Finite Group Databasebeta

← #31893023

Subgroups — #31893023

Order 18176.

694 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986052C211yesmaximal, characteristic, complemented
2C2×#17985733C212yesmaximal
3#22586567C211yesmaximal, characteristic, complemented
4#22586218C212yesmaximal, complemented
5C71×#2854C211yesmaximal, characteristic
6#17986052—24no—
7C2×C4⋊(C71⋊C8)—21no—
8C2×C22⋊(C71⋊C8)C2212yes—
9C2×C4×C71⋊C8—21no—
10C2×C71⋊#95—21no—
11C2×C71⋊#95—21no—
12C2×C71⋊#95C2212yes—
13#17985733—24no—
14C2×#123×C71—21no—
15C2×#121×C71C2211yescharacteristic
16C22×C42×C71C2211yescharacteristic
17#390×C71C411yescharacteristic, complemented
18#390×C71C411yescharacteristic, complemented
19#389×C71C411yescharacteristic, complemented
20#389×C71C411yescharacteristic, complemented
21#388×C71—21no—
22#388×C71—21no—
23#386×C71—21no—
24#380×C71C2211yescharacteristic
25C23×Dic71—22no—
26C22×C71⋊C8—22no—
27C4⋊(C71⋊C8)—24no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)D414yes—
30C4×C71⋊C8—24no—
31C71⋊#95—24no—
32C71⋊#95—22no—
33C71⋊#95—22no—
34C71⋊#95—42no—
35C22×D4×C71—21no—
36C22×D4×C71C2×C411yescharacteristic
37C23×C4×C71D411yescharacteristic
38C23×C4×C71D411yescharacteristic
39C23×C4×C71C2311yescharacteristic
40#123×C71—42no—
41#123×C71—42no—
42#121×C71—22no—
43#121×C71—22no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71—21no—
46C2×C4⋊C4×C71C2×C411yescharacteristic
47C2×C4⋊C4×C71C2×C411yescharacteristic
48C2×C4⋊C4×C71C2×C411yescharacteristic
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71—21no—
53C2×C22⋊C4×C71—21no—
54C2×C22⋊C4×C71—21no—
55C2×C42×C71—22no—
56C2×C42×C71—22no—
57C2×C42×C71C2×C411yescharacteristic
58C2×C42×C71C2×C411yescharacteristic
59C2×C42×C71D411yescharacteristic
60C2×C42×C71D411yescharacteristic
61#95×C71—21no—
62#95×C71—21no—
63#95×C71—21no—
64C22×Dic71—24no—
65C22×Dic71—42no—
66C22×Dic71—42no—
67C22×Dic71—42no—
68C22×Dic71—42no—
69C22×Dic71—42no—
70C22×Dic71—42no—
71C2×C71⋊C8—24no—
72C2×C71⋊C8—42no—
73C2×C71⋊C8—42no—
74C24×C71—21no—
75C24×C71C2×D411yescharacteristic
76C2×D4×C71—22no—
77C2×D4×C71—41no—
78C2×D4×C71—41no—
79C2×D4×C71—42no—
80C2×D4×C71—41no—
81C2×D4×C71—41no—
82C2×D4×C71—42no—
83C22×C4×C71—22no—
84C22×C4×C71—22no—
85C22×C4×C71—22no—
86C22×C4×C71—22no—
87C22×C4×C71—21no—
88C22×C4×C71—21no—
89C22×C4×C71—21no—
90C22×C4×C71—41no—
91C22×C4×C71—41no—
92C22×C4×C71—41no—
93C22×C4×C71C22⋊C411yescharacteristic
94C22×C4×C71C22⋊C411yescharacteristic
95C22×C4×C71C22⋊C411yescharacteristic
96C22×C4×C71C22⋊C411yescharacteristic
97C22×C4×C71D812yes—
98C22×C4×C71SD1612yes—
99C22×C4×C71C22×C411yescharacteristic
100C22×C4×C71C2×D412yes—
101C22×C4×C71C2×D411yescharacteristic
102C22×C4×C71#4112yes—
103C4⋊C4×C71—22no—
104C4⋊C4×C71—41no—
105C4⋊C4×C71—41no—
106C4⋊C4×C71—42no—
107C4⋊C4×C71—42no—
108C22⋊C4×C71—42no—
109C22⋊C4×C71—41no—
110C22⋊C4×C71—41no—
111C22⋊C4×C71—42no—
112C22⋊C4×C71—42no—
113C22⋊C4×C71—42no—
114C22⋊C4×C71—41no—
115C22⋊C4×C71—41no—
116C42×C71—22no—
117C42×C71—22no—
118C42×C71—21no—
119C42×C71—21no—
120C42×C71—21no—
121C42×C71—21no—
122C23×C71—21no—
123C23×C71—21no—
124C23×C71—21no—
125C23×C71—21no—
126C23×C71—41no—
127C23×C71—41no—
128C23×C71—41no—
129C23×C71—41no—
130C23×C71—41no—
131C23×C71—41no—
132C23×C71—41no—
133C23×C71C2×C22⋊C411yescharacteristic
134C23×C71#12112yes—
135C23×C71C2×D811yescharacteristic
136C23×C71C2×SD1611yescharacteristic
137C23×C71#13512yes—
138D4×C71—82no—
139D4×C71—82no—
140D4×C71—82no—
141C2×C4×C71—21no—
142C2×C4×C71—21no—
143C2×C4×C71—21no—
144C2×C4×C71—21no—
145C2×C4×C71—21no—
146C2×C4×C71—21no—
147C2×C4×C71—21no—
148C2×C4×C71—21no—
149C2×C4×C71—21no—
150C2×C4×C71—21no—
151C2×C4×C71—24no—
152C2×C4×C71—24no—
153C2×C4×C71—21no—
154C2×C4×C71—21no—
155C2×C4×C71—21no—
156C2×C4×C71—42no—
157C2×C4×C71—42no—
158C2×C4×C71—44no—
159C2×C4×C71—41no—
160C2×C4×C71—41no—
161C2×C4×C71—41no—
162C2×C4×C71—41no—
163C2×C4×C71—41no—
164C2×C4×C71—41no—
165C2×C4×C71—41no—
166C2×C4×C71—41no—
167C2×C4×C71—41no—
168C2×C4×C71#10212yes—
169C2×C4×C71#10212yes—
170C2×C4×C71#10412yes—
171C2×C4×C71C4×D412yes—
172C2×C4×C71#12011yescharacteristic
173C2×C4×C71#12311yescharacteristic
174C2×Dic71—44no—
175C2×Dic71—44no—
176C2×Dic71—44no—
177C2×Dic71—82no—
178C2×Dic71—82no—
179C2×Dic71—82no—
180C2×Dic71—82no—
181C71⋊C8—44no—
182C22×C71—22no—
183C22×C71—22no—
184C22×C71—22no—
185C22×C71—21no—
186C22×C71—21no—
187C22×C71—21no—
188C22×C71—21no—
189C22×C71—22no—
190C22×C71—81no—
191C22×C71—81no—
192C22×C71—81no—
193C22×C71—81no—
194C22×C71—81no—
195C22×C71—81no—
196C22×C71—81no—
197C22×C71#38611yescharacteristic
198C22×C71#41011yescharacteristic
199C22×C71#41011yescharacteristic
200C22×C71C2×#10211yescharacteristic
201C22×C71#41611yescharacteristic
202C22×C71C2×#10411yescharacteristic
203C22×C71#42111yescharacteristic
204C22×C71#46512yes—
205C22×C71#46612yes—
206C4×C71—22no—
207C4×C71—22no—
208C4×C71—22no—
209C4×C71—21no—
210C4×C71—22no—
211C4×C71—21no—
212C4×C71—41no—
213C4×C71—81no—
214C4×C71—81no—
215C4×C71—81no—
216Dic71—84no—
217#9114—711nomaximal, Sylow
218C2×C71—22no—
219C2×C71—22no—
220C2×C71—161no—
221C2×C71#150011yescharacteristic
222C2×C71#150211yescharacteristic
223C2×C71#151711yescharacteristic
224C2×C71#152111yescharacteristic
225C2×C71#187111yescharacteristic
226C2×C71#189311yescharacteristic
227C2×C71#189911yescharacteristic
228C2×#431—711no—
229#2854D7111yescharacteristic
230C2×#327—712no—
231#1434—711no—
232#1294—712no—
233C71#911411yescharacteristic, complemented, Sylow
234C2×#123—21no—
235C2×#121C2×D7111yescharacteristic
236C22×C42C2×D7111yescharacteristic
237#431—1424no—
238C2×C4⋊C8—1421no—
239C2×C22⋊C8—712no—
240C2×C4×C8—1421no—
241#390Dic7111yescharacteristic, complemented
242#390Dic7111yescharacteristic, complemented
243#389Dic7111yescharacteristic, complemented
244#389Dic7111yescharacteristic, complemented
245#388—21no—
246#388—21no—
247#386—21no—
248#380C2×D7111yescharacteristic
249C2×#95—712no—
250C2×#95—1421no—
251C2×#95—1421no—
252#327—1424no—
253C22×D4—21no—
254C22×D4C2×Dic7111yescharacteristic
255C23×C4—1422no—
256C23×C4C22⋊D7111yescharacteristic
257C23×C4C22⋊D7111yescharacteristic
258C23×C4C22×D7111yescharacteristic
259C22×C8—1422no—
260#123—42no—
261#123—42no—
262#121—22no—
263#121—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4C2×Dic7111yescharacteristic
267C2×C4⋊C4C2×Dic7111yescharacteristic
268C2×C4⋊C4C2×Dic7111yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42C2×Dic7111yescharacteristic
278C2×C42C2×Dic7111yescharacteristic
279C2×C42C22⋊D7111yescharacteristic
280C2×C42C22⋊D7111yescharacteristic
281C4⋊C8—1424no—
282C22⋊C8—714no—
283C22⋊C8—1422no—
284C4×C8—1424no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—2842no—
289#95—1424no—
290#95—1422no—
291#95—1422no—
292C24—21no—
293C24C2×C22⋊D7111yescharacteristic
294C2×D4—22no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—2842no—
312C22×C4—2842no—
313C22×C4—2842no—
314C22×C4—2842no—
315C22×C4—2842no—
316C22×C4—2842no—
317C22×C4—1424no—
318C22×C4C22⋊Dic7111yescharacteristic
319C22×C4C22⋊Dic7111yescharacteristic
320C22×C4C22⋊Dic7111yescharacteristic
321C22×C4C22⋊Dic7111yescharacteristic
322C22×C4#1176585012yes—
323C22×C4#1176585312yes—
324C22×C4C22×Dic7111yescharacteristic
325C22×C4D4×D7112yes—
326C22×C4C2×C22⋊D7111yescharacteristic
327C22×C4#1176586412yes—
328C2×C8—2842no—
329C2×C8—2842no—
330C2×C8—1424no—
331C4⋊C4—22no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C42—22no—
345C42—22no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23C2×C22⋊Dic7111yescharacteristic
362C23#1455610012yes—
363C23C2×#1176585011yescharacteristic
364C23C2×#1176585311yescharacteristic
365C23#1455614812yes—
366D4—82no—
367D4—82no—
368D4—82no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—24no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—24no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—2844no—
397C2×C4—2844no—
398C2×C4—2844no—
399C2×C4—5682no—
400C2×C4—5682no—
401C2×C4—5682no—
402C2×C4—5682no—
403C2×C4#1455604912yes—
404C2×C4#1455604912yes—
405C2×C4#1455605512yes—
406C2×C4D4×Dic7112yes—
407C2×C4#1455609611yescharacteristic
408C2×C4#1455610911yescharacteristic
409C8—2844no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22#1798587911yescharacteristic
426C22#1798597311yescharacteristic
427C22#1798597311yescharacteristic
428C22C2×#1455604911yescharacteristic
429C22#1798599511yescharacteristic
430C22C2×#1455605511yescharacteristic
431C22#1798601611yescharacteristic
432C22#1798622612yes—
433C22#1798623312yes—
434C4—22no—
435C4—22no—
436C4—21no—
437C4—22no—
438C4—21no—
439C4—22no—
440C4—41no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—5684no—
445C2—22no—
446C2—22no—
447C2—161no—
448C2#2258682611yescharacteristic
449C2#2258683511yescharacteristic
450C2#2258690011yescharacteristic
451C2#2258691711yescharacteristic
452C2#2258864311yescharacteristic
453C2#2258876911yescharacteristic
454C2#2258879911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2854C2
socleC23×C71C2×C22⋊C4
prime radicalC2×#121×C71C22
cosocleC23×C4×C71C23
hypercenter#2854D71
hypocenterC71#9114
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C2×#17986052C2semidirect
3. #22586567C2semidirect
4. #22586218C2semidirect
17. #390×C71C4semidirect
18. #390×C71C4semidirect
19. #389×C71C4semidirect
20. #389×C71C4semidirect
233. C71#9114semidirect
241. #390Dic71semidirect
242. #390Dic71semidirect
243. #389Dic71semidirect
244. #389Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860522
C2×#179857332
#225865672
#225862182
C71×#28542
#911471