Finite Group Databasebeta

← #30772341

Subgroups — #30772341

Order 17152.

442 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863223C211yesmaximal, characteristic, complemented
2C2×#16863221C211yesmaximal, characteristic, complemented
3C2×C22⋊(C67⋊C16)C211yesmaximal, characteristic
4#22560238C211yesmaximal, characteristic
5#22558300C211yesmaximal, characteristic, complemented
6#22558300C211yesmaximal, characteristic, complemented
7C2×#427×C67C211yesmaximal, characteristic
8C22×C67⋊C16C2211yescharacteristic
9C22×C67⋊C16C2211yescharacteristic
10C2×#14523977C2211yescharacteristic
11C2×#14523977C2211yescharacteristic
12#16863256—21no—
13#16863223—22no—
14#16863221—22no—
15C22⋊(C67⋊C16)C414yescomplemented
16#16863159—21no—
17C23×C8×C67C2211yescharacteristic
18C2×#117×C67—21no—
19#427×C67—22no—
20#427×C67—22no—
21C2×#107×C67C2211yescharacteristic
22C2×#107×C67C2211yescharacteristic
23C2×#106×C67—21no—
24C2×C4⋊Dic67—21no—
25C2×C4⋊Dic67—21no—
26C2×C22⋊Dic67—41no—
27C2×C67⋊C16—21no—
28C2×C67⋊C16—21no—
29C2×C67⋊C16—21no—
30C2×C67⋊C16—21no—
31C2×C67⋊C16C2×C412yes—
32C2×C67⋊C16C2×C412yes—
33#14523977—21no—
34#14523977—21no—
35#14523977—21no—
36#14523977—21no—
37C67⋊#95—41no—
38C23×C4×C67D411yescharacteristic
39C22×C8×C67—21no—
40C22×C8×C67—21no—
41C22×C8×C67—21no—
42C22×C8×C67—21no—
43C22×C8×C67C2×C412yes—
44C22×C8×C67D411yescharacteristic
45C22×C8×C67Q812yes—
46C22×C8×C67C2311yescharacteristic
47#117×C67—24no—
48#117×C67—42no—
49C2×C4⋊C4×C67—21no—
50C2×C4⋊C4×C67—21no—
51C2×C22⋊C4×C67—41no—
52C2×C42×C67—41no—
53#107×C67—21no—
54#107×C67—21no—
55#107×C67D412yes—
56#107×C67D412yes—
57#106×C67—24no—
58C22×Dic67—41no—
59C22×Dic67—41no—
60C4⋊Dic67—41no—
61C4⋊Dic67—41no—
62C4⋊Dic67—41no—
63C4⋊Dic67—41no—
64C22⋊Dic67—82no—
65C67⋊C16—22no—
66C67⋊C16—22no—
67C24×C67D811yescharacteristic
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—21no—
71C22×C4×C67—21no—
72C22×C4×C67—41no—
73C22×C4×C67—41no—
74C22×C4×C67C4⋊C412yes—
75C22×C4×C67D811yescharacteristic
76C22×C4×C67Q1612yes—
77C22×C4×C67C2×D411yescharacteristic
78C2×C8×C67—21no—
79C2×C8×C67—21no—
80C2×C8×C67—22no—
81C2×C8×C67—22no—
82C2×C8×C67—22no—
83C2×C8×C67—22no—
84C2×C8×C67C4⋊C412yes—
85C2×C8×C67C22×C411yescharacteristic
86C2×C8×C67C2×D411yescharacteristic
87C2×C8×C67C2×D411yescharacteristic
88C2×C8×C67C2×Q811yescharacteristic
89C2×C8×C67#4111yescharacteristic
90C2×C8×C67#4111yescharacteristic
91C4⋊C4×C67—22no—
92C4⋊C4×C67—22no—
93C4⋊C4×C67—41no—
94C4⋊C4×C67—41no—
95C22⋊C4×C67—44no—
96C42×C67—44no—
97C23×C67—21no—
98C23×C67—21no—
99C23×C67—21no—
100C23×C67—21no—
101C23×C67—21no—
102C23×C67—21no—
103C23×C67#10712yes—
104C23×C67C2×D811yescharacteristic
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—22no—
108C2×C4×C67—22no—
109C2×C4×C67—22no—
110C2×C4×C67—22no—
111C2×C4×C67—42no—
112C2×C4×C67—42no—
113C2×C4×C67—81no—
114C2×C4×C67—81no—
115C2×C4×C67—81no—
116C2×C4×C67—81no—
117C2×C4×C67#10712yes—
118C2×C4×C67C2×C4⋊C411yescharacteristic
119C2×C4×C67#12011yescharacteristic
120C2×C4×C67#12311yescharacteristic
121C2×C4×C67C2×Q1611yescharacteristic
122C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
123C2×C4×C67#13711yescharacteristic
124C2×Dic67—81no—
125C2×Dic67—81no—
126C2×Dic67—81no—
127C2×Dic67—81no—
128C2×Dic67—81no—
129C2×Dic67—81no—
130C8×C67—22no—
131C8×C67C4×D411yescharacteristic
132C8×C67C4×D411yescharacteristic
133C8×C67C22⋊Q811yescharacteristic
134C8×C67C22⋊Q811yescharacteristic
135C22×C67—22no—
136C22×C67—22no—
137C22×C67—22no—
138C22×C67—41no—
139C22×C67—41no—
140C22×C67—41no—
141C22×C67—41no—
142C22×C67C2×#10711yescharacteristic
143C22×C67#44711yescharacteristic
144C22×C67#48011yescharacteristic
145C22×C67#50811yescharacteristic
146C22×C67#50811yescharacteristic
147C22×C67#50911yescharacteristic
148C22×C67#51011yescharacteristic
149C4×C67—22no—
150C4×C67—82no—
151C4×C67—82no—
152C4×C67#38511yescharacteristic
153C4×C67#42811yescharacteristic
154C4×C67#45111yescharacteristic
155C4×C67#48411yescharacteristic
156Dic67—161no—
157Dic67—161no—
158#12921—671nomaximal, Sylow
159C2×C67—42no—
160C2×C67#185211yescharacteristic
161C2×C67#215811yescharacteristic
162C2×C67#215911yescharacteristic
163C2×C67#218611yescharacteristic
164C2×C67#218911yescharacteristic
165C2×C67#223411yescharacteristic
166C2×C67#223711yescharacteristic
167C2×#427D6711yescharacteristic
168C2×#367—671no—
169C2×#366—671no—
170C2×C22⋊C16—671no—
171#1808—671no—
172#1380—671no—
173#1380—671no—
174C67#1292111yescharacteristic, complemented, Sylow
175C23×C8C2×D6711yescharacteristic
176C2×#117—21no—
177C22×C16—671no—
178C22×C16—671no—
179#427—22no—
180#427—22no—
181C2×#107—671no—
182C2×#107—671no—
183C2×#107C2×D6711yescharacteristic
184C2×#107C2×D6711yescharacteristic
185C2×#106—21no—
186#380—1341no—
187#367—1342no—
188#366—1342no—
189C22⋊C16—674no—
190#340—1341no—
191C23×C4C22⋊D6711yescharacteristic
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8Dic13412yes—
197C22×C8C4×D6712yes—
198C22×C8C22⋊D6711yescharacteristic
199C22×C8C22×D6711yescharacteristic
200#117—24no—
201#117—42no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—1341no—
205C2×C4⋊C4—1341no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—2681no—
208C2×C42—41no—
209C2×C16—672no—
210C2×C16—672no—
211C2×C16—1341no—
212C2×C16—1341no—
213C2×C16—1341no—
214C2×C16—1341no—
215#107—21no—
216#107—21no—
217#107—1341no—
218#107—1341no—
219#107—1341no—
220#107—1341no—
221#107C22⋊D6712yes—
222#107C22⋊D6712yes—
223#106—24no—
224#95—2681no—
225C24#1176161311yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—2681no—
233C22×C4—2681no—
234C22×C4Dic67⋊C412yes—
235C22×C4#1176161311yescharacteristic
236C22×C4#1176161812yes—
237C22×C4C2×C22⋊D6711yescharacteristic
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8Dic67⋊C412yes—
245C2×C8C2×C4×D6711yescharacteristic
246C2×C8C2×C22⋊D6711yescharacteristic
247C2×C8C2×C22⋊D6711yescharacteristic
248C2×C8C2×Dic13411yescharacteristic
249C2×C8#1176162811yescharacteristic
250C2×C8#1176162811yescharacteristic
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—2681no—
256C4⋊C4—2681no—
257C4⋊C4—2681no—
258C4⋊C4—2681no—
259C22⋊C4—44no—
260C22⋊C4—5362no—
261C42—44no—
262C16—1342no—
263C16—1342no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23#1452397612yes—
271C23C2×#1176161311yescharacteristic
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—5361no—
285C2×C4—5361no—
286C2×C4—5361no—
287C2×C4—5361no—
288C2×C4—5361no—
289C2×C4—5361no—
290C2×C4#1452397612yes—
291C2×C4C2×Dic67⋊C411yescharacteristic
292C2×C4#1452401211yescharacteristic
293C2×C4#1452402611yescharacteristic
294C2×C4C2×#1176161811yescharacteristic
295C2×C4#1452406911yescharacteristic
296C2×C4#1452407411yescharacteristic
297C8—22no—
298C8C4×C22⋊D6711yescharacteristic
299C8C4×C22⋊D6711yescharacteristic
300C8#1452402211yescharacteristic
301C8#1452402211yescharacteristic
302C22—22no—
303C22—22no—
304C22—22no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22C2×#1452397611yescharacteristic
310C22#1686353011yescharacteristic
311C22#1686370911yescharacteristic
312C22#1686382611yescharacteristic
313C22#1686382611yescharacteristic
314C22#1686383111yescharacteristic
315C22#1686383611yescharacteristic
316C4—22no—
317C4—82no—
318C4—82no—
319C4—10721no—
320C4—10721no—
321C4#1686326911yescharacteristic
322C4#1686343611yescharacteristic
323C4#1686355211yescharacteristic
324C4#1686373111yescharacteristic
325C2—42no—
326C2#2256039711yescharacteristic
327C2#2256199111yescharacteristic
328C2#2256199811yescharacteristic
329C2#2256213211yescharacteristic
330C2#2256214911yescharacteristic
331C2#2256240011yescharacteristic
332C2#2256241711yescharacteristic
subgroupG/H
centerC23C2×#11761613
commutatorC2×C8×C67C22×C4
FrattiniC22×C8C22×D67
FittingC2×#427×C67C2
socleC23×C67C2×D8
prime radicalC24×C67D8
cosocleC22×C8×C67C23
hypercenterC2×#427D67
hypocenterC67#12921
LCSG ▷ C2×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
1. C2×#16863223C2semidirect
2. C2×#16863221C2semidirect
5. #22558300C2semidirect
6. #22558300C2semidirect
15. C22⋊(C67⋊C16)C4semidirect
174. C67#12921semidirect

Maximal subgroups

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

subgroupindex
C2×#168632232
C2×#168632212
C2×C22⋊(C67⋊C16)2
#225602382
#225583002
C2×#427×C672
#1292167