Finite Group Databasebeta

← #31892507

Subgroups — #31892507

Order 18176.

616 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 135 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#22590943C211yesmaximal, characteristic
2#22588426C212yesmaximal
3C2×#17985725C211yesmaximal, characteristic
4#22586219C212yesmaximal
5C2×#398×C71C211yesmaximal, characteristic
6C2×C22⋊(C71⋊C8)C2212yes—
7#17985885—24no—
8#17985862—21no—
9#17985859C2212yes—
10C4×C22⋊Dic71—21no—
11#17985761—24no—
12#17985725—22no—
13C22×C4⋊C4×C71—21no—
14C22×C4⋊C4×C71C2211yescharacteristic
15C22×C4⋊C4×C71C2211yescharacteristic
16#398×C71—22no—
17#398×C71C414yescomplemented
18C2×#95×C71—21no—
19C2×#95×C71C2211yescharacteristic
20C22×C71⋊C8—22no—
21C2×C4⋊Dic71D414yes—
22C2×C22⋊Dic71—22no—
23C2×C4×Dic71—22no—
24C22⋊(C71⋊C8)—22no—
25C22⋊(C71⋊C8)—22no—
26C71⋊#95—22no—
27C71⋊#95—24no—
28C71⋊#95—22no—
29C23×C4×C71—21no—
30C23×C4×C71—21no—
31C23×C4×C71D411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C23×C4×C71C2311yescharacteristic
34C2×C4⋊C4×C71—22no—
35C2×C4⋊C4×C71—22no—
36C2×C4⋊C4×C71—22no—
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71—24no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—44no—
41C2×C4⋊C4×C71C2×C412yes—
42C2×C4⋊C4×C71C2×C412yes—
43C2×C4⋊C4×C71D412yes—
44#95×C71—24no—
45#95×C71—21no—
46#95×C71C2×C412yes—
47C22×Dic71—24no—
48C2×C71⋊C8—44no—
49C2×C71⋊C8—42no—
50C4⋊Dic71—44no—
51C22⋊Dic71—42no—
52C22⋊Dic71—42no—
53C4×Dic71—82no—
54C24×C71C2×D411yescharacteristic
55C22×C4×C71—24no—
56C22×C4×C71—22no—
57C22×C4×C71—21no—
58C22×C4×C71—21no—
59C22×C4×C71—21no—
60C22×C4×C71—22no—
61C22×C4×C71—22no—
62C22×C4×C71—21no—
63C22×C4×C71—22no—
64C22×C4×C71—42no—
65C22×C4×C71—42no—
66C22×C4×C71—42no—
67C22×C4×C71—42no—
68C22×C4×C71—41no—
69C22×C4×C71—41no—
70C22×C4×C71—42no—
71C22×C4×C71—42no—
72C22×C4×C71C22⋊C412yes—
73C22×C4×C71C22⋊C412yes—
74C22×C4×C71C22×C411yescharacteristic
75C22×C4×C71C2×D412yes—
76C22×C4×C71C2×D411yescharacteristic
77C22×C4×C71#4112yes—
78C4⋊C4×C71—24no—
79C4⋊C4×C71—24no—
80C4⋊C4×C71—24no—
81C4⋊C4×C71—48no—
82C4⋊C4×C71—42no—
83C23×C71—22no—
84C23×C71—22no—
85C23×C71C2×C22⋊C411yescharacteristic
86C23×C71#12011yescharacteristic
87C23×C71#12311yescharacteristic
88C23×C71#13512yes—
89C23×C71AGL(1,ℤ/8)12yes—
90C2×C4×C71—24no—
91C2×C4×C71—24no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—41no—
97C2×C4×C71—41no—
98C2×C4×C71—42no—
99C2×C4×C71—48no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71—42no—
105C2×C4×C71—42no—
106C2×C4×C71—42no—
107C2×C4×C71—42no—
108C2×C4×C71—42no—
109C2×C4×C71—81no—
110C2×C4×C71—82no—
111C2×C4×C71—82no—
112C2×C4×C71C4×D412yes—
113C2×C4×C71#12112yes—
114C2×Dic71—44no—
115C2×Dic71—44no—
116C2×Dic71—44no—
117C71⋊C8—82no—
118C22×C71—24no—
119C22×C71—24no—
120C22×C71—21no—
121C22×C71—21no—
122C22×C71—21no—
123C22×C71—21no—
124C22×C71#38611yescharacteristic
125C22×C71#41611yescharacteristic
126C22×C71#41811yescharacteristic
127C22×C71#44912yes—
128C22×C71#45612yes—
129C22×C71#45812yes—
130C22×C71#48212yes—
131C4×C71—22no—
132C4×C71—22no—
133C4×C71—44no—
134C4×C71—42no—
135C4×C71—42no—
136C4×C71—82no—
137Dic71—84no—
138#9026—711nomaximal, Sylow
139C2×C71—21no—
140C2×C71—21no—
141C2×C71—22no—
142C2×C71#160914yes—
143C2×C71#187511yescharacteristic
144C2×C71#188611yescharacteristic
145C2×C71#189011yescharacteristic
146C2×#398D7111yescharacteristic
147#2313—711no—
148#1823—712no—
149C2×#324—711no—
150#1294—712no—
151C71#902611yescharacteristic, complemented, Sylow
152C22×C4⋊C4—21no—
153C22×C4⋊C4C2×D7111yescharacteristic
154C22×C4⋊C4C2×D7111yescharacteristic
155C2×C22⋊C8—712no—
156#398—22no—
157#398Dic7114yescomplemented
158#387—1424no—
159#381—1421no—
160#380—712no—
161C4×C22⋊C4—1421no—
162C2×#95—21no—
163C2×#95C2×D7111yescharacteristic
164#340—1424no—
165#324—1422no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4C22⋊D7111yescharacteristic
169C23×C4C22⋊D7111yescharacteristic
170C23×C4C22×D7111yescharacteristic
171C22×C8—1422no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—714no—
180C2×C4⋊C4C2×Dic7112yes—
181C2×C4⋊C4C2×Dic7112yes—
182C2×C4⋊C4C22⋊D7112yes—
183C2×C22⋊C4—1422no—
184C2×C42—1422no—
185C22⋊C8—1422no—
186C22⋊C8—1422no—
187#95—24no—
188#95—21no—
189#95—1422no—
190#95—1424no—
191#95—1422no—
192#95C2×Dic7112yes—
193C24C2×C22⋊D7111yescharacteristic
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—1424no—
212C22×C4C22⋊Dic7112yes—
213C22×C4C22⋊Dic7112yes—
214C22×C4C22×Dic7111yescharacteristic
215C22×C4D4×D7112yes—
216C22×C4C2×C22⋊D7111yescharacteristic
217C22×C4#1176586412yes—
218C2×C8—2844no—
219C2×C8—2842no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—48no—
224C4⋊C4—42no—
225C4⋊C4—2844no—
226C22⋊C4—2842no—
227C22⋊C4—2842no—
228C42—5682no—
229C23—22no—
230C23—22no—
231C23C2×C22⋊Dic7111yescharacteristic
232C23#1455609611yescharacteristic
233C23#1455610911yescharacteristic
234C23#1455614812yes—
235C23#1455615412yes—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—48no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—2844no—
259C2×C4—2844no—
260C2×C4—2844no—
261C2×C4D4×Dic7112yes—
262C2×C4#1455610012yes—
263C8—5682no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—24no—
269C22—24no—
270C22#1798587911yescharacteristic
271C22#1798599511yescharacteristic
272C22#1798600411yescharacteristic
273C22#1798614212yes—
274C22#1798618112yes—
275C22#1798618812yes—
276C22#1798632012yes—
277C4—22no—
278C4—22no—
279C4—44no—
280C4—42no—
281C4—42no—
282C4—82no—
283C4—5684no—
284C2—21no—
285C2—22no—
286C2—21no—
287C2#2258733514yes—
288C2#2258866311yescharacteristic
289C2#2258873411yescharacteristic
290C2#2258875411yescharacteristic
subgroupG/H
centerC23#14556096
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#398×C71C2
socleC23×C71#120
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenterC2×#398D71
hypocenterC71#9026
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
17. #398×C71C4semidirect
151. C71#9026semidirect
157. #398Dic71semidirect

Maximal subgroups

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

subgroupindex
#225909432
#225884262
C2×#179857252
#225862192
C2×#398×C712
#902671