Finite Group Databasebeta

← #31907125

Subgroups — #31907125

Order 18176.

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

normal subgroups only

This group has 554 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
1#22586681C211yesmaximal, characteristic
2#22586658C211yesmaximal, characteristic
3#22586288C214yesmaximal, complemented
4C71×#2409C211yesmaximal, characteristic
5#17986056—21no—
6C2×C22⋊(C71⋊C8)—21no—
7#17985725—21no—
8#17985723—21no—
9#17985723C2212yes—
10#17985723C2212yes—
11C23×Q8×C71C411yescharacteristic
12C2×C22⋊Q8×C71—21no—
13C2×C4×D4×C71C2211yescharacteristic
14#395×C71—21no—
15#393×C71—21no—
16#391×C71—21no—
17#386×C71C412yes—
18#385×C71C2212yes—
19#380×C71C411yescharacteristic
20C22×C71⋊C8—41no—
21C4⋊(C71⋊C8)—41no—
22C22⋊(C71⋊C8)—22no—
23C22⋊(C71⋊C8)—22no—
24C22⋊(C71⋊C8)—41no—
25#14556039—41no—
26C22×Q8×C71—21no—
27C22×Q8×C71—21no—
28C22×Q8×C71—41no—
29C22×Q8×C71—41no—
30C22×Q8×C71—44no—
31C22×D4×C71C2×C411yescharacteristic
32C23×C4×C71C2×C411yescharacteristic
33C23×C4×C71C2×C412yes—
34C23×C4×C71D411yescharacteristic
35C22⋊Q8×C71—44no—
36C4×D4×C71—41no—
37C4×D4×C71—41no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—22no—
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C22⋊C4×C71—22no—
42C2×C22⋊C4×C71C2×C412yes—
43C2×C22⋊C4×C71D411yescharacteristic
44C2×C22⋊C4×C71C2311yescharacteristic
45C2×C42×C71D411yescharacteristic
46#95×C71—21no—
47#95×C71—22no—
48#95×C71—22no—
49C2×C71⋊C8—42no—
50C2×C71⋊C8—42no—
51C2×C71⋊C8—82no—
52C24×C71C22⋊C411yescharacteristic
53C24×C71C22×C411yescharacteristic
54C2×Q8×C71—24no—
55C2×Q8×C71—42no—
56C2×Q8×C71—44no—
57C2×Q8×C71—42no—
58C2×Q8×C71—44no—
59C2×Q8×C71—48no—
60C2×Q8×C71—44no—
61C2×Q8×C71—41no—
62C2×Q8×C71—41no—
63C2×D4×C71—22no—
64C2×D4×C71—22no—
65C2×D4×C71—41no—
66C22×C4×C71—22no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—22no—
72C22×C4×C71—42no—
73C22×C4×C71—44no—
74C22×C4×C71—42no—
75C22×C4×C71—42no—
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C4⋊C4×C71—41no—
82C4⋊C4×C71—42no—
83C4⋊C4×C71—44no—
84C22⋊C4×C71—24no—
85C22⋊C4×C71—41no—
86C22⋊C4×C71—41no—
87C22⋊C4×C71—44no—
88C42×C71—21no—
89C42×C71—21no—
90C23×C71—21no—
91C23×C71—21no—
92C23×C71—21no—
93C23×C71—21no—
94C23×C71—41no—
95C23×C71—41no—
96C23×C71—41no—
97C23×C71—41no—
98C23×C71#10112yes—
99C23×C71C2×C22⋊C411yescharacteristic
100C23×C71#13112yes—
101Q8×C71—44no—
102Q8×C71—48no—
103Q8×C71—44no—
104D4×C71—44no—
105C2×C4×C71—22no—
106C2×C4×C71—24no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—42no—
114C2×C4×C71—42no—
115C2×C4×C71—44no—
116C2×C4×C71—42no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—41no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
122C2×C4×C71—44no—
123C2×C4×C71—42no—
124C2×C4×C71—42no—
125C2×C4×C71—42no—
126C2×C4×C71—81no—
127C2×C4×C71#9911yescharacteristic
128C2×C4×C71#9911yescharacteristic
129C71⋊C8—82no—
130C71⋊C8—82no—
131C22×C71—21no—
132C22×C71—21no—
133C22×C71—22no—
134C22×C71—22no—
135C22×C71—42no—
136C22×C71—42no—
137C22×C71—42no—
138C22×C71—41no—
139C22×C71—41no—
140C22×C71—41no—
141C22×C71—41no—
142C22×C71—81no—
143C22×C71#40811yescharacteristic
144C22×C71C2×#9911yescharacteristic
145C22×C71C2×#10111yescharacteristic
146C4×C71—21no—
147C4×C71—41no—
148C4×C71—44no—
149C4×C71—41no—
150C4×C71—41no—
151C4×C71—41no—
152C4×C71—82no—
153C4×C71AΣL(1,𝔽16)12yes—
154#11575—711nomaximal, Sylow
155C2×C71—21no—
156C2×C71—21no—
157C2×C71—42no—
158C2×C71—42no—
159C2×C71#147211yescharacteristic
160C2×C71C2×AΣL(1,𝔽16)11yescharacteristic
161C2×C71#212311yescharacteristic
162#2409D7111yescharacteristic
163#1466—711no—
164#1459—711no—
165#1326—714no—
166C71#1157511yescharacteristic, complemented, Sylow
167C23×Q8Dic7111yescharacteristic
168C2×C22⋊Q8—21no—
169C2×C4×D4C2×D7111yescharacteristic
170#432—1421no—
171C2×C22⋊C8—1421no—
172#395—21no—
173#393—21no—
174#391—21no—
175#386Dic7112yes—
176#385C2×D7112yes—
177#380Dic7111yescharacteristic
178#324—1421no—
179#323—712no—
180#323—712no—
181#323—1421no—
182C22×Q8—21no—
183C22×Q8—21no—
184C22×Q8—41no—
185C22×Q8—41no—
186C22×Q8—44no—
187C22×D4C2×Dic7111yescharacteristic
188C23×C4C2×Dic7111yescharacteristic
189C23×C4C2×Dic7112yes—
190C23×C4C22⋊D7111yescharacteristic
191C22×C8—2841no—
192C22⋊Q8—44no—
193C4×D4—41no—
194C4×D4—41no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C22⋊D7111yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4C2×Dic7112yes—
200C2×C22⋊C4C22⋊D7111yescharacteristic
201C2×C22⋊C4C22×D7111yescharacteristic
202C2×C42C22⋊D7111yescharacteristic
203C4⋊C8—2841no—
204C22⋊C8—2841no—
205C22⋊C8—1422no—
206C22⋊C8—1422no—
207#97—2841no—
208#95—21no—
209#95—22no—
210#95—22no—
211C24C22⋊Dic7111yescharacteristic
212C24C22×Dic7111yescharacteristic
213C2×Q8—24no—
214C2×Q8—42no—
215C2×Q8—44no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—48no—
219C2×Q8—44no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—41no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4C22⋊Dic7111yescharacteristic
236C22×C4C22⋊Dic7111yescharacteristic
237C22×C4C22⋊Dic7111yescharacteristic
238C22×C4C2×C22⋊D7111yescharacteristic
239C22×C4C2×C22⋊D7111yescharacteristic
240C2×C8—2842no—
241C2×C8—2842no—
242C2×C8—5682no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C22⋊C4—24no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—44no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23#1455604712yes—
261C23C2×C22⋊Dic7111yescharacteristic
262C23#1455613512yes—
263Q8—44no—
264Q8—48no—
265Q8—44no—
266D4—44no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4#1455604311yescharacteristic
290C2×C4#1455604311yescharacteristic
291C8—5682no—
292C8—5682no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22#1798596611yescharacteristic
306C22C2×#1455604311yescharacteristic
307C22C2×#1455604711yescharacteristic
308C4—21no—
309C4—41no—
310C4—44no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—82no—
315C4#1798578612yes—
316C2—21no—
317C2—21no—
318C2—42no—
319C2—42no—
320C2#2258670411yescharacteristic
321C2C2×#1798578611yescharacteristic
322C2#2258999411yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC24×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2409C2
socleC22×C71C2×#99
prime radicalC22×D4×C71C2×C4
cosocleC2×C22⋊C4×C71C23
hypercenter#2409D71
hypocenterC71#11575
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. #22586288C2semidirect
166. C71#11575semidirect

Maximal subgroups

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

subgroupindex
#225866812
#225866582
#225862882
C71×#24092
#1157571