Finite Group Databasebeta

← #31889170

Subgroups — #31889170

Order 18176.

802 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 151 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#22588336C211yesmaximal, characteristic, complemented
2#22587008C212yesmaximal
3#22586769C212yesmaximal, complemented
4#22586664C211yesmaximal, characteristic, complemented
5C2×#559×C71C211yesmaximal, characteristic
6#17986052C2212yes—
7#17986023C2212yes—
8C2×#14556047—21no—
9#17985768—21no—
10#17985743—24no—
11Q8⋊(C71⋊C8)—24no—
12#17985725—22no—
13#559×C71—24no—
14#559×C71—24no—
15C2×#127×C71C2211yescharacteristic
16C2×#124×C71—21no—
17C2×#124×C71C412yes—
18C2×#123×C71—22no—
19C2×C4×Q8×C71C2211yescharacteristic
20C2×C4×Q8×C71C2211yescharacteristic
21C2×C4×D4×C71—21no—
22C2×C4×D4×C71C412yes—
23C2×#11765849—22no—
24C4⋊(C71⋊C8)—22no—
25C4⋊(C71⋊C8)—22no—
26C4⋊(C71⋊C8)—22no—
27#14556047—42no—
28C22⋊(C71⋊C8)—22no—
29C4×C71⋊C8—22no—
30C22×Q8×C71C2×C411yescharacteristic
31C22×Q8×C71D411yescharacteristic
32C22×D4×C71—21no—
33C22×D4×C71—21no—
34C22×D4×C71C2×C412yes—
35C23×C4×C71—22no—
36#127×C71—22no—
37#127×C71D414yes—
38#124×C71—28no—
39#124×C71—24no—
40#124×C71—24no—
41#123×C71—216no—
42C4×Q8×C71—22no—
43C4×Q8×C71—22no—
44C4×Q8×C71—22no—
45C4×Q8×C71D414yes—
46C4×D4×C71—28no—
47C4×D4×C71—24no—
48C4×D4×C71—24no—
49C2×C4⋊C4×C71—21no—
50C2×C4⋊C4×C71C2×C411yescharacteristic
51C2×C4⋊C4×C71D411yescharacteristic
52C2×C4⋊C4×C71D412yes—
53C2×C22⋊C4×C71—22no—
54C2×C22⋊C4×C71—22no—
55C2×C22⋊C4×C71—22no—
56C2×C42×C71—21no—
57C2×C42×C71C2×C412yes—
58C2×C42×C71C2311yescharacteristic
59#11765849—42no—
60#11765849—42no—
61C2×C71⋊C8—42no—
62C2×C71⋊C8—42no—
63C24×C71—22no—
64C2×Q8×C71—21no—
65C2×Q8×C71—22no—
66C2×Q8×C71—22no—
67C2×Q8×C71—21no—
68C2×Q8×C71—21no—
69C2×Q8×C71—42no—
70C2×Q8×C71C22⋊C412yes—
71C2×D4×C71—22no—
72C2×D4×C71—24no—
73C2×D4×C71—24no—
74C2×D4×C71—44no—
75C2×D4×C71—42no—
76C2×D4×C71—41no—
77C2×D4×C71—41no—
78C2×D4×C71—82no—
79C2×D4×C71C22⋊C414yes—
80C22×C4×C71—21no—
81C22×C4×C71—21no—
82C22×C4×C71—28no—
83C22×C4×C71—84no—
84C22×C4×C71C22⋊C412yes—
85C22×C4×C71C22⋊C412yes—
86C22×C4×C71C22×C411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C22×C4×C71C2×D411yescharacteristic
89C4⋊C4×C71—21no—
90C4⋊C4×C71—24no—
91C4⋊C4×C71—22no—
92C4⋊C4×C71—24no—
93C4⋊C4×C71C22⋊C412yes—
94C22⋊C4×C71—28no—
95C22⋊C4×C71—28no—
96C22⋊C4×C71—28no—
97C42×C71—22no—
98C42×C71—22no—
99C42×C71—24no—
100C42×C71C2×D412yes—
101C42×C71C2×D412yes—
102C23×C71—24no—
103C23×C71—44no—
104C23×C71—44no—
105C23×C71—42no—
106C23×C71—42no—
107C23×C71C2×C22⋊C411yescharacteristic
108Q8×C71—24no—
109Q8×C71—42no—
110Q8×C71—42no—
111Q8×C71—42no—
112D4×C71—44no—
113D4×C71—44no—
114D4×C71—44no—
115D4×C71—44no—
116D4×C71—84no—
117C2×C4×C71—22no—
118C2×C4×C71—22no—
119C2×C4×C71—22no—
120C2×C4×C71—22no—
121C2×C4×C71—22no—
122C2×C4×C71—22no—
123C2×C4×C71—22no—
124C2×C4×C71—41no—
125C2×C4×C71—42no—
126C2×C4×C71—42no—
127C2×C4×C71—42no—
128C2×C4×C71—88no—
129C2×C4×C71C2×C22⋊C411yescharacteristic
130C2×C4×C71C2×C22⋊C411yescharacteristic
131C2×C4×C71#12012yes—
132C2×C4×C71#12012yes—
133C2×C4×C71#13512yes—
134C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C71#13711yescharacteristic
136C71⋊C8—82no—
137C71⋊C8—82no—
138C22×C71—21no—
139C22×C71—21no—
140C22×C71—44no—
141C22×C71—44no—
142C22×C71—44no—
143C22×C71—84no—
144C22×C71—84no—
145C22×C71#37911yescharacteristic
146C22×C71#41611yescharacteristic
147C22×C71#41711yescharacteristic
148C4×C71—22no—
149C4×C71—21no—
150C4×C71—42no—
151C4×C71—42no—
152C4×C71—42no—
153C4×C71—42no—
154C4×C71#41312yes—
155C4×C71#44912yes—
156C4×C71#45212yes—
157#8386—711nomaximal, Sylow
158C2×C71—21no—
159C2×C71—21no—
160C2×C71—84no—
161C2×C71#178511yescharacteristic
162C2×C71#178811yescharacteristic
163C2×C71#179511yescharacteristic
164C2×#559D7111yescharacteristic
165#1801—711no—
166#1545—712no—
167#1487—712no—
168#1461—711no—
169C71#838611yescharacteristic, complemented, Sylow
170#559—24no—
171#559—24no—
172C2×#127C2×D7111yescharacteristic
173C2×#124—21no—
174C2×#124Dic7112yes—
175C2×#123—22no—
176C2×C4×Q8C2×D7111yescharacteristic
177C2×C4×Q8C2×D7111yescharacteristic
178C2×C4×D4—21no—
179C2×C4×D4Dic7112yes—
180#431—712no—
181#423—712no—
182C2×#101—1421no—
183#343—1421no—
184#331—1424no—
185Q8⋊C8—1424no—
186#324—1422no—
187C22×Q8C2×Dic7111yescharacteristic
188C22×Q8C22⋊D7111yescharacteristic
189C22×D4—21no—
190C22×D4—21no—
191C22×D4C2×Dic7112yes—
192C23×C4—22no—
193C2×M16—1422no—
194#127—22no—
195#127C22⋊D7114yes—
196#124—28no—
197#124—24no—
198#124—24no—
199#123—216no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8C22⋊D7114yes—
204C4×D4—28no—
205C4×D4—24no—
206C4×D4—24no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C2×Dic7111yescharacteristic
209C2×C4⋊C4C22⋊D7111yescharacteristic
210C2×C4⋊C4C22⋊D7112yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C42—21no—
215C2×C42C2×Dic7112yes—
216C2×C42C22×D7111yescharacteristic
217C4⋊C8—1422no—
218C4⋊C8—1422no—
219C4⋊C8—1422no—
220#101—2842no—
221C22⋊C8—1422no—
222C4×C8—1422no—
223C24—22no—
224C2×Q8—21no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—42no—
230C2×Q8C22⋊Dic7112yes—
231C2×D4—22no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—44no—
235C2×D4—42no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—82no—
239C2×D4C22⋊Dic7114yes—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—28no—
243C22×C4—84no—
244C22×C4C22⋊Dic7112yes—
245C22×C4C22⋊Dic7112yes—
246C22×C4C22×Dic7111yescharacteristic
247C22×C4C2×C22⋊D7111yescharacteristic
248C22×C4C2×C22⋊D7111yescharacteristic
249M16—2842no—
250M16—2842no—
251C2×C8—2842no—
252C2×C8—2842no—
253C4⋊C4—21no—
254C4⋊C4—24no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4C22⋊Dic7112yes—
258C22⋊C4—28no—
259C22⋊C4—28no—
260C22⋊C4—28no—
261C42—22no—
262C42—22no—
263C42—24no—
264C42C2×C22⋊D7112yes—
265C42C2×C22⋊D7112yes—
266C23—24no—
267C23—44no—
268C23—44no—
269C23—42no—
270C23—42no—
271C23C2×C22⋊Dic7111yescharacteristic
272Q8—24no—
273Q8—42no—
274Q8—42no—
275Q8—42no—
276D4—44no—
277D4—44no—
278D4—44no—
279D4—44no—
280D4—84no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—88no—
293C2×C4C2×C22⋊Dic7111yescharacteristic
294C2×C4C2×C22⋊Dic7111yescharacteristic
295C2×C4#1455609712yes—
296C2×C4#1455609712yes—
297C2×C4#1455614812yes—
298C2×C4#1455615311yescharacteristic
299C2×C4#1455615811yescharacteristic
300C8—5682no—
301C8—5682no—
302C22—21no—
303C22—21no—
304C22—44no—
305C22—44no—
306C22—44no—
307C22—84no—
308C22—84no—
309C22#1798585511yescharacteristic
310C22#1798599511yescharacteristic
311C22#1798600011yescharacteristic
312C4—21no—
313C4—22no—
314C4—42no—
315C4—42no—
316C4—42no—
317C4—42no—
318C4#1798598312yes—
319C4#1798614412yes—
320C4#1798616112yes—
321C2—21no—
322C2—21no—
323C2—84no—
324C2#2258826811yescharacteristic
325C2#2258828011yescharacteristic
326C2#2258831011yescharacteristic
subgroupG/H
centerC22#17985855
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC2×#559×C71C2
socleC22×C71#379
prime radicalC2×#127×C71C22
cosocleC2×C42×C71C23
hypercenterC2×#559D71
hypocenterC71#8386
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
1. #22588336C2semidirect
3. #22586769C2semidirect
4. #22586664C2semidirect
169. C71#8386semidirect

Maximal subgroups

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

subgroupindex
#225883362
#225870082
#225867692
#225866642
C2×#559×C712
#838671