Finite Group Databasebeta

← #31892437

Subgroups — #31892437

Order 18176.

558 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 111 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#22597381C211yesmaximal, characteristic
2#22588439C212yesmaximal, complemented
3C4×#14556043C211yesmaximal, characteristic
4#22586226C212yesmaximal, complemented
5C71×#2395C211yesmaximal, characteristic
6#17986077—24no—
7#17986052—21no—
8C2×#14556043—21no—
9#17985963—21no—
10#17985963C2212yes—
11C4×C22⋊Dic71C2212yes—
12#17985766—21no—
13#17985758—24no—
14C2×C4×D4×C71C2211yescharacteristic
15C2×#117×C71C2211yescharacteristic
16#400×C71C411yescharacteristic, complemented
17#399×C71C411yescharacteristic, complemented
18#398×C71C411yescharacteristic, complemented
19#397×C71—21no—
20#397×C71C411yescharacteristic, complemented
21#394×C71—21no—
22C4×C4⋊C4×C71—21no—
23C4×C22⋊C4×C71—21no—
24C4×C22⋊C4×C71C2211yescharacteristic
25C2×#11765849—22no—
26C2×#11765849D414yes—
27C2×C22⋊Dic71—22no—
28C2×C4×Dic71—22no—
29C4⋊(C71⋊C8)—22no—
30#14556043—24no—
31C22⋊(C71⋊C8)—22no—
32C22⋊(C71⋊C8)—22no—
33C22⋊(C71⋊C8)—22no—
34C4×C71⋊C8—22no—
35C71⋊#95—22no—
36C22×D4×C71D411yescharacteristic
37C23×C4×C71D411yescharacteristic
38C23×C4×C71C2311yescharacteristic
39C4×D4×C71—22no—
40C4×D4×C71—22no—
41#117×C71—22no—
42#117×C71—22no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C4⋊C4×C71C2×C411yescharacteristic
47C2×C4⋊C4×C71C2×C411yescharacteristic
48C2×C22⋊C4×C71—21no—
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71C2×C411yescharacteristic
53C2×C22⋊C4×C71D411yescharacteristic
54C2×C42×C71—21no—
55C2×C42×C71—21no—
56C2×C42×C71D411yescharacteristic
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60#95×C71—21no—
61#95×C71C2×C411yescharacteristic
62#95×C71C2×C411yescharacteristic
63C22×Dic71—42no—
64#11765849—44no—
65#11765849—82no—
66C2×C71⋊C8—24no—
67C22⋊Dic71—24no—
68C22⋊Dic71—42no—
69C4×Dic71—44no—
70C24×C71C22⋊C411yescharacteristic
71C24×C71C2×D411yescharacteristic
72C2×D4×C71—22no—
73C2×D4×C71—22no—
74C2×D4×C71—41no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—21no—
81C22×C4×C71—42no—
82C22×C4×C71—42no—
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71C22×C411yescharacteristic
87C22×C4×C71C2×D412yes—
88C22×C4×C71C2×D411yescharacteristic
89C22×C4×C71#4112yes—
90C4⋊C4×C71—22no—
91C4⋊C4×C71—22no—
92C4⋊C4×C71—22no—
93C4⋊C4×C71—42no—
94C4⋊C4×C71—42no—
95C22⋊C4×C71—22no—
96C22⋊C4×C71—22no—
97C22⋊C4×C71—42no—
98C22⋊C4×C71—42no—
99C22⋊C4×C71—42no—
100C22⋊C4×C71—42no—
101C42×C71—21no—
102C42×C71—21no—
103C42×C71—42no—
104C42×C71—41no—
105C42×C71—41no—
106C23×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×C71C2×C22⋊C411yescharacteristic
116C23×C71#12112yes—
117D4×C71—44no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—22no—
121C2×C4×C71—24no—
122C2×C4×C71—22no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—41no—
126C2×C4×C71—42no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—42no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—81no—
142C2×C4×C71#10412yes—
143C2×C4×C71C4×D412yes—
144C2×C4×C71#12011yescharacteristic
145C2×C4×C71#12311yescharacteristic
146C2×Dic71—44no—
147C2×Dic71—82no—
148C2×Dic71—82no—
149C71⋊C8—44no—
150C22×C71—21no—
151C22×C71—21no—
152C22×C71—22no—
153C22×C71—22no—
154C22×C71—22no—
155C22×C71—42no—
156C22×C71—41no—
157C22×C71—41no—
158C22×C71—41no—
159C22×C71—81no—
160C22×C71—81no—
161C22×C71—81no—
162C22×C71#38611yescharacteristic
163C22×C71C2×#10411yescharacteristic
164C22×C71#42111yescharacteristic
165C4×C71—22no—
166C4×C71—21no—
167C4×C71—22no—
168C4×C71—42no—
169C4×C71—41no—
170C4×C71—41no—
171C4×C71—81no—
172C4×C71—81no—
173C4×C71—81no—
174C4×C71#45812yes—
175Dic71—84no—
176#9012—711nomaximal, Sylow
177C2×C71—21no—
178C2×C71—21no—
179C2×C71—42no—
180C2×C71—81no—
181C2×C71#187111yescharacteristic
182C2×C71#188411yescharacteristic
183C2×C71#189011yescharacteristic
184#2975—711no—
185#2395D7111yescharacteristic
186#1826—712no—
187C4×#99—711no—
188#1297—712no—
189C71#901211yescharacteristic, complemented, Sylow
190C2×C4×D4C2×D7111yescharacteristic
191C2×#117C2×D7111yescharacteristic
192#436—1424no—
193#431—1421no—
194C2×#99—1421no—
195#407—712no—
196#407—1421no—
197#400Dic7111yescharacteristic, complemented
198#399Dic7111yescharacteristic, complemented
199#398Dic7111yescharacteristic, complemented
200#397—21no—
201#397Dic7111yescharacteristic, complemented
202#394—21no—
203C4×C4⋊C4—21no—
204C4×C22⋊C4—21no—
205C4×C22⋊C4—712no—
206C4×C22⋊C4C2×D7111yescharacteristic
207#342—1421no—
208#339—1424no—
209C22×D4C22⋊D7111yescharacteristic
210C23×C4C22⋊D7111yescharacteristic
211C23×C4C22×D7111yescharacteristic
212C2×M16—714no—
213C2×M16—1422no—
214C4×D4—22no—
215C4×D4—22no—
216#117—22no—
217#117—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C2×Dic7111yescharacteristic
221C2×C4⋊C4C2×Dic7111yescharacteristic
222C2×C4⋊C4C2×Dic7111yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—1422no—
228C2×C22⋊C4C2×Dic7111yescharacteristic
229C2×C22⋊C4C22⋊D7111yescharacteristic
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—1422no—
233C2×C42C22⋊D7111yescharacteristic
234C4⋊C8—1422no—
235#99—1424no—
236C22⋊C8—1422no—
237C22⋊C8—1422no—
238C22⋊C8—1422no—
239C4×C8—1422no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—1422no—
245#95C2×Dic7111yescharacteristic
246#95C2×Dic7111yescharacteristic
247C24C22⋊Dic7111yescharacteristic
248C24C2×C22⋊D7111yescharacteristic
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—2842no—
261C22×C4C22⋊Dic7111yescharacteristic
262C22×C4C22⋊Dic7111yescharacteristic
263C22×C4C22⋊Dic7111yescharacteristic
264C22×C4C22×Dic7111yescharacteristic
265C22×C4D4×D7112yes—
266C22×C4C2×C22⋊D7111yescharacteristic
267C22×C4#1176586412yes—
268M16—2844no—
269M16—5682no—
270C2×C8—1424no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—2842no—
283C22⋊C4—1424no—
284C42—21no—
285C42—21no—
286C42—42no—
287C42—41no—
288C42—41no—
289C42—2844no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23C2×C22⋊Dic7111yescharacteristic
300C23#1455610012yes—
301D4—44no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—2844no—
327C2×C4—5682no—
328C2×C4—5682no—
329C2×C4#1455605512yes—
330C2×C4D4×Dic7112yes—
331C2×C4#1455609611yescharacteristic
332C2×C4#1455610911yescharacteristic
333C8—2844no—
334C22—21no—
335C22—21no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22#1798587911yescharacteristic
347C22C2×#1455605511yescharacteristic
348C22#1798601611yescharacteristic
349C4—21no—
350C4—22no—
351C4—22no—
352C4—42no—
353C4—41no—
354C4—41no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—5684no—
359C4#1798618812yes—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—81no—
364C2#2258864311yescharacteristic
365C2#2258872411yescharacteristic
366C2#2258875411yescharacteristic
subgroupG/H
centerC2×C4#14556096
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2395C2
socleC22×C71#386
prime radicalC22×D4×C71D4
cosocleC23×C4×C71C23
hypercenter#2395D71
hypocenterC71#9012
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2395

Complements

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

NH ≅ G/Ntype
2. #22588439C2semidirect
4. #22586226C2semidirect
16. #400×C71C4semidirect
17. #399×C71C4semidirect
18. #398×C71C4semidirect
20. #397×C71C4semidirect
189. C71#9012semidirect
197. #400Dic71semidirect
198. #399Dic71semidirect
199. #398Dic71semidirect
201. #397Dic71semidirect

Maximal subgroups

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

subgroupindex
#225973812
#225884392
C4×#145560432
#225862262
C71×#23952
#901271