Finite Group Databasebeta

← #31884665

Subgroups — #31884665

Order 18176.

402 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 75 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#22589139C211yesmaximal, characteristic
2C4×#14556043C211yesmaximal, characteristic
3C4×#99×C71C211yesmaximal, characteristic
4C2×C4⋊(C71⋊C8)C411yescharacteristic, complemented
5C2×#14556043—21no—
6C2×C22⋊(C71⋊C8)—21no—
7C2×C22⋊(C71⋊C8)—21no—
8C2×#14556039C411yescharacteristic, complemented
9C4×C22⋊Dic71—21no—
10#17985766C412yescomplemented
11C2×C4×D4×C71C2211yescharacteristic
12C2×#99×C71—21no—
13C4×C22⋊C4×C71—21no—
14#342×C71C412yescomplemented
15C22×C71⋊C8—21no—
16C22×C71⋊C8—21no—
17C2×C22⋊Dic71—22no—
18C2×C4×Dic71—41no—
19C4⋊(C71⋊C8)—41no—
20#14556043—24no—
21C22⋊(C71⋊C8)—42no—
22C22⋊(C71⋊C8)—42no—
23#14556039—41no—
24C71⋊#95—41no—
25C22×D4×C71D411yescharacteristic
26C23×C4×C71D411yescharacteristic
27C23×C4×C71D411yescharacteristic
28C4×D4×C71—41no—
29C4×D4×C71—41no—
30C2×C4⋊C4×C71Q811yescharacteristic
31C2×C22⋊C4×C71—22no—
32C2×C22⋊C4×C71C2×C411yescharacteristic
33C2×C22⋊C4×C71C2×C411yescharacteristic
34C2×C42×C71—41no—
35C2×C42×C71C2×C411yescharacteristic
36#99×C71—24no—
37#95×C71—41no—
38C22×Dic71—42no—
39C2×C71⋊C8—22no—
40C2×C71⋊C8—41no—
41C2×C71⋊C8—41no—
42C2×C71⋊C8—41no—
43C2×C71⋊C8—41no—
44C2×C71⋊C8—41no—
45C22⋊Dic71—24no—
46C22⋊Dic71—42no—
47C4×Dic71—81no—
48C4×Dic71—81no—
49C24×C71C22⋊C411yescharacteristic
50C24×C71C22⋊C411yescharacteristic
51C2×D4×C71—22no—
52C2×D4×C71—22no—
53C2×D4×C71—41no—
54C22×C4×C71—21no—
55C22×C4×C71—21no—
56C22×C4×C71—21no—
57C22×C4×C71—21no—
58C22×C4×C71—41no—
59C22×C4×C71—41no—
60C22×C4×C71—41no—
61C22×C4×C71—41no—
62C22×C4×C71—42no—
63C22×C4×C71C4211yescharacteristic
64C22×C4×C71C22⋊C411yescharacteristic
65C22×C4×C71C4⋊C411yescharacteristic
66C22×C4×C71C4⋊C411yescharacteristic
67C22×C4×C71C4⋊C411yescharacteristic
68C4⋊C4×C71—41no—
69C22⋊C4×C71—24no—
70C22⋊C4×C71—41no—
71C22⋊C4×C71—41no—
72C22⋊C4×C71—42no—
73C42×C71—21no—
74C42×C71—21no—
75C42×C71—81no—
76C42×C71—81no—
77C23×C71—21no—
78C23×C71—21no—
79C23×C71—21no—
80C23×C71—21no—
81C23×C71—41no—
82C23×C71—41no—
83C23×C71—41no—
84C23×C71—41no—
85C23×C71#9511yescharacteristic
86C23×C71#10412yes—
87C23×C71#10412yes—
88D4×C71—44no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—41no—
98C2×C4×C71—41no—
99C2×C4×C71—41no—
100C2×C4×C71—41no—
101C2×C4×C71—41no—
102C2×C4×C71—41no—
103C2×C4×C71—41no—
104C2×C4×C71—44no—
105C2×C4×C71—81no—
106C2×C4×C71—81no—
107C2×C4×C71—82no—
108C2×C4×C71—82no—
109C2×C4×C71#9911yescharacteristic
110C2×C4×C71#9911yescharacteristic
111C2×Dic71—44no—
112C2×Dic71—82no—
113C2×Dic71—82no—
114C71⋊C8—42no—
115C71⋊C8—81no—
116C22×C71—21no—
117C22×C71—21no—
118C22×C71—22no—
119C22×C71—22no—
120C22×C71—42no—
121C22×C71—42no—
122C22×C71—41no—
123C22×C71—41no—
124C22×C71—41no—
125C22×C71—41no—
126C22×C71—81no—
127C22×C71—81no—
128C22×C71#33911yescharacteristic
129C22×C71#33911yescharacteristic
130C22×C71#34211yescharacteristic
131C4×C71—21no—
132C4×C71—41no—
133C4×C71—41no—
134C4×C71—41no—
135C4×C71—41no—
136C4×C71—84no—
137C4×C71#35311yescharacteristic
138C4×C71#35411yescharacteristic
139Dic71—84no—
140#7283—711nomaximal, Sylow
141C2×C71—21no—
142C2×C71—21no—
143C2×C71—42no—
144C2×C71—42no—
145C2×C71#129711yescharacteristic
146C2×C71#139111yescharacteristic
147C2×C71#139311yescharacteristic
148#1955—711no—
149C4×#99—711no—
150C4×#99D7111yescharacteristic
151C71#728311yescharacteristic, complemented, Sylow
152C2×C4×D4C2×D7111yescharacteristic
153C2×C4⋊C8—711no—
154C2×#99—21no—
155C2×#99—1421no—
156C2×C22⋊C8—1421no—
157C2×C22⋊C8—1421no—
158C2×#97—711no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—1421no—
161#342—712no—
162#342Dic7112yescomplemented
163C22×D4C22⋊D7111yescharacteristic
164C23×C4D28411yescharacteristic
165C23×C4C22⋊D7111yescharacteristic
166C22×C8—1421no—
167C22×C8—1421no—
168C4×D4—41no—
169C4×D4—41no—
170C2×C4⋊C4Dic14211yescharacteristic
171C2×C22⋊C4—22no—
172C2×C22⋊C4—1422no—
173C2×C22⋊C4C4×D7111yescharacteristic
174C2×C22⋊C4C2×Dic7111yescharacteristic
175C2×C42—41no—
176C2×C42—2841no—
177C2×C42C4×D7111yescharacteristic
178C4⋊C8—2841no—
179#99—24no—
180#99—1424no—
181C22⋊C8—2842no—
182C22⋊C8—2842no—
183#97—2841no—
184#95—41no—
185#95—2841no—
186C24(C2×D71)⋊C411yescharacteristic
187C24C22⋊Dic7111yescharacteristic
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—41no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—2842no—
201C22×C4C4×Dic7111yescharacteristic
202C22×C4(C2×D71)⋊C411yescharacteristic
203C22×C4Dic71⋊C411yescharacteristic
204C22×C4Dic71⋊C411yescharacteristic
205C22×C4C4⋊Dic7111yescharacteristic
206C2×C8—2841no—
207C2×C8—2841no—
208C2×C8—2841no—
209C2×C8—2841no—
210C2×C8—2841no—
211C2×C8—1422no—
212C4⋊C4—41no—
213C22⋊C4—24no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—42no—
217C22⋊C4—2842no—
218C22⋊C4—1424no—
219C42—21no—
220C42—21no—
221C42—81no—
222C42—81no—
223C42—5681no—
224C42—5681no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23C71⋊#9511yescharacteristic
234C23#1455605512yes—
235C23#1455605612yes—
236D4—44no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—44no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—2844no—
258C2×C4—5682no—
259C2×C4—5682no—
260C2×C4#1455604211yescharacteristic
261C2×C4#1455604211yescharacteristic
262C8—2842no—
263C8—5681no—
264C22—21no—
265C22—21no—
266C22—22no—
267C22—22no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22—81no—
276C22#1798575811yescharacteristic
277C22#1798575911yescharacteristic
278C22#1798576511yescharacteristic
279C4—21no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—84no—
285C4—5684no—
286C4#1798579111yescharacteristic
287C4#1798579311yescharacteristic
288C2—21no—
289C2—21no—
290C2—42no—
291C2—42no—
292C2#2258622611yescharacteristic
293C2#2258646011yescharacteristic
294C2#2258646411yescharacteristic
subgroupG/H
centerC2×C4#14556042
commutatorC22×C4×C71C42
FrattiniC2×C4×D4C2×D71
FittingC4×#99×C71C2
socleC22×C71#342
prime radicalC22×D4×C71D4
cosocleC2×C4×D4×C71C22
hypercenterC4×#99D71
hypocenterC71#7283
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊(C71⋊C8)C4semidirect
8. C2×#14556039C4semidirect
10. #17985766C4semidirect
14. #342×C71C4semidirect
151. C71#7283semidirect
162. #342Dic71semidirect

Maximal subgroups

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

subgroupindex
#225891392
C4×#145560432
C4×#99×C712
#728371