Finite Group Databasebeta

← #31890752

Subgroups — #31890752

Order 18176.

358 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 103 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#22597315C211yesmaximal, characteristic
2#22587103C211yesmaximal, characteristic, complemented
3#22586945C212yesmaximal
4#22586212C212yesmaximal, complemented
5C4×#108×C71C211yesmaximal, characteristic, complemented
6#17986533C2211yescharacteristic
7#17986129C411yescharacteristic
8C8×Dic142C411yescharacteristic
9#17986078C411yescharacteristic
10#17986070—21no—
11C8×D284C411yescharacteristic
12C8×C22⋊D71—21no—
13#17986058—21no—
14#17986056C2212yes—
15#17986056C2211yescharacteristic
16#17985953—21no—
17#17985752—21no—
18#17985750—21no—
19#17985740—22no—
20#17985738—22no—
21C2×#108×C71—21no—
22C4×M16×C71C2212yes—
23C2×C4×C8×C71C2211yescharacteristic
24#341×C71—21no—
25C2×#11765848—21no—
26C2×C8×D71—21no—
27#14556122—21no—
28#14556120D411yescharacteristic
29#14556116D411yescharacteristic
30#14556111—21no—
31#14556107Q811yescharacteristic
32#14556102Q811yescharacteristic
33C4×Dic142C2×C411yescharacteristic
34C4×D284C2×C411yescharacteristic
35C4×C22⋊D71—21no—
36#14556085—21no—
37C4⋊(C71⋊C8)—22no—
38C4⋊(C71⋊C8)C2×C411yescharacteristic
39C4⋊(C71⋊C8)C2×C411yescharacteristic
40Dic71⋊C8—21no—
41Dic71⋊C8—21no—
42C22⋊(C71⋊C8)—21no—
43C22⋊(C71⋊C8)—22no—
44(C2×D71)⋊C8—21no—
45(C2×D71)⋊C8—21no—
46#14556039—21no—
47#14556039—22no—
48#14556038—21no—
49C8×Dic71—21no—
50C2×M16×C71—22no—
51C22×C8×C71D411yescharacteristic
52C22×C8×C71D411yescharacteristic
53C2×C42×C71C2311yescharacteristic
54#108×C71—24no—
55#97×C71—22no—
56C4×C8×C71—22no—
57C4×C8×C71C2×C411yescharacteristic
58C4×C8×C71C2×C411yescharacteristic
59C4×C8×C71D411yescharacteristic
60C4×C8×C71D411yescharacteristic
61C2×Dic142C4⋊C411yescharacteristic
62C2×D284C4⋊C411yescharacteristic
63C2×C22⋊D71—21no—
64C2×C4×D71—21no—
65#11765848—42no—
66C2×C71⋊C8—21no—
67C2×C71⋊C8—21no—
68C2×C71⋊C8—42no—
69C8×D71—42no—
70C4⋊Dic71C4⋊C411yescharacteristic
71C4⋊Dic71C4⋊C411yescharacteristic
72Dic71⋊C4—21no—
73Dic71⋊C4—21no—
74Dic71⋊C4—21no—
75Dic71⋊C4—21no—
76C22⋊Dic71—21no—
77(C2×D71)⋊C4—21no—
78(C2×D71)⋊C4—21no—
79(C2×D71)⋊C4—21no—
80(C2×D71)⋊C4—21no—
81C4×Dic71—21no—
82C22×C4×C71C2×D411yescharacteristic
83C22×C4×C71C2×Q811yescharacteristic
84C22×C4×C71#4111yescharacteristic
85M16×C71—24no—
86M16×C71—42no—
87C2×C8×C71—22no—
88C2×C8×C71—41no—
89C2×C8×C71—42no—
90C2×C8×C71C22⋊C411yescharacteristic
91C2×C8×C71C22⋊C411yescharacteristic
92C2×C8×C71C22⋊C411yescharacteristic
93C2×C8×C71C22⋊C411yescharacteristic
94C42×C71C22×C411yescharacteristic
95C42×C71C2×D411yescharacteristic
96C42×C71C2×D411yescharacteristic
97C42×C71#4111yescharacteristic
98C23×C71C22⋊Q811yescharacteristic
99C22×D71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—41no—
103C2×C4×C71—41no—
104C2×C4×C71C2×C22⋊C411yescharacteristic
105C2×C4×C71C2×C4⋊C411yescharacteristic
106C2×C4×C71#11711yescharacteristic
107C2×C4×C71#12111yescharacteristic
108C2×C4×C71#12111yescharacteristic
109C2×C4×C71C22⋊Q811yescharacteristic
110C22⋊D71—42no—
111C2×Dic71—21no—
112C2×Dic71—21no—
113C2×Dic71—21no—
114D284—41no—
115C4×D71—41no—
116C4×D71—41no—
117Dic142—41no—
118C8×C71—21no—
119C8×C71—21no—
120C8×C71—21no—
121C8×C71—44no—
122C8×C71#10412yes—
123C71⋊C8—41no—
124C71⋊C8—41no—
125C71⋊C8—82no—
126C22×C71—21no—
127C22×C71—21no—
128C22×C71—21no—
129C22×C71#38011yescharacteristic
130C2×D71—41no—
131C2×D71—41no—
132C2×D71—41no—
133C4×C71—21no—
134C4×C71—21no—
135C4×C71—21no—
136C4×C71—21no—
137C4×C71—21no—
138C4×C71C2×#10411yescharacteristic
139C4×C71#42111yescharacteristic
140Dic71—41no—
141Dic71—41no—
142Dic71—41no—
143#8693—711nomaximal, Sylow
144C2×C71—41no—
145C2×C71#180611yescharacteristic
146C2×C71#181311yescharacteristic
147C2×C71#181411yescharacteristic
148D71—81no—
149#2966—711no—
150C4×#108D7111yescharacteristic, complemented
151#1568—711no—
152#1529—712no—
153#1291—712no—
154C71#869311yescharacteristic, complemented, Sylow
155#529—711no—
156#446—711no—
157C8×Q8—711no—
158#436—711no—
159#435—1421no—
160C8×D4—711no—
161C8×D4—1421no—
162#433—1421no—
163#432—711no—
164#432—712no—
165C2×#108—21no—
166#405—1421no—
167C4×M16C2×D7112yes—
168C2×C4×C8C2×D7111yescharacteristic
169#341—21no—
170#335—1421no—
171#334—1421no—
172#330—1422no—
173#329—1422no—
174C2×M16—22no—
175C2×M16—1421no—
176C22×C8—1421no—
177C22×C8C22⋊D7111yescharacteristic
178C22×C8C22⋊D7111yescharacteristic
179#126—1421no—
180#125—711no—
181#124—711no—
182#123—1421no—
183C22⋊Q8—711no—
184#121—711no—
185C4×Q8—711no—
186C4×D4—711no—
187C4×D4—1421no—
188#117—1421no—
189C2×C42C22×D7111yescharacteristic
190#108—24no—
191C4⋊C8—711no—
192C4⋊C8—711no—
193C4⋊C8—1421no—
194C4⋊C8—1421no—
195C4⋊C8—1422no—
196C22⋊C8—1422no—
197C22⋊C8—1421no—
198C22⋊C8—1421no—
199C22⋊C8—1421no—
200#97—22no—
201#97—1421no—
202#97—1422no—
203#97—1421no—
204C4×C8—22no—
205C4×C8—1421no—
206C4×C8C4×D7111yescharacteristic
207C4×C8C4×D7111yescharacteristic
208C4×C8D28411yescharacteristic
209C4×C8D28411yescharacteristic
210C2×Q8—711no—
211C2×D4—711no—
212C2×D4—1421no—
213C22×C4—1421no—
214C22×C4C2×C22⋊D7111yescharacteristic
215C22×C4Q8×D7111yescharacteristic
216C22×C4#1176586611yescharacteristic
217M16—24no—
218M16—42no—
219M16—2842no—
220C2×C8—22no—
221C2×C8—41no—
222C2×C8—42no—
223C2×C8—2842no—
224C2×C8—2842no—
225C2×C8—1421no—
226C2×C8—1421no—
227C2×C8(C2×D71)⋊C411yescharacteristic
228C2×C8(C2×D71)⋊C411yescharacteristic
229C2×C8(C2×D71)⋊C411yescharacteristic
230C2×C8(C2×D71)⋊C411yescharacteristic
231C4⋊C4—711no—
232C4⋊C4—711no—
233C4⋊C4—1421no—
234C4⋊C4—1421no—
235C4⋊C4—1421no—
236C4⋊C4—1421no—
237C22⋊C4—1421no—
238C22⋊C4—1421no—
239C22⋊C4—1421no—
240C22⋊C4—1421no—
241C22⋊C4—1421no—
242C42—1421no—
243C42C2×C4×D7111yescharacteristic
244C42D4×D7111yescharacteristic
245C42C2×D28411yescharacteristic
246C42#1176586411yescharacteristic
247C23—1421no—
248C23#1455610411yescharacteristic
249Q8—2841no—
250D4—2841no—
251D4—2842no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—2841no—
257C2×C4—2841no—
258C2×C4—1421no—
259C2×C4—1421no—
260C2×C4—1421no—
261C2×C4C2×(C2×D71)⋊C411yescharacteristic
262C2×C4C4⋊C4×D7111yescharacteristic
263C2×C4#1455608611yescharacteristic
264C2×C4#1455609911yescharacteristic
265C2×C4#1455610311yescharacteristic
266C2×C4#1455610811yescharacteristic
267C8—21no—
268C8—21no—
269C8—21no—
270C8—44no—
271C8—2841no—
272C8—2841no—
273C8—5682no—
274C8#1455605612yes—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—2841no—
279C22—2841no—
280C22—2841no—
281C22#1798585611yescharacteristic
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—21no—
287C4—2841no—
288C4—2841no—
289C4—2841no—
290C4C2×#1455605611yescharacteristic
291C4#1798601711yescharacteristic
292C2—41no—
293C2—5681no—
294C2#2258836111yescharacteristic
295C2#2258838711yescharacteristic
296C2#2258839511yescharacteristic
subgroupG/H
centerC22#17985856
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC4×#108×C71C2
socleC22×C71#380
prime radical#14556102Q8
cosocleC2×C42×C71C23
hypercenterC4×#108D71
hypocenterC71#8693
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#108

Complements

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

NH ≅ G/Ntype
2. #22587103C2semidirect
4. #22586212C2semidirect
5. C4×#108×C71C2semidirect
150. C4×#108D71semidirect
154. C71#8693semidirect

Maximal subgroups

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

subgroupindex
#225973152
#225871032
#225869452
#225862122
C4×#108×C712
#869371