Finite Group Databasebeta

← #31891054

Subgroups — #31891054

Order 18176.

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

normal subgroups only

This group has 426 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#22591313C211yesmaximal, characteristic
2#22588332C211yesmaximal, characteristic
3#22586242C212yesmaximal, complemented
4#22586199C212yesmaximal, complemented
5C71×#1460C211yesmaximal, characteristic
6#17986023C2212yes—
7C2×#14556045—21no—
8#17985902—21no—
9#17985892—22no—
10#17985865C2211yescharacteristic
11#17985862C412yes—
12C4⋊C4×Dic71—21no—
13C4×C4⋊Dic71C2211yescharacteristic
14C4×C22⋊Dic71C412yes—
15#17985768—21no—
16C2×C4×D4×C71C2211yescharacteristic
17#431×C71C2212yes—
18#323×C71—22no—
19C2×#11765849—22no—
20C2×C4⋊Dic71—21no—
21C2×C4⋊Dic71—21no—
22C2×C22⋊Dic71—22no—
23C2×C22⋊Dic71—22no—
24C2×C4×Dic71—22no—
25C2×C4×Dic71—21no—
26C4⋊(C71⋊C8)—22no—
27C4⋊(C71⋊C8)—22no—
28#14556045—42no—
29C71⋊#95—22no—
30C71⋊#95—21no—
31C71⋊#95C2×C411yescharacteristic
32C71⋊#95C2×C412yes—
33C71⋊#95C2×C411yescharacteristic
34C22×D4×C71D411yescharacteristic
35C23×C4×C71C2×C412yes—
36C4×D4×C71—22no—
37C4×D4×C71—22no—
38C2×C4⋊C4×C71D411yescharacteristic
39C2×C22⋊C4×C71D412yes—
40C2×C42×C71C2311yescharacteristic
41C4⋊C8×C71—22no—
42C22⋊C8×C71—22no—
43C4×C8×C71—22no—
44C22×Dic71—21no—
45C22×Dic71—22no—
46C22×Dic71—21no—
47#11765849—42no—
48#11765849—42no—
49C2×C71⋊C8—42no—
50C4⋊Dic71—42no—
51C4⋊Dic71—41no—
52C4⋊Dic71—41no—
53C22⋊Dic71—44no—
54C22⋊Dic71—82no—
55C4×Dic71—42no—
56C4×Dic71—82no—
57C24×C71C22⋊C412yes—
58C2×D4×C71—22no—
59C2×D4×C71—22no—
60C2×D4×C71—21no—
61C2×D4×C71—21no—
62C22×C4×C71—22no—
63C22×C4×C71—22no—
64C22×C4×C71—82no—
65C22×C4×C71C22⋊C412yes—
66C22×C4×C71C22×C411yescharacteristic
67C22×C4×C71C2×D411yescharacteristic
68C22×C4×C71C2×D411yescharacteristic
69C2×C8×C71—42no—
70C4⋊C4×C71—22no—
71C22⋊C4×C71—24no—
72C42×C71#4112yes—
73C42×C71#4112yes—
74C23×C71—22no—
75C23×C71—22no—
76C23×C71—22no—
77C23×C71—42no—
78C23×C71—42no—
79C23×C71C2×C22⋊C411yescharacteristic
80D4×C71—42no—
81D4×C71—42no—
82C2×C4×C71—22no—
83C2×C4×C71—41no—
84C2×C4×C71—41no—
85C2×C4×C71—42no—
86C2×C4×C71—82no—
87C2×C4×C71—82no—
88C2×C4×C71#11711yescharacteristic
89C2×C4×C71#11711yescharacteristic
90C2×C4×C71#12312yes—
91C2×C4×C71#12312yes—
92C2×C4×C71#13512yes—
93C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
94C2×C4×C71#13711yescharacteristic
95C2×Dic71—22no—
96C2×Dic71—41no—
97C2×Dic71—41no—
98C2×Dic71—42no—
99C2×Dic71—42no—
100C2×Dic71—41no—
101C2×Dic71—41no—
102C2×Dic71—41no—
103C2×Dic71—42no—
104C8×C71—82no—
105C71⋊C8—82no—
106C22×C71—21no—
107C22×C71—21no—
108C22×C71—42no—
109C22×C71—42no—
110C22×C71—42no—
111C22×C71—82no—
112C22×C71—82no—
113C22×C71#38111yescharacteristic
114C22×C71#41611yescharacteristic
115C22×C71#41711yescharacteristic
116C4×C71—22no—
117C4×C71—21no—
118C4×C71—21no—
119C4×C71—42no—
120C4×C71#48212yes—
121C4×C71#48512yes—
122Dic71—42no—
123Dic71—81no—
124Dic71—82no—
125#8771—711nomaximal, Sylow
126C2×C71—21no—
127C2×C71—21no—
128C2×C71—82no—
129C2×C71#182511yescharacteristic
130C2×C71#182711yescharacteristic
131C2×C71#182911yescharacteristic
132#2357—711no—
133#1800—711no—
134#1460D7111yescharacteristic
135#1303—712no—
136#1286—712no—
137C71#877111yescharacteristic, complemented, Sylow
138C2×C4×D4C2×D7111yescharacteristic
139#431C2×D7112yes—
140#423—712no—
141C2×#100—1421no—
142#390—1421no—
143#388—1422no—
144#382—711no—
145#381—712no—
146C4×C4⋊C4—711no—
147C4×C4⋊C4—1421no—
148C4×C22⋊C4—712no—
149#343—1421no—
150#323—22no—
151C22×D4D28411yescharacteristic
152C23×C4C4×D7112yes—
153C2×M16—1422no—
154C4×D4—22no—
155C4×D4—22no—
156C2×C4⋊C4—1421no—
157C2×C4⋊C4—1421no—
158C2×C4⋊C4D28411yescharacteristic
159C2×C22⋊C4—1422no—
160C2×C22⋊C4—1422no—
161C2×C22⋊C4C22⋊D7112yes—
162C2×C42—1421no—
163C2×C42—1422no—
164C2×C42C22×D7111yescharacteristic
165C4⋊C8—22no—
166C4⋊C8—1422no—
167C4⋊C8—1422no—
168#100—2842no—
169C22⋊C8—22no—
170C4×C8—22no—
171#95—711no—
172#95—712no—
173#95—711no—
174#95—1422no—
175#95—1421no—
176C24(C2×D71)⋊C412yes—
177C2×D4—21no—
178C2×D4—22no—
179C2×D4—22no—
180C2×D4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—82no—
184C22×C4—1422no—
185C22×C4—1421no—
186C22×C4—1421no—
187C22×C4(C2×D71)⋊C412yes—
188C22×C4C2×C4×D7111yescharacteristic
189C22×C4C2×C22⋊D7111yescharacteristic
190C22×C4C2×D28411yescharacteristic
191M16—2842no—
192M16—2842no—
193C2×C8—42no—
194C2×C8—2842no—
195C4⋊C4—22no—
196C4⋊C4—2842no—
197C4⋊C4—2841no—
198C4⋊C4—2841no—
199C22⋊C4—24no—
200C22⋊C4—2844no—
201C22⋊C4—5682no—
202C42—2842no—
203C42—5682no—
204C42#1176586412yes—
205C42#1176586412yes—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23C2×(C2×D71)⋊C411yescharacteristic
212D4—42no—
213D4—42no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—2841no—
221C2×C4—2841no—
222C2×C4—2842no—
223C2×C4—2842no—
224C2×C4—2841no—
225C2×C4—2841no—
226C2×C4—2841no—
227C2×C4—2842no—
228C2×C4—1422no—
229C2×C4#1455608411yescharacteristic
230C2×C4#1455608411yescharacteristic
231C2×C4#1455610912yes—
232C2×C4#1455611212yes—
233C2×C4#1455615012yes—
234C2×C4#1455615611yescharacteristic
235C2×C4#1455616111yescharacteristic
236C8—82no—
237C8—5682no—
238C22—21no—
239C22—21no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22—82no—
245C22#1798586011yescharacteristic
246C22#1798599611yescharacteristic
247C22#1798600111yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—2842no—
253C4—5681no—
254C4—5682no—
255C4#1798632412yes—
256C4#1798634112yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#2258843611yescharacteristic
261C2#2258844111yescharacteristic
262C2#2258844911yescharacteristic
subgroupG/H
centerC22#17985860
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1460C2
socleC22×C71#381
prime radicalC22×D4×C71D4
cosocleC2×C42×C71C23
hypercenter#1460D71
hypocenterC71#8771
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #22586242C2semidirect
4. #22586199C2semidirect
137. C71#8771semidirect

Maximal subgroups

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

subgroupindex
#225913132
#225883322
#225862422
#225861992
C71×#14602
#877171