Finite Group Databasebeta

← #30766972

Subgroups — #30766972

Order 17152.

452 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 103 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#22560197C211yesmaximal, characteristic
2#22558530C211yesmaximal, characteristic
3#22558233C212yesmaximal, complemented
4#22558230C212yesmaximal, complemented
5C67×#2564C211yesmaximal, characteristic
6#16863449—21no—
7C2×#14523958—21no—
8#16863256—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×C4⋊Dic67—41no—
24C2×C22⋊Dic67—22no—
25C4⋊(C67⋊C8)—41no—
26#14523958—42no—
27C22⋊(C67⋊C8)—22no—
28C4×C67⋊C8—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—
59C4⋊Dic67—82no—
60C22⋊Dic67—42no—
61C22⋊Dic67—42no—
62C24×C67C22⋊C411yescharacteristic
63C24×C67C2×D411yescharacteristic
64C2×Q8×C67—41no—
65C2×Q8×C67—81no—
66C2×D4×C67—41no—
67C2×D4×C67—41no—
68C2×D4×C67SD1612yes—
69C2×D4×C67Q1612yes—
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—21no—
78C22×C4×C67—42no—
79C22×C4×C67—42no—
80C22×C4×C67—81no—
81C22×C4×C67—81no—
82C22×C4×C67—81no—
83C22×C4×C67C22⋊C411yescharacteristic
84C22×C4×C67C22⋊C411yescharacteristic
85C22×C4×C67C22⋊C411yescharacteristic
86C22×C4×C67C22×C411yescharacteristic
87C22×C4×C67C2×D411yescharacteristic
88C4⋊C4×C67—41no—
89C4⋊C4×C67—41no—
90C4⋊C4×C67—42no—
91C4⋊C4×C67—41no—
92C4⋊C4×C67—41no—
93C4⋊C4×C67—81no—
94C22⋊C4×C67—41no—
95C22⋊C4×C67—41no—
96C22⋊C4×C67—42no—
97C22⋊C4×C67—42no—
98C22⋊C4×C67—81no—
99C42×C67—21no—
100C42×C67—21no—
101C23×C67—21no—
102C23×C67—21no—
103C23×C67—21no—
104C23×C67—21no—
105C23×C67—41no—
106C23×C67—41no—
107C23×C67—41no—
108C23×C67—41no—
109C23×C67#10312yes—
110C23×C67C2×C22⋊C411yescharacteristic
111C23×C67C2×SD1611yescharacteristic
112C23×C67C2×Q1611yescharacteristic
113Q8×C67—82no—
114D4×C67—42no—
115D4×C67—42no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—22no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—41no—
123C2×C4×C67—42no—
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—41no—
137C2×C4×C67—41no—
138C2×C4×C67—81no—
139C2×C4×C67—81no—
140C2×C4×C67—81no—
141C2×C4×C67—84no—
142C2×C4×C67#9911yescharacteristic
143C2×C4×C67#9911yescharacteristic
144C2×C4×C67#10312yes—
145C2×Dic67—82no—
146C2×Dic67—82no—
147C2×Dic67—82no—
148C67⋊C8—82no—
149C22×C67—21no—
150C22×C67—21no—
151C22×C67—21no—
152C22×C67—22no—
153C22×C67—21no—
154C22×C67—42no—
155C22×C67—41no—
156C22×C67—41no—
157C22×C67—42no—
158C22×C67—41no—
159C22×C67—41no—
160C22×C67—41no—
161C22×C67—41no—
162C22×C67—81no—
163C22×C67C2×#9911yescharacteristic
164C22×C67C2×#10311yescharacteristic
165C22×C67#42111yescharacteristic
166C4×C67—21no—
167C4×C67—21no—
168C4×C67—22no—
169C4×C67—42no—
170C4×C67—41no—
171C4×C67—41no—
172C4×C67—41no—
173C4×C67—81no—
174C4×C67—81no—
175C4×C67—81no—
176Dic67—162no—
177#11902—671nomaximal, Sylow
178C2×C67—21no—
179C2×C67—21no—
180C2×C67—41no—
181C2×C67—42no—
182C2×C67—41no—
183C2×C67#152011yescharacteristic
184C2×C67#212111yescharacteristic
185C2×C67#212611yescharacteristic
186#2564D6711yescharacteristic
187#1799—671no—
188#1460—671no—
189#1352—672no—
190#1351—672no—
191C67#1190211yescharacteristic, complemented, Sylow
192C2×C22⋊Q8Dic6711yescharacteristic, complemented
193C2×C4×D4C2×D6711yescharacteristic
194#431—1341no—
195C2×#99—1341no—
196#400Dic6711yescharacteristic, complemented
197#399C2×D6711yescharacteristic
198#395Dic6711yescharacteristic, complemented
199#393C2×D6711yescharacteristic
200#387—21no—
201#386—21no—
202#385—21no—
203#380—21no—
204#380—1341no—
205C2×#95Dic6711yescharacteristic, complemented
206#342—672no—
207#323—672no—
208#323—1341no—
209C22×Q8C2×Dic6711yescharacteristic
210C22×D4C22⋊D6711yescharacteristic
211C23×C4—21no—
212C23×C4C2×Dic6711yescharacteristic
213C23×C4C22⋊D6711yescharacteristic
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216C4×D4—41no—
217C4×D4—41no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—2681no—
222C2×C4⋊C4C2×Dic6711yescharacteristic
223C2×C4⋊C4C2×Dic6711yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—1342no—
228C2×C22⋊C4C22⋊D6711yescharacteristic
229C2×C22⋊C4C22×D6711yescharacteristic
230C2×C42C22⋊D6711yescharacteristic
231C4⋊C8—2681no—
232#99—2682no—
233C22⋊C8—1342no—
234C4×C8—2681no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—2681no—
241#95C2×Dic6711yescharacteristic
242#95C2×Dic6711yescharacteristic
243C24C22⋊Dic6711yescharacteristic
244C24C2×C22⋊D6711yescharacteristic
245C2×Q8—41no—
246C2×Q8—81no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4#1176161512yes—
250C2×D4#1176161812yes—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—2682no—
265C22×C4C22⋊Dic6711yescharacteristic
266C22×C4C22⋊Dic6711yescharacteristic
267C22×C4C22⋊Dic6711yescharacteristic
268C22×C4C22×Dic6711yescharacteristic
269C22×C4C2×C22⋊D6711yescharacteristic
270C2×C8—2682no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—81no—
277C4⋊C4—5362no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—81no—
283C22⋊C4—2682no—
284C22⋊C4—2682no—
285C42—21no—
286C42—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23#1452396712yes—
296C23C2×C22⋊Dic6711yescharacteristic
297C23C2×#1176161511yescharacteristic
298C23C2×#1176161811yescharacteristic
299Q8—82no—
300D4—42no—
301D4—42no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—84no—
328C2×C4—5362no—
329C2×C4—5362no—
330C2×C4—5362no—
331C2×C4#1452395811yescharacteristic
332C2×C4#1452395811yescharacteristic
333C2×C4#1452396712yes—
334C8—5362no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—21no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—81no—
349C22C2×#1452395811yescharacteristic
350C22C2×#1452396711yescharacteristic
351C22#1686341311yescharacteristic
352C4—22no—
353C4—21no—
354C4—21no—
355C4—42no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—10722no—
363C2—21no—
364C2—21no—
365C2—41no—
366C2—42no—
367C2—41no—
368C2#2255878111yescharacteristic
369C2#2256185511yescharacteristic
370C2#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#11902
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
191. C67#11902semidirect
192. C2×C22⋊Q8Dic67semidirect
196. #400Dic67semidirect
198. #395Dic67semidirect
205. C2×#95Dic67semidirect

Maximal subgroups

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

subgroupindex
#225601972
#225585302
#225582332
#225582302
C67×#25642
#1190267