Finite Group Databasebeta

← #30750150

Subgroups — #30750150

Order 17152.

392 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22568875C211yesmaximal, characteristic
2#22558985C211yesmaximal, characteristic, complemented
3#22558964C211yesmaximal, characteristic, complemented
4#22558591C211yesmaximal, characteristic
5#22558112C211yesmaximal, characteristic, complemented
6#22558050C211yesmaximal, characteristic, complemented
7C67×#1806C211yesmaximal, characteristic
8#16863457—22no—
9#16863453—21no—
10#16863449—21no—
11#16863449C2211yescharacteristic
12#16863420C2211yescharacteristic
13C2×C4⋊(C67⋊C8)C2211yescharacteristic
14#16863363—21no—
15#16863352—21no—
16C2×C4×C67⋊C8C2211yescharacteristic
17#16863148—21no—
18#16863148—21no—
19#16863146—22no—
20Q8⋊(C67⋊C8)C411yescharacteristic, complemented
21Q8⋊(C67⋊C8)C411yescharacteristic, complemented
22D4⋊(C67⋊C8)C411yescharacteristic, complemented
23D4⋊(C67⋊C8)C411yescharacteristic, complemented
24C4×#41×C67C2211yescharacteristic
25#423×C67C2211yescharacteristic
26C2×#104×C67—21no—
27C4×C4⋊C4×C67C2211yescharacteristic
28#339×C67—21no—
29C2×#11761612—21no—
30C22×C67⋊C8—21no—
31C22×C67⋊C8—21no—
32C4⋊(C67⋊C8)—21no—
33C4⋊(C67⋊C8)—21no—
34C4⋊(C67⋊C8)C2×C411yescharacteristic
35C4⋊(C67⋊C8)C2×C411yescharacteristic
36C4⋊(C67⋊C8)D412yes—
37C4⋊(C67⋊C8)D411yescharacteristic
38C4⋊(C67⋊C8)D411yescharacteristic
39C22⋊(C67⋊C8)—22no—
40C22⋊(C67⋊C8)—22no—
41C22⋊(C67⋊C8)—21no—
42#14523954—22no—
43C4×C67⋊C8—22no—
44C4×C67⋊C8—21no—
45C4×C67⋊C8—21no—
46C4×C67⋊C8C2×C411yescharacteristic
47C4×C67⋊C8C2×C411yescharacteristic
48C2×#41×C67D411yescharacteristic
49C2×M16×C67—21no—
50C4×Q8×C67C2×C411yescharacteristic
51C4×Q8×C67Q811yescharacteristic
52C4×D4×C67—21no—
53C4×D4×C67—21no—
54C4×D4×C67C2×C411yescharacteristic
55C4×D4×C67Q811yescharacteristic
56#117×C67—21no—
57#117×C67D411yescharacteristic
58C2×C4⋊C4×C67—21no—
59C2×C42×C67—21no—
60C2×C42×C67—21no—
61C2×C42×C67C2311yescharacteristic
62C4⋊C8×C67—21no—
63C4⋊C8×C67—21no—
64#104×C67—42no—
65#95×C67—21no—
66#11761612—41no—
67#11761612—41no—
68C2×C67⋊C8—21no—
69C2×C67⋊C8—21no—
70C2×C67⋊C8—21no—
71C2×C67⋊C8—21no—
72C2×C67⋊C8—21no—
73C2×C67⋊C8—22no—
74C2×C67⋊C8—21no—
75C2×C67⋊C8—42no—
76C2×C67⋊C8—41no—
77C2×C67⋊C8—41no—
78#41×C67—21no—
79#41×C67—21no—
80#41×C67—21no—
81#41×C67—21no—
82C2×Q8×C67C4⋊C411yescharacteristic
83C2×D4×C67—21no—
84C2×D4×C67C4⋊C411yescharacteristic
85C22×C4×C67—21no—
86C22×C4×C67—21no—
87C22×C4×C67—21no—
88C22×C4×C67—41no—
89C22×C4×C67C2×D411yescharacteristic
90C22×C4×C67#4111yescharacteristic
91C22×C4×C67#4111yescharacteristic
92M16×C67—41no—
93M16×C67—41no—
94C2×C8×C67—41no—
95C4⋊C4×C67—21no—
96C4⋊C4×C67—21no—
97C4⋊C4×C67—22no—
98C4⋊C4×C67—41no—
99C4⋊C4×C67C4⋊C411yescharacteristic
100C4⋊C4×C67C4⋊C411yescharacteristic
101C22⋊C4×C67—21no—
102C22⋊C4×C67—21no—
103C22⋊C4×C67—21no—
104C42×C67—21no—
105C42×C67—21no—
106C42×C67—21no—
107C42×C67—41no—
108C42×C67—41no—
109C42×C67C22×C411yescharacteristic
110C42×C67C2×D411yescharacteristic
111C42×C67C2×D411yescharacteristic
112C42×C67C2×Q811yescharacteristic
113C23×C67—21no—
114C23×C67#12311yescharacteristic
115Q8×C67—21no—
116Q8×C67—21no—
117D4×C67—21no—
118D4×C67—21no—
119D4×C67—41no—
120D4×C67—41no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67—41no—
131C2×C4×C67—42no—
132C2×C4×C67—81no—
133C2×C4×C67—81no—
134C2×C4×C67—81no—
135C2×C4×C67C2×C4⋊C411yescharacteristic
136C2×C4×C67C4×D411yescharacteristic
137C2×C4×C67C4×D411yescharacteristic
138C2×C4×C67#12011yescharacteristic
139C2×C4×C67C22⋊Q811yescharacteristic
140C2×C4×C67C22⋊Q811yescharacteristic
141C2×C4×C67#13511yescharacteristic
142C2×C4×C67#13511yescharacteristic
143C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C67#13711yescharacteristic
145C8×C67—81no—
146C67⋊C8—22no—
147C67⋊C8—41no—
148C67⋊C8—41no—
149C67⋊C8—41no—
150C67⋊C8—41no—
151C67⋊C8—41no—
152C67⋊C8—42no—
153C22×C67—21no—
154C22×C67—21no—
155C22×C67—41no—
156C22×C67—41no—
157C22×C67—41no—
158C22×C67#38511yescharacteristic
159C22×C67#48211yescharacteristic
160C22×C67#48511yescharacteristic
161C4×C67—21no—
162C4×C67—21no—
163C4×C67—21no—
164C4×C67—41no—
165C4×C67—41no—
166C4×C67—41no—
167C4×C67—82no—
168C4×C67#42711yescharacteristic
169C4×C67#42811yescharacteristic
170C4×C67#44311yescharacteristic
171C4×C67#44411yescharacteristic
172C4×C67#44911yescharacteristic
173C4×C67#45211yescharacteristic
174#8905—671nomaximal, Sylow
175C2×C67—21no—
176C2×C67—21no—
177C2×C67—81no—
178C2×C67#185411yescharacteristic
179C2×C67#186111yescharacteristic
180C2×C67#186611yescharacteristic
181#2925—671no—
182#1806D6711yescharacteristic
183#1571—671no—
184#1566—671no—
185#1476—671no—
186#1303—671no—
187#1279—671no—
188C67#890511yescharacteristic, complemented, Sylow
189C4×#41C2×D6711yescharacteristic
190#433—1342no—
191#432—1341no—
192#431—671no—
193#431—1341no—
194#423—671no—
195#423C2×D6711yescharacteristic
196C2×C4⋊C8—671no—
197C2×#104—21no—
198#408—1341no—
199#405—1341no—
200C2×C4×C8—671no—
201C4×C4⋊C4C2×D6711yescharacteristic
202#339—21no—
203#335—1341no—
204#335—1341no—
205#334—1342no—
206Q8⋊C8—671no—
207Q8⋊C8—671no—
208D4⋊C8—671no—
209D4⋊C8—671no—
210C2×#41C22⋊D6711yescharacteristic
211C2×M16—21no—
212C2×M16—1341no—
213C22×C8—1341no—
214C22×C8—1341no—
215C4×Q8Dic13411yescharacteristic
216C4×Q8C4×D6711yescharacteristic
217C4×D4—21no—
218C4×D4—21no—
219C4×D4Dic13411yescharacteristic
220C4×D4C4×D6711yescharacteristic
221#117—21no—
222#117C22⋊D6711yescharacteristic
223C2×C4⋊C4—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42C22×D6711yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—671no—
230C4⋊C8—671no—
231C4⋊C8—672no—
232C4⋊C8—671no—
233C4⋊C8—671no—
234C4⋊C8—1341no—
235C4⋊C8—1341no—
236#104—42no—
237C22⋊C8—1342no—
238C22⋊C8—1342no—
239C22⋊C8—1341no—
240#97—1342no—
241C4×C8—671no—
242C4×C8—671no—
243C4×C8—1342no—
244C4×C8—1341no—
245C4×C8—1341no—
246#95—21no—
247#41—21no—
248#41—21no—
249#41—21no—
250#41—21no—
251C2×Q8Dic67⋊C411yescharacteristic
252C2×D4—21no—
253C2×D4Dic67⋊C411yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4C2×C22⋊D6711yescharacteristic
259C22×C4#1176162711yescharacteristic
260C22×C4#1176162711yescharacteristic
261M16—41no—
262M16—41no—
263M16—2681no—
264M16—2681no—
265C2×C8—41no—
266C2×C8—2681no—
267C2×C8—2681no—
268C2×C8—2682no—
269C2×C8—1341no—
270C2×C8—1341no—
271C2×C8—1341no—
272C2×C8—1341no—
273C2×C8—1342no—
274C2×C8—1341no—
275C2×C8—1341no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4Dic67⋊C411yescharacteristic
281C4⋊C4Dic67⋊C411yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—41no—
289C42—41no—
290C42C2×C4×D6711yescharacteristic
291C42D4×D6711yescharacteristic
292C42D4×D6711yescharacteristic
293C42C2×Dic13411yescharacteristic
294C23—21no—
295C23#1452402411yescharacteristic
296Q8—21no—
297Q8—21no—
298D4—21no—
299D4—21no—
300D4—41no—
301D4—41no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4C2×Dic67⋊C411yescharacteristic
317C2×C4(C22⋊D67)⋊C411yescharacteristic
318C2×C4(C22⋊D67)⋊C411yescharacteristic
319C2×C4#1452401111yescharacteristic
320C2×C4#1452402011yescharacteristic
321C2×C4#1452402011yescharacteristic
322C2×C4#1452406311yescharacteristic
323C2×C4#1452406311yescharacteristic
324C2×C4#1452406911yescharacteristic
325C2×C4#1452407411yescharacteristic
326C8—81no—
327C8—2681no—
328C8—2681no—
329C8—2681no—
330C8—2681no—
331C8—2681no—
332C8—2682no—
333C8—1342no—
334C22—21no—
335C22—21no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22#1686326811yescharacteristic
340C22#1686371711yescharacteristic
341C22#1686373411yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—82no—
349C4#1686343211yescharacteristic
350C4#1686343611yescharacteristic
351C4#1686351011yescharacteristic
352C4#1686351511yescharacteristic
353C4#1686353911yescharacteristic
354C4#1686355611yescharacteristic
355C2—21no—
356C2—21no—
357C2—81no—
358C2#2256040611yescharacteristic
359C2#2256044711yescharacteristic
360C2#2256048211yescharacteristic
subgroupG/H
centerC2×C4#14524011
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1806C2
socleC22×C67#385
prime radicalC2×#41×C67D4
cosocleC2×C42×C67C23
hypercenter#1806D67
hypocenterC67#8905
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #22558985C2semidirect
3. #22558964C2semidirect
5. #22558112C2semidirect
6. #22558050C2semidirect
20. Q8⋊(C67⋊C8)C4semidirect
21. Q8⋊(C67⋊C8)C4semidirect
22. D4⋊(C67⋊C8)C4semidirect
23. D4⋊(C67⋊C8)C4semidirect
188. C67#8905semidirect

Maximal subgroups

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

subgroupindex
#225688752
#225589852
#225589642
#225585912
#225581122
#225580502
C67×#18062
#890567