Finite Group Databasebeta

← #30766457

Subgroups — #30766457

Order 17152.

462 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C4×#14523958C211yesmaximal, characteristic
2#22558550C211yesmaximal, characteristic
3#22558233C212yesmaximal, complemented
4#22558230C212yesmaximal, complemented
5C67×#2564C211yesmaximal, characteristic
6#16863453—21no—
7C2×#14523958—21no—
8C4×C22⋊Dic67—21no—
9#16863163C2212yes—
10#16863120—21no—
11#16863120C2212yes—
12C2×C22⋊Q8×C67C411yescharacteristic, complemented
13C2×C4×D4×C67C2211yescharacteristic
14#400×C67C411yescharacteristic, complemented
15#399×C67C2211yescharacteristic
16#395×C67C411yescharacteristic, complemented
17#393×C67C2211yescharacteristic
18#387×C67—21no—
19#386×C67—21no—
20#385×C67—21no—
21#380×C67—21no—
22C2×#95×C67C411yescharacteristic, complemented
23C2×C22⋊Dic67—22no—
24C2×C4×Dic67—41no—
25C4⋊(C67⋊C8)—41no—
26#14523958—24no—
27C22⋊(C67⋊C8)—22no—
28#14523954—41no—
29C67⋊#95—41no—
30C22×Q8×C67C2×C411yescharacteristic
31C22×D4×C67D411yescharacteristic
32C23×C4×C67—21no—
33C23×C4×C67C2×C411yescharacteristic
34C23×C4×C67D411yescharacteristic
35C22⋊Q8×C67—41no—
36C22⋊Q8×C67—41no—
37C4×D4×C67—41no—
38C4×D4×C67—41no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—21no—
41C2×C4⋊C4×C67—21no—
42C2×C4⋊C4×C67C2×C411yescharacteristic
43C2×C4⋊C4×C67C2×C411yescharacteristic
44C2×C22⋊C4×C67—21no—
45C2×C22⋊C4×C67—21no—
46C2×C22⋊C4×C67—21no—
47C2×C22⋊C4×C67D411yescharacteristic
48C2×C22⋊C4×C67C2311yescharacteristic
49C2×C42×C67D411yescharacteristic
50#95×C67—21no—
51#95×C67—21no—
52#95×C67—21no—
53#95×C67—21no—
54#95×C67—21no—
55#95×C67C2×C411yescharacteristic
56#95×C67C2×C411yescharacteristic
57C22×Dic67—42no—
58C2×C67⋊C8—42no—
59C22⋊Dic67—24no—
60C22⋊Dic67—42no—
61C4×Dic67—82no—
62C24×C67C22⋊C411yescharacteristic
63C24×C67C2×D411yescharacteristic
64C2×Q8×C67—41no—
65C2×Q8×C67—81no—
66C2×D4×C67—22no—
67C2×D4×C67—22no—
68C2×D4×C67—41no—
69C22×C4×C67—21no—
70C22×C4×C67—21no—
71C22×C4×C67—21no—
72C22×C4×C67—21no—
73C22×C4×C67—21no—
74C22×C4×C67—21no—
75C22×C4×C67—21no—
76C22×C4×C67—21no—
77C22×C4×C67—42no—
78C22×C4×C67—42no—
79C22×C4×C67—81no—
80C22×C4×C67—81no—
81C22×C4×C67—81no—
82C22×C4×C67C22⋊C411yescharacteristic
83C22×C4×C67C22⋊C411yescharacteristic
84C22×C4×C67C22⋊C411yescharacteristic
85C22×C4×C67C22×C411yescharacteristic
86C22×C4×C67C2×D411yescharacteristic
87C4⋊C4×C67—41no—
88C4⋊C4×C67—41no—
89C4⋊C4×C67—42no—
90C4⋊C4×C67—41no—
91C4⋊C4×C67—41no—
92C4⋊C4×C67—81no—
93C22⋊C4×C67—41no—
94C22⋊C4×C67—41no—
95C22⋊C4×C67—42no—
96C22⋊C4×C67—42no—
97C22⋊C4×C67—81no—
98C42×C67—21no—
99C42×C67—21no—
100C23×C67—21no—
101C23×C67—21no—
102C23×C67—21no—
103C23×C67—21no—
104C23×C67—41no—
105C23×C67—41no—
106C23×C67—41no—
107C23×C67—41no—
108C23×C67#10412yes—
109C23×C67C2×C22⋊C411yescharacteristic
110C23×C67#13712yes—
111Q8×C67—82no—
112D4×C67—44no—
113C2×C4×C67—22no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—41no—
121C2×C4×C67—42no—
122C2×C4×C67—41no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67—41no—
131C2×C4×C67—41no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—41no—
135C2×C4×C67—41no—
136C2×C4×C67—81no—
137C2×C4×C67—81no—
138C2×C4×C67—81no—
139C2×C4×C67—84no—
140C2×C4×C67#9911yescharacteristic
141C2×C4×C67#9911yescharacteristic
142C2×Dic67—44no—
143C2×Dic67—82no—
144C2×Dic67—82no—
145C67⋊C8—82no—
146C22×C67—21no—
147C22×C67—21no—
148C22×C67—22no—
149C22×C67—22no—
150C22×C67—42no—
151C22×C67—42no—
152C22×C67—42no—
153C22×C67—41no—
154C22×C67—41no—
155C22×C67—41no—
156C22×C67—41no—
157C22×C67—81no—
158C22×C67C2×#9911yescharacteristic
159C22×C67#41911yescharacteristic
160C22×C67C2×#10411yescharacteristic
161C4×C67—21no—
162C4×C67—21no—
163C4×C67—42no—
164C4×C67—41no—
165C4×C67—41no—
166C4×C67—41no—
167C4×C67—41no—
168C4×C67—81no—
169C4×C67—81no—
170C4×C67—81no—
171Dic67—84no—
172#11789—671nomaximal, Sylow
173C2×C67—21no—
174C2×C67—21no—
175C2×C67—42no—
176C2×C67—42no—
177C2×C67#150811yescharacteristic
178C2×C67#212111yescharacteristic
179C2×C67#212611yescharacteristic
180#2564D6711yescharacteristic
181C4×#99—671no—
182#1466—671no—
183#1352—672no—
184#1351—672no—
185C67#1178911yescharacteristic, complemented, Sylow
186C2×C22⋊Q8Dic6711yescharacteristic, complemented
187C2×C4×D4C2×D6711yescharacteristic
188#432—1341no—
189C2×#99—1341no—
190#400Dic6711yescharacteristic, complemented
191#399C2×D6711yescharacteristic
192#395Dic6711yescharacteristic, complemented
193#393C2×D6711yescharacteristic
194#387—21no—
195#386—21no—
196#385—21no—
197#380—21no—
198C4×C22⋊C4—1341no—
199C2×#95Dic6711yescharacteristic, complemented
200#342—672no—
201#323—672no—
202#323—1341no—
203C22×Q8C2×Dic6711yescharacteristic
204C22×D4C22⋊D6711yescharacteristic
205C23×C4—21no—
206C23×C4C2×Dic6711yescharacteristic
207C23×C4C22⋊D6711yescharacteristic
208C22⋊Q8—41no—
209C22⋊Q8—41no—
210C4×D4—41no—
211C4×D4—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×Dic6711yescharacteristic
216C2×C4⋊C4C2×Dic6711yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—1342no—
221C2×C22⋊C4C22⋊D6711yescharacteristic
222C2×C22⋊C4C22×D6711yescharacteristic
223C2×C42—2681no—
224C2×C42C22⋊D6711yescharacteristic
225C4⋊C8—2681no—
226#99—1344no—
227C22⋊C8—1342no—
228#97—2681no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—2681no—
235#95C2×Dic6711yescharacteristic
236#95C2×Dic6711yescharacteristic
237C24C22⋊Dic6711yescharacteristic
238C24C2×C22⋊D6711yescharacteristic
239C2×Q8—41no—
240C2×Q8—81no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—2682no—
258C22×C4C22⋊Dic6711yescharacteristic
259C22×C4C22⋊Dic6711yescharacteristic
260C22×C4C22⋊Dic6711yescharacteristic
261C22×C4C22×Dic6711yescharacteristic
262C22×C4C2×C22⋊D6711yescharacteristic
263C2×C8—2682no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—81no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—81no—
275C22⋊C4—2682no—
276C22⋊C4—1344no—
277C42—21no—
278C42—21no—
279C42—5362no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#1452397012yes—
289C23C2×C22⋊Dic6711yescharacteristic
290C23#1452407412yes—
291Q8—82no—
292D4—44no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—84no—
320C2×C4—2684no—
321C2×C4—5362no—
322C2×C4—5362no—
323C2×C4#1452395811yescharacteristic
324C2×C4#1452395811yescharacteristic
325C8—5362no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22C2×#1452395811yescharacteristic
339C22#1686340511yescharacteristic
340C22C2×#1452397011yescharacteristic
341C4—21no—
342C4—21no—
343C4—42no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—5364no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—42no—
356C2#2255872911yescharacteristic
357C2#2256185511yescharacteristic
358C2#2256187411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2564C2
socleC22×C67C2×#99
prime radicalC22×D4×C67D4
cosocleC2×C22⋊C4×C67C23
hypercenter#2564D67
hypocenterC67#11789
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2564

Complements

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

NH ≅ G/Ntype
3. #22558233C2semidirect
4. #22558230C2semidirect
12. C2×C22⋊Q8×C67C4semidirect
14. #400×C67C4semidirect
16. #395×C67C4semidirect
22. C2×#95×C67C4semidirect
185. C67#11789semidirect
186. C2×C22⋊Q8Dic67semidirect
190. #400Dic67semidirect
192. #395Dic67semidirect
199. C2×#95Dic67semidirect

Maximal subgroups

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

subgroupindex
C4×#145239582
#225585502
#225582332
#225582302
C67×#25642
#1178967