Finite Group Databasebeta

← #31889190

Subgroups — #31889190

Order 18176.

706 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 159 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
2#22587012C212yesmaximal
3#22586769C212yesmaximal
4#22586664C211yesmaximal, characteristic
5C2×#557×C71C211yesmaximal, characteristic
6#17986052C2212yes—
7#17986023C2212yes—
8C2×#14556047—21no—
9#17985768—21no—
10#17985745—24no—
11Q8⋊(C71⋊C8)—24no—
12#17985725—22no—
13#557×C71—24no—
14#557×C71—24no—
15C2×C4⋊Q8×C71—21no—
16C2×C4⋊Q8×C71C2211yescharacteristic
17C2×#125×C71—22no—
18C2×#125×C71C412yes—
19C2×C4×Q8×C71—21no—
20C2×C4×Q8×C71C412yes—
21C2×C4×Q8×C71C2211yescharacteristic
22C2×C4×Q8×C71C2211yescharacteristic
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×C71—21no—
31C22×Q8×C71C2×C411yescharacteristic
32C22×Q8×C71D411yescharacteristic
33C4⋊Q8×C71—28no—
34C4⋊Q8×C71—22no—
35C4⋊Q8×C71D414yes—
36#125×C71—216no—
37#125×C71—24no—
38#125×C71—24no—
39C4×Q8×C71—28no—
40C4×Q8×C71—24no—
41C4×Q8×C71—24no—
42C4×Q8×C71—22no—
43C4×Q8×C71—22no—
44C4×Q8×C71—22no—
45C4×Q8×C71D414yes—
46C2×C4⋊C4×C71—22no—
47C2×C4⋊C4×C71—22no—
48C2×C4⋊C4×C71—22no—
49C2×C4⋊C4×C71—21no—
50C2×C4⋊C4×C71—21no—
51C2×C4⋊C4×C71C2×C412yes—
52C2×C4⋊C4×C71C2×C411yescharacteristic
53C2×C4⋊C4×C71D411yescharacteristic
54C2×C4⋊C4×C71D412yes—
55C2×C42×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—
63C2×Q8×C71—24no—
64C2×Q8×C71—21no—
65C2×Q8×C71—22no—
66C2×Q8×C71—24no—
67C2×Q8×C71—22no—
68C2×Q8×C71—21no—
69C2×Q8×C71—21no—
70C2×Q8×C71—42no—
71C2×Q8×C71—41no—
72C2×Q8×C71—41no—
73C2×Q8×C71C22⋊C412yes—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—22no—
77C22×C4×C71C22⋊C412yes—
78C22×C4×C71C22⋊C412yes—
79C22×C4×C71C22×C411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C4⋊C4×C71—28no—
83C4⋊C4×C71—28no—
84C4⋊C4×C71—28no—
85C4⋊C4×C71—21no—
86C4⋊C4×C71—22no—
87C4⋊C4×C71—24no—
88C4⋊C4×C71—24no—
89C4⋊C4×C71—22no—
90C4⋊C4×C71—24no—
91C4⋊C4×C71C22⋊C414yes—
92C4⋊C4×C71C22⋊C412yes—
93C42×C71—28no—
94C42×C71—22no—
95C42×C71—22no—
96C42×C71—24no—
97C42×C71C2×D412yes—
98C42×C71C2×D412yes—
99C23×C71C2×C22⋊C411yescharacteristic
100Q8×C71—24no—
101Q8×C71—22no—
102Q8×C71—42no—
103Q8×C71—44no—
104Q8×C71—42no—
105Q8×C71—44no—
106Q8×C71#10114yescomplemented
107C2×C4×C71—24no—
108C2×C4×C71—22no—
109C2×C4×C71—22no—
110C2×C4×C71—22no—
111C2×C4×C71—22no—
112C2×C4×C71—22no—
113C2×C4×C71—24no—
114C2×C4×C71—22no—
115C2×C4×C71—22no—
116C2×C4×C71—41no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71C2×C22⋊C411yescharacteristic
122C2×C4×C71C2×C22⋊C411yescharacteristic
123C2×C4×C71#12012yes—
124C2×C4×C71#12012yes—
125C2×C4×C71#13512yes—
126C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C71#13711yescharacteristic
128C71⋊C8—82no—
129C71⋊C8—82no—
130C22×C71—21no—
131C22×C71—21no—
132C22×C71#37911yescharacteristic
133C22×C71#41611yescharacteristic
134C22×C71#41711yescharacteristic
135C4×C71—22no—
136C4×C71—24no—
137C4×C71—21no—
138C4×C71—42no—
139C4×C71—42no—
140C4×C71—42no—
141C4×C71—44no—
142C4×C71C2×#10112yes—
143C4×C71#44912yes—
144C4×C71#45212yes—
145#8390—711nomaximal, Sylow
146C2×C71—21no—
147C2×C71—21no—
148C2×C71#178411yescharacteristic
149C2×C71#178811yescharacteristic
150C2×C71#179611yescharacteristic
151C2×#557D7111yescharacteristic
152#1801—711no—
153#1546—712no—
154#1487—712no—
155#1461—711no—
156C71#839011yescharacteristic, complemented, Sylow
157#557—24no—
158#557—24no—
159C2×C4⋊Q8—21no—
160C2×C4⋊Q8C2×D7111yescharacteristic
161C2×#125—22no—
162C2×#125Dic7112yes—
163C2×C4×Q8—21no—
164C2×C4×Q8Dic7112yes—
165C2×C4×Q8C2×D7111yescharacteristic
166C2×C4×Q8C2×D7111yescharacteristic
167#431—712no—
168#423—712no—
169C2×#101—1421no—
170#343—1421no—
171#332—1424no—
172Q8⋊C8—1424no—
173#324—1422no—
174C22×Q8—21no—
175C22×Q8C2×Dic7111yescharacteristic
176C22×Q8C22⋊D7111yescharacteristic
177C2×M16—1422no—
178C4⋊Q8—28no—
179C4⋊Q8—22no—
180C4⋊Q8C22⋊D7114yes—
181#125—216no—
182#125—24no—
183#125—24no—
184C4×Q8—28no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8C22⋊D7114yes—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C2×Dic7112yes—
197C2×C4⋊C4C2×Dic7111yescharacteristic
198C2×C4⋊C4C22⋊D7111yescharacteristic
199C2×C4⋊C4C22⋊D7112yes—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42C2×Dic7112yes—
203C2×C42C22×D7111yescharacteristic
204C4⋊C8—1422no—
205C4⋊C8—1422no—
206C4⋊C8—1422no—
207#101—2842no—
208C22⋊C8—1422no—
209C4×C8—1422no—
210C2×Q8—24no—
211C2×Q8—21no—
212C2×Q8—22no—
213C2×Q8—24no—
214C2×Q8—22no—
215C2×Q8—21no—
216C2×Q8—21no—
217C2×Q8—42no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8C22⋊Dic7112yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4C22⋊Dic7112yes—
225C22×C4C22⋊Dic7112yes—
226C22×C4C22×Dic7111yescharacteristic
227C22×C4C2×C22⋊D7111yescharacteristic
228C22×C4C2×C22⋊D7111yescharacteristic
229M16—2842no—
230M16—2842no—
231C2×C8—2842no—
232C2×C8—2842no—
233C4⋊C4—28no—
234C4⋊C4—28no—
235C4⋊C4—28no—
236C4⋊C4—21no—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—22no—
241C4⋊C4—24no—
242C4⋊C4C22⋊Dic7114yes—
243C4⋊C4C22⋊Dic7112yes—
244C42—28no—
245C42—22no—
246C42—22no—
247C42—24no—
248C42C2×C22⋊D7112yes—
249C42C2×C22⋊D7112yes—
250C23C2×C22⋊Dic7111yescharacteristic
251Q8—24no—
252Q8—22no—
253Q8—42no—
254Q8—44no—
255Q8—42no—
256Q8—44no—
257Q8#1455604714yescomplemented
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4C2×C22⋊Dic7111yescharacteristic
273C2×C4C2×C22⋊Dic7111yescharacteristic
274C2×C4#1455609712yes—
275C2×C4#1455609712yes—
276C2×C4#1455614812yes—
277C2×C4#1455615311yescharacteristic
278C2×C4#1455615811yescharacteristic
279C8—5682no—
280C8—5682no—
281C22—21no—
282C22—21no—
283C22#1798585511yescharacteristic
284C22#1798599511yescharacteristic
285C22#1798600011yescharacteristic
286C4—24no—
287C4—21no—
288C4—22no—
289C4—42no—
290C4—44no—
291C4—42no—
292C4—42no—
293C4C2×#1455604712yes—
294C4#1798614412yes—
295C4#1798616112yes—
296C2—21no—
297C2—21no—
298C2#2258826411yescharacteristic
299C2#2258828011yescharacteristic
300C2#2258831511yescharacteristic
subgroupG/H
centerC22#17985855
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC2×#557×C71C2
socleC22×C71#379
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenterC2×#557D71
hypocenterC71#8390
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
106. Q8×C71#101semidirect
156. C71#8390semidirect
257. Q8#14556047semidirect

Maximal subgroups

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

subgroupindex
#225883362
#225870122
#225867692
#225866642
C2×#557×C712
#839071