Finite Group Databasebeta

← #30766963

Subgroups — #30766963

Order 17152.

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

normal subgroups only

This group has 452 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#22558230C214yesmaximal, complemented
4C67×#2563C211yesmaximal, 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×C4×D4×C67C2211yescharacteristic
13#393×C67C412yescomplemented
14#393×C67C2212yes—
15#390×C67—21no—
16#382×C67—21no—
17#380×C67—21no—
18#380×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×C67C2×C412yes—
28C22×D4×C67D411yescharacteristic
29C23×C4×C67D411yescharacteristic
30C23×C4×C67D411yescharacteristic
31C4⋊Q8×C67—41no—
32C4⋊Q8×C67—41no—
33C4×D4×C67—41no—
34C4×D4×C67—41no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67D411yescharacteristic
39C2×C22⋊C4×C67—22no—
40C2×C22⋊C4×C67—22no—
41C2×C22⋊C4×C67C2×C411yescharacteristic
42C2×C22⋊C4×C67C2311yescharacteristic
43C2×C42×C67—21no—
44C2×C42×C67C2×C411yescharacteristic
45#95×C67—22no—
46#95×C67—22no—
47#95×C67C2×C412yes—
48C22×Dic67—42no—
49C2×C67⋊C8—42no—
50C2×C67⋊C8—82no—
51C4⋊Dic67—82no—
52C22⋊Dic67—42no—
53C22⋊Dic67—42no—
54C24×C67C22⋊C411yescharacteristic
55C24×C67C2×D411yescharacteristic
56C2×Q8×C67—42no—
57C2×Q8×C67—82no—
58C2×D4×C67—41no—
59C2×D4×C67—41no—
60C2×D4×C67SD1612yes—
61C2×D4×C67Q1612yes—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—22no—
67C22×C4×C67—22no—
68C22×C4×C67—42no—
69C22×C4×C67—42no—
70C22×C4×C67C22⋊C411yescharacteristic
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22×C411yescharacteristic
74C22×C4×C67C2×D411yescharacteristic
75C4⋊C4×C67—41no—
76C4⋊C4×C67—42no—
77C4⋊C4×C67—42no—
78C4⋊C4×C67—41no—
79C4⋊C4×C67—41no—
80C22⋊C4×C67—41no—
81C22⋊C4×C67—41no—
82C22⋊C4×C67—82no—
83C22⋊C4×C67—82no—
84C42×C67—21no—
85C42×C67—21no—
86C42×C67—42no—
87C23×C67—21no—
88C23×C67—21no—
89C23×C67—21no—
90C23×C67—21no—
91C23×C67—41no—
92C23×C67—41no—
93C23×C67—41no—
94C23×C67—41no—
95C23×C67#10312yes—
96C23×C67C2×C22⋊C411yescharacteristic
97C23×C67C2×SD1611yescharacteristic
98C23×C67C2×Q1611yescharacteristic
99Q8×C67—84no—
100D4×C67—42no—
101D4×C67—42no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—22no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—42no—
109C2×C4×C67—42no—
110C2×C4×C67—41no—
111C2×C4×C67—41no—
112C2×C4×C67—41no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67—42no—
116C2×C4×C67—42no—
117C2×C4×C67—42no—
118C2×C4×C67—42no—
119C2×C4×C67—81no—
120C2×C4×C67—81no—
121C2×C4×C67#9911yescharacteristic
122C2×C4×C67#9911yescharacteristic
123C2×C4×C67#10312yes—
124C2×Dic67—82no—
125C2×Dic67—82no—
126C2×Dic67—82no—
127C67⋊C8—82no—
128C22×C67—21no—
129C22×C67—21no—
130C22×C67—21no—
131C22×C67—22no—
132C22×C67—21no—
133C22×C67—42no—
134C22×C67—41no—
135C22×C67—41no—
136C22×C67—41no—
137C22×C67—41no—
138C22×C67—41no—
139C22×C67—41no—
140C22×C67—81no—
141C22×C67—81no—
142C22×C67C2×#9911yescharacteristic
143C22×C67C2×#10311yescharacteristic
144C22×C67#42111yescharacteristic
145C4×C67—21no—
146C4×C67—22no—
147C4×C67—41no—
148C4×C67—41no—
149C4×C67—41no—
150C4×C67—82no—
151C4×C67—82no—
152C4×C67#35412yes—
153Dic67—162no—
154#11900—671nomaximal, Sylow
155C2×C67—21no—
156C2×C67—21no—
157C2×C67—41no—
158C2×C67—42no—
159C2×C67—41no—
160C2×C67#152011yescharacteristic
161C2×C67#212211yescharacteristic
162C2×C67C2×#35411yescharacteristic
163#2563D6711yescharacteristic
164#1799—671no—
165#1459—671no—
166#1351—674no—
167C67#1190011yescharacteristic, complemented, Sylow
168C2×C4⋊Q8Dic6711yescharacteristic, complemented
169C2×C4×D4C2×D6711yescharacteristic
170C2×#99—1341no—
171C2×C22⋊C8—1341no—
172#393Dic6712yescomplemented
173#393C2×D6712yes—
174#390—21no—
175#382—21no—
176#380—21no—
177#380—21no—
178#380—1341no—
179C4×C4⋊C4Dic6711yescharacteristic, complemented
180#342—672no—
181#324—1341no—
182#323—672no—
183C22×Q8C2×Dic6712yes—
184C22×D4C22⋊D6711yescharacteristic
185C23×C4C22⋊D6711yescharacteristic
186C23×C4C22⋊D6711yescharacteristic
187C22×C8—2681no—
188C4⋊Q8—41no—
189C4⋊Q8—41no—
190C4×D4—41no—
191C4×D4—41no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—2681no—
196C2×C4⋊C4C22⋊D6711yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—1342no—
200C2×C22⋊C4C2×Dic6711yescharacteristic
201C2×C22⋊C4C22×D6711yescharacteristic
202C2×C42—21no—
203C2×C42C2×Dic6711yescharacteristic
204#99—2682no—
205C22⋊C8—2681no—
206C22⋊C8—1342no—
207#95—22no—
208#95—22no—
209#95—2681no—
210#95C2×Dic6712yes—
211C24C22⋊Dic6711yescharacteristic
212C24C2×C22⋊D6711yescharacteristic
213C2×Q8—42no—
214C2×Q8—82no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4#1176161512yes—
218C2×D4#1176161812yes—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—2682no—
228C22×C4C22⋊Dic6711yescharacteristic
229C22×C4C22⋊Dic6711yescharacteristic
230C22×C4C22⋊Dic6711yescharacteristic
231C22×C4C22×Dic6711yescharacteristic
232C22×C4C2×C22⋊D6711yescharacteristic
233C2×C8—2682no—
234C2×C8—5362no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—5362no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—82no—
244C22⋊C4—82no—
245C22⋊C4—2682no—
246C22⋊C4—2682no—
247C42—21no—
248C42—21no—
249C42—42no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23#1452396712yes—
259C23C2×C22⋊Dic6711yescharacteristic
260C23C2×#1176161511yescharacteristic
261C23C2×#1176161811yescharacteristic
262Q8—84no—
263D4—42no—
264D4—42no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—5362no—
285C2×C4—5362no—
286C2×C4—5362no—
287C2×C4#1452395811yescharacteristic
288C2×C4#1452395811yescharacteristic
289C2×C4#1452396712yes—
290C8—5362no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—21no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22—81no—
305C22C2×#1452395811yescharacteristic
306C22C2×#1452396711yescharacteristic
307C22#1686341311yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—82no—
314C4—82no—
315C4—10722no—
316C4#1686319112yes—
317C2—21no—
318C2—21no—
319C2—41no—
320C2—42no—
321C2—41no—
322C2#2255878111yescharacteristic
323C2#2256185911yescharacteristic
324C2C2×#1686319111yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2563C2
socleC22×C67C2×#99
prime radicalC22×D4×C67D4
cosocleC2×C22⋊C4×C67C23
hypercenter#2563D67
hypocenterC67#11900
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2563

Complements

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

NH ≅ G/Ntype
3. #22558230C2semidirect
11. C2×C4⋊Q8×C67C4semidirect
13. #393×C67C4semidirect
19. C4×C4⋊C4×C67C4semidirect
167. C67#11900semidirect
168. C2×C4⋊Q8Dic67semidirect
172. #393Dic67semidirect
179. 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
#225582302
C67×#25632
#1190067