Finite Group Databasebeta

← #31913479

Subgroups — #31913479

Order 18176.

420 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 97 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#22597058C211yesmaximal, characteristic
2#22590005C211yesmaximal, characteristic
3#22588384C212yesmaximal, complemented
4#22586460C212yesmaximal, complemented
5C71×#2126C211yesmaximal, characteristic
6C2×#14556135C411yescharacteristic, complemented
7C2×#14556135C2211yescharacteristic
8#17986071—22no—
9#17986071—22no—
10#17986035—22no—
11#17986023C411yescharacteristic, complemented
12#17985973C2212yes—
13#17985966C411yescharacteristic, complemented
14#17985966C2211yescharacteristic
15C2×#14556039C411yescharacteristic, complemented
16#17985794—21no—
17#17985792—21no—
18#17985755—22no—
19#532×C71C2211yescharacteristic
20#410×C71C2212yes—
21#354×C71—21no—
22#353×C71—21no—
23#14556135—42no—
24#14556135—42no—
25C2×#11765849—21no—
26C2×#11765849C2×C411yescharacteristic
27C2×#11765849C2×C411yescharacteristic
28C2×#11765849D412yes—
29C22×C71⋊C8—21no—
30C22×C71⋊C8C2×C411yescharacteristic
31C22×C71⋊C8C2×C411yescharacteristic
32C22×C71⋊C8D412yes—
33#14556087—22no—
34#14556061—42no—
35#14556059—42no—
36C4⋊(C71⋊C8)—22no—
37#14556043—22no—
38#14556043—22no—
39C22⋊(C71⋊C8)—22no—
40C22⋊(C71⋊C8)—22no—
41#14556039—21no—
42#14556039—21no—
43C2×#41×C71C2×C411yescharacteristic
44C2×#41×C71C2311yescharacteristic
45C4⋊Q8×C71Q811yescharacteristic
46#127×C71D411yescharacteristic
47#124×C71D411yescharacteristic
48#124×C71Q811yescharacteristic
49#121×C71—22no—
50C4×D4×C71—22no—
51#117×C71—22no—
52C2×C42×C71C2×C411yescharacteristic
53#99×C71—22no—
54#99×C71—22no—
55#11765849—41no—
56#11765849—42no—
57#11765849—42no—
58#11765849—41no—
59C2×C71⋊C8—22no—
60C2×C71⋊C8—22no—
61C2×C71⋊C8—22no—
62C2×C71⋊C8—21no—
63C2×C71⋊C8—21no—
64C2×C71⋊C8—42no—
65C2×C71⋊C8—41no—
66C2×C71⋊C8C22⋊C411yescharacteristic
67C2×C71⋊C8C22⋊C412yes—
68C2×C71⋊C8C22⋊C411yescharacteristic
69C4⋊Dic71—42no—
70C22⋊Dic71—42no—
71C4×Dic71—42no—
72#41×C71—41no—
73#41×C71—41no—
74#41×C71—41no—
75#41×C71—41no—
76C2×Q8×C71C4⋊C411yescharacteristic
77C2×Q8×C71C2×Q811yescharacteristic
78C2×D4×C71—21no—
79C2×D4×C71—22no—
80C2×D4×C71C4⋊C411yescharacteristic
81C2×D4×C71C2×D411yescharacteristic
82C2×D4×C71#4112yes—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71C22×C411yescharacteristic
86C22×C4×C71C2×D412yes—
87C4⋊C4×C71—22no—
88C4⋊C4×C71—42no—
89C22⋊C4×C71—22no—
90C22⋊C4×C71—22no—
91C22⋊C4×C71—42no—
92C42×C71—21no—
93C42×C71—42no—
94C42×C71C4⋊C411yescharacteristic
95C42×C71C4⋊C411yescharacteristic
96C23×C71—21no—
97C23×C71#11711yescharacteristic
98C23×C71#12112yes—
99Q8×C71—41no—
100Q8×C71—41no—
101D4×C71—41no—
102D4×C71—42no—
103D4×C71—41no—
104D4×C71—42no—
105D4×C71—82no—
106C2×C4×C71—22no—
107C2×C4×C71—21no—
108C2×C4×C71—22no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—81no—
116C2×C4×C71—82no—
117C2×C4×C71C2×C22⋊C411yescharacteristic
118C2×C4×C71C2×C4⋊C411yescharacteristic
119C2×C4×C71C22⋊Q812yes—
120C2×Dic71—82no—
121C71⋊C8—21no—
122C71⋊C8—22no—
123C71⋊C8—41no—
124C71⋊C8—41no—
125C71⋊C8#9912yes—
126C22×C71—22no—
127C22×C71—21no—
128C22×C71—41no—
129C22×C71—42no—
130C22×C71—42no—
131C22×C71—41no—
132C22×C71#38011yescharacteristic
133C4×C71—21no—
134C4×C71—22no—
135C4×C71—41no—
136C4×C71—41no—
137C4×C71—41no—
138C4×C71—162no—
139C4×C71C2×#9911yescharacteristic
140C4×C71#41011yescharacteristic
141Dic71—162no—
142#12808—711nomaximal, Sylow
143C2×C71—21no—
144C2×C71—42no—
145C2×C71—41no—
146C2×C71—81no—
147C2×C71#179911yescharacteristic
148#2932—711no—
149#2126—711no—
150#2126D7111yescharacteristic
151#1812—712no—
152#1391—712no—
153C71#1280811yescharacteristic, complemented, Sylow
154C2×#131—711no—
155C2×#131—711no—
156#532C2×D7111yescharacteristic
157#435—1422no—
158#435—1422no—
159#427—1422no—
160#423—711no—
161#410—712no—
162#410C2×D7112yes—
163#408—711no—
164#408—711no—
165C2×#97—711no—
166#354—21no—
167#354—1421no—
168#353—21no—
169#353—1421no—
170H(3,ℤ/4)—1422no—
171C2×#41C4×D7111yescharacteristic
172C2×#41C22×D7111yescharacteristic
173#131—2842no—
174#131—2842no—
175C2×M16—711no—
176C2×M16—712no—
177C2×M16—711no—
178C2×M16—1421no—
179C22×C8—711no—
180C22×C8—711no—
181C22×C8—712no—
182C22×C8—1421no—
183C4⋊Q8Dic14211yescharacteristic
184#127C22⋊D7111yescharacteristic
185#124Dic14211yescharacteristic
186#124C22⋊D7111yescharacteristic
187#121—22no—
188C4×D4—22no—
189#117—22no—
190#117—1422no—
191C2×C42C4×D7111yescharacteristic
192#107—2842no—
193#106—2842no—
194C4⋊C8—1422no—
195#99—22no—
196#99—22no—
197#99—1422no—
198#99—1422no—
199C22⋊C8—1422no—
200C22⋊C8—1422no—
201#97—1421no—
202#97—1421no—
203#41—41no—
204#41—41no—
205#41—41no—
206#41—41no—
207C2×Q8Dic71⋊C411yescharacteristic
208C2×Q8C2×Dic14211yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4Dic71⋊C411yescharacteristic
212C2×D4C2×C22⋊D7111yescharacteristic
213C2×D4#1176586412yes—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4C2×C4×D7111yescharacteristic
217C22×C4D4×D7112yes—
218M16—2842no—
219M16—2841no—
220M16—2842no—
221M16—2841no—
222C2×C8—2842no—
223C2×C8—2841no—
224C2×C8—711no—
225C2×C8—711no—
226C2×C8—712no—
227C2×C8—1421no—
228C2×C8—1421no—
229C2×C8—1422no—
230C2×C8—1422no—
231C2×C8—1422no—
232C4⋊C4—22no—
233C4⋊C4—42no—
234C4⋊C4—2842no—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—42no—
238C22⋊C4—2842no—
239C42—21no—
240C42—42no—
241C42—2842no—
242C42Dic71⋊C411yescharacteristic
243C42Dic71⋊C411yescharacteristic
244C23—21no—
245C23#1455608411yescharacteristic
246C23#1455610012yes—
247Q8—41no—
248Q8—41no—
249D4—41no—
250D4—42no—
251D4—41no—
252D4—42no—
253D4—82no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—81no—
264C2×C4—82no—
265C2×C4—5682no—
266C2×C4C22⋊C4×D7111yescharacteristic
267C2×C4C2×Dic71⋊C411yescharacteristic
268C2×C4#1455610512yes—
269C8—2841no—
270C8—2841no—
271C8—712no—
272C8—1421no—
273C8—1422no—
274C22—22no—
275C22—21no—
276C22—41no—
277C22—42no—
278C22—42no—
279C22—41no—
280C22#1798585711yescharacteristic
281C4—21no—
282C4—22no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—162no—
287C4—11362no—
288C4#99×D7111yescharacteristic
289C4#1798597111yescharacteristic
290C2—21no—
291C2—42no—
292C2—41no—
293C2—81no—
294C2#2258832611yescharacteristic
subgroupG/H
centerC4#17985971
commutatorC22×C4×C71C22×C4
FrattiniC2×#41C22×D71
FittingC71×#2126C2
socleC2×C71#1799
prime radical#532×C71C22
cosocleC2×#41×C71C23
hypercenter#2126D71
hypocenterC71#12808
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2126

Complements

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

NH ≅ G/Ntype
3. #22588384C2semidirect
4. #22586460C2semidirect
6. C2×#14556135C4semidirect
11. #17986023C4semidirect
13. #17985966C4semidirect
15. C2×#14556039C4semidirect
153. C71#12808semidirect

Maximal subgroups

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

subgroupindex
#225970582
#225900052
#225883842
#225864602
C71×#21262
#1280871