Finite Group Databasebeta

← #31908018

Subgroups — #31908018

Order 18176.

532 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C4×#14556043C211yesmaximal, characteristic, complemented
2#22586681C211yesmaximal, characteristic, complemented
3#22586340C212yesmaximal, complemented
4#22586337C212yesmaximal, complemented
5C71×#2543C211yesmaximal, characteristic
6#17986056—21no—
7C2×#14556043—21no—
8C4×C22⋊Dic71—21no—
9#17985766C2212yes—
10#17985723—21no—
11#17985723C2212yes—
12C2×#121×C71—21no—
13C2×#121×C71C411yescharacteristic, complemented
14C2×C4×D4×C71C2211yescharacteristic
15#398×C71C411yescharacteristic, complemented
16#397×C71C2211yescharacteristic
17#394×C71C411yescharacteristic, complemented
18#392×C71C2211yescharacteristic
19#388×C71—21no—
20#386×C71—21no—
21#386×C71—21no—
22C2×#95×C71C411yescharacteristic, complemented
23C2×C22⋊Dic71—22no—
24C2×C4×Dic71—41no—
25C4⋊(C71⋊C8)—41no—
26#14556043—24no—
27C22⋊(C71⋊C8)—22no—
28#14556039—41no—
29C71⋊#95—41no—
30C22×D4×C71—21no—
31C22×D4×C71—21no—
32C22×D4×C71C2×C411yescharacteristic
33C22×D4×C71D411yescharacteristic
34C23×C4×C71—21no—
35C23×C4×C71C2×C411yescharacteristic
36C23×C4×C71D411yescharacteristic
37#121×C71—44no—
38#121×C71—41no—
39#121×C71—41no—
40C4×D4×C71—41no—
41C4×D4×C71—41no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C4⋊C4×C71C2×C411yescharacteristic
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71—21no—
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71D411yescharacteristic
51C2×C22⋊C4×C71C2311yescharacteristic
52C2×C42×C71D411yescharacteristic
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56#95×C71C2×C411yescharacteristic
57#95×C71C2×C411yescharacteristic
58C22×Dic71—42no—
59C2×C71⋊C8—42no—
60C22⋊Dic71—24no—
61C22⋊Dic71—42no—
62C4×Dic71—82no—
63C24×C71—21no—
64C24×C71C22⋊C411yescharacteristic
65C24×C71C2×D411yescharacteristic
66C2×D4×C71—22no—
67C2×D4×C71—22no—
68C2×D4×C71—41no—
69C2×D4×C71—41no—
70C2×D4×C71—42no—
71C2×D4×C71—42no—
72C2×D4×C71—42no—
73C2×D4×C71—42no—
74C2×D4×C71—42no—
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—42no—
85C22×C4×C71—42no—
86C22×C4×C71—81no—
87C22×C4×C71—81no—
88C22×C4×C71—81no—
89C22×C4×C71C22⋊C411yescharacteristic
90C22×C4×C71C22⋊C411yescharacteristic
91C22×C4×C71C22⋊C411yescharacteristic
92C22×C4×C71C22×C411yescharacteristic
93C22×C4×C71C2×D411yescharacteristic
94C4⋊C4×C71—41no—
95C4⋊C4×C71—41no—
96C4⋊C4×C71—42no—
97C22⋊C4×C71—41no—
98C22⋊C4×C71—41no—
99C22⋊C4×C71—42no—
100C22⋊C4×C71—42no—
101C22⋊C4×C71—42no—
102C22⋊C4×C71—41no—
103C22⋊C4×C71—41no—
104C22⋊C4×C71—81no—
105C42×C71—21no—
106C42×C71—21no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71—41no—
112C23×C71—41no—
113C23×C71—41no—
114C23×C71—41no—
115C23×C71—41no—
116C23×C71—41no—
117C23×C71—41no—
118C23×C71—41no—
119C23×C71—41no—
120C23×C71—41no—
121C23×C71—41no—
122C23×C71#10412yes—
123C23×C71C2×C22⋊C411yescharacteristic
124C23×C71AGL(1,ℤ/8)12yes—
125D4×C71—44no—
126D4×C71—84no—
127D4×C71—82no—
128D4×C71—84no—
129C2×C4×C71—22no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—21no—
136C2×C4×C71—41no—
137C2×C4×C71—42no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71—41no—
148C2×C4×C71—41no—
149C2×C4×C71—81no—
150C2×C4×C71—81no—
151C2×C4×C71—81no—
152C2×C4×C71—84no—
153C2×C4×C71#9911yescharacteristic
154C2×C4×C71#9911yescharacteristic
155C2×Dic71—44no—
156C2×Dic71—82no—
157C2×Dic71—82no—
158C71⋊C8—82no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—22no—
162C22×C71—22no—
163C22×C71—42no—
164C22×C71—42no—
165C22×C71—42no—
166C22×C71—41no—
167C22×C71—41no—
168C22×C71—41no—
169C22×C71—41no—
170C22×C71—81no—
171C22×C71—81no—
172C22×C71—81no—
173C22×C71—81no—
174C22×C71—81no—
175C22×C71—81no—
176C22×C71—81no—
177C22×C71—81no—
178C22×C71C2×#9911yescharacteristic
179C22×C71#41811yescharacteristic
180C22×C71C2×#10411yescharacteristic
181C4×C71—21no—
182C4×C71—21no—
183C4×C71—42no—
184C4×C71—41no—
185C4×C71—41no—
186C4×C71—41no—
187C4×C71—41no—
188C4×C71—81no—
189C4×C71—81no—
190Dic71—84no—
191#11775—711nomaximal, Sylow
192C2×C71—21no—
193C2×C71—21no—
194C2×C71—42no—
195C2×C71—42no—
196C2×C71—161no—
197C2×C71#150711yescharacteristic
198C2×C71#212111yescharacteristic
199C2×C71#212611yescharacteristic
200#2543D7111yescharacteristic
201C4×#99—711no—
202#1466—711no—
203#1344—712no—
204#1343—712no—
205C71#1177511yescharacteristic, complemented, Sylow
206C2×#121—21no—
207C2×#121Dic7111yescharacteristic, complemented
208C2×C4×D4C2×D7111yescharacteristic
209#432—1421no—
210C2×#99—1421no—
211#398Dic7111yescharacteristic, complemented
212#397C2×D7111yescharacteristic
213#394Dic7111yescharacteristic, complemented
214#392C2×D7111yescharacteristic
215#388—21no—
216#386—21no—
217#386—21no—
218C4×C22⋊C4—1421no—
219C2×#95Dic7111yescharacteristic, complemented
220#342—712no—
221#323—712no—
222#323—1421no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4C2×Dic7111yescharacteristic
226C22×D4C22⋊D7111yescharacteristic
227C23×C4—21no—
228C23×C4C2×Dic7111yescharacteristic
229C23×C4C22⋊D7111yescharacteristic
230#121—44no—
231#121—41no—
232#121—41no—
233C4×D4—41no—
234C4×D4—41no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C2×Dic7111yescharacteristic
237C2×C4⋊C4C2×Dic7111yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—1422no—
244C2×C22⋊C4C22⋊D7111yescharacteristic
245C2×C22⋊C4C22×D7111yescharacteristic
246C2×C42—2841no—
247C2×C42C22⋊D7111yescharacteristic
248C4⋊C8—2841no—
249#99—1424no—
250C22⋊C8—1422no—
251#97—2841no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—2841no—
256#95C2×Dic7111yescharacteristic
257#95C2×Dic7111yescharacteristic
258C24—21no—
259C24C22⋊Dic7111yescharacteristic
260C24C2×C22⋊D7111yescharacteristic
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—81no—
271C2×D4—81no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—2842no—
285C22×C4C22⋊Dic7111yescharacteristic
286C22×C4C22⋊Dic7111yescharacteristic
287C22×C4C22⋊Dic7111yescharacteristic
288C22×C4C22×Dic7111yescharacteristic
289C22×C4C2×C22⋊D7111yescharacteristic
290C2×C8—2842no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—81no—
302C22⋊C4—2842no—
303C22⋊C4—1424no—
304C42—21no—
305C42—21no—
306C42—5682no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23#1455605512yes—
323C23C2×C22⋊Dic7111yescharacteristic
324C23#1455615412yes—
325D4—44no—
326D4—84no—
327D4—82no—
328D4—84no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—84no—
353C2×C4—2844no—
354C2×C4—5682no—
355C2×C4—5682no—
356C2×C4#1455604311yescharacteristic
357C2×C4#1455604311yescharacteristic
358C8—5682no—
359C22—21no—
360C22—21no—
361C22—22no—
362C22—22no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22C2×#1455604311yescharacteristic
379C22#1798600411yescharacteristic
380C22C2×#1455605511yescharacteristic
381C4—21no—
382C4—21no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4—81no—
390C4—5684no—
391C2—21no—
392C2—21no—
393C2—42no—
394C2—42no—
395C2—161no—
396C2#2258685611yescharacteristic
397C2#2258998611yescharacteristic
398C2#2259000511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
Fitting, prime radicalC71×#2543C2
socleC22×C71C2×#99
cosocleC2×C22⋊C4×C71C23
hypercenter#2543D71
hypocenterC71#11775
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2543

Complements

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

NH ≅ G/Ntype
1. C4×#14556043C2semidirect
2. #22586681C2semidirect
3. #22586340C2semidirect
4. #22586337C2semidirect
13. C2×#121×C71C4semidirect
15. #398×C71C4semidirect
17. #394×C71C4semidirect
22. C2×#95×C71C4semidirect
205. C71#11775semidirect
207. C2×#121Dic71semidirect
211. #398Dic71semidirect
213. #394Dic71semidirect
219. C2×#95Dic71semidirect

Maximal subgroups

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

subgroupindex
C4×#145560432
#225866812
#225863402
#225863372
C71×#25432
#1177571