Finite Group Databasebeta

← #30766185

Subgroups — #30766185

Order 17152.

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

normal subgroups only

This group has 436 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#22558530C211yesmaximal, characteristic
3#22558230C212yesmaximal, complemented
4#22558209C212yesmaximal, complemented
5C67×#2638C211yesmaximal, characteristic
6#16863449—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#398×C67C411yescharacteristic, complemented
15#397×C67—21no—
16#397×C67C2211yescharacteristic
17#395×C67C411yescharacteristic, complemented
18#394×C67—21no—
19#393×C67C2211yescharacteristic
20#387×C67—21no—
21#385×C67—21no—
22#381×C67C411yescharacteristic, complemented
23C2×C22⋊Dic67—22no—
24C2×C4×Dic67—41no—
25C4⋊(C67⋊C8)—41no—
26#14523958—24no—
27C22⋊(C67⋊C8)—22no—
28C4×C67⋊C8—41no—
29C67⋊#95—41no—
30C22×Q8×C67C2×C411yescharacteristic
31C22×D4×C67D411yescharacteristic
32C23×C4×C67C2×C411yescharacteristic
33C23×C4×C67D411yescharacteristic
34C22⋊Q8×C67—41no—
35C22⋊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×C67C2×C411yescharacteristic
42C2×C4⋊C4×C67C2×C411yescharacteristic
43C2×C22⋊C4×C67—21no—
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—81no—
78C22×C4×C67—81no—
79C22×C4×C67C22⋊C411yescharacteristic
80C22×C4×C67C22⋊C411yescharacteristic
81C22×C4×C67C22⋊C411yescharacteristic
82C22×C4×C67C22×C411yescharacteristic
83C22×C4×C67C2×D411yescharacteristic
84C4⋊C4×C67—41no—
85C4⋊C4×C67—41no—
86C4⋊C4×C67—41no—
87C4⋊C4×C67—41no—
88C4⋊C4×C67—42no—
89C4⋊C4×C67—81no—
90C22⋊C4×C67—41no—
91C22⋊C4×C67—41no—
92C22⋊C4×C67—42no—
93C22⋊C4×C67—42no—
94C22⋊C4×C67—81no—
95C22⋊C4×C67—81no—
96C42×C67—21no—
97C42×C67—21no—
98C23×C67—21no—
99C23×C67—21no—
100C23×C67—21no—
101C23×C67—21no—
102C23×C67—21no—
103C23×C67—41no—
104C23×C67—41no—
105C23×C67—41no—
106C23×C67—41no—
107C23×C67C2×C22⋊C411yescharacteristic
108C23×C67#13512yes—
109Q8×C67—82no—
110D4×C67—44no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—41no—
117C2×C4×C67—42no—
118C2×C4×C67—41no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
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—81no—
134C2×C4×C67—81no—
135C2×C4×C67—81no—
136C2×C4×C67#9911yescharacteristic
137C2×C4×C67#9911yescharacteristic
138C2×C4×C67#10412yes—
139C2×Dic67—44no—
140C2×Dic67—82no—
141C2×Dic67—82no—
142C67⋊C8—82no—
143C22×C67—21no—
144C22×C67—21no—
145C22×C67—22no—
146C22×C67—42no—
147C22×C67—41no—
148C22×C67—41no—
149C22×C67—41no—
150C22×C67—41no—
151C22×C67—41no—
152C22×C67—81no—
153C22×C67—81no—
154C22×C67—81no—
155C22×C67C2×#9911yescharacteristic
156C22×C67#41611yescharacteristic
157C22×C67C2×#10411yescharacteristic
158C4×C67—21no—
159C4×C67—21no—
160C4×C67—22no—
161C4×C67—41no—
162C4×C67—41no—
163C4×C67—41no—
164C4×C67—81no—
165C4×C67—81no—
166C4×C67—81no—
167C4×C67—81no—
168Dic67—84no—
169#11731—671nomaximal, Sylow
170C2×C67—21no—
171C2×C67—21no—
172C2×C67—42no—
173C2×C67—81no—
174C2×C67#150111yescharacteristic
175C2×C67#212111yescharacteristic
176C2×C67#212911yescharacteristic
177#2638D6711yescharacteristic
178C4×#99—671no—
179#1460—671no—
180#1351—672no—
181#1344—672no—
182C67#1173111yescharacteristic, complemented, Sylow
183C2×C22⋊Q8Dic6711yescharacteristic, complemented
184C2×C4×D4C2×D6711yescharacteristic
185#431—1341no—
186C2×#99—1341no—
187#398Dic6711yescharacteristic, complemented
188#397—21no—
189#397C2×D6711yescharacteristic
190#395Dic6711yescharacteristic, complemented
191#394—21no—
192#393C2×D6711yescharacteristic
193#387—21no—
194#385—21no—
195#381Dic6711yescharacteristic, complemented
196C4×C22⋊C4—1341no—
197#342—672no—
198#323—672no—
199#323—1341no—
200C22×Q8C2×Dic6711yescharacteristic
201C22×D4C22⋊D6711yescharacteristic
202C23×C4C2×Dic6711yescharacteristic
203C23×C4C22⋊D6711yescharacteristic
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C4×D4—41no—
207C4×D4—41no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C2×Dic6711yescharacteristic
212C2×C4⋊C4C2×Dic6711yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—1342no—
218C2×C22⋊C4C22⋊D6711yescharacteristic
219C2×C22⋊C4C22×D6711yescharacteristic
220C2×C42—2681no—
221C2×C42C22⋊D6711yescharacteristic
222C4⋊C8—2681no—
223#99—1344no—
224C22⋊C8—1342no—
225C4×C8—2681no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—2681no—
232#95C2×Dic6711yescharacteristic
233#95C2×Dic6711yescharacteristic
234C24C22⋊Dic6711yescharacteristic
235C24C2×C22⋊D6711yescharacteristic
236C2×Q8—41no—
237C2×Q8—81no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—2682no—
252C22×C4C22⋊Dic6711yescharacteristic
253C22×C4C22⋊Dic6711yescharacteristic
254C22×C4C22⋊Dic6711yescharacteristic
255C22×C4C22×Dic6711yescharacteristic
256C22×C4C2×C22⋊D6711yescharacteristic
257C2×C8—2682no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—81no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C22⋊C4—2682no—
271C22⋊C4—1344no—
272C42—21no—
273C42—21no—
274C42—5362no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23C2×C22⋊Dic6711yescharacteristic
285C23#1452406312yes—
286Q8—82no—
287D4—44no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
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—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—2684no—
314C2×C4—5362no—
315C2×C4—5362no—
316C2×C4#1452395811yescharacteristic
317C2×C4#1452395811yescharacteristic
318C2×C4#1452397012yes—
319C8—5362no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22C2×#1452395811yescharacteristic
333C22#1686339211yescharacteristic
334C22C2×#1452397011yescharacteristic
335C4—21no—
336C4—21no—
337C4—22no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—5364no—
346C2—21no—
347C2—21no—
348C2—42no—
349C2—81no—
350C2#2255870011yescharacteristic
351C2#2256185511yescharacteristic
352C2#2256188411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2638C2
socleC22×C67C2×#99
prime radicalC22×D4×C67D4
cosocleC2×C22⋊C4×C67C23
hypercenter#2638D67
hypocenterC67#11731
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2638

Complements

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

NH ≅ G/Ntype
3. #22558230C2semidirect
4. #22558209C2semidirect
12. C2×C22⋊Q8×C67C4semidirect
14. #398×C67C4semidirect
17. #395×C67C4semidirect
22. #381×C67C4semidirect
182. C67#11731semidirect
183. C2×C22⋊Q8Dic67semidirect
187. #398Dic67semidirect
190. #395Dic67semidirect
195. #381Dic67semidirect

Maximal subgroups

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

subgroupindex
C4×#145239582
#225585302
#225582302
#225582092
C67×#26382
#1173167