Finite Group Databasebeta

← #30766622

Subgroups — #30766622

Order 17152.

414 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 87 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#22558550C211yesmaximal, characteristic
3#22558230C212yesmaximal, complemented
4#22558209C212yesmaximal, complemented
5C67×#2638C211yesmaximal, characteristic
6#16863453—21no—
7C2×#14523958—21no—
8#16863256—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×C4⋊Dic67—41no—
24C2×C22⋊Dic67—22no—
25C4⋊(C67⋊C8)—41no—
26#14523958—42no—
27C22⋊(C67⋊C8)—22no—
28#14523954—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—
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—21no—
67C2×D4×C67—21no—
68C2×D4×C67—41no—
69C2×D4×C67—41no—
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—81no—
79C22×C4×C67—81no—
80C22×C4×C67C22⋊C411yescharacteristic
81C22×C4×C67C22⋊C411yescharacteristic
82C22×C4×C67C22⋊C411yescharacteristic
83C22×C4×C67C22×C411yescharacteristic
84C22×C4×C67C2×D411yescharacteristic
85C4⋊C4×C67—41no—
86C4⋊C4×C67—41no—
87C4⋊C4×C67—42no—
88C4⋊C4×C67—41no—
89C4⋊C4×C67—41no—
90C4⋊C4×C67—81no—
91C22⋊C4×C67—41no—
92C22⋊C4×C67—41no—
93C22⋊C4×C67—42no—
94C22⋊C4×C67—41no—
95C22⋊C4×C67—41no—
96C22⋊C4×C67—81no—
97C22⋊C4×C67—81no—
98C42×C67—21no—
99C42×C67—21no—
100C23×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×C67C2×C22⋊C411yescharacteristic
110C23×C67AGL(1,ℤ/8)11yescharacteristic
111C23×C67#13711yescharacteristic
112Q8×C67—82no—
113D4×C67—42no—
114D4×C67—42no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—41no—
122C2×C4×C67—42no—
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—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#9911yescharacteristic
142C2×C4×C67#9911yescharacteristic
143C2×Dic67—82no—
144C2×Dic67—82no—
145C2×Dic67—82no—
146C67⋊C8—82no—
147C22×C67—21no—
148C22×C67—21no—
149C22×C67—21no—
150C22×C67—21no—
151C22×C67—41no—
152C22×C67—41no—
153C22×C67—41no—
154C22×C67—41no—
155C22×C67—41no—
156C22×C67—41no—
157C22×C67—41no—
158C22×C67—81no—
159C22×C67—81no—
160C22×C67—81no—
161C22×C67C2×#9911yescharacteristic
162C22×C67#41711yescharacteristic
163C22×C67#42111yescharacteristic
164C4×C67—21no—
165C4×C67—21no—
166C4×C67—41no—
167C4×C67—41no—
168C4×C67—41no—
169C4×C67—41no—
170C4×C67—81no—
171C4×C67—81no—
172C4×C67—81no—
173C4×C67—81no—
174Dic67—162no—
175#11826—671nomaximal, Sylow
176C2×C67—21no—
177C2×C67—21no—
178C2×C67—41no—
179C2×C67—41no—
180C2×C67—81no—
181C2×C67#151311yescharacteristic
182C2×C67#212111yescharacteristic
183C2×C67#212911yescharacteristic
184#2638D6711yescharacteristic
185#1799—671no—
186#1466—671no—
187#1351—672no—
188#1344—672no—
189C67#1182611yescharacteristic, complemented, Sylow
190C2×C22⋊Q8Dic6711yescharacteristic, complemented
191C2×C4×D4C2×D6711yescharacteristic
192#432—1341no—
193C2×#99—1341no—
194#398Dic6711yescharacteristic, complemented
195#397—21no—
196#397C2×D6711yescharacteristic
197#395Dic6711yescharacteristic, complemented
198#394—21no—
199#393C2×D6711yescharacteristic
200#387—21no—
201#385—21no—
202#381Dic6711yescharacteristic, complemented
203#380—1341no—
204#342—672no—
205#323—672no—
206#323—1341no—
207C22×Q8C2×Dic6711yescharacteristic
208C22×D4C22⋊D6711yescharacteristic
209C23×C4C2×Dic6711yescharacteristic
210C23×C4C22⋊D6711yescharacteristic
211C22⋊Q8—41no—
212C22⋊Q8—41no—
213C4×D4—41no—
214C4×D4—41no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—2681no—
219C2×C4⋊C4C2×Dic6711yescharacteristic
220C2×C4⋊C4C2×Dic6711yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—1342no—
226C2×C22⋊C4C22⋊D6711yescharacteristic
227C2×C22⋊C4C22×D6711yescharacteristic
228C2×C42C22⋊D6711yescharacteristic
229C4⋊C8—2681no—
230#99—2682no—
231C22⋊C8—1342no—
232#97—2681no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—2681no—
239#95C2×Dic6711yescharacteristic
240#95C2×Dic6711yescharacteristic
241C24C22⋊Dic6711yescharacteristic
242C24C2×C22⋊D6711yescharacteristic
243C2×Q8—41no—
244C2×Q8—81no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—2682no—
260C22×C4C22⋊Dic6711yescharacteristic
261C22×C4C22⋊Dic6711yescharacteristic
262C22×C4C22⋊Dic6711yescharacteristic
263C22×C4C22×Dic6711yescharacteristic
264C22×C4C2×C22⋊D6711yescharacteristic
265C2×C8—2682no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—81no—
272C4⋊C4—5362no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—42no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—81no—
279C22⋊C4—81no—
280C22⋊C4—2682no—
281C22⋊C4—2682no—
282C42—21no—
283C42—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23C2×C22⋊Dic6711yescharacteristic
294C23#1452406811yescharacteristic
295C23#1452407311yescharacteristic
296Q8—82no—
297D4—42no—
298D4—42no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
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—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—5362no—
326C2×C4—5362no—
327C2×C4—5362no—
328C2×C4#1452395811yescharacteristic
329C2×C4#1452395811yescharacteristic
330C8—5362no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22C2×#1452395811yescharacteristic
346C22#1686339711yescharacteristic
347C22#1686341311yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—10722no—
359C2—21no—
360C2—21no—
361C2—41no—
362C2—41no—
363C2—81no—
364C2#2255875211yescharacteristic
365C2#2256185511yescharacteristic
366C2#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#11826
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
189. C67#11826semidirect
190. C2×C22⋊Q8Dic67semidirect
194. #398Dic67semidirect
197. #395Dic67semidirect
202. #381Dic67semidirect

Maximal subgroups

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

subgroupindex
#225601972
#225585502
#225582302
#225582092
C67×#26382
#1182667