Finite Group Databasebeta

← #31892504

Subgroups — #31892504

Order 18176.

616 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 127 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×#17985929C211yesmaximal, characteristic
2#22590943C211yesmaximal, characteristic
3#22588427C212yesmaximal
4#22586220C212yesmaximal
5C2×#324×C71C211yesmaximal, characteristic
6C22×C4⋊Dic71C2211yescharacteristic
7C22×Dic71⋊C4—21no—
8#17985929—22no—
9#17985929C414yescomplemented
10#17985885—24no—
11#17985862—21no—
12#17985859C2212yes—
13C4×C22⋊Dic71—21no—
14C2×C71⋊#95—21no—
15C2×C71⋊#95C2211yescharacteristic
16#17985762—24no—
17C22×C4⋊C4×C71C2211yescharacteristic
18C2×C22⋊C8×C71C2212yes—
19#324×C71—22no—
20C23×Dic71—21no—
21C23×Dic71—21no—
22C2×C4⋊Dic71—22no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—22no—
25C2×C4⋊Dic71C2×C412yes—
26C2×C4⋊Dic71D414yes—
27C2×Dic71⋊C4—24no—
28C2×Dic71⋊C4—44no—
29C2×C22⋊Dic71—22no—
30C2×C4×Dic71—22no—
31C71⋊#95—22no—
32C71⋊#95—24no—
33C71⋊#95—24no—
34C71⋊#95—21no—
35C71⋊#95—22no—
36C71⋊#95C2×C412yes—
37C23×C4×C71D411yescharacteristic
38C23×C4×C71D411yescharacteristic
39C23×C4×C71C2311yescharacteristic
40C22×C8×C71—22no—
41C2×C4⋊C4×C71—22no—
42C2×C4⋊C4×C71—22no—
43C2×C4⋊C4×C71C2×C412yes—
44C2×C4⋊C4×C71D412yes—
45C22⋊C8×C71—22no—
46C22⋊C8×C71—22no—
47C22×Dic71—24no—
48C22×Dic71—22no—
49C22×Dic71—22no—
50C22×Dic71—24no—
51C22×Dic71—42no—
52C22×Dic71—42no—
53C22×Dic71—41no—
54C22×Dic71—41no—
55C22×Dic71—42no—
56C22×Dic71—42no—
57C4⋊Dic71—24no—
58C4⋊Dic71—42no—
59C4⋊Dic71—44no—
60Dic71⋊C4—48no—
61C22⋊Dic71—42no—
62C22⋊Dic71—42no—
63C4×Dic71—82no—
64C24×C71C2×D411yescharacteristic
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—22no—
69C22×C4×C71—22no—
70C22×C4×C71—21no—
71C22×C4×C71—42no—
72C22×C4×C71—42no—
73C22×C4×C71C22⋊C412yes—
74C22×C4×C71C22⋊C412yes—
75C22×C4×C71C22×C411yescharacteristic
76C22×C4×C71C2×D412yes—
77C22×C4×C71C2×D411yescharacteristic
78C22×C4×C71#4112yes—
79C2×C8×C71—44no—
80C2×C8×C71—42no—
81C4⋊C4×C71—24no—
82C4⋊C4×C71—24no—
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—22no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—41no—
96C2×C4×C71—41no—
97C2×C4×C71—42no—
98C2×C4×C71—42no—
99C2×C4×C71—42no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71—81no—
103C2×C4×C71C4×D412yes—
104C2×C4×C71#12112yes—
105C2×Dic71—24no—
106C2×Dic71—48no—
107C2×Dic71—42no—
108C2×Dic71—42no—
109C2×Dic71—42no—
110C2×Dic71—42no—
111C2×Dic71—42no—
112C2×Dic71—44no—
113C2×Dic71—44no—
114C2×Dic71—44no—
115C2×Dic71—82no—
116C2×Dic71—82no—
117C8×C71—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—42no—
134C4×C71—42no—
135Dic71—44no—
136Dic71—84no—
137Dic71—82no—
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×#398—711no—
147#2313—711no—
148#1823—712no—
149C2×#324D7111yescharacteristic
150#1294—712no—
151C71#902611yescharacteristic, complemented, Sylow
152C22×C4⋊C4—711no—
153C22×C4⋊C4—1421no—
154C22×C4⋊C4C2×D7111yescharacteristic
155C2×C22⋊C8C2×D7112yes—
156#398—714no—
157#398—1422no—
158#387—1424no—
159#381—1421no—
160#380—712no—
161C4×C22⋊C4—1421no—
162C2×#95—711no—
163C2×#95—1421no—
164#340—1424no—
165#324—22no—
166C23×C4—1421no—
167C23×C4—1421no—
168C23×C4D28411yescharacteristic
169C23×C4D28411yescharacteristic
170C23×C4C22×D7111yescharacteristic
171C22×C8—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—2844no—
175C2×C4⋊C4—714no—
176C2×C4⋊C4—712no—
177C2×C4⋊C4—1424no—
178C2×C4⋊C4—1421no—
179C2×C4⋊C4—1422no—
180C2×C4⋊C4—1422no—
181C2×C4⋊C4C4×D7112yes—
182C2×C4⋊C4C22⋊D7112yes—
183C2×C22⋊C4—1422no—
184C2×C42—1422no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187#95—712no—
188#95—1422no—
189#95—1424no—
190#95—1424no—
191#95—1422no—
192#95—1421no—
193C24C2×D28411yescharacteristic
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—2842no—
203C22×C4—2842no—
204C22×C4—2841no—
205C22×C4—2841no—
206C22×C4—2842no—
207C22×C4—2842no—
208C22×C4—1422no—
209C22×C4—1424no—
210C22×C4—1424no—
211C22×C4—1422no—
212C22×C4(C2×D71)⋊C412yes—
213C22×C4(C2×D71)⋊C412yes—
214C22×C4C2×C4×D7111yescharacteristic
215C22×C4D4×D7112yes—
216C22×C4C2×C22⋊D7111yescharacteristic
217C22×C4#1176586412yes—
218C2×C8—44no—
219C2×C8—42no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—2844no—
223C4⋊C4—2848no—
224C4⋊C4—2842no—
225C4⋊C4—1424no—
226C22⋊C4—2842no—
227C22⋊C4—2842no—
228C42—5682no—
229C23—22no—
230C23—22no—
231C23C2×(C2×D71)⋊C411yescharacteristic
232C23#1455609811yescharacteristic
233C23#1455611211yescharacteristic
234C23#1455615012yes—
235C23#1455615612yes—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—81no—
249C2×C4—2842no—
250C2×C4—2848no—
251C2×C4—2842no—
252C2×C4—2842no—
253C2×C4—2844no—
254C2×C4—2844no—
255C2×C4—2844no—
256C2×C4—2842no—
257C2×C4—2842no—
258C2×C4—5682no—
259C2×C4—5682no—
260C2×C4—1424no—
261C2×C4(C22⋊D71)⋊C412yes—
262C2×C4#1455610012yes—
263C8—82no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—24no—
269C22—24no—
270C22#1798587611yescharacteristic
271C22#1798599611yescharacteristic
272C22#1798600511yescharacteristic
273C22#1798614612yes—
274C22#1798618412yes—
275C22#1798619012yes—
276C22#1798632412yes—
277C4—22no—
278C4—22no—
279C4—42no—
280C4—42no—
281C4—2844no—
282C4—5684no—
283C4—5682no—
284C2—21no—
285C2—22no—
286C2—21no—
287C2#2258733914yes—
288C2#2258866011yescharacteristic
289C2#2258873111yescharacteristic
290C2#2258875111yescharacteristic
subgroupG/H
centerC23#14556098
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#324×C71C2
socleC23×C71#120
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenterC2×#324D71
hypocenterC71#9026
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
9. #17985929C4semidirect
151. C71#9026semidirect

Maximal subgroups

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

subgroupindex
C2×#179859292
#225909432
#225884272
#225862202
C2×#324×C712
#902671