Finite Group Databasebeta

← #31895327

Subgroups — #31895327

Order 18176.

522 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 119 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#22597425C211yesmaximal, characteristic, complemented
2#22588336C211yesmaximal, characteristic, complemented
3#22586765C212yesmaximal
4#22586198C212yesmaximal, complemented
5C71×#2536C211yesmaximal, characteristic
6#17986130—24no—
7#17986056—22no—
8#17986052C2212yes—
9#17986023C2212yes—
10C2×#14556047—21no—
11#17985768—21no—
12D4⋊(C71⋊C8)—24no—
13C2×C22⋊Q8×C71—21no—
14C2×C4×Q8×C71C2211yescharacteristic
15C2×C4×D4×C71C2211yescharacteristic
16#398×C71C412yes—
17#394×C71C412yes—
18#388×C71—22no—
19#382×C71C2211yescharacteristic
20#380×C71—21no—
21C2×#11765849—22no—
22C4⋊(C71⋊C8)—24no—
23C4⋊(C71⋊C8)—22no—
24C4⋊(C71⋊C8)—22no—
25C4⋊(C71⋊C8)—22no—
26#14556047—42no—
27C22⋊(C71⋊C8)—22no—
28#14556039—24no—
29C4×C71⋊C8D414yes—
30C22×Q8×C71D411yescharacteristic
31C22×D4×C71C2×C411yescharacteristic
32C23×C4×C71—21no—
33C22⋊Q8×C71—44no—
34C4×Q8×C71—22no—
35C4×Q8×C71—22no—
36C4×D4×C71—22no—
37C4×D4×C71—22no—
38C2×C4⋊C4×C71—22no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71C2×C412yes—
42C2×C4⋊C4×C71D411yescharacteristic
43C2×C22⋊C4×C71—22no—
44C2×C22⋊C4×C71—22no—
45C2×C22⋊C4×C71—21no—
46C2×C42×C71D412yes—
47C2×C42×C71C2311yescharacteristic
48#95×C71—22no—
49#95×C71—21no—
50#95×C71C2×C412yes—
51#11765849—42no—
52#11765849—42no—
53C2×C71⋊C8—24no—
54C2×C71⋊C8—42no—
55C24×C71—21no—
56C2×Q8×C71—22no—
57C2×Q8×C71—22no—
58C2×Q8×C71—21no—
59C2×Q8×C71—21no—
60C2×D4×C71—22no—
61C2×D4×C71—42no—
62C22×C4×C71—22no—
63C22×C4×C71—21no—
64C22×C4×C71—42no—
65C22×C4×C71—42no—
66C22×C4×C71—41no—
67C22×C4×C71—41no—
68C22×C4×C71C22⋊C412yes—
69C22×C4×C71C22⋊C412yes—
70C22×C4×C71C22×C411yescharacteristic
71C22×C4×C71C2×D411yescharacteristic
72C22×C4×C71C2×D411yescharacteristic
73C4⋊C4×C71—24no—
74C4⋊C4×C71—22no—
75C4⋊C4×C71—22no—
76C4⋊C4×C71—44no—
77C4⋊C4×C71—42no—
78C22⋊C4×C71—44no—
79C22⋊C4×C71—44no—
80C22⋊C4×C71—41no—
81C22⋊C4×C71—41no—
82C42×C71—22no—
83C42×C71—22no—
84C42×C71C2×D412yes—
85C42×C71#4112yes—
86C23×C71—22no—
87C23×C71—41no—
88C23×C71—41no—
89C23×C71—41no—
90C23×C71—41no—
91C23×C71—41no—
92C23×C71—41no—
93C23×C71C2×C22⋊C411yescharacteristic
94Q8×C71—42no—
95Q8×C71—42no—
96D4×C71—24no—
97D4×C71—42no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—22no—
102C2×C4×C71—41no—
103C2×C4×C71—41no—
104C2×C4×C71—42no—
105C2×C4×C71—42no—
106C2×C4×C71—42no—
107C2×C4×C71—41no—
108C2×C4×C71—41no—
109C2×C4×C71—42no—
110C2×C4×C71—41no—
111C2×C4×C71—82no—
112C2×C4×C71—82no—
113C2×C4×C71C4×D412yes—
114C2×C4×C71#12011yescharacteristic
115C2×C4×C71#12112yes—
116C2×C4×C71#12311yescharacteristic
117C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C71AGL(1,ℤ/8)12yes—
119C2×C4×C71#13711yescharacteristic
120C71⋊C8—44no—
121C71⋊C8—82no—
122C22×C71—21no—
123C22×C71—21no—
124C22×C71—42no—
125C22×C71—42no—
126C22×C71—42no—
127C22×C71—81no—
128C22×C71—81no—
129C22×C71—81no—
130C22×C71—81no—
131C22×C71#38611yescharacteristic
132C22×C71#41711yescharacteristic
133C22×C71#41811yescharacteristic
134C4×C71—22no—
135C4×C71—21no—
136C4×C71—21no—
137C4×C71—42no—
138C4×C71—42no—
139C4×C71—82no—
140C4×C71—81no—
141C4×C71#46812yes—
142C4×C71#46912yes—
143#9490—711nomaximal, Sylow
144C2×C71—21no—
145C2×C71—21no—
146C2×C71—82no—
147C2×C71#189111yescharacteristic
148C2×C71#189511yescharacteristic
149C2×C71#191211yescharacteristic
150#2983—711no—
151#2536D7111yescharacteristic
152#1801—711no—
153#1486—712no—
154#1286—712no—
155C71#949011yescharacteristic, complemented, Sylow
156C2×C22⋊Q8—21no—
157C2×C4×Q8C2×D7111yescharacteristic
158C2×C4×D4C2×D7111yescharacteristic
159#446—1424no—
160#432—1422no—
161#431—712no—
162#423—712no—
163C2×#101—1421no—
164#398Dic7112yes—
165#394Dic7112yes—
166#388—22no—
167#382C2×D7111yescharacteristic
168#380—21no—
169#343—1421no—
170D4⋊C8—1424no—
171C22×Q8C22⋊D7111yescharacteristic
172C22×D4C2×Dic7111yescharacteristic
173C23×C4—21no—
174C2×M16—1422no—
175C22⋊Q8—44no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—22no—
179C4×D4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C2×Dic7111yescharacteristic
183C2×C4⋊C4C2×Dic7112yes—
184C2×C4⋊C4C22⋊D7111yescharacteristic
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C42C22⋊D7112yes—
189C2×C42C22×D7111yescharacteristic
190C4⋊C8—1424no—
191C4⋊C8—1422no—
192C4⋊C8—1422no—
193C4⋊C8—1422no—
194#101—2842no—
195C22⋊C8—1422no—
196#97—1424no—
197C4×C8—714no—
198#95—22no—
199#95—21no—
200#95C2×Dic7112yes—
201C24—21no—
202C2×Q8—22no—
203C2×Q8—21no—
204C2×Q8—22no—
205C2×Q8—21no—
206C2×D4—22no—
207C2×D4—42no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4C22⋊Dic7112yes—
215C22×C4C22⋊Dic7112yes—
216C22×C4C22×Dic7111yescharacteristic
217C22×C4C2×C22⋊D7111yescharacteristic
218C22×C4C2×C22⋊D7111yescharacteristic
219M16—2842no—
220M16—2842no—
221C2×C8—2842no—
222C2×C8—1424no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C42—22no—
233C42—22no—
234C42D4×D7112yes—
235C42#1176586412yes—
236C23—22no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23C2×C22⋊Dic7111yescharacteristic
244Q8—42no—
245Q8—42no—
246D4—24no—
247D4—42no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4D4×Dic7112yes—
264C2×C4#1455609611yescharacteristic
265C2×C4#1455610012yes—
266C2×C4#1455610911yescharacteristic
267C2×C4#1455615311yescharacteristic
268C2×C4#1455615412yes—
269C2×C4#1455615811yescharacteristic
270C8—2844no—
271C8—5682no—
272C22—21no—
273C22—21no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22#1798587911yescharacteristic
282C22#1798600011yescharacteristic
283C22#1798600411yescharacteristic
284C4—22no—
285C4—21no—
286C4—21no—
287C4—42no—
288C4—42no—
289C4—82no—
290C4—81no—
291C4#1798624312yes—
292C4#1798625012yes—
293C2—21no—
294C2—21no—
295C2—82no—
296C2#2258875911yescharacteristic
297C2#2258877911yescharacteristic
298C2#2258887811yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2536C2
socleC22×C71#386
prime radicalC22×D4×C71C2×C4
cosocleC2×C42×C71C23
hypercenter#2536D71
hypocenterC71#9490
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. #22597425C2semidirect
2. #22588336C2semidirect
4. #22586198C2semidirect
155. C71#9490semidirect

Maximal subgroups

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

subgroupindex
#225974252
#225883362
#225867652
#225861982
C71×#25362
#949071