Finite Group Databasebeta

← #31908187

Subgroups — #31908187

Order 18176.

434 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#17985938C211yesmaximal, characteristic
2#22588420C211yesmaximal, characteristic
3#22586368C214yesmaximal, complemented
4C71×#1463C211yesmaximal, characteristic
5C22×C4⋊Dic71C411yescharacteristic
6#17985963—21no—
7#17985938—21no—
8#17985938—21no—
9#17985938C412yes—
10#17985938C2212yes—
11C2×C71⋊#95—21no—
12C2×C71⋊#95—21no—
13C2×C71⋊#95C411yescharacteristic
14#17985768—21no—
15#17985768C2212yes—
16C22×C4⋊C4×C71C2211yescharacteristic
17#407×C71—21no—
18#324×C71—21no—
19#324×C71C2212yes—
20C23×Dic71—21no—
21C23×Dic71—21no—
22C2×#11765849—22no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—41no—
25C2×C4⋊Dic71—41no—
26C2×C4⋊Dic71C2×C412yes—
27C22⋊(C71⋊C8)—41no—
28C22⋊(C71⋊C8)—41no—
29C71⋊#95—22no—
30C71⋊#95—22no—
31C71⋊#95—21no—
32C71⋊#95—22no—
33C71⋊#95—22no—
34C71⋊#95C2×C412yes—
35C23×C4×C71C2×C411yescharacteristic
36C23×C4×C71D411yescharacteristic
37C23×C4×C71D411yescharacteristic
38C2×M16×C71—41no—
39C2×C4⋊C4×C71—41no—
40C2×C4⋊C4×C71—41no—
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C4⋊C4×C71D411yescharacteristic
43C2×C4⋊C4×C71D411yescharacteristic
44C2×C4⋊C4×C71C2311yescharacteristic
45C22⋊C8×C71—22no—
46C22⋊C8×C71—41no—
47C22×Dic71—22no—
48C22×Dic71—22no—
49C22×Dic71—42no—
50C22×Dic71—42no—
51C22×Dic71—44no—
52C22×Dic71—42no—
53C22×Dic71—41no—
54C22×Dic71—41no—
55#11765849—42no—
56#11765849—42no—
57C2×C71⋊C8—42no—
58C4⋊Dic71—42no—
59C4⋊Dic71—42no—
60C24×C71C22⋊C411yescharacteristic
61C22×C4×C71—21no—
62C22×C4×C71—21no—
63C22×C4×C71—21no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—22no—
68C22×C4×C71—41no—
69C22×C4×C71—42no—
70C22×C4×C71—42no—
71C22×C4×C71C22⋊C411yescharacteristic
72C22×C4×C71C22⋊C411yescharacteristic
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22×C411yescharacteristic
75C22×C4×C71C2×D411yescharacteristic
76C22×C4×C71C2×D411yescharacteristic
77M16×C71—82no—
78C2×C8×C71—42no—
79C4⋊C4×C71—41no—
80C4⋊C4×C71—41no—
81C4⋊C4×C71—41no—
82C4⋊C4×C71—41no—
83C23×C71—22no—
84C23×C71—22no—
85C23×C71—21no—
86C23×C71—21no—
87C23×C71#9911yescharacteristic
88C23×C71#9911yescharacteristic
89C23×C71C2×C22⋊C411yescharacteristic
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—21no—
98C2×C4×C71—21no—
99C2×C4×C71—42no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71—41no—
105C2×C4×C71—41no—
106C2×C4×C71—41no—
107C2×C4×C71—41no—
108C2×C4×C71—81no—
109C2×C4×C71#13712yes—
110C2×Dic71—42no—
111C2×Dic71—42no—
112C2×Dic71—44no—
113C2×Dic71—42no—
114C2×Dic71—42no—
115C2×Dic71—84no—
116C2×Dic71—82no—
117C2×Dic71—82no—
118C8×C71—82no—
119C71⋊C8—82no—
120C22×C71—24no—
121C22×C71—24no—
122C22×C71—21no—
123C22×C71—21no—
124C22×C71—21no—
125C22×C71—21no—
126C22×C71—21no—
127C22×C71—21no—
128C22×C71#35212yes—
129C22×C71#35512yes—
130C22×C71C2×#9911yescharacteristic
131C22×C71#41911yescharacteristic
132C22×C71#42111yescharacteristic
133C4×C71—22no—
134C4×C71—41no—
135C4×C71—41no—
136C4×C71—41no—
137C4×C71—41no—
138C4×C71—41no—
139Dic71—82no—
140Dic71—82no—
141#11814—711nomaximal, Sylow
142C2×C71—21no—
143C2×C71—22no—
144C2×C71—22no—
145C2×C71—21no—
146C2×C71#151111yescharacteristic
147C2×C71C2×#35211yescharacteristic
148C2×C71C2×#35511yescharacteristic
149C2×#400—711no—
150#1821—711no—
151#1463D7111yescharacteristic
152#1353—714no—
153C71#1181411yescharacteristic, complemented, Sylow
154C22×C4⋊C4—711no—
155C22×C4⋊C4C2×D7111yescharacteristic
156#407—21no—
157#407—1421no—
158#400—712no—
159#400—712no—
160#400—1421no—
161#400—1421no—
162C2×#95—711no—
163C2×#95—1421no—
164C2×#95—1421no—
165#343—712no—
166#343—1421no—
167#324—21no—
168#324C2×D7112yes—
169C23×C4—1421no—
170C23×C4—1421no—
171C23×C4C4×D7111yescharacteristic
172C23×C4D28411yescharacteristic
173C23×C4C22⋊D7111yescharacteristic
174C2×M16—41no—
175C2×M16—1422no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—2841no—
179C2×C4⋊C4—2841no—
180C2×C4⋊C4—712no—
181C2×C4⋊C4—1421no—
182C2×C4⋊C4C4×D7111yescharacteristic
183C2×C4⋊C4D28411yescharacteristic
184C2×C4⋊C4C22⋊D7111yescharacteristic
185C2×C4⋊C4C22×D7111yescharacteristic
186C22⋊C8—22no—
187C22⋊C8—41no—
188C22⋊C8—2841no—
189C22⋊C8—2841no—
190#95—712no—
191#95—1421no—
192#95—1422no—
193#95—1422no—
194#95—1422no—
195#95—1422no—
196C24(C2×D71)⋊C411yescharacteristic
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—2842no—
208C22×C4—2842no—
209C22×C4—2844no—
210C22×C4—2842no—
211C22×C4—2841no—
212C22×C4—2841no—
213C22×C4—1422no—
214C22×C4—1422no—
215C22×C4(C2×D71)⋊C411yescharacteristic
216C22×C4(C2×D71)⋊C411yescharacteristic
217C22×C4(C2×D71)⋊C411yescharacteristic
218C22×C4C2×C4×D7111yescharacteristic
219C22×C4C2×C22⋊D7111yescharacteristic
220C22×C4C2×D28411yescharacteristic
221M16—82no—
222M16—2842no—
223M16—2842no—
224C2×C8—42no—
225C2×C8—2842no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—2842no—
231C4⋊C4—2842no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23#1455604211yescharacteristic
237C23#1455604211yescharacteristic
238C23C2×(C2×D71)⋊C411yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—81no—
258C2×C4—2842no—
259C2×C4—2842no—
260C2×C4—2844no—
261C2×C4—2842no—
262C2×C4—2842no—
263C2×C4—5684no—
264C2×C4—5682no—
265C2×C4—5682no—
266C2×C4#1455616112yes—
267C8—82no—
268C8—5682no—
269C22—24no—
270C22—24no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22#1798578812yes—
278C22#1798579512yes—
279C22C2×#1455604211yescharacteristic
280C22#1798600911yescharacteristic
281C22#1798601511yescharacteristic
282C4—22no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—5682no—
289C4—5682no—
290C2—21no—
291C2—22no—
292C2—21no—
293C2—22no—
294C2#2258687411yescharacteristic
295C2C2×#1798578811yescharacteristic
296C2C2×#1798579511yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#1463C2
socleC22×C71C2×#99
prime radicalC24×C71C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#1463D71
hypocenterC71#11814
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #22586368C2semidirect
153. C71#11814semidirect

Maximal subgroups

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

subgroupindex
C2×#179859382
#225884202
#225863682
C71×#14632
#1181471