Finite Group Databasebeta

← #31894793

Subgroups — #31894793

Order 18176.

532 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 111 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#22596759C211yesmaximal, characteristic
2#22586942C212yesmaximal, complemented
3#22586674C211yesmaximal, characteristic
4#22586233C212yesmaximal, complemented
5C71×#2611C211yesmaximal, characteristic
6C22×#11765849—21no—
7#17986056C2212yes—
8#17985955—24no—
9C4×#11765849—21no—
10C4×#11765849C2212yes—
11#17985741—24no—
12#17985725—21no—
13#17985723—21no—
14C2×#125×C71C2211yescharacteristic
15C2×#117×C71C2211yescharacteristic
16#400×C71C411yescharacteristic
17#399×C71C411yescharacteristic
18#398×C71C411yescharacteristic
19#397×C71C411yescharacteristic
20#389×C71—21no—
21#388×C71—21no—
22#387×C71—21no—
23#385×C71—21no—
24#376×C71C2211yescharacteristic
25C2×#11765849—22no—
26C2×#11765849—24no—
27C2×#11765849—24no—
28C2×#11765849—22no—
29C22×C71⋊C8—22no—
30C4⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)—22no—
32#14556039—22no—
33#14556039—22no—
34#14556039D414yes—
35C4×C71⋊C8—22no—
36C4×C71⋊C8—22no—
37C23×C4×C71D411yescharacteristic
38#125×C71—22no—
39#125×C71—22no—
40#117×C71—22no—
41#117×C71—22no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71—21no—
46C2×C4⋊C4×C71C2×C411yescharacteristic
47C2×C4⋊C4×C71C2×C411yescharacteristic
48C2×C4⋊C4×C71C2×C411yescharacteristic
49C2×C4⋊C4×C71D411yescharacteristic
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71C2×C411yescharacteristic
53C2×C22⋊C4×C71D411yescharacteristic
54C2×C42×C71—21no—
55C2×C42×C71D411yescharacteristic
56C2×C42×C71C2311yescharacteristic
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60#95×C71—21no—
61#95×C71—21no—
62#95×C71C2×C411yescharacteristic
63#95×C71C2×C411yescharacteristic
64#11765849—28no—
65#11765849—24no—
66#11765849—24no—
67C2×C71⋊C8—28no—
68C2×C71⋊C8—24no—
69C2×C71⋊C8—42no—
70C24×C71C22⋊C411yescharacteristic
71C22×C4×C71—22no—
72C22×C4×C71—22no—
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×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22×C411yescharacteristic
83C22×C4×C71C2×D411yescharacteristic
84C22×C4×C71C2×D411yescharacteristic
85C4⋊C4×C71—22no—
86C4⋊C4×C71—22no—
87C4⋊C4×C71—22no—
88C4⋊C4×C71—22no—
89C4⋊C4×C71—42no—
90C4⋊C4×C71—42no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—42no—
93C4⋊C4×C71—41no—
94C22⋊C4×C71—22no—
95C22⋊C4×C71—22no—
96C22⋊C4×C71—42no—
97C22⋊C4×C71—42no—
98C42×C71—21no—
99C42×C71—21no—
100C42×C71—42no—
101C42×C71C2×D412yes—
102C42×C71#4112yes—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—41no—
106C23×C71—41no—
107C23×C71#10412yes—
108C23×C71C2×C22⋊C411yescharacteristic
109C2×C4×C71—24no—
110C2×C4×C71—22no—
111C2×C4×C71—22no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—22no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—41no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
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—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71#10412yes—
135C2×C4×C71C4×D412yes—
136C2×C4×C71#12011yescharacteristic
137C2×C4×C71#12112yes—
138C2×C4×C71#12311yescharacteristic
139C71⋊C8—28no—
140C71⋊C8—82no—
141C22×C71—21no—
142C22×C71—21no—
143C22×C71—22no—
144C22×C71—42no—
145C22×C71—41no—
146C22×C71—41no—
147C22×C71—81no—
148C22×C71#38611yescharacteristic
149C22×C71C2×#10411yescharacteristic
150C22×C71C2×#10411yescharacteristic
151C4×C71—22no—
152C4×C71—21no—
153C4×C71—22no—
154C4×C71—21no—
155C4×C71—22no—
156C4×C71—42no—
157C4×C71—41no—
158C4×C71—41no—
159C4×C71—81no—
160C4×C71—81no—
161C4×C71—81no—
162#9388—711nomaximal, Sylow
163C2×C71—21no—
164C2×C71—21no—
165C2×C71—42no—
166C2×C71#188411yescharacteristic
167C2×C71#189611yescharacteristic
168C2×C71#191511yescharacteristic
169#2873—711no—
170#2611D7111yescharacteristic
171#1528—712no—
172#1464—711no—
173#1300—712no—
174C71#938811yescharacteristic, complemented, Sylow
175C22×M16—1421no—
176C2×#125C2×D7111yescharacteristic
177C2×#117C2×D7111yescharacteristic
178#432—712no—
179#405—1424no—
180C4×M16—712no—
181C4×M16—1421no—
182#400Dic7111yescharacteristic
183#399Dic7111yescharacteristic
184#398Dic7111yescharacteristic
185#397Dic7111yescharacteristic
186#389—21no—
187#388—21no—
188#387—21no—
189#385—21no—
190#376C2×D7111yescharacteristic
191#330—1424no—
192#324—1421no—
193#323—1421no—
194C23×C4C22⋊D7111yescharacteristic
195C2×M16—1422no—
196C2×M16—1424no—
197C2×M16—1424no—
198C2×M16—1422no—
199C22×C8—1422no—
200#125—22no—
201#125—22no—
202#117—22no—
203#117—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C2×Dic7111yescharacteristic
209C2×C4⋊C4C2×Dic7111yescharacteristic
210C2×C4⋊C4C2×Dic7111yescharacteristic
211C2×C4⋊C4C22⋊D7111yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4C2×Dic7111yescharacteristic
215C2×C22⋊C4C22⋊D7111yescharacteristic
216C2×C42—21no—
217C2×C42C22⋊D7111yescharacteristic
218C2×C42C22×D7111yescharacteristic
219C4⋊C8—1422no—
220C22⋊C8—1422no—
221#97—714no—
222#97—1422no—
223#97—1422no—
224C4×C8—1422no—
225C4×C8—1422no—
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—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4C22⋊Dic7111yescharacteristic
243C22×C4C22⋊Dic7111yescharacteristic
244C22×C4C22⋊Dic7111yescharacteristic
245C22×C4C22×Dic7111yescharacteristic
246C22×C4C2×C22⋊D7111yescharacteristic
247C22×C4C2×C22⋊D7111yescharacteristic
248M16—1428no—
249M16—1424no—
250M16—1424no—
251C2×C8—2842no—
252C2×C8—1428no—
253C2×C8—1424no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C42—21no—
268C42—21no—
269C42—42no—
270C42D4×D7112yes—
271C42#1176586412yes—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23#1455605512yes—
277C23C2×C22⋊Dic7111yescharacteristic
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4#1455605512yes—
304C2×C4D4×Dic7112yes—
305C2×C4#1455609611yescharacteristic
306C2×C4#1455610012yes—
307C2×C4#1455610911yescharacteristic
308C8—5682no—
309C8—1428no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22#1798587911yescharacteristic
318C22C2×#1455605511yescharacteristic
319C22C2×#1455605511yescharacteristic
320C4—22no—
321C4—21no—
322C4—22no—
323C4—21no—
324C4—22no—
325C4—42no—
326C4—41no—
327C4—41no—
328C4—81no—
329C4—81no—
330C4—81no—
331C2—21no—
332C2—21no—
333C2—42no—
334C2#2258872411yescharacteristic
335C2#2258878411yescharacteristic
336C2#2258889311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2611C2
socleC22×C71#386
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2611D71
hypocenterC71#9388
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2611

Complements

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

NH ≅ G/Ntype
2. #22586942C2semidirect
4. #22586233C2semidirect
174. C71#9388semidirect

Maximal subgroups

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

subgroupindex
#225967592
#225869422
#225866742
#225862332
C71×#26112
#938871