67—41no—
50C2×D268—22no—
51C2×D268—42no—
52C2×D268—42no—
53C2×D268—41no—
54C2×C22⋊D67—42no—
55C2×C22⋊D67—42no—
56C2×C22⋊D67—84no—
57C22×Dic67—41no—
58D536—42no—
59D536—42no—
60C4⋊Dic67—22no—
61C4⋊Dic67—41no—
62(C2×D67)⋊C4—42no—
63(C2×D67)⋊C4—42no—
64C67⋊C16—42no—
65C24×C67SD1611yescharacteristic
66C22×C4×C67—41no—
67C22×C4×C67—41no—
68C22×C4×C67C22⋊C411yescharacteristic
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67D811yescharacteristic
71C22×C4×C67D811yescharacteristic
72C22×C4×C67SD1611yescharacteristic
73C22×C4×C67C2×D411yescharacteristic
74M16×C67—42no—
75M16×C67—42no—
76C2×C8×C67—22no—
77C2×C8×C67—22no—
78C2×C8×C67—22no—
79C2×C8×C67—41no—
80C2×C8×C67—41no—
81C2×C8×C67—41no—
82C2×C8×C67—42no—
83C2×C8×C67C22⋊C411yescharacteristic
84C2×C8×C67C22⋊C411yescharacteristic
85C2×C8×C67C22×C411yescharacteristic
86C2×C8×C67C2×D411yescharacteristic
87C23×C67—22no—
88C23×C67—41no—
89C23×C67—41no—
90C23×C67#10211yescharacteristic
91C23×C67#10211yescharacteristic
92C23×C67C2×SD1611yescharacteristic
93C22×D67—42no—
94C22×D67—82no—
95C22×D67—82no—
96C22×D67—82no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—41no—
104C2×C4×C67—41no—
105C2×C4×C67—41no—
106C2×C4×C67#10211yescharacteristic
107C2×C4×C67#10211yescharacteristic
108C2×C4×C67C2×C22⋊C411yescharacteristic
109C2×C4×C67C2×D811yescharacteristic
110C22⋊D67—84no—
111C22⋊D67—84no—
112C2×Dic67—42no—
113C2×Dic67—82no—
114D268—42no—
115D268—42no—
116D268—82no—
117C8×C67—21no—
118C8×C67—21no—
119C8×C67—21no—
120C8×C67—82no—
121C8×C67#9912yes—
122C22×C67—21no—
123C22×C67—21no—
124C22×C67—21no—
125C22×C67—21no—
126C22×C67—21no—
127C22×C67—21no—
128C22×C67—21no—
129C22×C67—21no—
130C22×C67—44no—
131C22×C67#36112yes—
132C22×C67C2×#10211yescharacteristic
133C2×D67—82no—
134C2×D67—82no—
135C2×D67—82no—
136C2×D67—164no—
137C4×C67—21no—
138C4×C67—21no—
139C4×C67—21no—
140C4×C67C2×#9911yescharacteristic
141C4×C67#42111yescharacteristic
142Dic67—82no—
143#11895—671nomaximal, Sylow
144C2×C67—22no—
145C2×C67—41no—
146C2×C67—41no—
147C2×C67#151911yescharacteristic
148C2×C67#214311yescharacteristic
149C2×C67C2×#36111yescharacteristic
150D67—162no—
151C2×#466—671no—
152#2112—671no—
153#1810D6711yescharacteristic, complemented
154#1349—674no—
155C67#1189511yescharacteristic, complemented, Sylow
156C22×D8—671no—
157C23×C8C2×D6711yescharacteristic
158C2×#121—1341no—
159C2×M32—1341no—
160#466—672no—
161#466—672no—
162#466—1341no—
163#466—1341no—
164C2×#108—21no—
165C2×#107—671no—
166C2×#102—1341no—
167#407—21no—
168C22⋊C16—672no—
169C22⋊C16—1341no—
170#341C2×D6712yescomplemented
171C22×D4—2681no—
172C22×D4—1341no—
173C23×C4C22⋊D6711yescharacteristic
174C2×D8—2682no—
175C2×D8—672no—
176C2×D8—1341no—
177C2×M16—22no—
178C22×C8—42no—
179C22×C8C4×D6711yescharacteristic
180C22×C8C4×D6711yescharacteristic
181C22×C8D26811yescharacteristic
182C22×C8D26811yescharacteristic
183C22×C8C22⋊D6711yescharacteristic
184C22×C8C22×D6711yescharacteristic
185#121—2681no—
186#121—2681no—
187#121—1342no—
188#121—1342no—
189C2×C4⋊C4—1341no—
190C2×C22⋊C4—2681no—
191M32—2682no—
192C2×C16—1342no—
193#108—42no—
194#107—672no—
195#107—1341no—
196#102—1342no—
197#102—1342no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C24—2681no—
201C24#1176161511yescharacteristic
202C2×D4—2682no—
203C2×D4—2682no—
204C2×D4—2682no—
205C2×D4—2682no—
206C2×D4—2681no—
207C2×D4—5364no—
208C2×D4—1342no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—2681no—
212C22×C4(C2×D67)⋊C411yescharacteristic
213C22×C4(C2×D67)⋊C411yescharacteristic
214C22×C4#1176161311yescharacteristic
215C22×C4#1176161311yescharacteristic
216C22×C4#1176161511yescharacteristic
217C22×C4C2×C22⋊D6711yescharacteristic
218D8—2682no—
219D8—2682no—
220M16—42no—
221M16—42no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—42no—
229C2×C8(C2×D67)⋊C411yescharacteristic
230C2×C8(C2×D67)⋊C411yescharacteristic
231C2×C8C2×C4×D6711yescharacteristic
232C2×C8C2×D26811yescharacteristic
233C4⋊C4—2681no—
234C4⋊C4—1342no—
235C22⋊C4—2682no—
236C22⋊C4—2682no—
237C16—2682no—
238C23—22no—
239C23—41no—
240C23—41no—
241C23—2682no—
242C23—5362no—
243C23—5362no—
244C23—5362no—
245C23#1452396311yescharacteristic
246C23#1452396311yescharacteristic
247C23C2×#1176161511yescharacteristic
248D4—2682no—
249D4—2682no—
250D4—5362no—
251D4—5364no—
252D4—5364no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—2682no—
263C2×C4—5362no—
264C2×C4#1452396311yescharacteristic
265C2×C4#1452396311yescharacteristic
266C2×C4C2×(C2×D67)⋊C411yescharacteristic
267C2×C4C2×#1176161311yescharacteristic
268C8—21no—
269C8—21no—
270C8—21no—
271C8—82no—
272C8#1452395712yes—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—44no—
282C22—5362no—
283C22—5362no—
284C22—5362no—
285C22—10724no—
286C22#1686320812yes—
287C22C2×#1452396311yescharacteristic
288C4—21no—
289C4—21no—
290C4—21no—
291C4—5362no—
292C4C2×#1452395711yescharacteristic
293C4#1686341411yescharacteristic
294C2—22no—
295C2—41no—
296C2—41no—
297C2—10722no—
298C2#2255877611yescharacteristic
299C2#2256193611yescharacteristic
300C2C2×#1686320811yescharacteristic
subgroupG/H
centerC22C2×#14523963
commutatorC2×C8×C67C22×C4
FrattiniC22×C8C22×D67
FittingC67×#1810C2
socleC22×C67C2×#102
prime radicalC2×#16863632C2
cosocleC22×C8×C67C23
hypercenter#1810D67
hypocenterC67#11895
LCSG ▷ C2×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #1810

Complements

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

NH ≅ G/Ntype
2. #22561820C2semidirect
3. #22558223C2semidirect
4. C67×#1810C2semidirect
15. C22⋊(C67⋊C16)C22semidirect
19. #341×C67C22semidirect
153. #1810D67semidirect
155. C67#11895semidirect
170. #341C2×D67semidirect

Maximal subgroups

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

subgroupindex
C2×#168636322
#225618202
#225582232
C67×#18102
#1189567