Finite Group Databasebeta

← #31906910

Subgroups — #31906910

Order 18176.

530 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22586681C211yesmaximal, characteristic, complemented
2#22586678C211yesmaximal, characteristic, complemented
3#22586288C212yesmaximal, complemented
4#22586268C212yesmaximal, complemented
5C71×#2321C211yesmaximal, characteristic
6#17986056—21no—
7#17985963—21no—
8#17985725—21no—
9#17985723—21no—
10#17985723C2212yes—
11#17985723C2212yes—
12C2×#120×C71C411yescharacteristic
13C2×C4×D4×C71C2211yescharacteristic
14C22×C22⋊C4×C71C411yescharacteristic
15#390×C71—21no—
16#388×C71—21no—
17#388×C71—21no—
18#386×C71—21no—
19#386×C71C411yescharacteristic
20#385×C71C2211yescharacteristic
21#379×C71C2211yescharacteristic
22C4×C22⋊C4×C71C411yescharacteristic
23C2×#11765849—41no—
24C4⋊(C71⋊C8)—41no—
25C22⋊(C71⋊C8)—22no—
26C22⋊(C71⋊C8)—22no—
27C22⋊(C71⋊C8)—41no—
28#14556039—41no—
29C25×C71C2×C411yescharacteristic
30C22×D4×C71—21no—
31C22×D4×C71C2×C411yescharacteristic
32C23×C4×C71C2×C411yescharacteristic
33C23×C4×C71C2×C411yescharacteristic
34C23×C4×C71D411yescharacteristic
35#120×C71—22no—
36#120×C71—22no—
37C4×D4×C71—41no—
38C4×D4×C71—41no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C22⋊C4×C71—21no—
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71—41no—
48C2×C22⋊C4×C71—41no—
49C2×C22⋊C4×C71C2×C411yescharacteristic
50C2×C22⋊C4×C71C2×C411yescharacteristic
51C2×C22⋊C4×C71D411yescharacteristic
52C2×C22⋊C4×C71C2311yescharacteristic
53C2×C42×C71—21no—
54C2×C42×C71D411yescharacteristic
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58#11765849—82no—
59C2×C71⋊C8—42no—
60C2×C71⋊C8—42no—
61C24×C71—21no—
62C24×C71—21no—
63C24×C71—41no—
64C24×C71—41no—
65C24×C71—81no—
66C24×C71—81no—
67C24×C71C22⋊C411yescharacteristic
68C24×C71M1612yescomplemented
69C24×C71C22×C411yescharacteristic
70C2×D4×C71—21no—
71C2×D4×C71—21no—
72C2×D4×C71—24no—
73C2×D4×C71—41no—
74C2×D4×C71—41no—
75C2×D4×C71—81no—
76C2×D4×C71—81no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—21no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—21no—
84C22×C4×C71—41no—
85C22×C4×C71—81no—
86C22×C4×C71—81no—
87C22×C4×C71—81no—
88C22×C4×C71C22⋊C411yescharacteristic
89C22×C4×C71C22⋊C411yescharacteristic
90C22×C4×C71C22⋊C411yescharacteristic
91C22×C4×C71C2×D411yescharacteristic
92C22×C4×C71C2×D411yescharacteristic
93C4⋊C4×C71—41no—
94C4⋊C4×C71—81no—
95C22⋊C4×C71—24no—
96C22⋊C4×C71—21no—
97C22⋊C4×C71—41no—
98C22⋊C4×C71—41no—
99C22⋊C4×C71—42no—
100C22⋊C4×C71—41no—
101C22⋊C4×C71—41no—
102C22⋊C4×C71—41no—
103C22⋊C4×C71—81no—
104C22⋊C4×C71—81no—
105C22⋊C4×C71—81no—
106C22⋊C4×C71M1612yescomplemented
107C42×C71—21no—
108C42×C71—21no—
109C42×C71—81no—
110C23×C71—21no—
111C23×C71—21no—
112C23×C71—22no—
113C23×C71—21no—
114C23×C71—21no—
115C23×C71—21no—
116C23×C71—41no—
117C23×C71—41no—
118C23×C71—41no—
119C23×C71—41no—
120C23×C71—41no—
121C23×C71—41no—
122C23×C71—41no—
123C23×C71—41no—
124C23×C71—42no—
125C23×C71—42no—
126C23×C71—41no—
127C23×C71—41no—
128C23×C71—82no—
129C23×C71—82no—
130C23×C71—82no—
131C23×C71—82no—
132C23×C71—84no—
133C23×C71C2×C22⋊C411yescharacteristic
134C23×C71C2×M1611yescharacteristic
135C23×C71C2×M1611yescharacteristic
136D4×C71—42no—
137D4×C71—42no—
138D4×C71—84no—
139C2×C4×C71—21no—
140C2×C4×C71—22no—
141C2×C4×C71—21no—
142C2×C4×C71—21no—
143C2×C4×C71—21no—
144C2×C4×C71—21no—
145C2×C4×C71—21no—
146C2×C4×C71—41no—
147C2×C4×C71—42no—
148C2×C4×C71—41no—
149C2×C4×C71—41no—
150C2×C4×C71—41no—
151C2×C4×C71—41no—
152C2×C4×C71—41no—
153C2×C4×C71—41no—
154C2×C4×C71—41no—
155C2×C4×C71—41no—
156C2×C4×C71—41no—
157C2×C4×C71—41no—
158C2×C4×C71—41no—
159C2×C4×C71—81no—
160C2×C4×C71—81no—
161C2×C4×C71—81no—
162C2×C4×C71—82no—
163C2×C4×C71#9911yescharacteristic
164C2×C4×C71#9911yescharacteristic
165C71⋊C8—82no—
166C71⋊C8—82no—
167C22×C71—21no—
168C22×C71—21no—
169C22×C71—21no—
170C22×C71—21no—
171C22×C71—41no—
172C22×C71—41no—
173C22×C71—42no—
174C22×C71—42no—
175C22×C71—42no—
176C22×C71—42no—
177C22×C71—41no—
178C22×C71—41no—
179C22×C71—41no—
180C22×C71—41no—
181C22×C71—41no—
182C22×C71—42no—
183C22×C71—81no—
184C22×C71—81no—
185C22×C71—81no—
186C22×C71—84no—
187C22×C71—84no—
188C22×C71—81no—
189C22×C71—81no—
190C22×C71#40711yescharacteristic
191C22×C71C2×#9911yescharacteristic
192C22×C71#41311yescharacteristic
193C4×C71—21no—
194C4×C71—21no—
195C4×C71—42no—
196C4×C71—41no—
197C4×C71—41no—
198C4×C71—41no—
199C4×C71—41no—
200C4×C71—81no—
201C4×C71—81no—
202#11527—711nomaximal, Sylow
203C2×C71—21no—
204C2×C71—21no—
205C2×C71—41no—
206C2×C71—41no—
207C2×C71—81no—
208C2×C71—82no—
209C2×C71#146511yescharacteristic
210C2×C71#212111yescharacteristic
211C2×C71#212211yescharacteristic
212#2321D7111yescharacteristic
213#1466—711no—
214#1465—711no—
215#1326—712no—
216#1316—712no—
217C71#1152711yescharacteristic, complemented, Sylow
218C2×#120Dic7111yescharacteristic
219C2×C4×D4C2×D7111yescharacteristic
220C22×C22⋊C4Dic7111yescharacteristic
221#432—1421no—
222#407—1421no—
223#390—21no—
224#388—21no—
225#388—21no—
226#386—21no—
227#386Dic7111yescharacteristic
228#385C2×D7111yescharacteristic
229#379C2×D7111yescharacteristic
230C4×C22⋊C4Dic7111yescharacteristic
231#324—1421no—
232#323—712no—
233#323—712no—
234#323—1421no—
235C25C2×Dic7111yescharacteristic
236C22×D4—21no—
237C22×D4C2×Dic7111yescharacteristic
238C23×C4C2×Dic7111yescharacteristic
239C23×C4C2×Dic7111yescharacteristic
240C23×C4C22⋊D7111yescharacteristic
241C2×M16—2841no—
242#120—22no—
243#120—22no—
244C4×D4—41no—
245C4×D4—41no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C22⋊D7111yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4C2×Dic7111yescharacteristic
257C2×C22⋊C4C2×Dic7111yescharacteristic
258C2×C22⋊C4C22⋊D7111yescharacteristic
259C2×C22⋊C4C22×D7111yescharacteristic
260C2×C42—21no—
261C2×C42C22⋊D7111yescharacteristic
262C4⋊C8—2841no—
263C22⋊C8—2841no—
264C22⋊C8—1422no—
265C22⋊C8—1422no—
266#97—2841no—
267#95—21no—
268#95—21no—
269#95—21no—
270C24—21no—
271C24—21no—
272C24—41no—
273C24—41no—
274C24—81no—
275C24—81no—
276C24C22⋊Dic7111yescharacteristic
277C24#1176584912yescomplemented
278C24C22×Dic7111yescharacteristic
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—24no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—81no—
285C2×D4—81no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4C22⋊Dic7111yescharacteristic
298C22×C4C22⋊Dic7111yescharacteristic
299C22×C4C22⋊Dic7111yescharacteristic
300C22×C4C2×C22⋊D7111yescharacteristic
301C22×C4C2×C22⋊D7111yescharacteristic
302M16—5682no—
303C2×C8—2842no—
304C2×C8—2842no—
305C4⋊C4—41no—
306C4⋊C4—81no—
307C22⋊C4—24no—
308C22⋊C4—21no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—81no—
316C22⋊C4—81no—
317C22⋊C4—81no—
318C22⋊C4#1176584912yescomplemented
319C42—21no—
320C42—21no—
321C42—81no—
322C23—21no—
323C23—21no—
324C23—22no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—82no—
341C23—82no—
342C23—82no—
343C23—82no—
344C23—84no—
345C23C2×C22⋊Dic7111yescharacteristic
346C23C2×#1176584911yescharacteristic
347C23C2×#1176584911yescharacteristic
348D4—42no—
349D4—42no—
350D4—84no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4#1455604311yescharacteristic
376C2×C4#1455604311yescharacteristic
377C8—5682no—
378C8—5682no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—84no—
399C22—84no—
400C22—81no—
401C22—81no—
402C22#1798596311yescharacteristic
403C22C2×#1455604311yescharacteristic
404C22#1798598311yescharacteristic
405C4—21no—
406C4—21no—
407C4—42no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—81no—
413C4—81no—
414C2—21no—
415C2—21no—
416C2—41no—
417C2—41no—
418C2—81no—
419C2—82no—
420C2#2258667811yescharacteristic
421C2#2258998611yescharacteristic
422C2#2258999011yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC24×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2321C2
socleC22×C71C2×#99
prime radicalC2×#120×C71C4
cosocleC2×C22⋊C4×C71C23
hypercenter#2321D71
hypocenterC71#11527
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2321

Complements

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

NH ≅ G/Ntype
1. #22586681C2semidirect
2. #22586678C2semidirect
3. #22586288C2semidirect
4. #22586268C2semidirect
68. C24×C71M16semidirect
106. C22⋊C4×C71M16semidirect
217. C71#11527semidirect
277. C24#11765849semidirect
318. C22⋊C4#11765849semidirect

Maximal subgroups

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

subgroupindex
#225866812
#225866782
#225862882
#225862682
C71×#23212
#1152771