Finite Group Databasebeta

← #31884669

Subgroups — #31884669

Order 18176.

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

normal subgroups only

This group has 324 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#22588638C211yesmaximal, characteristic
2#22586661C211yesmaximal, characteristic
3C71×#1459C211yesmaximal, characteristic
4C22×#11765849C411yescharacteristic
5#17986052—21no—
6C2×C22⋊(C71⋊C8)—21no—
7C2×C22⋊(C71⋊C8)C411yescharacteristic
8#17985754—21no—
9#17985723—21no—
10#17985723C412yes—
11C2×C4×D4×C71C2211yescharacteristic
12C2×C22⋊C8×C71—21no—
13#324×C71C412yes—
14#323×C71—21no—
15C2×#11765849—41no—
16C2×#11765849—42no—
17C22×C71⋊C8—21no—
18C22×C71⋊C8—21no—
19C4⋊(C71⋊C8)—41no—
20C22⋊(C71⋊C8)—22no—
21C22⋊(C71⋊C8)—41no—
22C22⋊(C71⋊C8)—42no—
23C4×C71⋊C8—41no—
24C22×D4×C71Q811yescharacteristic
25C23×C4×C71C2×C411yescharacteristic
26C23×C4×C71D411yescharacteristic
27C22×C8×C71—41no—
28C4×D4×C71—41no—
29C4×D4×C71—41no—
30C2×C4⋊C4×C71C2×C411yescharacteristic
31C2×C22⋊C4×C71C2×C411yescharacteristic
32C2×C22⋊C4×C71D411yescharacteristic
33C2×C42×C71D411yescharacteristic
34C22⋊C8×C71—22no—
35C22⋊C8×C71—41no—
36#11765849—42no—
37#11765849—42no—
38C2×C71⋊C8—22no—
39C2×C71⋊C8—41no—
40C2×C71⋊C8—41no—
41C2×C71⋊C8—42no—
42C2×C71⋊C8—41no—
43C2×C71⋊C8—41no—
44C2×C71⋊C8—41no—
45C24×C71C4⋊C411yescharacteristic
46C24×C71C4⋊C411yescharacteristic
47C2×D4×C71—22no—
48C2×D4×C71—22no—
49C2×D4×C71—41no—
50C22×C4×C71—21no—
51C22×C4×C71—21no—
52C22×C4×C71—21no—
53C22×C4×C71—21no—
54C22×C4×C71—41no—
55C22×C4×C71—41no—
56C22×C4×C71—41no—
57C22×C4×C71—41no—
58C22×C4×C71C4211yescharacteristic
59C22×C4×C71C22⋊C411yescharacteristic
60C22×C4×C71C22⋊C411yescharacteristic
61C22×C4×C71C22⋊C411yescharacteristic
62C22×C4×C71C4⋊C411yescharacteristic
63C2×C8×C71—42no—
64C2×C8×C71—81no—
65C2×C8×C71—81no—
66C4⋊C4×C71—41no—
67C22⋊C4×C71—41no—
68C22⋊C4×C71—41no—
69C42×C71—21no—
70C42×C71—21no—
71C23×C71—21no—
72C23×C71—21no—
73C23×C71—21no—
74C23×C71—21no—
75C23×C71—21no—
76C23×C71—41no—
77C23×C71—41no—
78C23×C71—41no—
79C23×C71—41no—
80C23×C71#9511yescharacteristic
81C23×C71#10812yes—
82D4×C71—44no—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—42no—
91C2×C4×C71—41no—
92C2×C4×C71—41no—
93C2×C4×C71—41no—
94C2×C4×C71—41no—
95C2×C4×C71—41no—
96C2×C4×C71—41no—
97C2×C4×C71—41no—
98C2×C4×C71—81no—
99C2×C4×C71#9911yescharacteristic
100C2×C4×C71#9911yescharacteristic
101C2×C4×C71#10812yes—
102C8×C71—82no—
103C71⋊C8—42no—
104C71⋊C8—81no—
105C71⋊C8—82no—
106C22×C71—21no—
107C22×C71—21no—
108C22×C71—22no—
109C22×C71—42no—
110C22×C71—41no—
111C22×C71—41no—
112C22×C71—41no—
113C22×C71—41no—
114C22×C71—41no—
115C22×C71—81no—
116C22×C71—81no—
117C22×C71—81no—
118C22×C71#34111yescharacteristic
119C22×C71#34111yescharacteristic
120C22×C71#34211yescharacteristic
121C4×C71—22no—
122C4×C71—21no—
123C4×C71—41no—
124C4×C71—41no—
125C4×C71—41no—
126C4×C71AΣL(1,𝔽16)11yescharacteristic
127C4×C71#35211yescharacteristic
128#7285—711nomaximal, Sylow
129C2×C71—21no—
130C2×C71—21no—
131C2×C71—42no—
132C2×C71—81no—
133C2×C71#129811yescharacteristic
134C2×C71#139011yescharacteristic
135C2×C71#139511yescharacteristic
136#1870—711no—
137#1460—711no—
138#1459D7111yescharacteristic
139C71#728511yescharacteristic, complemented, Sylow
140C22×M16—711no—
141C2×C4×D4C2×D7111yescharacteristic
142#431—1421no—
143C2×C22⋊C8—21no—
144C2×C22⋊C8—711no—
145C2×C22⋊C8—1421no—
146#336—1421no—
147#324Dic7112yes—
148#323—21no—
149#323—712no—
150#323—1421no—
151C22×D4Dic14211yescharacteristic
152C23×C4C4×D7111yescharacteristic
153C23×C4C22⋊D7111yescharacteristic
154C2×M16—2842no—
155C2×M16—2841no—
156C22×C8—41no—
157C22×C8—1421no—
158C22×C8—1421no—
159C4×D4—41no—
160C4×D4—41no—
161C2×C4⋊C4C2×Dic7111yescharacteristic
162C2×C22⋊C4C4×D7111yescharacteristic
163C2×C22⋊C4C22⋊D7111yescharacteristic
164C2×C42D28411yescharacteristic
165C4⋊C8—2841no—
166C22⋊C8—22no—
167C22⋊C8—41no—
168C22⋊C8—2842no—
169C22⋊C8—2841no—
170C22⋊C8—1422no—
171C4×C8—2841no—
172C24Dic71⋊C411yescharacteristic
173C24Dic71⋊C411yescharacteristic
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—41no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4C4×Dic7111yescharacteristic
186C22×C4(C2×D71)⋊C411yescharacteristic
187C22×C4(C2×D71)⋊C411yescharacteristic
188C22×C4C22⋊Dic7111yescharacteristic
189C22×C4C4⋊Dic7111yescharacteristic
190M16—2842no—
191M16—2842no—
192C2×C8—42no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—2841no—
196C2×C8—2841no—
197C2×C8—2842no—
198C2×C8—2841no—
199C2×C8—2841no—
200C2×C8—2841no—
201C2×C8—1422no—
202C4⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C42—21no—
206C42—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23C71⋊#9511yescharacteristic
217C23#1455606312yes—
218D4—44no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—81no—
235C2×C4#1455604211yescharacteristic
236C2×C4#1455604211yescharacteristic
237C2×C4#1455606412yes—
238C8—82no—
239C8—2842no—
240C8—5681no—
241C8—5682no—
242C22—21no—
243C22—21no—
244C22—22no—
245C22—42no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22#1798576311yescharacteristic
255C22#1798576411yescharacteristic
256C22#1798576511yescharacteristic
257C4—21no—
258C4—22no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4#1798578511yescharacteristic
263C4#1798578811yescharacteristic
264C2—21no—
265C2—21no—
266C2—42no—
267C2—81no—
268C2#2258622811yescharacteristic
269C2#2258645711yescharacteristic
270C2#2258647011yescharacteristic
subgroupG/H
centerC2×C4#14556042
commutatorC22×C4×C71C42
FrattiniC2×C4×D4C2×D71
FittingC71×#1459C2
socleC22×C71#342
prime radicalC22×D4×C71Q8
cosocleC2×C4×D4×C71C22
hypercenter#1459D71
hypocenterC71#7285
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
139. C71#7285semidirect

Maximal subgroups

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

subgroupindex
#225886382
#225866612
C71×#14592
#728571