Finite Group Databasebeta

← #31908616

Subgroups — #31908616

Order 18176.

372 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 99 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#22593552C211yesmaximal, characteristic
2#22588336C211yesmaximal, characteristic
3#22586368C214yesmaximal
4C71×#1461C211yesmaximal, characteristic
5C2×#14556128C411yescharacteristic
6#17986023—21no—
7C2×#14556047—21no—
8#17985938C412yes—
9#17985938C2212yes—
10#17985906—21no—
11#17985905—21no—
12#17985905—21no—
13#17985864—21no—
14C4×C4⋊Dic71C411yescharacteristic
15#17985768C2212yes—
16C2×C4×Q8×C71C2211yescharacteristic
17#431×C71—21no—
18#324×C71—21no—
19#324×C71C2212yes—
20C22×Dic142—21no—
21C2×#11765849—22no—
22#14556128—41no—
23#14556128—41no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71C2×C412yes—
27C2×C4×Dic71—21no—
28C4⋊(C71⋊C8)—41no—
29C4⋊(C71⋊C8)—41no—
30#14556047—42no—
31C71⋊#95—22no—
32C71⋊#95—22no—
33C71⋊#95—22no—
34C71⋊#95—22no—
35C71⋊#95C2×C412yes—
36C22×Q8×C71D411yescharacteristic
37C4×Q8×C71—41no—
38C4×Q8×C71—41no—
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C4⋊C4×C71C2311yescharacteristic
42C2×C42×C71C2×C411yescharacteristic
43C2×C42×C71D411yescharacteristic
44C2×C42×C71D411yescharacteristic
45C4⋊C8×C71—41no—
46C22⋊C8×C71—22no—
47C4×C8×C71—41no—
48C2×Dic142—44no—
49C2×Dic142—41no—
50C2×Dic142—41no—
51C22×Dic71—22no—
52C22×Dic71—22no—
53#11765849—42no—
54#11765849—42no—
55C2×C71⋊C8—42no—
56C4⋊Dic71—42no—
57C4⋊Dic71—42no—
58C4⋊Dic71—42no—
59C4×Dic71—42no—
60C2×Q8×C71—41no—
61C2×Q8×C71—41no—
62C2×Q8×C71SD1612yes—
63C2×Q8×C71Q1612yes—
64C22×C4×C71C22⋊C411yescharacteristic
65C22×C4×C71C22⋊C411yescharacteristic
66C22×C4×C71C22⋊C411yescharacteristic
67C22×C4×C71C22⋊C411yescharacteristic
68C22×C4×C71C22×C411yescharacteristic
69C22×C4×C71C2×D411yescharacteristic
70C22×C4×C71C2×D411yescharacteristic
71C2×C8×C71—42no—
72C4⋊C4×C71—41no—
73C4⋊C4×C71—41no—
74C4⋊C4×C71—41no—
75C42×C71—21no—
76C42×C71—21no—
77C42×C71—21no—
78C42×C71—21no—
79C42×C71—21no—
80C42×C71—21no—
81C23×C71C2×C22⋊C411yescharacteristic
82Q8×C71—42no—
83Q8×C71—42no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—22no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—41no—
95C2×C4×C71—41no—
96C2×C4×C71—41no—
97C2×C4×C71#9911yescharacteristic
98C2×C4×C71#9911yescharacteristic
99C2×C4×C71#10312yes—
100C2×C4×C71#10312yes—
101C2×C4×C71C2×SD1611yescharacteristic
102C2×C4×C71C2×Q1611yescharacteristic
103C2×Dic71—42no—
104C2×Dic71—42no—
105C2×Dic71—44no—
106C2×Dic71—42no—
107C2×Dic71—42no—
108Dic142—82no—
109Dic142—82no—
110C8×C71—82no—
111C71⋊C8—82no—
112C22×C71—21no—
113C22×C71—21no—
114C22×C71C2×#9911yescharacteristic
115C22×C71C2×#10311yescharacteristic
116C22×C71#42111yescharacteristic
117C4×C71—21no—
118C4×C71—21no—
119C4×C71—22no—
120C4×C71—22no—
121C4×C71—21no—
122C4×C71—41no—
123C4×C71—41no—
124C4×C71—41no—
125C4×C71#35512yes—
126Dic71—82no—
127Dic71—82no—
128#11907—711nomaximal, Sylow
129C2×C71—21no—
130C2×C71—21no—
131C2×C71#152011yescharacteristic
132C2×C71#212311yescharacteristic
133C2×C71C2×#35511yescharacteristic
134#2568—711no—
135#1801—711no—
136#1461D7111yescharacteristic
137#1353—714no—
138C71#1190711yescharacteristic, complemented, Sylow
139C2×C4⋊Q8—711no—
140C2×C4×Q8C2×D7111yescharacteristic
141#431—21no—
142#423—1421no—
143C2×#101—1421no—
144#400—712no—
145#400—712no—
146#391—1421no—
147#391—1421no—
148#391—1421no—
149#382—1421no—
150C4×C4⋊C4—711no—
151#343—712no—
152#324—21no—
153#324C2×D7112yes—
154C22×Q8—1421no—
155C22×Q8D28411yescharacteristic
156C2×M16—1422no—
157C4⋊Q8—2841no—
158C4⋊Q8—2841no—
159C4×Q8—41no—
160C4×Q8—41no—
161C2×C4⋊C4—712no—
162C2×C4⋊C4—1421no—
163C2×C4⋊C4—1421no—
164C2×C4⋊C4C4×D7111yescharacteristic
165C2×C4⋊C4C22⋊D7111yescharacteristic
166C2×C4⋊C4C22×D7111yescharacteristic
167C2×C42—1421no—
168C2×C42C4×D7111yescharacteristic
169C2×C42D28411yescharacteristic
170C2×C42C22⋊D7111yescharacteristic
171C4⋊C8—41no—
172C4⋊C8—2841no—
173C4⋊C8—2841no—
174#101—2842no—
175C22⋊C8—22no—
176C4×C8—41no—
177#95—712no—
178#95—1422no—
179#95—1422no—
180#95—1422no—
181#95—1422no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—2844no—
185C2×Q8—2841no—
186C2×Q8—2841no—
187C2×Q8#1176585412yes—
188C2×Q8Dic28412yes—
189C22×C4—1422no—
190C22×C4—1422no—
191C22×C4(C2×D71)⋊C411yescharacteristic
192C22×C4(C2×D71)⋊C411yescharacteristic
193C22×C4(C2×D71)⋊C411yescharacteristic
194C22×C4(C2×D71)⋊C411yescharacteristic
195C22×C4C2×C4×D7111yescharacteristic
196C22×C4C2×C22⋊D7111yescharacteristic
197C22×C4C2×D28411yescharacteristic
198M16—2842no—
199M16—2842no—
200C2×C8—42no—
201C2×C8—2842no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—2842no—
206C4⋊C4—2842no—
207C4⋊C4—2842no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—2842no—
215C23C2×(C2×D71)⋊C411yescharacteristic
216Q8—42no—
217Q8—42no—
218Q8—5682no—
219Q8—5682no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—2842no—
234C2×C4—2844no—
235C2×C4—2842no—
236C2×C4—2842no—
237C2×C4—2842no—
238C2×C4#1455604211yescharacteristic
239C2×C4#1455604211yescharacteristic
240C2×C4#1455605312yes—
241C2×C4#1455605312yes—
242C2×C4C2×#1176585411yescharacteristic
243C2×C4C2×Dic28411yescharacteristic
244C8—82no—
245C8—5682no—
246C22—21no—
247C22—21no—
248C22C2×#1455604211yescharacteristic
249C22C2×#1455605311yescharacteristic
250C22#1798601511yescharacteristic
251C4—21no—
252C4—22no—
253C4—22no—
254C4—21no—
255C4—21no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—5682no—
260C4—5682no—
261C4#1798579512yes—
262C2—21no—
263C2—21no—
264C2#2258691311yescharacteristic
265C2#2258999311yescharacteristic
266C2C2×#1798579511yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#1461C2
socleC22×C71C2×#99
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#1461D71
hypocenterC71#11907
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
138. C71#11907semidirect

Maximal subgroups

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

subgroupindex
#225935522
#225883362
#225863682
C71×#14612
#1190771