Finite Group Databasebeta

← #31736578

Subgroups — #31736578

Order 17152.

1674 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22574019C212yesmaximal
2#22573177C218yesmaximal, complemented
3#22573142C218yesmaximal, complemented
4#22570737C212yesmaximal
5#22570733C211yesmaximal, characteristic
6#22570047C214yesmaximal
7#22570014C212yesmaximal
8#22569149C211yesmaximal, characteristic
9#22569077C212yesmaximal
10C2×#441×C67C211yesmaximal, characteristic
11Q8×Dic134—22no—
12Q8×Dic134C2214yes—
13#16864125—22no—
14#16864125C2214yes—
15#16864122—24no—
16#16864122C2218yes—
17#16864027—24no—
18#16864027C2214yes—
19#16863953—24no—
20#16863953C2212yes—
21#16863953C2211yescharacteristic
22#16863874C2214yes—
23#16863805C2214yes—
24#16863805C2214yes—
25#16863803C2218yes—
26#16863796C2214yes—
27#16863793C2214yes—
28#16863705C2218yes—
29#16863703C2218yes—
30#16863701C2218yes—
31#16863691C2214yes—
32#16863689C2218yes—
33#16863681C2214yes—
34#16863679C2218yes—
35#16863524C2218yes—
36#16863503C2214yes—
37#16863503C2214yes—
38Q16×Dic67C2218yes—
39#16863436C2214yes—
40#16863434C2212yes—
41#16863434C2212yes—
42#16863422C2212yes—
43#16863420C2211yescharacteristic
44#16863405C2214yes—
45#16863351C2212yes—
46C22×Q16×C67C2211yescharacteristic
47C2×C4×Q8×C67C2212yes—
48#441×C67C2218yes—
49C2×#106×C67C2211yescharacteristic
50C2×#103×C67C2212yes—
51C2×#97×C67C2211yescharacteristic
52C2×#11761612C2312yes—
53#14524043—24no—
54#14524043—24no—
55#14524043D412yes—
56#14524043C2312yes—
57#14524043C2312yes—
58#14524042—28no—
59#14524042—24no—
60#14524042—28no—
61#14524042C2318yes—
62#14524035—28no—
63#14524035D412yes—
64#14524035C2314yes—
65#14524034—28no—
66#14524034—24no—
67#14524034—28no—
68#14524034C2318yes—
69#14524022—28no—
70#14524022—22no—
71#14524022—22no—
72#14524022D414yes—
73Q8×Dic67—24no—
74Q8×Dic67—24no—
75Q8×Dic67C2318yes—
76Q8×Dic67C2318yes—
77C4×Dic134—24no—
78C4×Dic134—24no—
79C4×Dic134—28no—
80C4×Dic134—22no—
81C4×Dic134—22no—
82C4×Dic134C2314yes—
83#14524002—28no—
84#14524002C2314yes—
85#14524002C2312yes—
86#14523977C2314yes—
87#14523977C2314yes—
88#14523976C2318yes—
89#14523975C2312yes—
90#14523975C2312yes—
91#14523975C2312yes—
92#14523975C2312yes—
93#14523974C2318yes—
94C4⋊(C67⋊C8)C2314yes—
95Dic67⋊C8C2318yes—
96#14523967C2318yes—
97#14523967C2318yes—
98#14523953C2314yes—
99C8×Dic67C2314yes—
100C22×Q8×C67C2312yes—
101C2×Q16×C67Q818yescomplemented
102C2×Q16×C67C2314yes—
103C22×C8×C67C2311yescharacteristic
104C22×C8×C67C2311yescharacteristic
105C4×Q8×C67—22no—
106C4×Q8×C67—22no—
107C4×Q8×C67C2314yes—
108C4×Q8×C67C2314yes—
109C2×C4⋊C4×C67—22no—
110C2×C4⋊C4×C67C2312yes—
111C2×C42×C67—22no—
112C2×C42×C67C2311yescharacteristic
113#106×C67C2314yes—
114#103×C67C2318yes—
115#97×C67C2314yes—
116C2×Dic134—22no—
117C2×Dic134—22no—
118C2×Dic134—28no—
119C2×Dic134C2×D414yes—
120#11761612—24no—
121C2×C67⋊C8C2414yes—
122C4⋊Dic67—22no—
123C4⋊Dic67—22no—
124C4⋊Dic67—28no—
125C4⋊Dic67—28no—
126C4⋊Dic67—28no—
127C4⋊Dic67C2×D414yes—
128C4⋊Dic67C2×D414yes—
129C4⋊Dic67C2414yes—
130C4⋊Dic67C2414yes—
131Dic67⋊C4—24no—
132Dic67⋊C4—24no—
133Dic67⋊C4—28no—
134Dic67⋊C4—28no—
135Dic67⋊C4C2×D418yes—
136C22⋊Dic67—24no—
137C22⋊Dic67C2×D414yes—
138C4×Dic67—24no—
139C4×Dic67—28no—
140C4×Dic67—28no—
141C4×Dic67C2414yes—
142C2×Q8×C67—22no—
143C2×Q8×C67—24no—
144C2×Q8×C67C2×Q818yes—
145C2×Q8×C67C2414yes—
146C22×C4×C67—22no—
147C22×C4×C67—22no—
148C22×C4×C67C2×D412yes—
149C22×C4×C67C2411yescharacteristic
150Q16×C67C2×Q8116yes—
151C2×C8×C67—22no—
152C2×C8×C67C2×Q814yes—
153C2×C8×C67C2412yes—
154C2×C8×C67C2412yes—
155C4⋊C4×C67—24no—
156C4⋊C4×C67—24no—
157C4⋊C4×C67—22no—
158C4⋊C4×C67C2414yes—
159C42×C67—24no—
160C42×C67—24no—
161C42×C67C2×D411yescharacteristic
162C42×C67C2×D411yescharacteristic
163C42×C67C2412yes—
164C23×C67C22×D411yescharacteristic
165Q8×C67—28no—
166Q8×C67—24no—
167Q8×C67C22×Q818yes—
168C2×C4×C67—28no—
169C2×C4×C67—24no—
170C2×C4×C67—24no—
171C2×C4×C67—42no—
172C2×C4×C67—82no—
173C2×C4×C67C22×D412yes—
174C2×C4×C67C22×D412yes—
175C2×C4×C67C22×D411yescharacteristic
176C2×C4×C67C22×Q812yes—
177C2×C4×C672−1+412yes—
178C2×C4×C67C2511yescharacteristic
179C2×Dic67—24no—
180C2×Dic67—24no—
181C2×Dic67C22×D418yes—
182Dic134—44no—
183Dic134—44no—
184Dic134—88no—
185Dic134—82no—
186C8×C67—22no—
187C8×C67C22×Q814yes—
188C67⋊C8—24no—
189C22×C67—22no—
190C22×C67D4×Q812yes—
191C22×C67C23×D411yescharacteristic
192C4×C67—22no—
193C4×C67—22no—
194C4×C67—28no—
195C4×C67—44no—
196C4×C67D4×Q812yes—
197C4×C67C23×Q811yescharacteristic
198C4×C67C2×2−1+411yescharacteristic
199Dic67—28no—
200Dic67—44no—
201Dic67—44no—
202#61379—671nomaximal, Sylow
203C2×C67—22no—
204C2×C67C2×D4×Q811yescharacteristic
205C2×C67#358612yes—
206#3502—672no—
207#3390—678no—
208#3387—678no—
209#3163—672no—
210#3162—671no—
211#3082—674no—
212#3078—672no—
213#2961—671no—
214#2950—672no—
215C2×#441D6711yescharacteristic
216C67#6137911yescharacteristic, complemented, Sylow
217C22×Q16C2×D6711yescharacteristic
218Q82—674no—
219Q82—1342no—
220#557—678no—
221#557—674no—
222#557—1344no—
223#557—1342no—
224#544—674no—
225#544—1344no—
226#533—671no—
227#533—672no—
228#533—1344no—
229#519—674no—
230C2×C4×Q8C2×D6712yes—
231#501—678no—
232#501—674no—
233#501—674no—
234#499—674no—
235#498—674no—
236#479—678no—
237#479—678no—
238#479—678no—
239#477—674no—
240#477—678no—
241#475—674no—
242#475—678no—
243#446—678no—
244#441—674no—
245#441—674no—
246#441C2×D6718yes—
247C4×Q16—678no—
248#428—674no—
249#427—672no—
250#427—672no—
251C2×#106C2×D6711yescharacteristic
252#424—672no—
253#423—671no—
254#419—674no—
255C2×#103C2×D6712yes—
256#405—672no—
257C2×#97C2×D6711yescharacteristic
258C22×Q8C22×D6712yes—
259C2×Q16Dic13418yescomplemented
260C2×Q16C22×D6714yes—
261C2×M16—672no—
262C22×C8C22×D6711yescharacteristic
263C22×C8C22×D6711yescharacteristic
264C4⋊Q8—678no—
265C4⋊Q8—672no—
266C4⋊Q8—672no—
267C4⋊Q8—672no—
268C4⋊Q8—1344no—
269C4⋊Q8—1344no—
270C4⋊Q8—1344no—
271C4⋊Q8—1348no—
272C4⋊Q8—1348no—
273#125—678no—
274#125—672no—
275#125—674no—
276#125—1348no—
277#125—1344no—
278#125—1348no—
279#125—1348no—
280C22⋊Q8—674no—
281C22⋊Q8—1348no—
282C22⋊Q8—1342no—
283C22⋊Q8—1342no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—674no—
287C4×Q8—678no—
288C4×Q8—678no—
289C4×Q8—1348no—
290C4×Q8—1344no—
291C4×Q8—1344no—
292C4×Q8—1344no—
293C4×Q8—1344no—
294C4×Q8—1342no—
295C4×Q8—1342no—
296C4×Q8C22×D6714yes—
297C4×Q8C22×D6714yes—
298#117—672no—
299#117—674no—
300#117—1348no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4C22×D6712yes—
303C2×C42—22no—
304C2×C42C22×D6711yescharacteristic
305#107—674no—
306#107—674no—
307#107—678no—
308#106—672no—
309#106—672no—
310#106—672no—
311#106—672no—
312#106—678no—
313#106C22×D6714yes—
314C4⋊C8—678no—
315C4⋊C8—674no—
316#103—678no—
317#103—678no—
318#103C22×D6718yes—
319#97—674no—
320#97C22×D6714yes—
321C4×C8—674no—
322C2×Q8—22no—
323C2×Q8—24no—
324C2×Q8—674no—
325C2×Q8—1348no—
326C2×Q8—1342no—
327C2×Q8—1342no—
328C2×Q8C2×Dic13418yes—
329C2×Q8C23×D6714yes—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4D4×D6712yes—
333C22×C4C23×D6711yescharacteristic
334Q16C2×Dic134116yes—
335M16—1344no—
336C2×C8—22no—
337C2×C8—674no—
338C2×C8C2×Dic13414yes—
339C2×C8C23×D6712yes—
340C2×C8C23×D6712yes—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—674no—
345C4⋊C4—678no—
346C4⋊C4—674no—
347C4⋊C4—674no—
348C4⋊C4—674no—
349C4⋊C4—1348no—
350C4⋊C4—1348no—
351C4⋊C4—1348no—
352C4⋊C4—1348no—
353C4⋊C4—1348no—
354C4⋊C4—1344no—
355C4⋊C4—1342no—
356C4⋊C4—1344no—
357C4⋊C4—1342no—
358C4⋊C4C23×D6714yes—
359C22⋊C4—674no—
360C22⋊C4—1344no—
361C42—24no—
362C42—24no—
363C42—674no—
364C42—1348no—
365C42—1348no—
366C42—1344no—
367C42D4×D6711yescharacteristic
368C42D4×D6711yescharacteristic
369C42C23×D6712yes—
370C23C2×D4×D6711yescharacteristic
371Q8—28no—
372Q8—24no—
373Q8—2684no—
374Q8—2684no—
375Q8—5362no—
376Q8—5368no—
377Q8C22×Dic13418yes—
378C2×C4—28no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—678no—
384C2×C4—1344no—
385C2×C4—1344no—
386C2×C4C2×D4×D6712yes—
387C2×C4C2×D4×D6712yes—
388C2×C4C2×D4×D6711yescharacteristic
389C2×C4C22×Dic13412yes—
390C2×C4#1452409012yes—
391C2×C4C24×D6711yescharacteristic
392C8—22no—
393C8—1344no—
394C8C22×Dic13414yes—
395C22—22no—
396C22D4×Dic13412yes—
397C22C22×D4×D6711yescharacteristic
398C4—28no—
399C4—22no—
400C4—22no—
401C4—44no—
402C4—2684no—
403C4—2684no—
404C4—1348no—
405C4D4×Dic13412yes—
406C4C23×Dic13411yescharacteristic
407C4C2×#1452409011yescharacteristic
408C2—22no—
409C2C2×D4×Dic13411yescharacteristic
410C2#2257499212yes—
subgroupG/H
centerC22C22×D4×D67
commutator, cosocleC2×C4×C67C25
FrattiniC2×C4C24×D67
FittingC2×#441×C67C2
socleC22×C67C23×D4
prime radicalC23×C67C22×D4
hypercenterC2×#441D67
hypocenterC67#61379
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
2. #22573177C2semidirect
3. #22573142C2semidirect
101. C2×Q16×C67Q8semidirect
216. C67#61379semidirect
259. C2×Q16Dic134semidirect

Maximal subgroups

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

subgroupindex
#225740192
#225731772
#225731422
#225707372
#225707332
#225700472
#225700142
#225691492
#225690772
C2×#441×C672
#6137967