Finite Group Databasebeta

← #31897561

Subgroups — #31897561

Order 18176.

424 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 127 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#22596703C211yesmaximal, characteristic
2#22588193C211yesmaximal, characteristic
3#22586222C214yesmaximal
4C71×#1745C211yesmaximal, characteristic
5C2×#14556120C412yes—
6C2×C4⋊(C71⋊C8)—21no—
7C2×#14556039C2212yes—
8#17985897—21no—
9#17985897—21no—
10#17985897—21no—
11#17985897—21no—
12#17985869C411yescharacteristic
13#17985869C411yescharacteristic
14#17985865C2212yes—
15#17985754—21no—
16C2×#97×C71C2212yes—
17C43×C71C2211yescharacteristic
18#336×C71—21no—
19#336×C71—21no—
20C22×C71⋊C8—22no—
21#14556120—44no—
22C2×C4⋊Dic71—21no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71C2×C412yes—
27C2×C4⋊Dic71C2×C412yes—
28C2×Dic71⋊C4—22no—
29C2×Dic71⋊C4—22no—
30C4⋊(C71⋊C8)—24no—
31#14556039—24no—
32C71⋊#95—22no—
33C71⋊#95—22no—
34C22×C8×C71—22no—
35C2×C42×C71C2×C411yescharacteristic
36C2×C42×C71C2×C411yescharacteristic
37C2×C42×C71D411yescharacteristic
38C2×C42×C71D411yescharacteristic
39C2×C42×C71D411yescharacteristic
40C2×C42×C71D411yescharacteristic
41C2×C42×C71C2311yescharacteristic
42#97×C71—22no—
43#97×C71Q814yes—
44C22×Dic71—22no—
45C22×Dic71—22no—
46C2×C71⋊C8—24no—
47C2×C71⋊C8—42no—
48C2×C71⋊C8—42no—
49C4⋊Dic71—42no—
50C4⋊Dic71—42no—
51C4⋊Dic71—42no—
52C4⋊Dic71—42no—
53C4⋊Dic71—42no—
54C4⋊Dic71—42no—
55Dic71⋊C4—44no—
56Dic71⋊C4—44no—
57C22×C4×C71C22⋊C411yescharacteristic
58C22×C4×C71C22⋊C411yescharacteristic
59C22×C4×C71C22⋊C411yescharacteristic
60C22×C4×C71C22⋊C411yescharacteristic
61C22×C4×C71C22×C411yescharacteristic
62C22×C4×C71C2×D411yescharacteristic
63C22×C4×C71C2×D411yescharacteristic
64C2×C8×C71—24no—
65C2×C8×C71—42no—
66C2×C8×C71—42no—
67C42×C71—22no—
68C42×C71—22no—
69C42×C71—22no—
70C42×C71—41no—
71C42×C71—41no—
72C42×C71—41no—
73C42×C71C2×Q812yes—
74C42×C71#4112yes—
75C23×C71C2×C22⋊C411yescharacteristic
76C2×C4×C71—21no—
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
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—21no—
91C2×C4×C71#10012yes—
92C2×C4×C71#10112yes—
93C2×C4×C71#10412yes—
94C2×C4×C71C4×Q812yes—
95C2×C4×C71C22⋊Q812yes—
96C2×C4×C71#12411yescharacteristic
97C2×C4×C71C4⋊Q811yescharacteristic
98C2×Dic71—42no—
99C2×Dic71—42no—
100C2×Dic71—42no—
101C2×Dic71—42no—
102C2×Dic71—42no—
103C2×Dic71—42no—
104C8×C71—44no—
105C71⋊C8—44no—
106C22×C71#39111yescharacteristic
107C22×C71C2×#10011yescharacteristic
108C22×C71C2×#10111yescharacteristic
109C22×C71C2×#10411yescharacteristic
110C22×C71#42111yescharacteristic
111C22×C71#42111yescharacteristic
112C22×C71#42111yescharacteristic
113C4×C71—21no—
114C4×C71—21no—
115C4×C71—22no—
116C4×C71—22no—
117C4×C71—22no—
118C4×C71—22no—
119C4×C71—22no—
120C4×C71—41no—
121Dic71—82no—
122Dic71—82no—
123#9894—711nomaximal, Sylow
124C2×C71#152811yescharacteristic
125C2×C71#153011yescharacteristic
126C2×C71#153411yescharacteristic
127C2×C71#153611yescharacteristic
128C2×C71#198611yescharacteristic
129C2×C71#199511yescharacteristic
130C2×C71#199711yescharacteristic
131#2859—711no—
132#1767—711no—
133#1745D7111yescharacteristic
134#1295—714no—
135C71#989411yescharacteristic, complemented, Sylow
136C2×#125—712no—
137C2×C4⋊C8—1421no—
138C2×#97—712no—
139C2×#97C2×D7112yes—
140#389—1421no—
141#389—1421no—
142#389—1421no—
143#389—1421no—
144#384—711no—
145#384—711no—
146#382—712no—
147C43C2×D7111yescharacteristic
148#336—21no—
149#336—21no—
150#336—1421no—
151C22×C8—22no—
152C22×C8—1422no—
153#125—2844no—
154C2×C4⋊C4—712no—
155C2×C4⋊C4—712no—
156C2×C4⋊C4—1422no—
157C2×C4⋊C4—1422no—
158C2×C4⋊C4—1421no—
159C2×C4⋊C4—1421no—
160C2×C4⋊C4—1421no—
161C2×C4⋊C4—1421no—
162C2×C42C4×D7111yescharacteristic
163C2×C42C4×D7111yescharacteristic
164C2×C42D28411yescharacteristic
165C2×C42D28411yescharacteristic
166C2×C42C22⋊D7111yescharacteristic
167C2×C42C22⋊D7111yescharacteristic
168C2×C42C22×D7111yescharacteristic
169C4⋊C8—1424no—
170#97—22no—
171#97—1424no—
172#97Dic14214yes—
173#95—1422no—
174#95—1422no—
175C22×C4—1422no—
176C22×C4—1422no—
177C22×C4(C2×D71)⋊C411yescharacteristic
178C22×C4(C2×D71)⋊C411yescharacteristic
179C22×C4(C2×D71)⋊C411yescharacteristic
180C22×C4(C2×D71)⋊C411yescharacteristic
181C22×C4C2×C4×D7111yescharacteristic
182C22×C4C2×C22⋊D7111yescharacteristic
183C22×C4C2×D28411yescharacteristic
184C2×C8—24no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—2842no—
188C2×C8—2842no—
189C2×C8—1424no—
190C4⋊C4—2842no—
191C4⋊C4—2842no—
192C4⋊C4—2844no—
193C4⋊C4—2842no—
194C4⋊C4—2842no—
195C4⋊C4—2842no—
196C4⋊C4—2844no—
197C4⋊C4—2842no—
198C42—22no—
199C42—22no—
200C42—22no—
201C42—41no—
202C42—41no—
203C42—41no—
204C42C2×Dic14212yes—
205C42#1176586512yes—
206C23C2×(C2×D71)⋊C411yescharacteristic
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—2842no—
223C2×C4—2842no—
224C2×C4—2842no—
225C2×C4—2842no—
226C2×C4—2842no—
227C2×C4—2842no—
228C2×C4#1455604412yes—
229C2×C4#1455604612yes—
230C2×C4#1455605412yes—
231C2×C4C4×Dic14212yes—
232C2×C4#1455610712yes—
233C2×C4#1455611611yescharacteristic
234C2×C4#1455612811yescharacteristic
235C8—44no—
236C8—2844no—
237C22#1798590511yescharacteristic
238C22C2×#1455604411yescharacteristic
239C22C2×#1455604611yescharacteristic
240C22C2×#1455605411yescharacteristic
241C22#1798601511yescharacteristic
242C22#1798601711yescharacteristic
243C22#1798601711yescharacteristic
244C4—22no—
245C4—22no—
246C4—22no—
247C4—21no—
248C4—21no—
249C4—22no—
250C4—22no—
251C4—41no—
252C4—5682no—
253C4—5682no—
254C2#2258694311yescharacteristic
255C2#2258695211yescharacteristic
256C2#2258696611yescharacteristic
257C2#2258697411yescharacteristic
258C2#2258928411yescharacteristic
259C2#2258932111yescharacteristic
260C2#2258932911yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1745C2
socle, prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1745D71
hypocenterC71#9894
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C43 ◁ #1745

Complements

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

NH ≅ G/Ntype
135. C71#9894semidirect

Maximal subgroups

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

subgroupindex
#225967032
#225881932
#225862222
C71×#17452
#989471