Finite Group Databasebeta

← #31895107

Subgroups — #31895107

Order 18176.

516 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 103 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#22596978C211yesmaximal, characteristic, complemented
2#22588417C211yesmaximal, characteristic, complemented
3#22586869C212yesmaximal
4#22586193C212yesmaximal, complemented
5C71×#2500C211yesmaximal, characteristic
6#17986060—24no—
7#17986023—21no—
8#17985963C2212yes—
9C4×#11765849—21no—
10#17985862C2212yes—
11#17985766—21no—
12#17985766—21no—
13#17985733—24no—
14C2×#121×C71C2211yescharacteristic
15C2×#117×C71C2211yescharacteristic
16#394×C71—21no—
17#394×C71—21no—
18#390×C71C411yescharacteristic, complemented
19#390×C71C411yescharacteristic, complemented
20#388×C71—21no—
21#388×C71—21no—
22#384×C71C411yescharacteristic, complemented
23#382×C71C411yescharacteristic, complemented
24C4×C22⋊C4×C71C2211yescharacteristic
25C2×#11765849—22no—
26C2×C22⋊Dic71—22no—
27C4⋊(C71⋊C8)—24no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)D414yes—
30#14556039—22no—
31C4×C71⋊C8—22no—
32C71⋊#95—22no—
33C71⋊#95—22no—
34C22×D4×C71—21no—
35C22×D4×C71C2×C411yescharacteristic
36C23×C4×C71C2311yescharacteristic
37#121×C71—22no—
38#121×C71—22no—
39#117×C71—22no—
40#117×C71—22no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C4⋊C4×C71C2×C411yescharacteristic
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71—21no—
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71D411yescharacteristic
53C2×C22⋊C4×C71D411yescharacteristic
54C2×C42×C71C2×C411yescharacteristic
55C2×C42×C71C2×C411yescharacteristic
56C2×C42×C71D411yescharacteristic
57C2×C42×C71D411yescharacteristic
58#95×C71—21no—
59#95×C71—21no—
60#95×C71—21no—
61C22×Dic71—42no—
62#11765849—42no—
63#11765849—42no—
64C2×C71⋊C8—24no—
65C22⋊Dic71—42no—
66C22⋊Dic71—42no—
67C24×C71—21no—
68C24×C71C2×D411yescharacteristic
69C2×D4×C71—22no—
70C2×D4×C71—41no—
71C2×D4×C71—41no—
72C2×D4×C71—42no—
73C2×D4×C71—81no—
74C2×D4×C71—81no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—41no—
79C22×C4×C71—41no—
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71C22×C411yescharacteristic
85C22×C4×C71C2×D412yes—
86C22×C4×C71C2×D411yescharacteristic
87C22×C4×C71#4112yes—
88C4⋊C4×C71—22no—
89C4⋊C4×C71—22no—
90C4⋊C4×C71—22no—
91C4⋊C4×C71—81no—
92C4⋊C4×C71—81no—
93C22⋊C4×C71—22no—
94C22⋊C4×C71—22no—
95C22⋊C4×C71—42no—
96C22⋊C4×C71—42no—
97C22⋊C4×C71—42no—
98C22⋊C4×C71—41no—
99C22⋊C4×C71—41no—
100C22⋊C4×C71—81no—
101C22⋊C4×C71—81no—
102C42×C71—21no—
103C42×C71—21no—
104C42×C71—21no—
105C42×C71—21no—
106C42×C71—41no—
107C42×C71—41no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—41no—
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×C71C2×C22⋊C411yescharacteristic
120C23×C71#12112yes—
121D4×C71—82no—
122D4×C71—84no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—22no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—42no—
133C2×C4×C71—42no—
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—41no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71C4×D412yes—
147C2×C4×C71#12011yescharacteristic
148C2×C4×C71#12311yescharacteristic
149C2×Dic71—82no—
150C2×Dic71—82no—
151C2×Dic71—82no—
152C71⋊C8—44no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—22no—
156C22×C71—22no—
157C22×C71—22no—
158C22×C71—81no—
159C22×C71—81no—
160C22×C71—81no—
161C22×C71—81no—
162C22×C71—81no—
163C22×C71—81no—
164C22×C71—81no—
165C22×C71—81no—
166C22×C71—81no—
167C22×C71#38611yescharacteristic
168C22×C71#41011yescharacteristic
169C22×C71#41011yescharacteristic
170C4×C71—21no—
171C4×C71—22no—
172C4×C71—21no—
173C4×C71—42no—
174C4×C71—41no—
175C4×C71—41no—
176C4×C71—41no—
177C4×C71—41no—
178C4×C71—81no—
179C4×C71—81no—
180Dic71—162no—
181#9446—711nomaximal, Sylow
182C2×C71—21no—
183C2×C71—21no—
184C2×C71—42no—
185C2×C71—161no—
186C2×C71#188411yescharacteristic
187C2×C71#189911yescharacteristic
188C2×C71#191011yescharacteristic
189#2921—711no—
190#2500D7111yescharacteristic
191#1820—711no—
192#1510—712no—
193#1284—712no—
194C71#944611yescharacteristic, complemented, Sylow
195C2×#121C2×D7111yescharacteristic
196C2×#117C2×D7111yescharacteristic
197#433—1424no—
198#423—1421no—
199#407—712no—
200C4×M16—1421no—
201#394—21no—
202#394—21no—
203#390Dic7111yescharacteristic, complemented
204#390Dic7111yescharacteristic, complemented
205#388—21no—
206#388—21no—
207#384Dic7111yescharacteristic, complemented
208#382Dic7111yescharacteristic, complemented
209#381—712no—
210C4×C22⋊C4C2×D7111yescharacteristic
211#342—1421no—
212#342—1421no—
213#327—1424no—
214C22×D4—21no—
215C22×D4C2×Dic7111yescharacteristic
216C23×C4C22×D7111yescharacteristic
217C2×M16—1422no—
218#121—22no—
219#121—22no—
220#117—22no—
221#117—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C2×Dic7111yescharacteristic
225C2×C4⋊C4C2×Dic7111yescharacteristic
226C2×C4⋊C4C2×Dic7111yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—1422no—
234C2×C22⋊C4C22⋊D7111yescharacteristic
235C2×C22⋊C4C22⋊D7111yescharacteristic
236C2×C42C2×Dic7111yescharacteristic
237C2×C42C2×Dic7111yescharacteristic
238C2×C42C22⋊D7111yescharacteristic
239C2×C42C22⋊D7111yescharacteristic
240C4⋊C8—1424no—
241C22⋊C8—714no—
242C22⋊C8—1422no—
243#97—1422no—
244C4×C8—1422no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—1422no—
249#95—1422no—
250C24—21no—
251C24C2×C22⋊D7111yescharacteristic
252C2×D4—22no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—42no—
256C2×D4—81no—
257C2×D4—81no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—2842no—
264C22×C4C22⋊Dic7111yescharacteristic
265C22×C4C22⋊Dic7111yescharacteristic
266C22×C4C22⋊Dic7111yescharacteristic
267C22×C4C22⋊Dic7111yescharacteristic
268C22×C4C22×Dic7111yescharacteristic
269C22×C4D4×D7112yes—
270C22×C4C2×C22⋊D7111yescharacteristic
271C22×C4#1176586412yes—
272M16—2842no—
273M16—2842no—
274C2×C8—1424no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—81no—
279C4⋊C4—81no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—81no—
288C22⋊C4—81no—
289C22⋊C4—2842no—
290C22⋊C4—2842no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—41no—
296C42—41no—
297C23—21no—
298C23—21no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23C2×C22⋊Dic7111yescharacteristic
309C23#1455610012yes—
310D4—82no—
311D4—84no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—5682no—
336C2×C4—5682no—
337C2×C4—5682no—
338C2×C4D4×Dic7112yes—
339C2×C4#1455609611yescharacteristic
340C2×C4#1455610911yescharacteristic
341C8—2844no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#1798587911yescharacteristic
357C22#1798597311yescharacteristic
358C22#1798597311yescharacteristic
359C4—21no—
360C4—22no—
361C4—21no—
362C4—42no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—81no—
368C4—81no—
369C4—11362no—
370C2—21no—
371C2—21no—
372C2—42no—
373C2—161no—
374C2#2258872411yescharacteristic
375C2#2258879911yescharacteristic
376C2#2258886811yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2500C2
socleC22×C71#386
prime radicalC2×#121×C71C22
cosocleC23×C4×C71C23
hypercenter#2500D71
hypocenterC71#9446
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2500

Complements

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

NH ≅ G/Ntype
1. #22596978C2semidirect
2. #22588417C2semidirect
4. #22586193C2semidirect
18. #390×C71C4semidirect
19. #390×C71C4semidirect
22. #384×C71C4semidirect
23. #382×C71C4semidirect
194. C71#9446semidirect
203. #390Dic71semidirect
204. #390Dic71semidirect
207. #384Dic71semidirect
208. #382Dic71semidirect

Maximal subgroups

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

subgroupindex
#225969782
#225884172
#225868692
#225861932
C71×#25002
#944671