Finite Group Databasebeta

← #31914512

Subgroups — #31914512

Order 18176.

590 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556069C211yesmaximal, characteristic
2C2×C4×D568C211yesmaximal, characteristic, complemented
3C2×#17985800C211yesmaximal, characteristic, complemented
4C2×#17985799C211yesmaximal, characteristic, complemented
5#22586431C211yesmaximal, characteristic, complemented
6#22586418C211yesmaximal, characteristic, complemented
7C71×#1970C211yesmaximal, characteristic, complemented
8C22×D568C2211yescharacteristic
9C2×C4×D284—21no—
10C2×#14556069—22no—
11C2×#14556069—22no—
12C2×#14556069C414yescomplemented
13C22×C71⋊C16C2211yescharacteristic
14C22×C71⋊C16C2211yescharacteristic
15C4×D568—22no—
16C4×D568—22no—
17C2×#14556062C2211yescharacteristic
18C2×#14556050—21no—
19#17985800—22no—
20#17985799—22no—
21C22×D8×C71C2211yescharacteristic
22C2×#107×C71C2211yescharacteristic
23C2×#102×C71—21no—
24C2×C4×C8×C71C2211yescharacteristic
25#390×C71—21no—
26C22×D284—21no—
27C22×C4×D71—41no—
28C2×D568—21no—
29C2×D568—21no—
30C2×D568—21no—
31C2×D568C2×C412yes—
32C2×D568D412yes—
33C2×D568D412yes—
34C4×D284—24no—
35C4×D284—42no—
36C2×C4⋊Dic71—21no—
37C2×(C2×D71)⋊C4—41no—
38#14556069—28no—
39C2×C71⋊C16—21no—
40C2×C71⋊C16—21no—
41C2×C71⋊C16—21no—
42C2×C71⋊C16—21no—
43C2×C71⋊C16—21no—
44C2×C71⋊C16C2×C412yescomplemented
45#14556062—21no—
46#14556062D412yes—
47#14556050—24no—
48C22×D4×C71—21no—
49C2×D8×C71—21no—
50C2×D8×C71—21no—
51C2×D8×C71—21no—
52C2×D8×C71—21no—
53C2×D8×C71—21no—
54C2×D8×C71C2×C412yescomplemented
55C22×C8×C71D411yescharacteristic
56C22×C8×C71C2311yescharacteristic
57C2×C4⋊C4×C71—21no—
58C2×C22⋊C4×C71—41no—
59C2×C22⋊C4×C71—41no—
60C2×C42×C71D411yescharacteristic
61#107×C71—21no—
62#107×C71—21no—
63#102×C71—42no—
64C4×C8×C71—21no—
65C4×C8×C71—21no—
66C23×D71—41no—
67C2×D284—22no—
68C2×D284—22no—
69C2×D284—22no—
70C2×D284—41no—
71C2×D284—41no—
72C2×D284—41no—
73C22×Dic71—41no—
74C2×C4×D71—44no—
75C2×C4×D71—82no—
76C2×C4×D71—82no—
77D568—22no—
78D568—22no—
79D568—22no—
80D568C22⋊C414yescomplemented
81C4⋊Dic71—22no—
82C4⋊Dic71—41no—
83(C2×D71)⋊C4—44no—
84C71⋊C16—22no—
85C71⋊C16—41no—
86C24×C71—41no—
87C2×D4×C71—22no—
88C2×D4×C71—41no—
89C2×D4×C71—41no—
90C2×D4×C71—41no—
91C2×D4×C71—41no—
92C2×D4×C71—41no—
93C22×C4×C71—41no—
94C22×C4×C71D811yescharacteristic
95C22×C4×C71D811yescharacteristic
96C22×C4×C71C2×D411yescharacteristic
97D8×C71—22no—
98D8×C71—22no—
99D8×C71—41no—
100D8×C71—41no—
101C2×C8×C71—21no—
102C2×C8×C71—21no—
103C2×C8×C71—21no—
104C2×C8×C71C22×C411yescharacteristic
105C2×C8×C71C2×D411yescharacteristic
106C2×C8×C71C2×D411yescharacteristic
107C2×C8×C71C2×D411yescharacteristic
108C2×C8×C71#4111yescharacteristic
109C2×C8×C71#4111yescharacteristic
110C4⋊C4×C71—41no—
111C4⋊C4×C71—41no—
112C22⋊C4×C71—82no—
113C22⋊C4×C71—82no—
114C42×C71—21no—
115C42×C71—21no—
116C23×C71—42no—
117C23×C71—81no—
118C23×C71—81no—
119C23×C71—81no—
120C23×C71—81no—
121C23×C71—81no—
122C23×C71—81no—
123C23×C71C2×D811yescharacteristic
124C22×D71—42no—
125C22×D71—42no—
126C22×D71—42no—
127C22×D71—81no—
128C22×D71—81no—
129C22×D71—81no—
130C22×D71—81no—
131D4×C71—42no—
132D4×C71—42no—
133D4×C71—81no—
134D4×C71—81no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
139C2×C4×C71—21no—
140C2×C4×C71—81no—
141C2×C4×C71—81no—
142C2×C4×C71—81no—
143C2×C4×C71D1612yescomplemented
144C2×C4×C71C4×D411yescharacteristic
145C2×C4×C71#12011yescharacteristic
146C2×C4×C71#12111yescharacteristic
147C2×C4×C71#13511yescharacteristic
148C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
150C2×Dic71—42no—
151C2×Dic71—81no—
152C2×Dic71—81no—
153D284—24no—
154D284—42no—
155D284—42no—
156D284—42no—
157C4×D71—88no—
158C8×C71—41no—
159C8×C71C2×C22⋊C411yescharacteristic
160C8×C71C4×D411yescharacteristic
161C8×C71#12111yescharacteristic
162C8×C71#12311yescharacteristic
163C22×C71—82no—
164C22×C71—82no—
165C22×C71—82no—
166C22×C71—161no—
167C22×C71—161no—
168C22×C71—161no—
169C22×C71—161no—
170C22×C71C4×D811yescharacteristic
171C22×C71#44711yescharacteristic
172C22×C71#45911yescharacteristic
173C22×C71C2×D1611yescharacteristic
174C22×C71#50811yescharacteristic
175C22×C71#50911yescharacteristic
176C22×C71#50911yescharacteristic
177C2×D71—44no—
178C2×D71—82no—
179C2×D71—82no—
180C2×D71—82no—
181C2×D71—82no—
182C2×D71—82no—
183C2×D71—82no—
184C4×C71—22no—
185C4×C71—41no—
186C4×C71—161no—
187C4×C71#38611yescharacteristic
188C4×C71#44211yescharacteristic
189C4×C71#44911yescharacteristic
190C4×C71#46311yescharacteristic
191Dic71—82no—
192#13012—711nomaximal, Sylow
193C2×C71—162no—
194C2×C71#187911yescharacteristic
195C2×C71C4×D1611yescharacteristic
196C2×C71#217711yescharacteristic
197C2×C71#218411yescharacteristic
198C2×C71#218611yescharacteristic
199C2×C71#220611yescharacteristic
200C2×C71#221011yescharacteristic
201D71—84no—
202C22×D16—711no—
203C2×C4×D8—711no—
204C2×#357—711no—
205C2×#357—711no—
206#1970D7111yescharacteristic, complemented
207#1380—711no—
208#1374—711no—
209C71#1301211yescharacteristic, complemented, Sylow
210C22×D8—711no—
211C22×D8C2×D7111yescharacteristic
212C2×C4×D4—1421no—
213C2×D16—714no—
214C2×D16—1422no—
215C2×D16—1422no—
216C22×C16—711no—
217C22×C16—711no—
218C4×D8—1422no—
219C4×D8—1422no—
220C2×#107—711no—
221C2×#107C2×D7111yescharacteristic
222C2×#102—21no—
223C2×#102—1421no—
224C2×C4×C8C2×D7111yescharacteristic
225#390—21no—
226#357—1422no—
227#357—1422no—
228C22×D4—21no—
229C22×D4—1421no—
230C23×C4—2841no—
231C2×D8—21no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8—21no—
235C2×D8—21no—
236C2×D8—712no—
237C2×D8—712no—
238C2×D8—712no—
239C2×D8—1421no—
240C2×D8—1421no—
241C2×D8—1421no—
242C2×D8C4×D7112yescomplemented
243C22×C8C22⋊D7111yescharacteristic
244C22×C8C22×D7111yescharacteristic
245C4×D4—2842no—
246C4×D4—1424no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—1421no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—2841no—
252C2×C42C22⋊D7111yescharacteristic
253D16—1428no—
254C2×C16—712no—
255C2×C16—1421no—
256C2×C16—1421no—
257C2×C16—1421no—
258C2×C16—1421no—
259C2×C16—1421no—
260#107—21no—
261#107—21no—
262#107—712no—
263#107—1421no—
264#102—42no—
265#102—1424no—
266C4×C8—21no—
267C4×C8—21no—
268C24—41no—
269C24—2841no—
270C2×D4—22no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—2841no—
277C2×D4—2841no—
278C2×D4—2841no—
279C2×D4—1422no—
280C2×D4—1422no—
281C2×D4—1422no—
282C22×C4—41no—
283C22×C4—2844no—
284C22×C4—2841no—
285C22×C4—5682no—
286C22×C4—5682no—
287C22×C4#1176585011yescharacteristic
288C22×C4#1176585011yescharacteristic
289C22×C4C2×C22⋊D7111yescharacteristic
290D8—22no—
291D8—22no—
292D8—41no—
293D8—41no—
294D8—714no—
295D8—1422no—
296D8—1422no—
297D8—1422no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8C2×C4×D7111yescharacteristic
302C2×C8D4×D7111yescharacteristic
303C2×C8D4×D7111yescharacteristic
304C2×C8D4×D7111yescharacteristic
305C2×C8#1176586411yescharacteristic
306C2×C8#1176586511yescharacteristic
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—2841no—
310C4⋊C4—1422no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—2844no—
314C42—21no—
315C42—21no—
316C16—2841no—
317C16—1422no—
318C23—42no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—2842no—
326C23—2842no—
327C23—2842no—
328C23—5681no—
329C23—5681no—
330C23—5681no—
331C23—5681no—
332C23C2×#1176585011yescharacteristic
333D4—42no—
334D4—42no—
335D4—81no—
336D4—81no—
337D4—2842no—
338D4—2842no—
339D4—2842no—
340D4—1424no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—2842no—
350C2×C4—5688no—
351C2×C4—5681no—
352C2×C4—5681no—
353C2×C4#1455606912yescomplemented
354C2×C4C4×C22⋊D7111yescharacteristic
355C2×C4#1455609611yescharacteristic
356C2×C4#1455609911yescharacteristic
357C2×C4#1455614811yescharacteristic
358C2×C4#1455615311yescharacteristic
359C2×C4#1455615411yescharacteristic
360C8—41no—
361C8C22⋊C4×D7111yescharacteristic
362C8(C22⋊D71)⋊C411yescharacteristic
363C8#1455610111yescharacteristic
364C8#1455611011yescharacteristic
365C22—82no—
366C22—82no—
367C22—82no—
368C22—161no—
369C22—161no—
370C22—161no—
371C22—161no—
372C22—2844no—
373C22—5682no—
374C22—5682no—
375C22—5682no—
376C22—5682no—
377C22—5682no—
378C22—5682no—
379C22C4×#1176585011yescharacteristic
380C22#1798613211yescharacteristic
381C22#1798619111yescharacteristic
382C22C2×#1455606911yescharacteristic
383C22#1798642911yescharacteristic
384C22#1798643411yescharacteristic
385C22#1798643511yescharacteristic
386C4—22no—
387C4—41no—
388C4—161no—
389C4—5682no—
390C4#1798587511yescharacteristic
391C4#1798610911yescharacteristic
392C4#1798614211yescharacteristic
393C4#1798621111yescharacteristic
394C2—162no—
395C2—5684no—
396C2#2258869111yescharacteristic
397C2C4×#1455606911yescharacteristic
398C2#2259021411yescharacteristic
399C2#2259025111yescharacteristic
400C2#2259026111yescharacteristic
401C2#2259037211yescharacteristic
402C2#2259039211yescharacteristic
subgroupG/H
centerC23C2×#11765850
commutatorC2×C8×C71C22×C4
FrattiniC22×C8C22×D71
FittingC71×#1970C2
socleC23×C71C2×D8
prime radicalC22×#14556069C2
cosocleC22×C8×C71C23
hypercenter#1970D71
hypocenterC71#13012
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1970

Complements

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

NH ≅ G/Ntype
2. C2×C4×D568C2semidirect
3. C2×#17985800C2semidirect
4. C2×#17985799C2semidirect
5. #22586431C2semidirect
6. #22586418C2semidirect
7. C71×#1970C2semidirect
12. C2×#14556069C4semidirect
44. C2×C71⋊C16C2×C4semidirect
54. C2×D8×C71C2×C4semidirect
80. D568C22⋊C4semidirect
143. C2×C4×C71D16semidirect
206. #1970D71semidirect
209. C71#13012semidirect
242. C2×D8C4×D71semidirect
353. C2×C4#14556069semidirect

Maximal subgroups

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

subgroupindex
C22×#145560692
C2×C4×D5682
C2×#179858002
C2×#179857992
#225864312
#225864182
C71×#19702
#1301271