Finite Group Databasebeta

← #31901304

Subgroups — #31901304

Order 18176.

474 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986286C211yesmaximal, characteristic
2#22589134C211yesmaximal, characteristic
3#22588337C211yesmaximal, characteristic
4C2×#17985752C211yesmaximal, characteristic, complemented
5C2×#17985748C211yesmaximal, characteristic, complemented
6#22586223C211yesmaximal, characteristic, complemented
7C71×#1990C211yesmaximal, characteristic, complemented
8C2×C4×D284C2211yescharacteristic
9#17986286—22no—
10#17986286—22no—
11C2×#14556059—21no—
12C2×C4⋊(C71⋊C8)C2211yescharacteristic
13C2×C4⋊(C71⋊C8)C2211yescharacteristic
14C2×C4⋊(C71⋊C8)C2211yescharacteristic
15C2×#14556048—21no—
16C2×#14556048—21no—
17C2×(C2×D71)⋊C8—21no—
18C2×(C2×D71)⋊C8—21no—
19#17985761—21no—
20#17985752—22no—
21#17985748—22no—
22C2×C4⋊Q8×C71C2211yescharacteristic
23C2×C4×C8×C71C2211yescharacteristic
24#384×C71C2211yescharacteristic
25#340×C71—21no—
26#340×C71—21no—
27C22×D284D411yescharacteristic
28C22×C4×D71—21no—
29C22×C71⋊C8—21no—
30C22×C71⋊C8—21no—
31C22×C71⋊C8—21no—
32C4×D284—21no—
33C4×D284—21no—
34C4×D284—21no—
35C4×D284—21no—
36C2×C4⋊Dic71D411yescharacteristic
37C2×(C2×D71)⋊C4—21no—
38#14556059—24no—
39C4⋊(C71⋊C8)—21no—
40C4⋊(C71⋊C8)—21no—
41C4⋊(C71⋊C8)—21no—
42C4⋊(C71⋊C8)—21no—
43C4⋊(C71⋊C8)—21no—
44C4⋊(C71⋊C8)D412yes—
45#14556048—24no—
46#14556048—42no—
47(C2×D71)⋊C8—42no—
48(C2×D71)⋊C8—42no—
49C22×Q8×C71—21no—
50C22×C8×C71D411yescharacteristic
51C22×C8×C71D411yescharacteristic
52C4⋊Q8×C71—21no—
53C4⋊Q8×C71—21no—
54C4⋊Q8×C71—21no—
55C4⋊Q8×C71D412yes—
56C2×C4⋊C4×C71—21no—
57C2×C4⋊C4×C71—21no—
58C2×C4⋊C4×C71—21no—
59C2×C4⋊C4×C71—21no—
60C2×C4⋊C4×C71—21no—
61C2×C42×C71C2311yescharacteristic
62C4×C8×C71—21no—
63C4×C8×C71—21no—
64C23×D71—21no—
65C2×D284—21no—
66C2×D284—21no—
67C2×D284—21no—
68C2×D284—21no—
69C2×D284—21no—
70C2×D284Q1612yes—
71C22×Dic71—21no—
72C2×C4×D71—41no—
73C2×C4×D71—41no—
74C2×C4×D71—41no—
75C2×C4×D71—41no—
76C2×C4×D71—41no—
77C2×C4×D71—41no—
78C2×C71⋊C8—22no—
79C2×C71⋊C8—41no—
80C2×C71⋊C8—41no—
81C2×C71⋊C8—41no—
82C2×C71⋊C8—41no—
83C2×C71⋊C8—41no—
84C2×C71⋊C8—41no—
85C2×C71⋊C8—41no—
86C2×C71⋊C8—41no—
87C4⋊Dic71—21no—
88C4⋊Dic71—21no—
89(C2×D71)⋊C4—41no—
90(C2×D71)⋊C4—41no—
91C2×Q8×C71—22no—
92C2×Q8×C71—41no—
93C2×Q8×C71—81no—
94C2×Q8×C71—81no—
95C22×C4×C71—41no—
96C22×C4×C71—41no—
97C22×C4×C71C2×D411yescharacteristic
98C22×C4×C71C2×D411yescharacteristic
99C22×C4×C71#4111yescharacteristic
100C2×C8×C71—21no—
101C2×C8×C71—21no—
102C2×C8×C71—21no—
103C2×C8×C71—21no—
104C2×C8×C71—21no—
105C2×C8×C71SD1612yes—
106C4⋊C4×C71—22no—
107C4⋊C4×C71—22no—
108C4⋊C4×C71—41no—
109C4⋊C4×C71—41no—
110C4⋊C4×C71—41no—
111C4⋊C4×C71—41no—
112C4⋊C4×C71—41no—
113C4⋊C4×C71—41no—
114C4⋊C4×C71—81no—
115C42×C71C2×D411yescharacteristic
116C42×C71C2×D411yescharacteristic
117C42×C71#4111yescharacteristic
118C42×C71#4111yescharacteristic
119C23×C71#12111yescharacteristic
120C22×D71—22no—
121C22×D71—41no—
122C22×D71—41no—
123C22×D71—41no—
124C22×D71—41no—
125C22×D71—41no—
126C22×D71—41no—
127Q8×C71—82no—
128Q8×C71—82no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—42no—
132C2×C4×C71—81no—
133C2×C4×C71—81no—
134C2×C4×C71—81no—
135C2×C4×C71—81no—
136C2×C4×C71—81no—
137C2×C4×C71#12011yescharacteristic
138C2×C4×C71#12111yescharacteristic
139C2×C4×C71#12111yescharacteristic
140C2×C4×C71#12311yescharacteristic
141C2×C4×C71#12311yescharacteristic
142C2×C4×C71#12411yescharacteristic
143C2×C4×C71C2×SD1611yescharacteristic
144C2×C4×C71C2×Q1611yescharacteristic
145C2×C4×C71#13511yescharacteristic
146C2×C4×C71#13511yescharacteristic
147C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C71#13711yescharacteristic
150C2×C4×C71#13711yescharacteristic
151C2×Dic71—41no—
152C2×Dic71—41no—
153C2×Dic71—41no—
154D284—22no—
155D284—22no—
156D284—41no—
157D284—41no—
158C4×D71—81no—
159C4×D71—81no—
160C4×D71—81no—
161C4×D71—81no—
162C8×C71—22no—
163C8×C71—41no—
164C71⋊C8—42no—
165C71⋊C8—81no—
166C71⋊C8—81no—
167C22×C71#39411yescharacteristic
168C22×C71#46111yescharacteristic
169C22×C71#46311yescharacteristic
170C22×C71#46711yescharacteristic
171C22×C71#46811yescharacteristic
172C22×C71#47111yescharacteristic
173C22×C71#47211yescharacteristic
174C2×D71—42no—
175C2×D71—42no—
176C2×D71—42no—
177C2×D71—81no—
178C2×D71—81no—
179C2×D71—81no—
180C2×D71—81no—
181C4×C71—21no—
182C4×C71—21no—
183C4×C71—82no—
184C4×C71—161no—
185C4×C71#44911yescharacteristic
186C4×C71#45111yescharacteristic
187C4×C71#46011yescharacteristic
188C4×C71#46411yescharacteristic
189C4×C71#46511yescharacteristic
190C4×C71#46811yescharacteristic
191C4×C71#48411yescharacteristic
192C4×C71#48511yescharacteristic
193Dic71—81no—
194#10481—711nomaximal, Sylow
195C2×C71#167511yescharacteristic
196C2×C71#168311yescharacteristic
197C2×C71#169111yescharacteristic
198C2×C71#169711yescharacteristic
199C2×C71#203611yescharacteristic
200C2×C71#205111yescharacteristic
201C2×C71#205511yescharacteristic
202D71—82no—
203C2×#476—711no—
204#1990D7111yescharacteristic, complemented
205#1954—711no—
206#1802—711no—
207C2×#335—711no—
208C2×#333—711no—
209#1296—711no—
210C71#1048111yescharacteristic, complemented, Sylow
211C2×C4⋊Q8C2×D7111yescharacteristic
212C2×C4×D4—711no—
213#476—1422no—
214#476—1422no—
215C2×#106—1421no—
216C2×C4⋊C8—711no—
217C2×C4⋊C8—711no—
218C2×C4⋊C8—711no—
219C2×#102—1421no—
220C2×#102—1421no—
221C2×C22⋊C8—1421no—
222C2×C22⋊C8—1421no—
223C2×C4×C8C2×D7111yescharacteristic
224#384C2×D7111yescharacteristic
225#340—21no—
226#340—21no—
227#340—1421no—
228#335—1422no—
229#333—1422no—
230C22×Q8—21no—
231C22×D4—711no—
232C23×C4—1421no—
233C22×C8—1421no—
234C22×C8—1421no—
235C22×C8—1421no—
236C22×C8C22⋊D7111yescharacteristic
237C22×C8C22⋊D7111yescharacteristic
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8D28412yes—
242C4×D4—1421no—
243C4×D4—1421no—
244C4×D4—1421no—
245C4×D4—1421no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—711no—
252C2×C22⋊C4—1421no—
253C2×C42C22×D7111yescharacteristic
254#106—1424no—
255C4⋊C8—712no—
256C4⋊C8—1421no—
257C4⋊C8—1421no—
258C4⋊C8—1421no—
259C4⋊C8—1421no—
260C4⋊C8—1421no—
261#102—2842no—
262#102—1424no—
263C22⋊C8—2842no—
264C22⋊C8—2842no—
265C4×C8—21no—
266C4×C8—21no—
267C24—1421no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—81no—
271C2×Q8—81no—
272C2×D4—712no—
273C2×D4—1421no—
274C2×D4—1421no—
275C2×D4—1421no—
276C2×D4—1421no—
277C2×D4—1421no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—2841no—
281C22×C4—2841no—
282C22×C4—2841no—
283C22×C4—2841no—
284C22×C4—2841no—
285C22×C4—2841no—
286C22×C4—1421no—
287C22×C4D4×D7111yescharacteristic
288C22×C4C2×C22⋊D7111yescharacteristic
289C22×C4#1176586411yescharacteristic
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—2841no—
296C2×C8—2841no—
297C2×C8—2841no—
298C2×C8—2841no—
299C2×C8—2841no—
300C2×C8—2841no—
301C2×C8—2841no—
302C2×C8—2841no—
303C2×C8—1422no—
304C2×C8#1176585212yes—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—81no—
314C4⋊C4—1421no—
315C4⋊C4—1421no—
316C22⋊C4—2841no—
317C22⋊C4—2841no—
318C42D4×D7111yescharacteristic
319C42C2×D28411yescharacteristic
320C42#1176586411yescharacteristic
321C42#1176586511yescharacteristic
322C23—2841no—
323C23—2841no—
324C23—2841no—
325C23—2841no—
326C23—2841no—
327C23—2841no—
328C23—1422no—
329C23#1455609911yescharacteristic
330Q8—82no—
331Q8—82no—
332D4—2841no—
333D4—2841no—
334D4—1422no—
335D4—1422no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—2841no—
345C2×C4—2841no—
346C2×C4—2841no—
347C2×C4—5681no—
348C2×C4—5681no—
349C2×C4—5681no—
350C2×C4—5681no—
351C2×C4#1455609811yescharacteristic
352C2×C4#1455610011yescharacteristic
353C2×C4#1455610211yescharacteristic
354C2×C4#1455611011yescharacteristic
355C2×C4#1455611211yescharacteristic
356C2×C4#1455611411yescharacteristic
357C2×C4C2×#1176585211yescharacteristic
358C2×C4Q16×D7111yescharacteristic
359C2×C4#1455614811yescharacteristic
360C2×C4#1455615111yescharacteristic
361C2×C4#1455615311yescharacteristic
362C2×C4#1455615511yescharacteristic
363C2×C4#1455615711yescharacteristic
364C2×C4#1455615911yescharacteristic
365C8—22no—
366C8—41no—
367C8—2842no—
368C8—5681no—
369C8—5681no—
370C22—2842no—
371C22—2842no—
372C22—2842no—
373C22—5681no—
374C22—5681no—
375C22—5681no—
376C22—5681no—
377C22#1798591511yescharacteristic
378C22#1798620111yescharacteristic
379C22#1798621111yescharacteristic
380C22#1798623711yescharacteristic
381C22#1798624211yescharacteristic
382C22#1798625911yescharacteristic
383C22#1798626611yescharacteristic
384C4—21no—
385C4—21no—
386C4—82no—
387C4—161no—
388C4—5681no—
389C4#1798614711yescharacteristic
390C4#1798615611yescharacteristic
391C4#1798619911yescharacteristic
392C4#1798622111yescharacteristic
393C4#1798623011yescharacteristic
394C4#1798624711yescharacteristic
395C4#1798633311yescharacteristic
396C4#1798633911yescharacteristic
397C2—5682no—
398C2#2258773611yescharacteristic
399C2#2258778811yescharacteristic
400C2#2258783611yescharacteristic
401C2#2258787011yescharacteristic
402C2#2258954211yescharacteristic
403C2#2258963311yescharacteristic
404C2#2258965511yescharacteristic
subgroupG/H
centerC23#14556099
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1990C2
socleC23×C71#121
prime radicalC22×D284D4
hypercenter#1990D71
hypocenterC71#10481
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
4. C2×#17985752C2semidirect
5. C2×#17985748C2semidirect
6. #22586223C2semidirect
7. C71×#1990C2semidirect
204. #1990D71semidirect
210. C71#10481semidirect

Maximal subgroups

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

subgroupindex
C2×#179862862
#225891342
#225883372
C2×#179857522
C2×#179857482
#225862232
C71×#19902
#1048171