Finite Group Databasebeta

← #31907666

Subgroups — #31907666

Order 18176.

452 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 107 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#22594106C211yesmaximal, characteristic
2#22586654C211yesmaximal, characteristic
3#22586345C214yesmaximal, complemented
4C71×#1742C211yesmaximal, characteristic
5C2×#14556128C411yescharacteristic
6C2×#14556107—21no—
7#17986052—21no—
8#17985929—21no—
9#17985929C2212yes—
10#17985924—21no—
11#17985924C412yes—
12#17985905—21no—
13#17985865C411yescharacteristic
14#17985725C2212yes—
15#17985723—21no—
16C22×M16×C71—21no—
17C2×#117×C71C2211yescharacteristic
18#343×C71—21no—
19#343×C71C2212yes—
20C22×Dic142—21no—
21#14556128—41no—
22#14556128—41no—
23#14556107—42no—
24#14556107—42no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71—21no—
27C2×C4⋊Dic71C2×C412yes—
28C2×Dic71⋊C4—22no—
29C2×C22⋊Dic71—22no—
30C4⋊(C71⋊C8)—41no—
31C22⋊(C71⋊C8)—22no—
32C4×C71⋊C8—41no—
33C71⋊#95—21no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C71⋊#95C2×C412yes—
37C23×C4×C71D411yescharacteristic
38C2×M16×C71—24no—
39C2×M16×C71—22no—
40C2×M16×C71—41no—
41C2×M16×C71—42no—
42C22×C8×C71—41no—
43#117×C71—41no—
44#117×C71—41no—
45C2×C4⋊C4×C71D411yescharacteristic
46C2×C4⋊C4×C71C2311yescharacteristic
47C2×C22⋊C4×C71C2×C411yescharacteristic
48C2×C22⋊C4×C71D411yescharacteristic
49C2×C42×C71C2×C411yescharacteristic
50C2×C42×C71D411yescharacteristic
51C2×Dic142—41no—
52C2×Dic142—41no—
53C2×Dic142—42no—
54C2×Dic142—42no—
55C22×Dic71—22no—
56C22×Dic71—22no—
57C2×C71⋊C8—42no—
58C4⋊Dic71—42no—
59C4⋊Dic71—42no—
60C4⋊Dic71—42no—
61Dic71⋊C4—42no—
62Dic71⋊C4—42no—
63C22⋊Dic71—44no—
64C24×C71C22⋊C411yescharacteristic
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—22no—
68C22×C4×C71C22⋊C411yescharacteristic
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71D812yes—
72C22×C4×C71SD1612yes—
73C22×C4×C71C22×C411yescharacteristic
74C22×C4×C71C2×D411yescharacteristic
75C22×C4×C71C2×D411yescharacteristic
76M16×C71—24no—
77M16×C71—44no—
78M16×C71—42no—
79C2×C8×C71—42no—
80C2×C8×C71—44no—
81C4⋊C4×C71—41no—
82C4⋊C4×C71—41no—
83C22⋊C4×C71—41no—
84C22⋊C4×C71—41no—
85C42×C71—21no—
86C42×C71—21no—
87C42×C71—21no—
88C42×C71—21no—
89C23×C71—21no—
90C23×C71—21no—
91C23×C71—41no—
92C23×C71—41no—
93C23×C71#10212yes—
94C23×C71C2×C22⋊C411yescharacteristic
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—41no—
106C2×C4×C71—41no—
107C2×C4×C71—41no—
108C2×C4×C71—41no—
109C2×C4×C71#9911yescharacteristic
110C2×C4×C71#9911yescharacteristic
111C2×C4×C71#10212yes—
112C2×C4×C71#10412yes—
113C2×C4×C71C2×D811yescharacteristic
114C2×C4×C71C2×SD1611yescharacteristic
115C2×Dic71—42no—
116C2×Dic71—42no—
117C2×Dic71—42no—
118C2×Dic71—42no—
119C2×Dic71—42no—
120C2×Dic71—42no—
121Dic142—82no—
122Dic142—82no—
123C8×C71—44no—
124C71⋊C8—82no—
125C22×C71—21no—
126C22×C71—21no—
127C22×C71—22no—
128C22×C71—42no—
129C22×C71—41no—
130C22×C71—41no—
131C22×C71—81no—
132C22×C71C2×#9911yescharacteristic
133C22×C71C2×#10211yescharacteristic
134C22×C71C2×#10411yescharacteristic
135C4×C71—21no—
136C4×C71—21no—
137C4×C71—41no—
138C4×C71—41no—
139C4×C71—41no—
140C4×C71—41no—
141C4×C71#32712yes—
142C4×C71#32712yes—
143Dic71—82no—
144Dic71—82no—
145#11700—711nomaximal, Sylow
146C2×C71—21no—
147C2×C71—21no—
148C2×C71—42no—
149C2×C71C2×#32711yescharacteristic
150C2×C71#212311yescharacteristic
151C2×C71#213511yescharacteristic
152#2621—711no—
153#1742D7111yescharacteristic
154#1458—711no—
155#1346—714no—
156C71#1170011yescharacteristic, complemented, Sylow
157C22×M16—21no—
158C2×C4⋊Q8—711no—
159C2×C22⋊Q8—1421no—
160C2×#117C2×D7111yescharacteristic
161#431—1421no—
162#398—712no—
163#398—1421no—
164#397—712no—
165#397—1421no—
166#391—1421no—
167#382—711no—
168#343—21no—
169#343C2×D7112yes—
170#324—712no—
171#323—1421no—
172C22×Q8—1421no—
173C23×C4C22⋊D7111yescharacteristic
174C2×M16—24no—
175C2×M16—22no—
176C2×M16—41no—
177C2×M16—42no—
178C22×C8—41no—
179C4⋊Q8—2841no—
180C4⋊Q8—2841no—
181C22⋊Q8—2842no—
182C22⋊Q8—2842no—
183#117—41no—
184#117—41no—
185C2×C4⋊C4—712no—
186C2×C4⋊C4—1421no—
187C2×C4⋊C4—1421no—
188C2×C4⋊C4—1422no—
189C2×C4⋊C4C22⋊D7111yescharacteristic
190C2×C4⋊C4C22×D7111yescharacteristic
191C2×C22⋊C4—1422no—
192C2×C22⋊C4C4×D7111yescharacteristic
193C2×C22⋊C4D28411yescharacteristic
194C2×C42C4×D7111yescharacteristic
195C2×C42D28411yescharacteristic
196C4⋊C8—2841no—
197C22⋊C8—1422no—
198C4×C8—2841no—
199#95—712no—
200#95—1421no—
201#95—1422no—
202#95—1422no—
203C24(C2×D71)⋊C411yescharacteristic
204C2×Q8—2841no—
205C2×Q8—2841no—
206C2×Q8—2842no—
207C2×Q8—2842no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—1422no—
212C22×C4—1422no—
213C22×C4(C2×D71)⋊C411yescharacteristic
214C22×C4(C2×D71)⋊C411yescharacteristic
215C22×C4(C2×D71)⋊C411yescharacteristic
216C22×C4#1176585012yes—
217C22×C4#1176585212yes—
218C22×C4C2×C4×D7111yescharacteristic
219C22×C4C2×C22⋊D7111yescharacteristic
220C22×C4C2×D28411yescharacteristic
221M16—24no—
222M16—44no—
223M16—42no—
224C2×C8—42no—
225C2×C8—44no—
226C2×C8—2842no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—2842no—
230C4⋊C4—2842no—
231C4⋊C4—2842no—
232C4⋊C4—2842no—
233C4⋊C4—2842no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C22⋊C4—2844no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23#1455604812yes—
246C23C2×(C2×D71)⋊C411yescharacteristic
247Q8—5682no—
248Q8—5682no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—2842no—
264C2×C4—2842no—
265C2×C4—2842no—
266C2×C4—2842no—
267C2×C4—2842no—
268C2×C4—2842no—
269C2×C4#1455604211yescharacteristic
270C2×C4#1455604211yescharacteristic
271C2×C4#1455604812yes—
272C2×C4#1455605612yes—
273C2×C4C2×#1176585011yescharacteristic
274C2×C4C2×#1176585211yescharacteristic
275C8—44no—
276C8—5682no—
277C22—21no—
278C22—21no—
279C22—22no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—81no—
284C22C2×#1455604211yescharacteristic
285C22C2×#1455604811yescharacteristic
286C22C2×#1455605611yescharacteristic
287C4—21no—
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—5682no—
294C4—5682no—
295C4#1798573212yes—
296C4#1798573212yes—
297C2—21no—
298C2—21no—
299C2—42no—
300C2C2×#1798573211yescharacteristic
301C2#2258999511yescharacteristic
302C2#2259003411yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#1742C2
socleC22×C71C2×#99
prime radicalC24×C71C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#1742D71
hypocenterC71#11700
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
3. #22586345C2semidirect
156. C71#11700semidirect

Maximal subgroups

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

subgroupindex
#225941062
#225866542
#225863452
C71×#17422
#1170071