Finite Group Databasebeta

← #31907118

Subgroups — #31907118

Order 18176.

358 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 87 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#22586664C211yesmaximal, characteristic
3#22586283C214yesmaximal
4C71×#1467C211yesmaximal, characteristic
5C2×#14556128C411yescharacteristic
6#17986052—21no—
7#17985938—21no—
8#17985938—21no—
9#17985906C412yes—
10#17985905—21no—
11#17985905—21no—
12#17985864C2212yes—
13C4×C4⋊Dic71C411yescharacteristic
14#17985725—21no—
15#17985725C2212yes—
16C2×C4×Q8×C71C2211yescharacteristic
17#432×C71—21no—
18#324×C71—21no—
19#324×C71C2212yes—
20C22×Dic142—21no—
21#14556128—41no—
22#14556128—41no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71C2×C412yes—
26C2×C4×Dic71C2×C412yes—
27C4⋊(C71⋊C8)—41no—
28C22⋊(C71⋊C8)—22no—
29C4×C71⋊C8—41no—
30C71⋊#95—21no—
31C71⋊#95—22no—
32C71⋊#95—22no—
33C71⋊#95—22no—
34C71⋊#95—22no—
35C22×Q8×C71C2×C411yescharacteristic
36C4×Q8×C71—41no—
37C4×Q8×C71—41no—
38C2×C4⋊C4×C71D411yescharacteristic
39C2×C4⋊C4×C71D411yescharacteristic
40C2×C4⋊C4×C71C2311yescharacteristic
41C2×C42×C71C2×C411yescharacteristic
42C2×C42×C71D411yescharacteristic
43C2×C42×C71D411yescharacteristic
44C4⋊C8×C71—41no—
45C22⋊C8×C71—22no—
46#97×C71—41no—
47C2×Dic142—44no—
48C2×Dic142—41no—
49C2×Dic142—41no—
50C22×Dic71—22no—
51C22×Dic71—22no—
52C2×C71⋊C8—42no—
53C4⋊Dic71—42no—
54C4⋊Dic71—42no—
55C4⋊Dic71—42no—
56C4×Dic71—24no—
57C2×Q8×C71—22no—
58C2×Q8×C71—22no—
59C2×Q8×C71—41no—
60C22×C4×C71C22⋊C411yescharacteristic
61C22×C4×C71C22⋊C411yescharacteristic
62C22×C4×C71C22⋊C411yescharacteristic
63C22×C4×C71C22⋊C411yescharacteristic
64C22×C4×C71C22×C411yescharacteristic
65C22×C4×C71C2×D411yescharacteristic
66C22×C4×C71C2×D411yescharacteristic
67C2×C8×C71—42no—
68C4⋊C4×C71—41no—
69C4⋊C4×C71—41no—
70C4⋊C4×C71—41no—
71C42×C71—21no—
72C42×C71—21no—
73C42×C71—21no—
74C42×C71—21no—
75C42×C71—21no—
76C42×C71—21no—
77C23×C71C2×C22⋊C411yescharacteristic
78Q8×C71—44no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—22no—
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—41no—
91C2×C4×C71—41no—
92C2×C4×C71—41no—
93C2×C4×C71#9911yescharacteristic
94C2×C4×C71#9911yescharacteristic
95C2×C4×C71#10112yes—
96C2×C4×C71#13112yes—
97C2×Dic71—42no—
98C2×Dic71—42no—
99C2×Dic71—44no—
100C2×Dic71—42no—
101C2×Dic71—42no—
102Dic142—82no—
103Dic142—82no—
104C8×C71—82no—
105C71⋊C8—82no—
106C22×C71—21no—
107C22×C71—21no—
108C22×C71#40811yescharacteristic
109C22×C71C2×#9911yescharacteristic
110C22×C71C2×#10111yescharacteristic
111C4×C71—21no—
112C4×C71—22no—
113C4×C71—22no—
114C4×C71—41no—
115C4×C71—41no—
116C4×C71—41no—
117C4×C71—41no—
118C4×C71#35512yes—
119Dic71—82no—
120Dic71—82no—
121#11573—711nomaximal, Sylow
122C2×C71—21no—
123C2×C71—21no—
124C2×C71#147211yescharacteristic
125C2×C71#212911yescharacteristic
126C2×C71C2×#35511yescharacteristic
127#2568—711no—
128#1467D7111yescharacteristic
129#1461—711no—
130#1324—714no—
131C71#1157311yescharacteristic, complemented, Sylow
132C2×C4⋊Q8—711no—
133C2×C4×Q8C2×D7111yescharacteristic
134#432—21no—
135#431—1421no—
136#400—1421no—
137#400—1421no—
138#391—712no—
139#391—1421no—
140#391—1421no—
141#382—712no—
142C4×C4⋊C4—711no—
143#324—21no—
144#324—712no—
145#324—1421no—
146#324C2×D7112yes—
147C22×Q8—1421no—
148C22×Q8C4×D7111yescharacteristic
149C4⋊Q8—2841no—
150C4⋊Q8—2841no—
151C4×Q8—41no—
152C4×Q8—41no—
153C2×C4⋊C4—712no—
154C2×C4⋊C4—1421no—
155C2×C4⋊C4—1421no—
156C2×C4⋊C4D28411yescharacteristic
157C2×C4⋊C4C22⋊D7111yescharacteristic
158C2×C4⋊C4C22×D7111yescharacteristic
159C2×C42—712no—
160C2×C42C4×D7111yescharacteristic
161C2×C42D28411yescharacteristic
162C2×C42C22⋊D7111yescharacteristic
163C4⋊C8—41no—
164C4⋊C8—2841no—
165C22⋊C8—22no—
166C22⋊C8—1422no—
167#97—41no—
168C4×C8—2841no—
169#95—1421no—
170#95—1422no—
171#95—1422no—
172#95—1422no—
173#95—1422no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—41no—
177C2×Q8—2844no—
178C2×Q8—2841no—
179C2×Q8—2841no—
180C22×C4—1422no—
181C22×C4—1422no—
182C22×C4(C2×D71)⋊C411yescharacteristic
183C22×C4(C2×D71)⋊C411yescharacteristic
184C22×C4(C2×D71)⋊C411yescharacteristic
185C22×C4(C2×D71)⋊C411yescharacteristic
186C22×C4C2×C4×D7111yescharacteristic
187C22×C4C2×C22⋊D7111yescharacteristic
188C22×C4C2×D28411yescharacteristic
189C2×C8—42no—
190C2×C8—2842no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—2842no—
195C4⋊C4—2842no—
196C4⋊C4—2842no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—1424no—
204C23C2×(C2×D71)⋊C411yescharacteristic
205Q8—44no—
206Q8—5682no—
207Q8—5682no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—2842no—
223C2×C4—2844no—
224C2×C4—2842no—
225C2×C4—2842no—
226C2×C4—2842no—
227C2×C4#1455604211yescharacteristic
228C2×C4#1455604211yescharacteristic
229C2×C4#1455604612yes—
230C2×C4#1455613612yes—
231C8—82no—
232C8—5682no—
233C22—21no—
234C22—21no—
235C22#1798596711yescharacteristic
236C22C2×#1455604211yescharacteristic
237C22C2×#1455604611yescharacteristic
238C4—21no—
239C4—22no—
240C4—22no—
241C4—41no—
242C4—41no—
243C4—41no—
244C4—41no—
245C4—5682no—
246C4—5682no—
247C4#1798579512yes—
248C2—21no—
249C2—21no—
250C2#2258670511yescharacteristic
251C2#2259001411yescharacteristic
252C2C2×#1798579511yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#1467C2
socleC22×C71C2×#99
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#1467D71
hypocenterC71#11573
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
131. C71#11573semidirect

Maximal subgroups

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

subgroupindex
#225935522
#225866642
#225862832
C71×#14672
#1157371