Finite Group Databasebeta

← #31908013

Subgroups — #31908013

Order 18176.

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

normal subgroups only

This group has 544 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#22586678C211yesmaximal, characteristic, complemented
3#22586340C212yesmaximal, complemented
4#22586337C212yesmaximal, complemented
5C71×#2535C211yesmaximal, characteristic
6C2×#14556043—21no—
7#17985963—21no—
8C4×C22⋊Dic71—21no—
9#17985766C2212yes—
10#17985725—21no—
11#17985723C2212yes—
12C2×#121×C71—21no—
13C2×C4×D4×C71C411yescharacteristic, complemented
14C2×C4×D4×C71C2211yescharacteristic
15#397×C71C411yescharacteristic, complemented
16#397×C71C2211yescharacteristic
17#392×C71C411yescharacteristic, complemented
18#392×C71C2211yescharacteristic
19#386×C71—21no—
20#386×C71—21no—
21#382×C71C411yescharacteristic, complemented
22#380×C71—21no—
23C2×#11765849—41no—
24C2×C22⋊Dic71—22no—
25C2×C4×Dic71—41no—
26#14556043—24no—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—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×C71D411yescharacteristic
36C23×C4×C71D411yescharacteristic
37#121×C71—44no—
38C4×D4×C71—41no—
39C4×D4×C71—41no—
40C4×D4×C71—41no—
41C4×D4×C71—41no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C4⋊C4×C71D411yescharacteristic
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×C71C2×C411yescharacteristic
51C2×C22⋊C4×C71C2311yescharacteristic
52C2×C42×C71C2×C411yescharacteristic
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56#95×C71C2×C411yescharacteristic
57#95×C71C2×C411yescharacteristic
58C22×Dic71—42no—
59#11765849—82no—
60C2×C71⋊C8—42no—
61C22⋊Dic71—24no—
62C22⋊Dic71—42no—
63C4×Dic71—82no—
64C24×C71—21no—
65C24×C71C22⋊C411yescharacteristic
66C24×C71C2×D411yescharacteristic
67C2×D4×C71—22no—
68C2×D4×C71—22no—
69C2×D4×C71—41no—
70C2×D4×C71—41no—
71C2×D4×C71—42no—
72C2×D4×C71—42no—
73C2×D4×C71—42no—
74C2×D4×C71—42no—
75C2×D4×C71—42no—
76C2×D4×C71—81no—
77C2×D4×C71—81no—
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—21no—
85C22×C4×C71—42no—
86C22×C4×C71—42no—
87C22×C4×C71—42no—
88C22×C4×C71—81no—
89C22×C4×C71—81no—
90C22×C4×C71C22⋊C411yescharacteristic
91C22×C4×C71C22⋊C411yescharacteristic
92C22×C4×C71C22⋊C411yescharacteristic
93C22×C4×C71C22×C411yescharacteristic
94C22×C4×C71C2×D411yescharacteristic
95C4⋊C4×C71—41no—
96C4⋊C4×C71—41no—
97C4⋊C4×C71—42no—
98C22⋊C4×C71—41no—
99C22⋊C4×C71—41no—
100C22⋊C4×C71—42no—
101C22⋊C4×C71—42no—
102C22⋊C4×C71—41no—
103C22⋊C4×C71—41no—
104C22⋊C4×C71—81no—
105C22⋊C4×C71—81no—
106C42×C71—21no—
107C42×C71—21no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—22no—
111C23×C71—21no—
112C23×C71—21no—
113C23×C71—21no—
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—41no—
123C23×C71—41no—
124C23×C71C2×C22⋊C411yescharacteristic
125C23×C71AGL(1,ℤ/8)12yes—
126D4×C71—44no—
127D4×C71—42no—
128D4×C71—42no—
129D4×C71—84no—
130D4×C71—84no—
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—42no—
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—41no—
150C2×C4×C71—41no—
151C2×C4×C71—81no—
152C2×C4×C71—81no—
153C2×C4×C71—84no—
154C2×C4×C71#9911yescharacteristic
155C2×C4×C71#9911yescharacteristic
156C2×C4×C71#10412yes—
157C2×Dic71—44no—
158C2×Dic71—82no—
159C2×Dic71—82no—
160C71⋊C8—82no—
161C22×C71—21no—
162C22×C71—21no—
163C22×C71—22no—
164C22×C71—42no—
165C22×C71—42no—
166C22×C71—42no—
167C22×C71—42no—
168C22×C71—41no—
169C22×C71—41no—
170C22×C71—42no—
171C22×C71—41no—
172C22×C71—41no—
173C22×C71—41no—
174C22×C71—81no—
175C22×C71—81no—
176C22×C71—81no—
177C22×C71—81no—
178C22×C71—81no—
179C22×C71—81no—
180C22×C71C2×#9911yescharacteristic
181C22×C71#41811yescharacteristic
182C22×C71C2×#10411yescharacteristic
183C4×C71—21no—
184C4×C71—21no—
185C4×C71—22no—
186C4×C71—41no—
187C4×C71—41no—
188C4×C71—41no—
189C4×C71—81no—
190C4×C71—81no—
191C4×C71—81no—
192Dic71—84no—
193#11774—711nomaximal, Sylow
194C2×C71—21no—
195C2×C71—21no—
196C2×C71—42no—
197C2×C71—81no—
198C2×C71—82no—
199C2×C71#150711yescharacteristic
200C2×C71#212011yescharacteristic
201C2×C71#212711yescharacteristic
202#2535D7111yescharacteristic
203C4×#99—711no—
204#1465—711no—
205#1344—712no—
206#1343—712no—
207C71#1177411yescharacteristic, complemented, Sylow
208C2×#121—21no—
209C2×C4×D4Dic7111yescharacteristic, complemented
210C2×C4×D4C2×D7111yescharacteristic
211C2×#99—1421no—
212#407—1421no—
213#397Dic7111yescharacteristic, complemented
214#397C2×D7111yescharacteristic
215#392Dic7111yescharacteristic, complemented
216#392C2×D7111yescharacteristic
217#386—21no—
218#386—21no—
219#382Dic7111yescharacteristic, complemented
220#380—21no—
221C4×C22⋊C4—1421no—
222#342—712no—
223#324—1421no—
224#323—712no—
225C22×D4—21no—
226C22×D4—21no—
227C22×D4C2×Dic7111yescharacteristic
228C22×D4C22⋊D7111yescharacteristic
229C23×C4—21no—
230C23×C4C22⋊D7111yescharacteristic
231C23×C4C22⋊D7111yescharacteristic
232C2×M16—2841no—
233#121—44no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—41no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C2×Dic7111yescharacteristic
240C2×C4⋊C4C22⋊D7111yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—1422no—
247C2×C22⋊C4C2×Dic7111yescharacteristic
248C2×C22⋊C4C22×D7111yescharacteristic
249C2×C42—2841no—
250C2×C42C2×Dic7111yescharacteristic
251#99—1424no—
252C22⋊C8—2841no—
253C22⋊C8—1422no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—2841no—
258#95C2×Dic7111yescharacteristic
259#95C2×Dic7111yescharacteristic
260C24—21no—
261C24C22⋊Dic7111yescharacteristic
262C24C2×C22⋊D7111yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—81no—
273C2×D4—81no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—2842no—
287C22×C4C22⋊Dic7111yescharacteristic
288C22×C4C22⋊Dic7111yescharacteristic
289C22×C4C22⋊Dic7111yescharacteristic
290C22×C4C22×Dic7111yescharacteristic
291C22×C4C2×C22⋊D7111yescharacteristic
292M16—5682no—
293C2×C8—2842no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—81no—
304C22⋊C4—81no—
305C22⋊C4—2842no—
306C22⋊C4—1424no—
307C42—21no—
308C42—21no—
309C42—5682no—
310C23—21no—
311C23—21no—
312C23—22no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23C2×C22⋊Dic7111yescharacteristic
327C23#1455615412yes—
328D4—44no—
329D4—42no—
330D4—42no—
331D4—84no—
332D4—84no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
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—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—84no—
356C2×C4—2844no—
357C2×C4—5682no—
358C2×C4—5682no—
359C2×C4#1455604311yescharacteristic
360C2×C4#1455604311yescharacteristic
361C2×C4#1455605512yes—
362C8—5682no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22C2×#1455604311yescharacteristic
383C22#1798600411yescharacteristic
384C22C2×#1455605511yescharacteristic
385C4—21no—
386C4—21no—
387C4—22no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—5684no—
395C2—21no—
396C2—21no—
397C2—42no—
398C2—81no—
399C2—82no—
400C2#2258685611yescharacteristic
401C2#2258998111yescharacteristic
402C2#2259000911yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
Fitting, prime radicalC71×#2535C2
socleC22×C71C2×#99
cosocleC2×C22⋊C4×C71C23
hypercenter#2535D71
hypocenterC71#11774
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. C4×#14556043C2semidirect
2. #22586678C2semidirect
3. #22586340C2semidirect
4. #22586337C2semidirect
13. C2×C4×D4×C71C4semidirect
15. #397×C71C4semidirect
17. #392×C71C4semidirect
21. #382×C71C4semidirect
207. C71#11774semidirect
209. C2×C4×D4Dic71semidirect
213. #397Dic71semidirect
215. #392Dic71semidirect
219. #382Dic71semidirect

Maximal subgroups

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

subgroupindex
C4×#145560432
#225866782
#225863402
#225863372
C71×#25352
#1177471