Finite Group Databasebeta

← #30766967

Subgroups — #30766967

Order 17152.

508 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 107 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#22558233C214yesmaximal, complemented
4C67×#2542C211yesmaximal, characteristic
5C2×#14523958—21no—
6C2×C22⋊(C67⋊C8)—21no—
7#16863256—21no—
8#16863163C2212yes—
9#16863122—21no—
10#16863120C2212yes—
11C2×C4⋊Q8×C67C411yescharacteristic, complemented
12C2×C22⋊Q8×C67—21no—
13C2×C22⋊Q8×C67—21no—
14C2×C4×D4×C67C2211yescharacteristic
15#399×C67C412yescomplemented
16#399×C67C2212yes—
17#390×C67—21no—
18#384×C67—21no—
19C4×C4⋊C4×C67C411yescharacteristic, complemented
20C22×C67⋊C8—41no—
21C2×C4⋊Dic67—41no—
22C2×C22⋊Dic67—22no—
23#14523958—42no—
24C22⋊(C67⋊C8)—22no—
25C22⋊(C67⋊C8)—41no—
26C67⋊#95—41no—
27C22×Q8×C67—21no—
28C22×D4×C67D411yescharacteristic
29C23×C4×C67D411yescharacteristic
30C23×C4×C67D411yescharacteristic
31C4⋊Q8×C67—41no—
32C4⋊Q8×C67—41no—
33C22⋊Q8×C67—44no—
34C22⋊Q8×C67—44no—
35C4×D4×C67—41no—
36C4×D4×C67—41no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—22no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—22no—
41C2×C4⋊C4×C67C2×C412yes—
42C2×C4⋊C4×C67D411yescharacteristic
43C2×C22⋊C4×C67—22no—
44C2×C22⋊C4×C67—22no—
45C2×C22⋊C4×C67C2×C411yescharacteristic
46C2×C22⋊C4×C67C2311yescharacteristic
47C2×C42×C67—21no—
48C2×C42×C67C2×C411yescharacteristic
49#95×C67C2×C412yes—
50C22×Dic67—42no—
51C2×C67⋊C8—42no—
52C2×C67⋊C8—82no—
53C4⋊Dic67—82no—
54C22⋊Dic67—42no—
55C22⋊Dic67—42no—
56C24×C67C22⋊C411yescharacteristic
57C24×C67C2×D411yescharacteristic
58C2×Q8×C67—42no—
59C2×Q8×C67—42no—
60C2×Q8×C67—41no—
61C2×Q8×C67—41no—
62C2×D4×C67—41no—
63C2×D4×C67—41no—
64C2×D4×C67SD1612yes—
65C2×D4×C67Q1612yes—
66C22×C4×C67—22no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—21no—
71C22×C4×C67—22no—
72C22×C4×C67—42no—
73C22×C4×C67—42no—
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C22⋊C411yescharacteristic
77C22×C4×C67C22×C411yescharacteristic
78C22×C4×C67C2×D411yescharacteristic
79C4⋊C4×C67—24no—
80C4⋊C4×C67—41no—
81C4⋊C4×C67—42no—
82C4⋊C4×C67—44no—
83C4⋊C4×C67—44no—
84C4⋊C4×C67—42no—
85C22⋊C4×C67—41no—
86C22⋊C4×C67—41no—
87C22⋊C4×C67—44no—
88C22⋊C4×C67—44no—
89C42×C67—21no—
90C42×C67—21no—
91C42×C67—42no—
92C23×C67—21no—
93C23×C67—21no—
94C23×C67—21no—
95C23×C67—21no—
96C23×C67—41no—
97C23×C67—41no—
98C23×C67—41no—
99C23×C67—41no—
100C23×C67#10312yes—
101C23×C67C2×C22⋊C411yescharacteristic
102C23×C67C2×SD1611yescharacteristic
103C23×C67C2×Q1611yescharacteristic
104Q8×C67—82no—
105Q8×C67—82no—
106D4×C67—42no—
107D4×C67—42no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—22no—
111C2×C4×C67—24no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—42no—
116C2×C4×C67—42no—
117C2×C4×C67—42no—
118C2×C4×C67—42no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
122C2×C4×C67—42no—
123C2×C4×C67—42no—
124C2×C4×C67—42no—
125C2×C4×C67—81no—
126C2×C4×C67—81no—
127C2×C4×C67#9911yescharacteristic
128C2×C4×C67#9911yescharacteristic
129C2×C4×C67#10312yes—
130C2×Dic67—82no—
131C2×Dic67—82no—
132C2×Dic67—82no—
133C67⋊C8—82no—
134C22×C67—21no—
135C22×C67—21no—
136C22×C67—21no—
137C22×C67—22no—
138C22×C67—21no—
139C22×C67—42no—
140C22×C67—41no—
141C22×C67—41no—
142C22×C67—41no—
143C22×C67—41no—
144C22×C67—41no—
145C22×C67—41no—
146C22×C67—81no—
147C22×C67—81no—
148C22×C67C2×#9911yescharacteristic
149C22×C67C2×#10311yescharacteristic
150C22×C67#42111yescharacteristic
151C4×C67—21no—
152C4×C67—22no—
153C4×C67—44no—
154C4×C67—41no—
155C4×C67—41no—
156C4×C67—41no—
157C4×C67—82no—
158C4×C67#35312yes—
159Dic67—162no—
160#11901—671nomaximal, Sylow
161C2×C67—21no—
162C2×C67—21no—
163C2×C67—41no—
164C2×C67—42no—
165C2×C67—41no—
166C2×C67#152011yescharacteristic
167C2×C67#212311yescharacteristic
168C2×C67C2×#35311yescharacteristic
169#2542D6711yescharacteristic
170#1799—671no—
171#1459—671no—
172#1352—674no—
173C67#1190111yescharacteristic, complemented, Sylow
174C2×C4⋊Q8Dic6711yescharacteristic, complemented
175C2×C22⋊Q8—21no—
176C2×C22⋊Q8—21no—
177C2×C4×D4C2×D6711yescharacteristic
178C2×#99—1341no—
179C2×C22⋊C8—1341no—
180#399Dic6712yescomplemented
181#399C2×D6712yes—
182#390—21no—
183#384—21no—
184#380—1341no—
185C4×C4⋊C4Dic6711yescharacteristic, complemented
186#342—672no—
187#324—1341no—
188#323—672no—
189C22×Q8—21no—
190C22×D4C22⋊D6711yescharacteristic
191C23×C4C22⋊D6711yescharacteristic
192C23×C4C22⋊D6711yescharacteristic
193C22×C8—2681no—
194C4⋊Q8—41no—
195C4⋊Q8—41no—
196C22⋊Q8—44no—
197C22⋊Q8—44no—
198C4×D4—41no—
199C4×D4—41no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—2681no—
205C2×C4⋊C4C2×Dic6712yes—
206C2×C4⋊C4C22⋊D6711yescharacteristic
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—1342no—
210C2×C22⋊C4C2×Dic6711yescharacteristic
211C2×C22⋊C4C22×D6711yescharacteristic
212C2×C42—21no—
213C2×C42C2×Dic6711yescharacteristic
214#99—2682no—
215C22⋊C8—2681no—
216C22⋊C8—1342no—
217#95—2681no—
218#95C2×Dic6712yes—
219C24C22⋊Dic6711yescharacteristic
220C24C2×C22⋊D6711yescharacteristic
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4#1176161512yes—
228C2×D4#1176161812yes—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—2682no—
238C22×C4C22⋊Dic6711yescharacteristic
239C22×C4C22⋊Dic6711yescharacteristic
240C22×C4C22⋊Dic6711yescharacteristic
241C22×C4C22×Dic6711yescharacteristic
242C22×C4C2×C22⋊D6711yescharacteristic
243C2×C8—2682no—
244C2×C8—5362no—
245C4⋊C4—24no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—5362no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—2682no—
257C22⋊C4—2682no—
258C42—21no—
259C42—21no—
260C42—42no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1452396712yes—
270C23C2×C22⋊Dic6711yescharacteristic
271C23C2×#1176161511yescharacteristic
272C23C2×#1176161811yescharacteristic
273Q8—82no—
274Q8—82no—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—5362no—
297C2×C4—5362no—
298C2×C4—5362no—
299C2×C4#1452395811yescharacteristic
300C2×C4#1452395811yescharacteristic
301C2×C4#1452396712yes—
302C8—5362no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22C2×#1452395811yescharacteristic
318C22C2×#1452396711yescharacteristic
319C22#1686341311yescharacteristic
320C4—22no—
321C4—21no—
322C4—44no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—82no—
327C4—10722no—
328C4#1686318912yes—
329C2—21no—
330C2—21no—
331C2—41no—
332C2—42no—
333C2—41no—
334C2#2255878111yescharacteristic
335C2#2256186311yescharacteristic
336C2C2×#1686318911yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2542C2
socleC22×C67C2×#99
prime radicalC22×D4×C67D4
cosocleC2×C22⋊C4×C67C23
hypercenter#2542D67
hypocenterC67#11901
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
3. #22558233C2semidirect
11. C2×C4⋊Q8×C67C4semidirect
15. #399×C67C4semidirect
19. C4×C4⋊C4×C67C4semidirect
173. C67#11901semidirect
174. C2×C4⋊Q8Dic67semidirect
180. #399Dic67semidirect
185. C4×C4⋊C4Dic67semidirect

Maximal subgroups

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

subgroupindex
#225601972
#225585272
#225582332
C67×#25422
#1190167