Finite Group Databasebeta

← #31908103

Subgroups — #31908103

Order 18176.

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

normal subgroups only

This group has 394 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#22593545C211yesmaximal, characteristic
2#22586684C211yesmaximal, characteristic
3#22586369C214yesmaximal
4C4×#101×C71C211yesmaximal, characteristic
5C2×#14556128C411yescharacteristic
6#17986056—21no—
7#17985919C412yes—
8#17985919C2212yes—
9#17985906—21no—
10#17985897—21no—
11#17985897—21no—
12#17985868—21no—
13C4×C4⋊Dic71C411yescharacteristic
14#17985725—21no—
15#17985725C2212yes—
16C2×C4×Q8×C71C2211yescharacteristic
17C2×#101×C71—21no—
18C4×M16×C71—21no—
19#343×C71C2212yes—
20C22×Dic142C2×C412yes—
21#14556128—41no—
22#14556128—41no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×Dic71⋊C4—22no—
27C2×Dic71⋊C4—22no—
28C2×C4×Dic71—21no—
29C4⋊(C71⋊C8)—41no—
30C22⋊(C71⋊C8)—22no—
31#14556039—41no—
32C71⋊#95—22no—
33C71⋊#95—22no—
34C71⋊#95C2×C412yes—
35C22×Q8×C71D411yescharacteristic
36C2×M16×C71—22no—
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
45#101×C71—24no—
46#97×C71—41no—
47C4×C8×C71—41no—
48C2×Dic142—42no—
49C2×Dic142—82no—
50C22×Dic71—22no—
51C22×Dic71—22no—
52C2×C71⋊C8—42no—
53C4⋊Dic71—24no—
54C4⋊Dic71—42no—
55C4⋊Dic71—41no—
56C4⋊Dic71—41no—
57Dic71⋊C4—82no—
58Dic71⋊C4—82no—
59C4×Dic71—42no—
60C2×Q8×C71—22no—
61C2×Q8×C71—22no—
62C2×Q8×C71—41no—
63C22×C4×C71C22⋊C411yescharacteristic
64C22×C4×C71C22⋊C411yescharacteristic
65C22×C4×C71C22⋊C411yescharacteristic
66C22×C4×C71C22⋊C411yescharacteristic
67C22×C4×C71C22×C411yescharacteristic
68C22×C4×C71C2×D411yescharacteristic
69C22×C4×C71C2×D411yescharacteristic
70M16×C71—24no—
71M16×C71—42no—
72C2×C8×C71—42no—
73C4⋊C4×C71—41no—
74C4⋊C4×C71—41no—
75C4⋊C4×C71—41no—
76C42×C71—21no—
77C42×C71—21no—
78C42×C71—21no—
79C42×C71—21no—
80C42×C71—21no—
81C42×C71—21no—
82C23×C71C2×C22⋊C411yescharacteristic
83Q8×C71—44no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—22no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—41no—
96C2×C4×C71—41no—
97C2×C4×C71—41no—
98C2×C4×C71#9911yescharacteristic
99C2×C4×C71#9911yescharacteristic
100C2×C4×C71#10412yes—
101C2×C4×C71#13712yes—
102C2×Dic71—24no—
103C2×Dic71—42no—
104C2×Dic71—42no—
105C2×Dic71—42no—
106C2×Dic71—42no—
107C2×Dic71—42no—
108Dic142—84no—
109C8×C71—44no—
110C71⋊C8—82no—
111C22×C71—21no—
112C22×C71—21no—
113C22×C71C2×#9911yescharacteristic
114C22×C71#41911yescharacteristic
115C22×C71C2×#10411yescharacteristic
116C4×C71—21no—
117C4×C71—22no—
118C4×C71—22no—
119C4×C71—41no—
120C4×C71—41no—
121C4×C71—41no—
122C4×C71—41no—
123C4×C71#35612yes—
124Dic71—44no—
125Dic71—82no—
126#11795—711nomaximal, Sylow
127C2×C71—21no—
128C2×C71—21no—
129C2×C71#150811yescharacteristic
130C2×C71#212211yescharacteristic
131C2×C71C2×#35611yescharacteristic
132#2567—711no—
133C4×#101D7111yescharacteristic
134#1467—711no—
135#1354—714no—
136C71#1179511yescharacteristic, complemented, Sylow
137C2×C4⋊Q8—711no—
138C2×C4×Q8C2×D7111yescharacteristic
139#432—1421no—
140C2×#101—21no—
141C4×M16—21no—
142#395—712no—
143#395—712no—
144#391—1421no—
145#389—1421no—
146#389—1421no—
147#384—1421no—
148C4×C4⋊C4—711no—
149#343C2×D7112yes—
150#324—712no—
151#324—1421no—
152C22×Q8—712no—
153C22×Q8C22⋊D7111yescharacteristic
154C2×M16—22no—
155C4⋊Q8—2841no—
156C4⋊Q8—2841no—
157C4×Q8—41no—
158C4×Q8—41no—
159C2×C4⋊C4—1421no—
160C2×C4⋊C4—1421no—
161C2×C4⋊C4—1422no—
162C2×C4⋊C4—1422no—
163C2×C4⋊C4—1421no—
164C2×C4⋊C4C4×D7111yescharacteristic
165C2×C4⋊C4D28411yescharacteristic
166C2×C4⋊C4C22×D7111yescharacteristic
167C2×C42—1421no—
168C2×C42C4×D7111yescharacteristic
169C2×C42D28411yescharacteristic
170C2×C42C22⋊D7111yescharacteristic
171C4⋊C8—2841no—
172#101—24no—
173C22⋊C8—1422no—
174#97—41no—
175#97—2841no—
176C4×C8—41no—
177#95—712no—
178#95—1422no—
179#95—1422no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—41no—
183C2×Q8—2842no—
184C2×Q8—5682no—
185C22×C4—1422no—
186C22×C4—1422no—
187C22×C4(C2×D71)⋊C411yescharacteristic
188C22×C4(C2×D71)⋊C411yescharacteristic
189C22×C4(C2×D71)⋊C411yescharacteristic
190C22×C4(C2×D71)⋊C411yescharacteristic
191C22×C4C2×C4×D7111yescharacteristic
192C22×C4C2×C22⋊D7111yescharacteristic
193C22×C4C2×D28411yescharacteristic
194M16—24no—
195M16—42no—
196C2×C8—42no—
197C2×C8—2842no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—2842no—
202C4⋊C4—2841no—
203C4⋊C4—2841no—
204C4⋊C4—5682no—
205C4⋊C4—5682no—
206C4⋊C4—1424no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—2842no—
214C23C2×(C2×D71)⋊C411yescharacteristic
215Q8—44no—
216Q8—5684no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—2842no—
232C2×C4—2842no—
233C2×C4—2842no—
234C2×C4—2842no—
235C2×C4—2842no—
236C2×C4—1424no—
237C2×C4#1455604211yescharacteristic
238C2×C4#1455604211yescharacteristic
239C2×C4#1455605612yes—
240C2×C4#1455615812yes—
241C8—44no—
242C8—5682no—
243C22—21no—
244C22—21no—
245C22C2×#1455604211yescharacteristic
246C22#1798600711yescharacteristic
247C22C2×#1455605611yescharacteristic
248C4—21no—
249C4—22no—
250C4—22no—
251C4—41no—
252C4—41no—
253C4—41no—
254C4—41no—
255C4—2844no—
256C4—5682no—
257C4#1798579712yes—
258C2—21no—
259C2—21no—
260C2#2258685911yescharacteristic
261C2#2258999111yescharacteristic
262C2C2×#1798579711yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC4×#101×C71C2
socleC22×C71C2×#99
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenterC4×#101D71
hypocenterC71#11795
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
136. C71#11795semidirect

Maximal subgroups

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

subgroupindex
#225935452
#225866842
#225863692
C4×#101×C712
#1179571