Finite Group Databasebeta

← #31891073

Subgroups — #31891073

Order 18176.

384 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C4×#14556047C211yesmaximal, characteristic
2#22586684C211yesmaximal, characteristic
3#22586233C212yesmaximal
4#22586204C212yesmaximal
5C71×#2356C211yesmaximal, characteristic
6#17986056C2212yes—
7C2×#14556047—21no—
8C4×#11765849C2212yes—
9#17985768—21no—
10#17985725—21no—
11#17985725—21no—
12C2×C4×Q8×C71C2211yescharacteristic
13#391×C71—21no—
14#387×C71—21no—
15#387×C71—21no—
16#383×C71C411yescharacteristic
17#383×C71C2211yescharacteristic
18#382×C71C411yescharacteristic
19C4×C4⋊C4×C71—21no—
20C4×C4⋊C4×C71C411yescharacteristic
21#376×C71C411yescharacteristic
22#376×C71C2211yescharacteristic
23C2×#11765849—22no—
24C4⋊(C71⋊C8)—22no—
25#14556047—24no—
26C22⋊(C71⋊C8)—22no—
27#14556039—22no—
28#14556039—22no—
29C4×C71⋊C8—22no—
30C22×Q8×C71D411yescharacteristic
31C4×Q8×C71—22no—
32C4×Q8×C71—22no—
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71D411yescharacteristic
36C2×C4⋊C4×C71D411yescharacteristic
37C2×C4⋊C4×C71D411yescharacteristic
38C2×C42×C71—21no—
39C2×C42×C71—21no—
40C2×C42×C71—21no—
41C2×C42×C71C2×C411yescharacteristic
42C2×C42×C71C2×C411yescharacteristic
43C2×C42×C71C2311yescharacteristic
44#95×C71—21no—
45#95×C71—21no—
46#95×C71—21no—
47#95×C71—21no—
48#95×C71—21no—
49#95×C71—21no—
50#95×C71—21no—
51#95×C71C2×C411yescharacteristic
52#95×C71C2×C411yescharacteristic
53#95×C71C2×C411yescharacteristic
54#95×C71C2×C411yescharacteristic
55#11765849—24no—
56#11765849—42no—
57C2×C71⋊C8—42no—
58C2×C71⋊C8—42no—
59C2×Q8×C71—22no—
60C2×Q8×C71—22no—
61C2×Q8×C71—41no—
62C22×C4×C71—21no—
63C22×C4×C71—21no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71C22⋊C411yescharacteristic
67C22×C4×C71C22⋊C411yescharacteristic
68C22×C4×C71C22⋊C411yescharacteristic
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71C22×C411yescharacteristic
71C22×C4×C71C2×D411yescharacteristic
72C22×C4×C71C2×D411yescharacteristic
73C4⋊C4×C71—22no—
74C4⋊C4×C71—22no—
75C4⋊C4×C71—22no—
76C4⋊C4×C71—42no—
77C4⋊C4×C71—42no—
78C42×C71—21no—
79C42×C71—21no—
80C42×C71—21no—
81C42×C71—21no—
82C42×C71—42no—
83C42×C71—42no—
84C42×C71—42no—
85C42×C71#4112yes—
86C42×C71#4112yes—
87C23×C71C2×C22⋊C411yescharacteristic
88Q8×C71—44no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—22no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—41no—
100C2×C4×C71—41no—
101C2×C4×C71—41no—
102C2×C4×C71—41no—
103C2×C4×C71—41no—
104C2×C4×C71—41no—
105C2×C4×C71—41no—
106C2×C4×C71—41no—
107C2×C4×C71—41no—
108C2×C4×C71—41no—
109C2×C4×C71—41no—
110C2×C4×C71—41no—
111C2×C4×C71—41no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71#10412yes—
115C2×C4×C71#11711yescharacteristic
116C2×C4×C71#11711yescharacteristic
117C2×C4×C71#12312yes—
118C2×C4×C71#12312yes—
119C71⋊C8—44no—
120C71⋊C8—82no—
121C22×C71—21no—
122C22×C71—21no—
123C22×C71#38111yescharacteristic
124C22×C71C2×#10411yescharacteristic
125C22×C71#42111yescharacteristic
126C4×C71—22no—
127C4×C71—22no—
128C4×C71—22no—
129C4×C71—21no—
130C4×C71—21no—
131C4×C71—42no—
132C4×C71—41no—
133C4×C71—41no—
134C4×C71—41no—
135C4×C71—81no—
136C4×C71—81no—
137C4×C71—81no—
138#8775—711nomaximal, Sylow
139C2×C71—21no—
140C2×C71—21no—
141C2×C71#182611yescharacteristic
142C2×C71#182711yescharacteristic
143C2×C71#183011yescharacteristic
144#2356D7111yescharacteristic
145C4×#101—711no—
146#1467—711no—
147#1300—712no—
148#1288—712no—
149C71#877511yescharacteristic, complemented, Sylow
150C2×C4×Q8C2×D7111yescharacteristic
151#432—712no—
152C2×#101—1421no—
153C4×M16—712no—
154#391—21no—
155#387—21no—
156#387—21no—
157#383Dic7111yescharacteristic
158#383C2×D7111yescharacteristic
159#382Dic7111yescharacteristic
160C4×C4⋊C4—21no—
161C4×C4⋊C4Dic7111yescharacteristic
162#376Dic7111yescharacteristic
163#376C2×D7111yescharacteristic
164#343—1421no—
165#324—1421no—
166#324—1421no—
167C22×Q8C22⋊D7111yescharacteristic
168C2×M16—1422no—
169C4×Q8—22no—
170C4×Q8—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4C22⋊D7111yescharacteristic
174C2×C4⋊C4C22⋊D7111yescharacteristic
175C2×C4⋊C4C22⋊D7111yescharacteristic
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42C2×Dic7111yescharacteristic
180C2×C42C2×Dic7111yescharacteristic
181C2×C42C22×D7111yescharacteristic
182C4⋊C8—1422no—
183#101—1424no—
184C22⋊C8—1422no—
185#97—1422no—
186#97—1422no—
187C4×C8—1422no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95C2×Dic7111yescharacteristic
196#95C2×Dic7111yescharacteristic
197#95C2×Dic7111yescharacteristic
198#95C2×Dic7111yescharacteristic
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—41no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4C22⋊Dic7111yescharacteristic
207C22×C4C22⋊Dic7111yescharacteristic
208C22×C4C22⋊Dic7111yescharacteristic
209C22×C4C22⋊Dic7111yescharacteristic
210C22×C4C22×Dic7111yescharacteristic
211C22×C4C2×C22⋊D7111yescharacteristic
212C22×C4C2×C22⋊D7111yescharacteristic
213M16—2842no—
214M16—1424no—
215C2×C8—2842no—
216C2×C8—2842no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—42no—
227C42—42no—
228C42—42no—
229C42#1176586512yes—
230C42#1176586512yes—
231C23C2×C22⋊Dic7111yescharacteristic
232Q8—44no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4#1455605512yes—
259C2×C4#1455608711yescharacteristic
260C2×C4#1455608711yescharacteristic
261C2×C4#1455611112yes—
262C2×C4#1455611112yes—
263C8—2844no—
264C8—5682no—
265C22—21no—
266C22—21no—
267C22#1798586211yescharacteristic
268C22C2×#1455605511yescharacteristic
269C22#1798601611yescharacteristic
270C4—22no—
271C4—22no—
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—81no—
280C4—81no—
281C4—81no—
282C2—21no—
283C2—21no—
284C2#2258843911yescharacteristic
285C2#2258844411yescharacteristic
286C2#2258845411yescharacteristic
subgroupG/H
centerC22#17985862
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2356C2
socleC22×C71#381
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2356D71
hypocenterC71#8775
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2356

Complements

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

NH ≅ G/Ntype
149. C71#8775semidirect

Maximal subgroups

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

subgroupindex
C4×#145560472
#225866842
#225862332
#225862042
C71×#23562
#877571