Finite Group Databasebeta

← #31010477

Subgroups — #31010477

Order 17152.

546 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570723C211yesmaximal, characteristic, complemented
2#22562373C212yesmaximal, complemented
3#22562366C212yesmaximal, complemented
4#22562361C211yesmaximal, characteristic, complemented
5#22562356C211yesmaximal, characteristic, complemented
6#22562351C211yesmaximal, characteristic, complemented
7#22562346C211yesmaximal, characteristic, complemented
8#22561991C212yesmaximal, complemented
9C2×C4⋊(C67⋊C16)C211yesmaximal, characteristic, complemented
10#22561916C212yesmaximal
11C4×#135×C67C211yesmaximal, characteristic
12#16863953—21no—
13C22×C67⋊C16C2212yes—
14#16863801C2211yescharacteristic
15#16863801C2211yescharacteristic
16#16863798—21no—
17#16863798C2212yes—
18#16863794—21no—
19#16863434—21no—
20#16863434C2212yes—
21#16863223C2212yes—
22#16863223C2212yes—
23#16863221C2212yes—
24#16863221C2212yes—
25C4⋊(C67⋊C16)C2211yescharacteristic
26C4⋊(C67⋊C16)C2211yescharacteristic
27C4⋊(C67⋊C16)C2212yes—
28#16863200C2212yes—
29#16863200C2212yes—
30#16863197C2212yes—
31#16863197C2212yes—
32C2×#135×C67C2211yescharacteristic
33C4×#41×C67—21no—
34C4×Q16×C67C2211yescharacteristic
35C4×Q16×C67C2211yescharacteristic
36C4×SD16×C67—21no—
37C4×SD16×C67—21no—
38C4×D8×C67C2211yescharacteristic
39C4×D8×C67C2211yescharacteristic
40#427×C67C2211yescharacteristic
41#416×C67—21no—
42C2×C4×C8×C67C2211yescharacteristic
43#14524043—21no—
44#14524043—21no—
45#14524035—22no—
46#14524022—41no—
47#14524022—41no—
48C4×Dic134—41no—
49C4×Dic134—41no—
50#14524002—22no—
51C2×C67⋊C16—24no—
52C2×C67⋊C16C2312yes—
53C2×C67⋊C16C2312yes—
54#14523977—21no—
55#14523977—21no—
56#14523977C2312yes—
57#14523977C2312yes—
58#14523975—22no—
59#14523975—22no—
60C2×#41×C67—21no—
61#135×C67—22no—
62#135×C67Q814yescomplemented
63C2×Q16×C67C2311yescharacteristic
64C2×SD16×C67—21no—
65C2×D8×C67C2311yescharacteristic
66C22×C8×C67D411yescharacteristic
67C22×C8×C67C2311yescharacteristic
68C4×Q8×C67—21no—
69C4×Q8×C67—21no—
70C4×D4×C67—21no—
71C4×D4×C67—21no—
72C4×D4×C67—41no—
73C4×D4×C67—41no—
74#117×C67—21no—
75#117×C67—41no—
76C2×C42×C67—41no—
77C2×C42×C67D411yescharacteristic
78#107×C67C2311yescharacteristic
79#107×C67C2311yescharacteristic
80#106×C67—21no—
81#103×C67—21no—
82#103×C67—21no—
83#102×C67—21no—
84#102×C67—21no—
85C4×C8×C67D411yescharacteristic
86C4×C8×C67D411yescharacteristic
87C4×C8×C67C2311yescharacteristic
88C4×C8×C67C2311yescharacteristic
89C2×Dic134—41no—
90C2×Dic134—41no—
91C4⋊Dic67—22no—
92C4⋊Dic67—22no—
93C4⋊Dic67—41no—
94C4⋊Dic67—41no—
95Dic67⋊C4—42no—
96Dic67⋊C4—42no—
97C22⋊Dic67—42no—
98C4×Dic67—42no—
99C67⋊C16—22no—
100C67⋊C16—22no—
101#41×C67—24no—
102#41×C67—42no—
103C2×Q8×C67—21no—
104C2×D4×C67—21no—
105C2×D4×C67—41no—
106C22×C4×C67—41no—
107C22×C4×C67—41no—
108C22×C4×C67—41no—
109C22×C4×C67D811yescharacteristic
110C22×C4×C67D811yescharacteristic
111C22×C4×C67C2×D411yescharacteristic
112Q16×C67—21no—
113Q16×C67C2×Q812yes—
114SD16×C67—24no—
115D8×C67—21no—
116D8×C67C2×Q812yes—
117C2×C8×C67—22no—
118C2×C8×C67C2×D411yescharacteristic
119C2×C8×C67C2×D411yescharacteristic
120C2×C8×C67C2×D411yescharacteristic
121C2×C8×C67C2×Q812yes—
122C2×C8×C67#4112yes—
123C2×C8×C67C2411yescharacteristic
124C4⋊C4×C67—21no—
125C4⋊C4×C67—21no—
126C4⋊C4×C67—41no—
127C4⋊C4×C67—41no—
128C22⋊C4×C67—41no—
129C22⋊C4×C67—41no—
130C22⋊C4×C67—41no—
131C42×C67—41no—
132C42×C67—41no—
133C42×C67—41no—
134C42×C67D811yescharacteristic
135C42×C67D811yescharacteristic
136C42×C67C2×D411yescharacteristic
137C42×C67C2×D411yescharacteristic
138C23×C67—41no—
139C23×C67C2×D811yescharacteristic
140Q8×C67—22no—
141Q8×C67—41no—
142D4×C67—22no—
143D4×C67—44no—
144D4×C67—41no—
145C2×C4×C67—22no—
146C2×C4×C67—41no—
147C2×C4×C67—41no—
148C2×C4×C67—44no—
149C2×C4×C67—41no—
150C2×C4×C67—41no—
151C2×C4×C67—82no—
152C2×C4×C67—82no—
153C2×C4×C67C22⋊Q812yes—
154C2×C4×C67C2×D811yescharacteristic
155C2×C4×C67C2×D811yescharacteristic
156C2×C4×C67C2×D811yescharacteristic
157C2×C4×C67C2×D811yescharacteristic
158C2×C4×C67C2×D811yescharacteristic
159C2×C4×C67#13712yes—
160C2×C4×C67C22×D411yescharacteristic
161C2×Dic67—42no—
162C2×Dic67—42no—
163Dic134—82no—
164Dic134—82no—
165C8×C67—21no—
166C8×C67—21no—
167C8×C67C22⋊Q812yes—
168C8×C67C22×Q811yescharacteristic
169C8×C67C2×#4111yescharacteristic
170C22×C67—21no—
171C22×C67—21no—
172C22×C67—42no—
173C22×C67—81no—
174C22×C67—81no—
175C22×C67#47412yes—
176C22×C67C22×D811yescharacteristic
177C4×C67—21no—
178C4×C67—21no—
179C4×C67—21no—
180C4×C67—42no—
181C4×C67—81no—
182C4×C67—81no—
183C4×C67#47412yes—
184C4×C67#50812yes—
185C4×C67C2×C22⋊Q811yescharacteristic
186C4×C67C2×#13711yescharacteristic
187Dic67—82no—
188Dic67—82no—
189#33626—671nomaximal, Sylow
190C2×C67—22no—
191C2×C67—82no—
192C2×C67C2×#47411yescharacteristic
193C2×C67C2×#50811yescharacteristic
194C2×C67#341411yescharacteristic
195#3160—671no—
196C4×#135D6711yescharacteristic
197#2229—672no—
198#2228—672no—
199#2227—671no—
200#2226—671no—
201#2225—671no—
202#2224—671no—
203#2158—672no—
204C2×C4⋊C16—671no—
205#2138—672no—
206C67#3362611yescharacteristic, complemented, Sylow
207C2×#135C2×D6711yescharacteristic
208#533—1341no—
209C4×#41—21no—
210C22×C16—672no—
211#500—671no—
212#500—671no—
213#499—672no—
214#499—1341no—
215#498—1341no—
216C4×Q16C2×D6711yescharacteristic
217C4×Q16C2×D6711yescharacteristic
218C4×SD16—21no—
219C4×SD16—21no—
220C4×D8C2×D6711yescharacteristic
221C4×D8C2×D6711yescharacteristic
222#427—672no—
223#427—1341no—
224#427C2×D6711yescharacteristic
225#416—21no—
226C2×C4×C8C2×D6711yescharacteristic
227#367—672no—
228#367—672no—
229#366—672no—
230#366—672no—
231C4⋊C16—671no—
232C4⋊C16—671no—
233C4⋊C16—672no—
234#358—672no—
235#358—672no—
236#357—672no—
237#357—672no—
238C2×#41—21no—
239#135—22no—
240#135Dic13414yescomplemented
241C2×Q16C22×D6711yescharacteristic
242C2×SD16—21no—
243C2×D8C22×D6711yescharacteristic
244C22×C8C22⋊D6711yescharacteristic
245C22×C8C22×D6711yescharacteristic
246C4⋊Q8—1341no—
247C4⋊Q8—1341no—
248#125—1342no—
249C22⋊Q8—2681no—
250C22⋊Q8—2681no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—2681no—
254C4×Q8—2681no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—41no—
258C4×D4—41no—
259#117—21no—
260#117—41no—
261#117—1342no—
262C2×C42—41no—
263C2×C42C22⋊D6711yescharacteristic
264C2×C16—672no—
265C2×C16—672no—
266C2×C16—1344no—
267#107—672no—
268#107—672no—
269#107—1341no—
270#107—1341no—
271#107C22×D6711yescharacteristic
272#107C22×D6711yescharacteristic
273#106—21no—
274#106—1342no—
275#106—1342no—
276#103—21no—
277#103—21no—
278#102—21no—
279#102—21no—
280C4×C8C22⋊D6711yescharacteristic
281C4×C8C22⋊D6711yescharacteristic
282C4×C8C22×D6711yescharacteristic
283C4×C8C22×D6711yescharacteristic
284#41—24no—
285#41—42no—
286C2×Q8—21no—
287C2×Q8—2681no—
288C2×Q8—2681no—
289C2×D4—21no—
290C2×D4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4#1176161311yescharacteristic
295C22×C4#1176161311yescharacteristic
296C22×C4C2×C22⋊D6711yescharacteristic
297Q16—21no—
298Q16C2×Dic13412yes—
299SD16—24no—
300D8—21no—
301D8C2×Dic13412yes—
302C2×C8—22no—
303C2×C8C2×C22⋊D6711yescharacteristic
304C2×C8C2×C22⋊D6711yescharacteristic
305C2×C8C2×C22⋊D6711yescharacteristic
306C2×C8C2×Dic13412yes—
307C2×C8#1176162812yes—
308C2×C8C23×D6711yescharacteristic
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—2682no—
314C4⋊C4—2682no—
315C4⋊C4—2681no—
316C4⋊C4—2681no—
317C4⋊C4—1342no—
318C4⋊C4—1342no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—2682no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—2682no—
327C42#1176161311yescharacteristic
328C42#1176161311yescharacteristic
329C42C2×C22⋊D6711yescharacteristic
330C42C2×C22⋊D6711yescharacteristic
331C16—1342no—
332C16—1342no—
333C23—41no—
334C23C2×#1176161311yescharacteristic
335Q8—22no—
336Q8—41no—
337Q8—5362no—
338Q8—5362no—
339D4—22no—
340D4—44no—
341D4—41no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—2682no—
351C2×C4—2682no—
352C2×C4#1452402212yes—
353C2×C4C2×#1176161311yescharacteristic
354C2×C4C2×#1176161311yescharacteristic
355C2×C4C2×#1176161311yescharacteristic
356C2×C4C2×#1176161311yescharacteristic
357C2×C4C2×#1176161311yescharacteristic
358C2×C4#1452407312yes—
359C2×C4C22×C22⋊D6711yescharacteristic
360C8—21no—
361C8—21no—
362C8#1452402212yes—
363C8C22×Dic13411yescharacteristic
364C8C2×#1176162811yescharacteristic
365C22—21no—
366C22—21no—
367C22—42no—
368C22—81no—
369C22—81no—
370C22#1686367612yes—
371C22C22×#1176161311yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4—5362no—
379C4—5362no—
380C4#1686367612yes—
381C4#1686382612yes—
382C4C2×#1452402211yescharacteristic
383C4C2×#1452407311yescharacteristic
384C2—22no—
385C2—82no—
386C2C2×#1686367611yescharacteristic
387C2C2×#1686382611yescharacteristic
388C2#2257334311yescharacteristic
subgroupG/H
centerC2×C4C2×#11761613
commutator, cosocleC2×C8×C67C24
FrattiniC2×C8C23×D67
FittingC4×#135×C67C2
socleC22×C67C22×D8
prime radicalC2×#135×C67C22
hypercenterC4×#135D67
hypocenterC67#33626
LCSG ▷ C2×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. #22570723C2semidirect
2. #22562373C2semidirect
3. #22562366C2semidirect
4. #22562361C2semidirect
5. #22562356C2semidirect
6. #22562351C2semidirect
7. #22562346C2semidirect
8. #22561991C2semidirect
9. C2×C4⋊(C67⋊C16)C2semidirect
62. #135×C67Q8semidirect
206. C67#33626semidirect
240. #135Dic134semidirect

Maximal subgroups

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

subgroupindex
#225707232
#225623732
#225623662
#225623612
#225623562
#225623512
#225623462
#225619912
C2×C4⋊(C67⋊C16)2
#225619162
C4×#135×C672
#3362667