Finite Group Databasebeta

← #31907013

Subgroups — #31907013

Order 18176.

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

normal subgroups only

This group has 408 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#22586284C212yesmaximal, complemented
4#22586272C212yesmaximal, complemented
5C71×#2500C211yesmaximal, characteristic
6#17986056—21no—
7#17986052—21no—
8#17985725C2212yes—
9#17985725C2212yes—
10#17985723—21no—
11#17985723—21no—
12C2×#121×C71C411yescharacteristic
13C2×#117×C71C2211yescharacteristic
14#394×C71—21no—
15#394×C71—21no—
16#390×C71C411yescharacteristic
17#390×C71C2211yescharacteristic
18#388×C71—21no—
19#388×C71—21no—
20#384×C71C411yescharacteristic
21#382×C71C2211yescharacteristic
22C4×C22⋊C4×C71C411yescharacteristic
23C4⋊(C71⋊C8)—41no—
24C4⋊(C71⋊C8)—41no—
25C22⋊(C71⋊C8)—22no—
26C22⋊(C71⋊C8)—22no—
27#14556039—41no—
28C4×C71⋊C8—41no—
29C22×D4×C71—21no—
30C22×D4×C71C2×C411yescharacteristic
31C23×C4×C71C2×C411yescharacteristic
32#121×C71—41no—
33#121×C71—41no—
34#117×C71—41no—
35#117×C71—41no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71C2×C411yescharacteristic
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71C2311yescharacteristic
41C2×C22⋊C4×C71—21no—
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71D411yescharacteristic
48C2×C22⋊C4×C71D411yescharacteristic
49C2×C42×C71C2×C411yescharacteristic
50C2×C42×C71C2×C411yescharacteristic
51C2×C42×C71D411yescharacteristic
52C2×C42×C71D411yescharacteristic
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56C2×C71⋊C8—42no—
57C2×C71⋊C8—42no—
58C24×C71—21no—
59C24×C71C22⋊C411yescharacteristic
60C2×D4×C71—22no—
61C2×D4×C71—41no—
62C2×D4×C71—41no—
63C2×D4×C71—42no—
64C2×D4×C71—81no—
65C2×D4×C71—81no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—22no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—41no—
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22×C411yescharacteristic
77C22×C4×C71C2×D411yescharacteristic
78C22×C4×C71C2×D411yescharacteristic
79C4⋊C4×C71—41no—
80C4⋊C4×C71—41no—
81C4⋊C4×C71—41no—
82C4⋊C4×C71—81no—
83C4⋊C4×C71—81no—
84C22⋊C4×C71—41no—
85C22⋊C4×C71—41no—
86C22⋊C4×C71—42no—
87C22⋊C4×C71—42no—
88C22⋊C4×C71—42no—
89C22⋊C4×C71—41no—
90C22⋊C4×C71—41no—
91C22⋊C4×C71—81no—
92C22⋊C4×C71—81no—
93C42×C71—21no—
94C42×C71—21no—
95C42×C71—21no—
96C42×C71—21no—
97C42×C71—22no—
98C42×C71—41no—
99C23×C71—21no—
100C23×C71—21no—
101C23×C71—21no—
102C23×C71—41no—
103C23×C71—41no—
104C23×C71—41no—
105C23×C71—41no—
106C23×C71—41no—
107C23×C71—41no—
108C23×C71—41no—
109C23×C71—41no—
110C23×C71—41no—
111C23×C71C2×C22⋊C411yescharacteristic
112D4×C71—82no—
113D4×C71—84no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—22no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—42no—
124C2×C4×C71—42no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
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#9911yescharacteristic
138C2×C4×C71#9911yescharacteristic
139C2×C4×C71#13112yes—
140C71⋊C8—82no—
141C71⋊C8—82no—
142C22×C71—21no—
143C22×C71—21no—
144C22×C71—41no—
145C22×C71—41no—
146C22×C71—41no—
147C22×C71—81no—
148C22×C71—81no—
149C22×C71—81no—
150C22×C71—81no—
151C22×C71—81no—
152C22×C71—81no—
153C22×C71—81no—
154C22×C71—81no—
155C22×C71—81no—
156C22×C71#40811yescharacteristic
157C22×C71C2×#9911yescharacteristic
158C22×C71#41311yescharacteristic
159C4×C71—21no—
160C4×C71—21no—
161C4×C71—22no—
162C4×C71—42no—
163C4×C71—41no—
164C4×C71—41no—
165C4×C71—41no—
166C4×C71—41no—
167C4×C71—81no—
168C4×C71—81no—
169#11550—711nomaximal, Sylow
170C2×C71—21no—
171C2×C71—21no—
172C2×C71—81no—
173C2×C71—161no—
174C2×C71#146811yescharacteristic
175C2×C71#212611yescharacteristic
176C2×C71#213411yescharacteristic
177#2500D7111yescharacteristic
178#1464—711no—
179#1458—711no—
180#1324—712no—
181#1318—712no—
182C71#1155011yescharacteristic, complemented, Sylow
183C2×#121Dic7111yescharacteristic
184C2×#117C2×D7111yescharacteristic
185#432—1421no—
186#431—1421no—
187#394—21no—
188#394—21no—
189#390Dic7111yescharacteristic
190#390C2×D7111yescharacteristic
191#388—21no—
192#388—21no—
193#384Dic7111yescharacteristic
194#382C2×D7111yescharacteristic
195C4×C22⋊C4Dic7111yescharacteristic
196#324—712no—
197#324—712no—
198#323—1421no—
199#323—1421no—
200C22×D4—21no—
201C22×D4C2×Dic7111yescharacteristic
202C23×C4C2×Dic7111yescharacteristic
203#121—41no—
204#121—41no—
205#117—41no—
206#117—41no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4C2×Dic7111yescharacteristic
210C2×C4⋊C4C2×Dic7111yescharacteristic
211C2×C4⋊C4C22×D7111yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4C22⋊D7111yescharacteristic
219C2×C22⋊C4C22⋊D7111yescharacteristic
220C2×C42C2×Dic7111yescharacteristic
221C2×C42C2×Dic7111yescharacteristic
222C2×C42C22⋊D7111yescharacteristic
223C2×C42C22⋊D7111yescharacteristic
224C4⋊C8—2841no—
225C4⋊C8—2841no—
226C22⋊C8—1422no—
227C22⋊C8—1422no—
228#97—2841no—
229C4×C8—2841no—
230#95—21no—
231#95—21no—
232#95—21no—
233C24—21no—
234C24C22⋊Dic7111yescharacteristic
235C2×D4—22no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—81no—
240C2×D4—81no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4C22⋊Dic7111yescharacteristic
249C22×C4C22⋊Dic7111yescharacteristic
250C22×C4C22⋊Dic7111yescharacteristic
251C22×C4C22×Dic7111yescharacteristic
252C22×C4C2×C22⋊D7111yescharacteristic
253C22×C4C2×C22⋊D7111yescharacteristic
254C2×C8—2842no—
255C2×C8—2842no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—81no—
260C4⋊C4—81no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—22no—
275C42—41no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23C2×C22⋊Dic7111yescharacteristic
289D4—82no—
290D4—84no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4#1455604311yescharacteristic
315C2×C4#1455604311yescharacteristic
316C2×C4#1455613512yes—
317C8—5682no—
318C8—5682no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22#1798596611yescharacteristic
334C22C2×#1455604311yescharacteristic
335C22#1798598311yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—42no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—81no—
345C4—81no—
346C2—21no—
347C2—21no—
348C2—81no—
349C2—161no—
350C2#2258668811yescharacteristic
351C2#2259000511yescharacteristic
352C2#2259003211yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2500C2
socleC22×C71C2×#99
prime radicalC2×#121×C71C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2500D71
hypocenterC71#11550
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2500

Complements

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

NH ≅ G/Ntype
1. #22586674C2semidirect
2. #22586654C2semidirect
3. #22586284C2semidirect
4. #22586272C2semidirect
182. C71#11550semidirect

Maximal subgroups

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

subgroupindex
#225866742
#225866542
#225862842
#225862722
C71×#25002
#1155071