Finite Group Databasebeta

← #31906993

Subgroups — #31906993

Order 18176.

460 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 87 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#22586674C211yesmaximal, characteristic, complemented
2#22586654C211yesmaximal, characteristic, complemented
3#22586288C212yesmaximal, complemented
4#22586268C212yesmaximal
5C71×#2482C211yesmaximal, characteristic
6#17986056—21no—
7#17986052—21no—
8#17985725—21no—
9#17985725—21no—
10#17985723C2212yes—
11#17985723C2212yes—
12C2×#117×C71C2211yescharacteristic
13C22×C22⋊C4×C71C411yescharacteristic
14#399×C71—21no—
15#397×C71—21no—
16#388×C71—21no—
17#388×C71—21no—
18#385×C71C411yescharacteristic
19#385×C71C2211yescharacteristic
20#381×C71C411yescharacteristic
21#379×C71C411yescharacteristic
22#379×C71C2211yescharacteristic
23C4⋊(C71⋊C8)—41no—
24C4⋊(C71⋊C8)—41no—
25C22⋊(C71⋊C8)—22no—
26C22⋊(C71⋊C8)—22no—
27#14556039—41no—
28C4×C71⋊C8—41no—
29C25×C71C2×C411yescharacteristic
30C23×C4×C71C2×C411yescharacteristic
31C23×C4×C71C2×C411yescharacteristic
32#117×C71—41no—
33#117×C71—41no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71D411yescharacteristic
37C2×C4⋊C4×C71D411yescharacteristic
38C2×C22⋊C4×C71—21no—
39C2×C22⋊C4×C71—21no—
40C2×C22⋊C4×C71—21no—
41C2×C22⋊C4×C71—21no—
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—41no—
45C2×C22⋊C4×C71—41no—
46C2×C22⋊C4×C71C2×C411yescharacteristic
47C2×C22⋊C4×C71C2×C411yescharacteristic
48C2×C22⋊C4×C71C2×C411yescharacteristic
49C2×C22⋊C4×C71C2311yescharacteristic
50C2×C42×C71D411yescharacteristic
51C2×C42×C71D411yescharacteristic
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56C2×C71⋊C8—42no—
57C2×C71⋊C8—42no—
58C24×C71—21no—
59C24×C71—22no—
60C24×C71—41no—
61C24×C71—42no—
62C24×C71—41no—
63C24×C71—81no—
64C24×C71C22×C411yescharacteristic
65C22×C4×C71—22no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—41no—
72C22×C4×C71—41no—
73C22×C4×C71—81no—
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C2×D411yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80C4⋊C4×C71—41no—
81C4⋊C4×C71—41no—
82C4⋊C4×C71—81no—
83C4⋊C4×C71—81no—
84C22⋊C4×C71—22no—
85C22⋊C4×C71—41no—
86C22⋊C4×C71—41no—
87C22⋊C4×C71—42no—
88C22⋊C4×C71—42no—
89C22⋊C4×C71—42no—
90C22⋊C4×C71—41no—
91C22⋊C4×C71—41no—
92C22⋊C4×C71—41no—
93C22⋊C4×C71—81no—
94C22⋊C4×C71—81no—
95C42×C71—21no—
96C42×C71—21no—
97C42×C71—21no—
98C42×C71—21no—
99C23×C71—21no—
100C23×C71—21no—
101C23×C71—22no—
102C23×C71—41no—
103C23×C71—41no—
104C23×C71—41no—
105C23×C71—41no—
106C23×C71—41no—
107C23×C71—42no—
108C23×C71—42no—
109C23×C71—41no—
110C23×C71—41no—
111C23×C71—41no—
112C23×C71—82no—
113C23×C71—84no—
114C23×C71—84no—
115C23×C71—82no—
116C23×C71C2×C22⋊C411yescharacteristic
117C23×C71#13112yes—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—42no—
128C2×C4×C71—42no—
129C2×C4×C71—41no—
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—41no—
142C2×C4×C71—82no—
143C2×C4×C71#9911yescharacteristic
144C2×C4×C71#9911yescharacteristic
145C71⋊C8—82no—
146C71⋊C8—82no—
147C22×C71—21no—
148C22×C71—21no—
149C22×C71—22no—
150C22×C71—42no—
151C22×C71—42no—
152C22×C71—42no—
153C22×C71—42no—
154C22×C71—41no—
155C22×C71—41no—
156C22×C71—42no—
157C22×C71—81no—
158C22×C71—81no—
159C22×C71—84no—
160C22×C71—84no—
161C22×C71—81no—
162C22×C71—81no—
163C22×C71#40811yescharacteristic
164C22×C71C2×#9911yescharacteristic
165C22×C71#41311yescharacteristic
166C4×C71—21no—
167C4×C71—21no—
168C4×C71—41no—
169C4×C71—41no—
170C4×C71—41no—
171C4×C71—41no—
172C4×C71—41no—
173C4×C71—81no—
174C4×C71—81no—
175C4×C71—81no—
176#11546—711nomaximal, Sylow
177C2×C71—21no—
178C2×C71—21no—
179C2×C71—42no—
180C2×C71—82no—
181C2×C71#146811yescharacteristic
182C2×C71#212011yescharacteristic
183C2×C71#212211yescharacteristic
184#2482D7111yescharacteristic
185#1464—711no—
186#1458—711no—
187#1326—712no—
188#1316—712no—
189C71#1154611yescharacteristic, complemented, Sylow
190C2×#117C2×D7111yescharacteristic
191C22×C22⋊C4Dic7111yescharacteristic
192#432—1421no—
193#431—1421no—
194#399—21no—
195#397—21no—
196#388—21no—
197#388—21no—
198#385Dic7111yescharacteristic
199#385C2×D7111yescharacteristic
200#381Dic7111yescharacteristic
201#379Dic7111yescharacteristic
202#379C2×D7111yescharacteristic
203#324—1421no—
204#324—1421no—
205#323—712no—
206#323—712no—
207C25C2×Dic7111yescharacteristic
208C23×C4C2×Dic7111yescharacteristic
209C23×C4C2×Dic7111yescharacteristic
210#117—41no—
211#117—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C22⋊D7111yescharacteristic
215C2×C4⋊C4C22⋊D7111yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4C2×Dic7111yescharacteristic
225C2×C22⋊C4C2×Dic7111yescharacteristic
226C2×C22⋊C4C2×Dic7111yescharacteristic
227C2×C22⋊C4C22×D7111yescharacteristic
228C2×C42C22⋊D7111yescharacteristic
229C2×C42C22⋊D7111yescharacteristic
230C4⋊C8—2841no—
231C4⋊C8—2841no—
232C22⋊C8—1422no—
233C22⋊C8—1422no—
234#97—2841no—
235C4×C8—2841no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240C24—21no—
241C24—22no—
242C24—41no—
243C24—42no—
244C24—41no—
245C24—81no—
246C24C22×Dic7111yescharacteristic
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—81no—
256C22×C4C22⋊Dic7111yescharacteristic
257C22×C4C22⋊Dic7111yescharacteristic
258C22×C4C22⋊Dic7111yescharacteristic
259C22×C4C22⋊Dic7111yescharacteristic
260C22×C4C2×C22⋊D7111yescharacteristic
261C22×C4C2×C22⋊D7111yescharacteristic
262C2×C8—2842no—
263C2×C8—2842no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—81no—
267C4⋊C4—81no—
268C22⋊C4—22no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—81no—
278C22⋊C4—81no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—21no—
283C23—21no—
284C23—21no—
285C23—22no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—42no—
292C23—42no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—82no—
297C23—84no—
298C23—84no—
299C23—82no—
300C23C2×C22⋊Dic7111yescharacteristic
301C23#1455613512yes—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
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×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—82no—
327C2×C4#1455604311yescharacteristic
328C2×C4#1455604311yescharacteristic
329C8—5682no—
330C8—5682no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—42no—
341C22—81no—
342C22—81no—
343C22—84no—
344C22—84no—
345C22—81no—
346C22—81no—
347C22#1798596611yescharacteristic
348C22C2×#1455604311yescharacteristic
349C22#1798598311yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—81no—
358C4—81no—
359C4—81no—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—82no—
364C2#2258668811yescharacteristic
365C2#2258998111yescharacteristic
366C2#2258999011yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC24×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2482C2
socleC22×C71C2×#99
prime radicalC25×C71C2×C4
cosocleC2×C22⋊C4×C71C23
hypercenter#2482D71
hypocenterC71#11546
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2482

Complements

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

NH ≅ G/Ntype
1. #22586674C2semidirect
2. #22586654C2semidirect
3. #22586288C2semidirect
189. C71#11546semidirect

Maximal subgroups

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

subgroupindex
#225866742
#225866542
#225862882
#225862682
C71×#24822
#1154671