Finite Group Databasebeta

← #31908509

Subgroups — #31908509

Order 18176.

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

normal subgroups only

This group has 408 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#22593325C211yesmaximal, characteristic
2#22586664C211yesmaximal, characteristic
3#22586345C214yesmaximal
4C71×#1801C211yesmaximal, characteristic
5C2×#14556128C411yescharacteristic
6C2×#14556126—21no—
7#17986052—21no—
8#17985929C412yes—
9#17985929C2212yes—
10#17985904—21no—
11#17985897—21no—
12#17985897—21no—
13C4×C4⋊Dic71C411yescharacteristic
14#17985725—21no—
15#17985725C2212yes—
16C2×C4×Q8×C71C2211yescharacteristic
17#423×C71—21no—
18C2×#101×C71—21no—
19#343×C71C2212yes—
20C22×Dic142—21no—
21#14556128—41no—
22#14556128—41no—
23#14556126—42no—
24#14556126—42no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71—21no—
27C2×C4⋊Dic71C2×C412yes—
28C2×Dic71⋊C4—22no—
29C2×Dic71⋊C4—22no—
30C2×C4×Dic71—21no—
31C4⋊(C71⋊C8)—41no—
32C22⋊(C71⋊C8)—22no—
33C4×C71⋊C8—41no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C71⋊#95C2×C412yes—
37C22×Q8×C71D411yescharacteristic
38C2×M16×C71—22no—
39C4×Q8×C71—41no—
40C4×Q8×C71—41no—
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C4⋊C4×C71D411yescharacteristic
43C2×C4⋊C4×C71C2311yescharacteristic
44C2×C42×C71C2×C411yescharacteristic
45C2×C42×C71D411yescharacteristic
46C2×C42×C71D411yescharacteristic
47C4⋊C8×C71—41no—
48C4⋊C8×C71—41no—
49#101×C71—42no—
50C2×Dic142—42no—
51C2×Dic142—42no—
52C2×Dic142—42no—
53C22×Dic71—22no—
54C22×Dic71—22no—
55C2×C71⋊C8—42no—
56C4⋊Dic71—24no—
57C4⋊Dic71—41no—
58C4⋊Dic71—41no—
59C4⋊Dic71—42no—
60Dic71⋊C4—44no—
61Dic71⋊C4—42no—
62Dic71⋊C4—42no—
63C4×Dic71—42no—
64C2×Q8×C71—41no—
65C2×Q8×C71—41no—
66C2×Q8×C71D812yes—
67C2×Q8×C71SD1612yes—
68C22×C4×C71C22⋊C411yescharacteristic
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71C22⋊C411yescharacteristic
72C22×C4×C71C22×C411yescharacteristic
73C22×C4×C71C2×D411yescharacteristic
74C22×C4×C71C2×D411yescharacteristic
75M16×C71—42no—
76M16×C71—42no—
77C2×C8×C71—42no—
78C4⋊C4×C71—41no—
79C4⋊C4×C71—41no—
80C4⋊C4×C71—41no—
81C42×C71—21no—
82C42×C71—21no—
83C42×C71—21no—
84C42×C71—21no—
85C42×C71—21no—
86C42×C71—21no—
87C23×C71C2×C22⋊C411yescharacteristic
88Q8×C71—42no—
89Q8×C71—42no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—22no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—41no—
101C2×C4×C71—41no—
102C2×C4×C71—41no—
103C2×C4×C71#9911yescharacteristic
104C2×C4×C71#9911yescharacteristic
105C2×C4×C71#10212yes—
106C2×C4×C71#10212yes—
107C2×C4×C71C2×D811yescharacteristic
108C2×C4×C71C2×SD1611yescharacteristic
109C2×Dic71—24no—
110C2×Dic71—42no—
111C2×Dic71—42no—
112C2×Dic71—42no—
113C2×Dic71—42no—
114C2×Dic71—42no—
115Dic142—84no—
116C8×C71—82no—
117C71⋊C8—82no—
118C22×C71—21no—
119C22×C71—21no—
120C22×C71C2×#9911yescharacteristic
121C22×C71C2×#10211yescharacteristic
122C22×C71#42111yescharacteristic
123C4×C71—21no—
124C4×C71—21no—
125C4×C71—22no—
126C4×C71—22no—
127C4×C71—21no—
128C4×C71—41no—
129C4×C71—41no—
130C4×C71—41no—
131C4×C71#35612yes—
132Dic71—44no—
133Dic71—82no—
134#11883—711nomaximal, Sylow
135C2×C71—21no—
136C2×C71—21no—
137C2×C71#151911yescharacteristic
138C2×C71#212311yescharacteristic
139C2×C71C2×#35611yescharacteristic
140#2546—711no—
141#1801D7111yescharacteristic
142#1461—711no—
143#1346—714no—
144C71#1188311yescharacteristic, complemented, Sylow
145C2×C4⋊Q8—711no—
146C2×C4⋊Q8—1421no—
147C2×C4×Q8C2×D7111yescharacteristic
148#431—1421no—
149#423—21no—
150C2×#101—21no—
151#398—712no—
152#398—712no—
153#391—1421no—
154#389—1421no—
155#389—1421no—
156C4×C4⋊C4—711no—
157#343C2×D7112yes—
158#324—712no—
159#324—1421no—
160C22×Q8—1421no—
161C22×Q8C22⋊D7111yescharacteristic
162C2×M16—22no—
163C4⋊Q8—2841no—
164C4⋊Q8—2841no—
165C4⋊Q8—2842no—
166C4⋊Q8—2842no—
167C4×Q8—41no—
168C4×Q8—41no—
169C2×C4⋊C4—712no—
170C2×C4⋊C4—1422no—
171C2×C4⋊C4—1421no—
172C2×C4⋊C4—1422no—
173C2×C4⋊C4—1421no—
174C2×C4⋊C4C4×D7111yescharacteristic
175C2×C4⋊C4D28411yescharacteristic
176C2×C4⋊C4C22×D7111yescharacteristic
177C2×C42—1421no—
178C2×C42C4×D7111yescharacteristic
179C2×C42D28411yescharacteristic
180C2×C42C22⋊D7111yescharacteristic
181C4⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8—2841no—
184#101—42no—
185C22⋊C8—1422no—
186C4×C8—2841no—
187#95—712no—
188#95—1422no—
189#95—1422no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—2842no—
193C2×Q8—2842no—
194C2×Q8—2842no—
195C2×Q8#1176585012yes—
196C2×Q8#1176585212yes—
197C22×C4—1422no—
198C22×C4—1422no—
199C22×C4(C2×D71)⋊C411yescharacteristic
200C22×C4(C2×D71)⋊C411yescharacteristic
201C22×C4(C2×D71)⋊C411yescharacteristic
202C22×C4(C2×D71)⋊C411yescharacteristic
203C22×C4C2×C4×D7111yescharacteristic
204C22×C4C2×C22⋊D7111yescharacteristic
205C22×C4C2×D28411yescharacteristic
206M16—42no—
207M16—42no—
208C2×C8—42no—
209C2×C8—2842no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—2842no—
214C4⋊C4—2841no—
215C4⋊C4—2841no—
216C4⋊C4—2844no—
217C4⋊C4—2842no—
218C4⋊C4—2842no—
219C4⋊C4—1424no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—2842no—
227C23C2×(C2×D71)⋊C411yescharacteristic
228Q8—42no—
229Q8—42no—
230Q8—5684no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—2842no—
245C2×C4—2842no—
246C2×C4—2842no—
247C2×C4—2842no—
248C2×C4—2842no—
249C2×C4—1424no—
250C2×C4#1455604211yescharacteristic
251C2×C4#1455604211yescharacteristic
252C2×C4#1455604812yes—
253C2×C4#1455604812yes—
254C2×C4C2×#1176585011yescharacteristic
255C2×C4C2×#1176585211yescharacteristic
256C8—82no—
257C8—5682no—
258C22—21no—
259C22—21no—
260C22C2×#1455604211yescharacteristic
261C22C2×#1455604811yescharacteristic
262C22#1798601711yescharacteristic
263C4—21no—
264C4—22no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—2844no—
272C4—5682no—
273C4#1798579712yes—
274C2—21no—
275C2—21no—
276C2#2258690711yescharacteristic
277C2#2258999511yescharacteristic
278C2C2×#1798579711yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#1801C2
socleC22×C71C2×#99
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#1801D71
hypocenterC71#11883
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
144. C71#11883semidirect

Maximal subgroups

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

subgroupindex
#225933252
#225866642
#225863452
C71×#18012
#1188371