Finite Group Databasebeta

← #31894798

Subgroups — #31894798

Order 18176.

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

normal subgroups only

This group has 496 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#22586938C212yesmaximal, complemented
3#22586674C211yesmaximal, characteristic, complemented
4#22586233C212yesmaximal, complemented
5C71×#2610C211yesmaximal, characteristic
6#17986060—24no—
7#17986056C2212yes—
8#17986023—21no—
9#17985963—21no—
10C4×#11765849C2212yes—
11#17985739—24no—
12#17985725—21no—
13#17985723—21no—
14C2×#124×C71C2211yescharacteristic
15C2×#117×C71C2211yescharacteristic
16#395×C71C411yescharacteristic
17#394×C71C411yescharacteristic
18#393×C71C411yescharacteristic
19#392×C71C411yescharacteristic
20#390×C71—21no—
21#388×C71—21no—
22#388×C71—21no—
23#386×C71—21no—
24#376×C71C2211yescharacteristic
25C2×#11765849—22no—
26C4⋊(C71⋊C8)—24no—
27C4⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—24no—
29C22⋊(C71⋊C8)—22no—
30#14556039—22no—
31#14556039D414yescomplemented
32C4×C71⋊C8—22no—
33C22×Q8×C71C2×C411yescharacteristic
34C22×D4×C71—21no—
35C22×D4×C71C2×C411yescharacteristic
36C23×C4×C71D411yescharacteristic
37#124×C71—22no—
38#124×C71—22no—
39#117×C71—22no—
40#117×C71—22no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71C2×C411yescharacteristic
43C2×C4⋊C4×C71D411yescharacteristic
44C2×C22⋊C4×C71—21no—
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×C71D411yescharacteristic
52C2×C42×C71—21no—
53C2×C42×C71D411yescharacteristic
54C2×C42×C71C2311yescharacteristic
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58#95×C71C2×C411yescharacteristic
59#95×C71C2×C411yescharacteristic
60#11765849—42no—
61#11765849—42no—
62C2×C71⋊C8—24no—
63C2×C71⋊C8—42no—
64C24×C71—21no—
65C24×C71C22⋊C411yescharacteristic
66C2×Q8×C71—22no—
67C2×Q8×C71—81no—
68C2×D4×C71—22no—
69C2×D4×C71—42no—
70C2×D4×C71—42no—
71C2×D4×C71—81no—
72C2×D4×C71—81no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
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×C71C2×D411yescharacteristic
85C22×C4×C71C2×D411yescharacteristic
86C4⋊C4×C71—22no—
87C4⋊C4×C71—22no—
88C4⋊C4×C71—81no—
89C22⋊C4×C71—22no—
90C22⋊C4×C71—22no—
91C22⋊C4×C71—42no—
92C22⋊C4×C71—42no—
93C22⋊C4×C71—41no—
94C22⋊C4×C71—42no—
95C22⋊C4×C71—41no—
96C22⋊C4×C71—81no—
97C22⋊C4×C71—81no—
98C42×C71—21no—
99C42×C71—21no—
100C42×C71—42no—
101C42×C71C2×D412yes—
102C42×C71#4112yes—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—21no—
106C23×C71—41no—
107C23×C71—41no—
108C23×C71—41no—
109C23×C71—41no—
110C23×C71—41no—
111C23×C71—41no—
112C23×C71—41no—
113C23×C71—41no—
114C23×C71—41no—
115C23×C71C2×C22⋊C411yescharacteristic
116Q8×C71—82no—
117D4×C71—82no—
118D4×C71—84no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—22no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—42no—
130C2×C4×C71—42no—
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×C71C4×D412yes—
142C2×C4×C71#12011yescharacteristic
143C2×C4×C71#12112yes—
144C2×C4×C71#12311yescharacteristic
145C71⋊C8—44no—
146C71⋊C8—82no—
147C22×C71—21no—
148C22×C71—21no—
149C22×C71—42no—
150C22×C71—41no—
151C22×C71—41no—
152C22×C71—41no—
153C22×C71—81no—
154C22×C71—81no—
155C22×C71—81no—
156C22×C71—81no—
157C22×C71—81no—
158C22×C71—81no—
159C22×C71—81no—
160C22×C71—81no—
161C22×C71#38611yescharacteristic
162C22×C71#42111yescharacteristic
163C22×C71#42111yescharacteristic
164C4×C71—22no—
165C4×C71—21no—
166C4×C71—21no—
167C4×C71—22no—
168C4×C71—42no—
169C4×C71—41no—
170C4×C71—41no—
171C4×C71—41no—
172C4×C71—81no—
173C4×C71—81no—
174#9389—711nomaximal, Sylow
175C2×C71—21no—
176C2×C71—21no—
177C2×C71—81no—
178C2×C71—161no—
179C2×C71#188411yescharacteristic
180C2×C71#189711yescharacteristic
181C2×C71#191411yescharacteristic
182#2921—711no—
183#2610D7111yescharacteristic
184#1527—712no—
185#1464—711no—
186#1300—712no—
187C71#938911yescharacteristic, complemented, Sylow
188C2×#124C2×D7111yescharacteristic
189C2×#117C2×D7111yescharacteristic
190#433—1424no—
191#432—712no—
192#423—1421no—
193#407—1421no—
194C4×M16—712no—
195#395Dic7111yescharacteristic
196#394Dic7111yescharacteristic
197#393Dic7111yescharacteristic
198#392Dic7111yescharacteristic
199#390—21no—
200#388—21no—
201#388—21no—
202#386—21no—
203#376C2×D7111yescharacteristic
204#329—1424no—
205#324—1421no—
206#323—1421no—
207C22×Q8C2×Dic7111yescharacteristic
208C22×D4—21no—
209C22×D4C2×Dic7111yescharacteristic
210C23×C4C22⋊D7111yescharacteristic
211C2×M16—1422no—
212#124—22no—
213#124—22no—
214#117—22no—
215#117—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C2×Dic7111yescharacteristic
218C2×C4⋊C4C22⋊D7111yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4C2×Dic7111yescharacteristic
226C2×C22⋊C4C22⋊D7111yescharacteristic
227C2×C42—21no—
228C2×C42C22⋊D7111yescharacteristic
229C2×C42C22×D7111yescharacteristic
230C4⋊C8—1424no—
231C4⋊C8—1422no—
232C22⋊C8—1422no—
233C22⋊C8—1424no—
234#97—714no—
235#97—1422no—
236C4×C8—1422no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95C2×Dic7111yescharacteristic
241#95C2×Dic7111yescharacteristic
242C24—21no—
243C24C22⋊Dic7111yescharacteristic
244C2×Q8—22no—
245C2×Q8—81no—
246C2×D4—22no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—81no—
250C2×D4—81no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C22⋊Dic7111yescharacteristic
259C22×C4C22⋊Dic7111yescharacteristic
260C22×C4C22⋊Dic7111yescharacteristic
261C22×C4C22×Dic7111yescharacteristic
262C22×C4C2×C22⋊D7111yescharacteristic
263C22×C4C2×C22⋊D7111yescharacteristic
264M16—2842no—
265M16—2842no—
266C2×C8—2842no—
267C2×C8—1424no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—81no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—41no—
278C22⋊C4—81no—
279C22⋊C4—81no—
280C42—21no—
281C42—21no—
282C42—42no—
283C42D4×D7112yes—
284C42#1176586412yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊Dic7111yescharacteristic
298Q8—82no—
299D4—82no—
300D4—84no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
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×C4D4×Dic7112yes—
324C2×C4#1455609611yescharacteristic
325C2×C4#1455610012yes—
326C2×C4#1455610911yescharacteristic
327C8—2844no—
328C8—5682no—
329C22—21no—
330C22—21no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22#1798587911yescharacteristic
344C22#1798601611yescharacteristic
345C22#1798601611yescharacteristic
346C4—22no—
347C4—21no—
348C4—21no—
349C4—22no—
350C4—42no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C2—21no—
357C2—21no—
358C2—81no—
359C2—161no—
360C2#2258872411yescharacteristic
361C2#2258878911yescharacteristic
362C2#2258888811yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2610C2
socleC22×C71#386
prime radical#392×C71C4
cosocleC2×C42×C71C23
hypercenter#2610D71
hypocenterC71#9389
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2610

Complements

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

NH ≅ G/Ntype
1. #22596978C2semidirect
2. #22586938C2semidirect
3. #22586674C2semidirect
4. #22586233C2semidirect
31. #14556039D4semidirect
187. C71#9389semidirect

Maximal subgroups

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

subgroupindex
#225969782
#225869382
#225866742
#225862332
C71×#26102
#938971