Finite Group Databasebeta

← #31908615

Subgroups — #31908615

Order 18176.

372 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 107 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#22586664C211yesmaximal, characteristic
3#22586367C214yesmaximal
4C71×#2568C211yesmaximal, characteristic
5#17986052—21no—
6#17986023—21no—
7C2×#14556047—21no—
8#17985768C2212yes—
9#17985725—21no—
10#17985725C2212yes—
11C2×C4⋊Q8×C71C411yescharacteristic
12C2×C4×Q8×C71C2211yescharacteristic
13#400×C71C412yes—
14#400×C71C2212yes—
15#391×C71—21no—
16#391×C71—21no—
17#391×C71—21no—
18#382×C71—21no—
19C4×C4⋊C4×C71C411yescharacteristic
20C2×#11765849—22no—
21C4⋊(C71⋊C8)—41no—
22C4⋊(C71⋊C8)—41no—
23C4⋊(C71⋊C8)—41no—
24#14556047—42no—
25C22⋊(C71⋊C8)—22no—
26C4×C71⋊C8—41no—
27C22×Q8×C71—21no—
28C22×Q8×C71D411yescharacteristic
29C4⋊Q8×C71—41no—
30C4⋊Q8×C71—41no—
31C4×Q8×C71—41no—
32C4×Q8×C71—41no—
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71C2×C412yes—
36C2×C4⋊C4×C71C2×C411yescharacteristic
37C2×C4⋊C4×C71D411yescharacteristic
38C2×C4⋊C4×C71C2311yescharacteristic
39C2×C42×C71—21no—
40C2×C42×C71C2×C411yescharacteristic
41C2×C42×C71D411yescharacteristic
42C2×C42×C71D411yescharacteristic
43#95×C71—22no—
44#95×C71—22no—
45#95×C71—22no—
46#95×C71—22no—
47#95×C71C2×C412yes—
48#11765849—42no—
49#11765849—42no—
50C2×C71⋊C8—42no—
51C2×C71⋊C8—42no—
52C2×Q8×C71—41no—
53C2×Q8×C71—41no—
54C2×Q8×C71—44no—
55C2×Q8×C71—41no—
56C2×Q8×C71—41no—
57C2×Q8×C71SD1612yes—
58C2×Q8×C71Q1612yes—
59C22×C4×C71—22no—
60C22×C4×C71—22no—
61C22×C4×C71C22⋊C411yescharacteristic
62C22×C4×C71C22⋊C411yescharacteristic
63C22×C4×C71C22⋊C411yescharacteristic
64C22×C4×C71C22⋊C411yescharacteristic
65C22×C4×C71C22×C411yescharacteristic
66C22×C4×C71C2×D411yescharacteristic
67C22×C4×C71C2×D411yescharacteristic
68C4⋊C4×C71—41no—
69C4⋊C4×C71—41no—
70C4⋊C4×C71—41no—
71C4⋊C4×C71—42no—
72C4⋊C4×C71—42no—
73C4⋊C4×C71—42no—
74C42×C71—21no—
75C42×C71—21no—
76C42×C71—21no—
77C42×C71—21no—
78C42×C71—21no—
79C42×C71—21no—
80C42×C71—42no—
81C23×C71C2×C22⋊C411yescharacteristic
82Q8×C71—42no—
83Q8×C71—42no—
84Q8×C71—82no—
85Q8×C71—82no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—22no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—41no—
97C2×C4×C71—41no—
98C2×C4×C71—41no—
99C2×C4×C71—42no—
100C2×C4×C71—42no—
101C2×C4×C71—44no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71#9911yescharacteristic
105C2×C4×C71#9911yescharacteristic
106C2×C4×C71#10312yes—
107C2×C4×C71#10312yes—
108C2×C4×C71C2×SD1611yescharacteristic
109C2×C4×C71C2×Q1611yescharacteristic
110C71⋊C8—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—82no—
126C4×C71—82no—
127C4×C71#35512yes—
128#11907—711nomaximal, Sylow
129C2×C71—21no—
130C2×C71—21no—
131C2×C71#152011yescharacteristic
132C2×C71#212311yescharacteristic
133C2×C71C2×#35511yescharacteristic
134#2568D7111yescharacteristic
135#1801—711no—
136#1461—711no—
137#1353—714no—
138C71#1190711yescharacteristic, complemented, Sylow
139C2×C4⋊Q8Dic7111yescharacteristic
140C2×C4×Q8C2×D7111yescharacteristic
141#431—1421no—
142#423—1421no—
143C2×#101—1421no—
144#400Dic7112yes—
145#400C2×D7112yes—
146#391—21no—
147#391—21no—
148#391—21no—
149#382—21no—
150C4×C4⋊C4Dic7111yescharacteristic
151#343—712no—
152#324—712no—
153#324—1421no—
154C22×Q8—21no—
155C22×Q8C22⋊D7111yescharacteristic
156C2×M16—1422no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159C4×Q8—41no—
160C4×Q8—41no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C2×Dic7112yes—
164C2×C4⋊C4C2×Dic7111yescharacteristic
165C2×C4⋊C4C22⋊D7111yescharacteristic
166C2×C4⋊C4C22×D7111yescharacteristic
167C2×C42—21no—
168C2×C42C2×Dic7111yescharacteristic
169C2×C42C22⋊D7111yescharacteristic
170C2×C42C22⋊D7111yescharacteristic
171C4⋊C8—2841no—
172C4⋊C8—2841no—
173C4⋊C8—2841no—
174#101—2842no—
175C22⋊C8—1422no—
176C4×C8—2841no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95C2×Dic7112yes—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—44no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8#1176585212yes—
188C2×Q8#1176585512yes—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4C22⋊Dic7111yescharacteristic
192C22×C4C22⋊Dic7111yescharacteristic
193C22×C4C22⋊Dic7111yescharacteristic
194C22×C4C22⋊Dic7111yescharacteristic
195C22×C4C22×Dic7111yescharacteristic
196C22×C4C2×C22⋊D7111yescharacteristic
197C22×C4C2×C22⋊D7111yescharacteristic
198M16—2842no—
199M16—2842no—
200C2×C8—2842no—
201C2×C8—2842no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—42no—
215C23C2×C22⋊Dic7111yescharacteristic
216Q8—42no—
217Q8—42no—
218Q8—82no—
219Q8—82no—
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—42no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4#1455604311yescharacteristic
239C2×C4#1455604311yescharacteristic
240C2×C4#1455605212yes—
241C2×C4#1455605212yes—
242C2×C4C2×#1176585211yescharacteristic
243C2×C4C2×#1176585511yescharacteristic
244C8—5682no—
245C8—5682no—
246C22—21no—
247C22—21no—
248C22C2×#1455604311yescharacteristic
249C22C2×#1455605211yescharacteristic
250C22#1798601611yescharacteristic
251C4—21no—
252C4—22no—
253C4—22no—
254C4—21no—
255C4—21no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—82no—
260C4—82no—
261C4#1798579612yes—
262C2—21no—
263C2—21no—
264C2#2258691211yescharacteristic
265C2#2258999411yescharacteristic
266C2C2×#1798579611yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2568C2
socleC22×C71C2×#99
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2568D71
hypocenterC71#11907
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2568

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
#225883362
#225866642
#225863672
C71×#25682
#1190771