Finite Group Databasebeta

← #31907963

Subgroups — #31907963

Order 18176.

446 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#17985768C211yesmaximal, characteristic
2#22586670C211yesmaximal, characteristic
3#22586367C212yesmaximal, complemented
4#22586346C212yesmaximal, complemented
5C71×#2484C211yesmaximal, characteristic
6C22×#11765849—21no—
7#17985963—21no—
8#17985768—21no—
9#17985768C2212yes—
10#17985725—21no—
11#17985725C2212yes—
12C2×#117×C71C411yescharacteristic
13C22×C4⋊C4×C71C2211yescharacteristic
14#400×C71C411yescharacteristic
15#400×C71C2211yescharacteristic
16#398×C71C411yescharacteristic
17#398×C71C2211yescharacteristic
18#387×C71—21no—
19#387×C71—21no—
20#385×C71—21no—
21#381×C71—21no—
22#381×C71C411yescharacteristic
23C2×#11765849—24no—
24C2×#11765849—22no—
25C2×#11765849—41no—
26C2×#11765849—42no—
27C2×#11765849—41no—
28C22×C71⋊C8—41no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—41no—
31C23×C4×C71C2×C411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C23×C4×C71D411yescharacteristic
34#117×C71—41no—
35#117×C71—41no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—41no—
39C2×C4⋊C4×C71—41no—
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C4⋊C4×C71C2×C411yescharacteristic
43C2×C4⋊C4×C71D411yescharacteristic
44C2×C4⋊C4×C71D411yescharacteristic
45C2×C4⋊C4×C71C2311yescharacteristic
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71—21no—
48C2×C42×C71—21no—
49#95×C71—21no—
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56#95×C71C2×C411yescharacteristic
57#95×C71C2×C411yescharacteristic
58#11765849—24no—
59#11765849—44no—
60#11765849—42no—
61#11765849—82no—
62C2×C71⋊C8—44no—
63C2×C71⋊C8—42no—
64C2×C71⋊C8—42no—
65C24×C71C22⋊C411yescharacteristic
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—22no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—42no—
80C22×C4×C71—42no—
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71C22×C411yescharacteristic
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71C2×D411yescharacteristic
87C4⋊C4×C71—41no—
88C4⋊C4×C71—41no—
89C4⋊C4×C71—41no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—41no—
93C4⋊C4×C71—42no—
94C4⋊C4×C71—41no—
95C4⋊C4×C71—41no—
96C22⋊C4×C71—42no—
97C22⋊C4×C71—41no—
98C22⋊C4×C71—41no—
99C42×C71—42no—
100C23×C71—22no—
101C23×C71—21no—
102C23×C71—21no—
103C23×C71—41no—
104C23×C71#9911yescharacteristic
105C23×C71#9911yescharacteristic
106C23×C71C2×C22⋊C411yescharacteristic
107C2×C4×C71—22no—
108C2×C4×C71—22no—
109C2×C4×C71—22no—
110C2×C4×C71—22no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
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—81no—
138C2×C4×C71#10412yes—
139C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C71#13711yescharacteristic
141C71⋊C8—44no—
142C71⋊C8—82no—
143C22×C71—21no—
144C22×C71—21no—
145C22×C71—21no—
146C22×C71—21no—
147C22×C71—21no—
148C22×C71—21no—
149C22×C71—21no—
150C22×C71—42no—
151C22×C71—42no—
152C22×C71#35212yes—
153C22×C71C2×#9911yescharacteristic
154C22×C71#41711yescharacteristic
155C22×C71C2×#10411yescharacteristic
156C4×C71—21no—
157C4×C71—22no—
158C4×C71—21no—
159C4×C71—41no—
160C4×C71—41no—
161C4×C71—41no—
162C4×C71—41no—
163C4×C71—81no—
164C4×C71—81no—
165C4×C71—81no—
166C4×C71—81no—
167#11764—711nomaximal, Sylow
168C2×C71—21no—
169C2×C71—21no—
170C2×C71—22no—
171C2×C71—41no—
172C2×C71#150411yescharacteristic
173C2×C71C2×#35211yescharacteristic
174C2×C71#213311yescharacteristic
175#2484D7111yescharacteristic
176C2×#343—711no—
177#1463—711no—
178#1353—712no—
179#1346—712no—
180C71#1176411yescharacteristic, complemented, Sylow
181C22×M16—1421no—
182C2×#117Dic7111yescharacteristic
183C22×C4⋊C4C2×D7111yescharacteristic
184#407—1421no—
185#400Dic7111yescharacteristic
186#400C2×D7111yescharacteristic
187#398Dic7111yescharacteristic
188#398C2×D7111yescharacteristic
189#387—21no—
190#387—21no—
191#385—21no—
192#381—21no—
193#381Dic7111yescharacteristic
194#343—712no—
195#343—1421no—
196#324—712no—
197#324—1421no—
198C23×C4C2×Dic7111yescharacteristic
199C23×C4C22⋊D7111yescharacteristic
200C23×C4C22⋊D7111yescharacteristic
201C2×M16—2841no—
202C2×M16—2842no—
203C2×M16—2841no—
204C2×M16—1424no—
205C2×M16—1422no—
206C22×C8—2841no—
207#117—41no—
208#117—41no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4C2×Dic7111yescharacteristic
214C2×C4⋊C4C2×Dic7111yescharacteristic
215C2×C4⋊C4C2×Dic7111yescharacteristic
216C2×C4⋊C4C22⋊D7111yescharacteristic
217C2×C4⋊C4C22⋊D7111yescharacteristic
218C2×C4⋊C4C22×D7111yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C42—21no—
222C22⋊C8—2841no—
223C22⋊C8—1422no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95C2×Dic7111yescharacteristic
232#95C2×Dic7111yescharacteristic
233C24C22⋊Dic7111yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4C22⋊Dic7111yescharacteristic
250C22×C4C22⋊Dic7111yescharacteristic
251C22×C4C22⋊Dic7111yescharacteristic
252C22×C4C22×Dic7111yescharacteristic
253C22×C4C2×C22⋊D7111yescharacteristic
254C22×C4C2×C22⋊D7111yescharacteristic
255M16—2844no—
256M16—2842no—
257M16—5682no—
258M16—1424no—
259C2×C8—2844no—
260C2×C8—2842no—
261C2×C8—2842no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C42—42no—
275C23—21no—
276C23—21no—
277C23—22no—
278C23—41no—
279C23#1455604311yescharacteristic
280C23#1455604311yescharacteristic
281C23C2×C22⋊Dic7111yescharacteristic
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4#1455605512yes—
314C2×C4#1455615311yescharacteristic
315C2×C4#1455615811yescharacteristic
316C8—2844no—
317C8—5682no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22#1798578912yes—
328C22C2×#1455604311yescharacteristic
329C22#1798600011yescharacteristic
330C22C2×#1455605511yescharacteristic
331C4—21no—
332C4—22no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C2—21no—
343C2—21no—
344C2—22no—
345C2—41no—
346C2#2258684411yescharacteristic
347C2C2×#1798578911yescharacteristic
348C2#2259002911yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2484C2
socleC22×C71C2×#99
prime radicalC24×C71C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2484D71
hypocenterC71#11764
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
3. #22586367C2semidirect
4. #22586346C2semidirect
180. C71#11764semidirect

Maximal subgroups

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

subgroupindex
C2×#179857682
#225866702
#225863672
#225863462
C71×#24842
#1176471