Finite Group Databasebeta

← #30766951

Subgroups — #30766951

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#22558527C211yesmaximal, characteristic
3#22558233C212yesmaximal, complemented
4#22558230C212yesmaximal, complemented
5C67×#2580C211yesmaximal, characteristic
6C2×#14523958—21no—
7C2×C22⋊(C67⋊C8)—21no—
8#16863256—21no—
9#16863163C2212yes—
10#16863122—21no—
11#16863120C2212yes—
12C2×C4×Q8×C67C411yescharacteristic, complemented
13C2×C4×D4×C67C2211yescharacteristic
14#399×C67C411yescharacteristic, complemented
15#399×C67C2211yescharacteristic
16#393×C67C411yescharacteristic, complemented
17#393×C67C2211yescharacteristic
18#390×C67—21no—
19#389×C67—21no—
20#385×C67—21no—
21#385×C67—21no—
22#382×C67C411yescharacteristic, complemented
23C22×C67⋊C8—41no—
24C2×C4⋊Dic67—41no—
25C2×C22⋊Dic67—22no—
26#14523958—42no—
27C22⋊(C67⋊C8)—22no—
28C22⋊(C67⋊C8)—41no—
29C67⋊#95—41no—
30C22×Q8×C67C2×C411yescharacteristic
31C22×D4×C67D411yescharacteristic
32C23×C4×C67D411yescharacteristic
33C23×C4×C67D411yescharacteristic
34C4×Q8×C67—41no—
35C4×Q8×C67—41no—
36C4×D4×C67—41no—
37C4×D4×C67—41no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—21no—
41C2×C4⋊C4×C67—21no—
42C2×C4⋊C4×C67C2×C411yescharacteristic
43C2×C4⋊C4×C67D411yescharacteristic
44C2×C22⋊C4×C67—21no—
45C2×C22⋊C4×C67—21no—
46C2×C22⋊C4×C67—21no—
47C2×C22⋊C4×C67—21no—
48C2×C22⋊C4×C67C2×C411yescharacteristic
49C2×C22⋊C4×C67C2311yescharacteristic
50C2×C42×C67—21no—
51C2×C42×C67C2×C411yescharacteristic
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—
59C2×C67⋊C8—82no—
60C4⋊Dic67—82no—
61C22⋊Dic67—42no—
62C22⋊Dic67—42no—
63C24×C67C22⋊C411yescharacteristic
64C24×C67C2×D411yescharacteristic
65C2×Q8×C67—42no—
66C2×Q8×C67—41no—
67C2×D4×C67—41no—
68C2×D4×C67—41no—
69C2×D4×C67SD1612yes—
70C2×D4×C67Q1612yes—
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—21no—
79C22×C4×C67—81no—
80C22×C4×C67—81no—
81C22×C4×C67C22⋊C411yescharacteristic
82C22×C4×C67C22⋊C411yescharacteristic
83C22×C4×C67C22⋊C411yescharacteristic
84C22×C4×C67C22×C411yescharacteristic
85C22×C4×C67C2×D411yescharacteristic
86C4⋊C4×C67—41no—
87C4⋊C4×C67—41no—
88C4⋊C4×C67—42no—
89C4⋊C4×C67—41no—
90C4⋊C4×C67—41no—
91C4⋊C4×C67—81no—
92C4⋊C4×C67—81no—
93C22⋊C4×C67—41no—
94C22⋊C4×C67—41no—
95C22⋊C4×C67—42no—
96C22⋊C4×C67—42no—
97C22⋊C4×C67—81no—
98C22⋊C4×C67—81no—
99C42×C67—21no—
100C42×C67—21no—
101C42×C67—42no—
102C23×C67—21no—
103C23×C67—21no—
104C23×C67—21no—
105C23×C67—21no—
106C23×C67—41no—
107C23×C67—41no—
108C23×C67—41no—
109C23×C67—41no—
110C23×C67#10312yes—
111C23×C67C2×C22⋊C411yescharacteristic
112C23×C67C2×SD1611yescharacteristic
113C23×C67C2×Q1611yescharacteristic
114Q8×C67—42no—
115Q8×C67—42no—
116D4×C67—42no—
117D4×C67—42no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—22no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—41no—
125C2×C4×C67—42no—
126C2×C4×C67—42no—
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—41no—
139C2×C4×C67—41no—
140C2×C4×C67—41no—
141C2×C4×C67—81no—
142C2×C4×C67—81no—
143C2×C4×C67#9911yescharacteristic
144C2×C4×C67#9911yescharacteristic
145C2×C4×C67#10312yes—
146C2×Dic67—82no—
147C2×Dic67—82no—
148C2×Dic67—82no—
149C67⋊C8—82no—
150C22×C67—21no—
151C22×C67—21no—
152C22×C67—21no—
153C22×C67—22no—
154C22×C67—21no—
155C22×C67—42no—
156C22×C67—41no—
157C22×C67—41no—
158C22×C67—41no—
159C22×C67—41no—
160C22×C67—41no—
161C22×C67—41no—
162C22×C67—81no—
163C22×C67—81no—
164C22×C67C2×#9911yescharacteristic
165C22×C67C2×#10311yescharacteristic
166C22×C67#42111yescharacteristic
167C4×C67—21no—
168C4×C67—21no—
169C4×C67—22no—
170C4×C67—42no—
171C4×C67—41no—
172C4×C67—41no—
173C4×C67—41no—
174C4×C67—81no—
175C4×C67—81no—
176C4×C67—81no—
177Dic67—162no—
178#11897—671nomaximal, Sylow
179C2×C67—21no—
180C2×C67—21no—
181C2×C67—41no—
182C2×C67—42no—
183C2×C67—41no—
184C2×C67#152011yescharacteristic
185C2×C67#212011yescharacteristic
186C2×C67#212711yescharacteristic
187#2580D6711yescharacteristic
188#1799—671no—
189#1459—671no—
190#1352—672no—
191#1351—672no—
192C67#1189711yescharacteristic, complemented, Sylow
193C2×C4×Q8Dic6711yescharacteristic, complemented
194C2×C4×D4C2×D6711yescharacteristic
195C2×#99—1341no—
196C2×C22⋊C8—1341no—
197#399Dic6711yescharacteristic, complemented
198#399C2×D6711yescharacteristic
199#393Dic6711yescharacteristic, complemented
200#393C2×D6711yescharacteristic
201#390—21no—
202#389—21no—
203#385—21no—
204#385—21no—
205#382Dic6711yescharacteristic, complemented
206#380—1341no—
207#342—672no—
208#324—1341no—
209#323—672no—
210C22×Q8C2×Dic6711yescharacteristic
211C22×D4C22⋊D6711yescharacteristic
212C23×C4C22⋊D6711yescharacteristic
213C23×C4C22⋊D6711yescharacteristic
214C22×C8—2681no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×D4—41no—
218C4×D4—41no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—2681no—
224C2×C4⋊C4C2×Dic6711yescharacteristic
225C2×C4⋊C4C22⋊D6711yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—1342no—
231C2×C22⋊C4C2×Dic6711yescharacteristic
232C2×C22⋊C4C22×D6711yescharacteristic
233C2×C42—21no—
234C2×C42C2×Dic6711yescharacteristic
235#99—2682no—
236C22⋊C8—2681no—
237C22⋊C8—1342no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—2681no—
242#95C2×Dic6711yescharacteristic
243#95C2×Dic6711yescharacteristic
244C24C22⋊Dic6711yescharacteristic
245C24C2×C22⋊D6711yescharacteristic
246C2×Q8—42no—
247C2×Q8—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4#1176161512yes—
251C2×D4#1176161812yes—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—2682no—
263C22×C4C22⋊Dic6711yescharacteristic
264C22×C4C22⋊Dic6711yescharacteristic
265C22×C4C22⋊Dic6711yescharacteristic
266C22×C4C22×Dic6711yescharacteristic
267C22×C4C2×C22⋊D6711yescharacteristic
268C2×C8—2682no—
269C2×C8—5362no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—81no—
276C4⋊C4—81no—
277C4⋊C4—5362no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—81no—
283C22⋊C4—81no—
284C22⋊C4—2682no—
285C22⋊C4—2682no—
286C42—21no—
287C42—21no—
288C42—42no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23#1452396712yes—
298C23C2×C22⋊Dic6711yescharacteristic
299C23C2×#1176161511yescharacteristic
300C23C2×#1176161811yescharacteristic
301Q8—42no—
302Q8—42no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
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—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—5362no—
331C2×C4—5362no—
332C2×C4—5362no—
333C2×C4#1452395811yescharacteristic
334C2×C4#1452395811yescharacteristic
335C2×C4#1452396712yes—
336C8—5362no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—21no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—81no—
350C22—81no—
351C22C2×#1452395811yescharacteristic
352C22C2×#1452396711yescharacteristic
353C22#1686341311yescharacteristic
354C4—22no—
355C4—21no—
356C4—21no—
357C4—42no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—10722no—
365C2—21no—
366C2—21no—
367C2—41no—
368C2—42no—
369C2—41no—
370C2#2255878111yescharacteristic
371C2#2256185011yescharacteristic
372C2#2256187811yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2580C2
socleC22×C67C2×#99
prime radicalC22×D4×C67D4
cosocleC2×C22⋊C4×C67C23
hypercenter#2580D67
hypocenterC67#11897
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2580

Complements

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

NH ≅ G/Ntype
3. #22558233C2semidirect
4. #22558230C2semidirect
12. C2×C4×Q8×C67C4semidirect
14. #399×C67C4semidirect
16. #393×C67C4semidirect
22. #382×C67C4semidirect
192. C67#11897semidirect
193. C2×C4×Q8Dic67semidirect
197. #399Dic67semidirect
199. #393Dic67semidirect
205. #382Dic67semidirect

Maximal subgroups

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

subgroupindex
#225601972
#225585272
#225582332
#225582302
C67×#25802
#1189767