Finite Group Databasebeta

← #31891039

Subgroups — #31891039

Order 18176.

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

normal subgroups only

This group has 378 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#22591305C211yesmaximal, characteristic
2#22588336C211yesmaximal, characteristic
3#22586242C212yesmaximal
4#22586199C212yesmaximal
5C71×#1461C211yesmaximal, characteristic
6#17986023C2212yes—
7C2×#14556047—21no—
8#17985906—21no—
9#17985885—22no—
10#17985867C412yes—
11#17985865C2211yescharacteristic
12C4⋊C4×Dic71—21no—
13C4×C4⋊Dic71C2211yescharacteristic
14#17985845C412yes—
15#17985768—21no—
16C2×C4×Q8×C71C2211yescharacteristic
17#431×C71C2212yes—
18#324×C71—22no—
19C2×#11765849—22no—
20C2×C4⋊Dic71—21no—
21C2×C4⋊Dic71—21no—
22C2×C4×Dic71—22no—
23C2×C4×Dic71—21no—
24C4⋊(C71⋊C8)—22no—
25C4⋊(C71⋊C8)—22no—
26#14556047—42no—
27C71⋊#95—22no—
28C71⋊#95—22no—
29C71⋊#95—22no—
30C71⋊#95—21no—
31C71⋊#95C2×C411yescharacteristic
32C71⋊#95C2×C412yes—
33C71⋊#95C2×C411yescharacteristic
34C22×Q8×C71D411yescharacteristic
35C4×Q8×C71—22no—
36C4×Q8×C71—22no—
37C2×C4⋊C4×C71D411yescharacteristic
38C2×C4⋊C4×C71D412yes—
39C2×C42×C71C2×C412yes—
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—42no—
52C4×Dic71—44no—
53C4×Dic71—41no—
54C4×Dic71—41no—
55C2×Q8×C71—22no—
56C2×Q8×C71—22no—
57C2×Q8×C71—21no—
58C2×Q8×C71—21no—
59C22×C4×C71C22⋊C412yes—
60C22×C4×C71C22⋊C412yes—
61C22×C4×C71C22×C411yescharacteristic
62C22×C4×C71C2×D411yescharacteristic
63C22×C4×C71C2×D411yescharacteristic
64C2×C8×C71—42no—
65C4⋊C4×C71—24no—
66C4⋊C4×C71—22no—
67C42×C71—22no—
68C42×C71—22no—
69C42×C71#4112yes—
70C42×C71#4112yes—
71C23×C71C2×C22⋊C411yescharacteristic
72Q8×C71—42no—
73Q8×C71—42no—
74C2×C4×C71—22no—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71—41no—
79C2×C4×C71—41no—
80C2×C4×C71—42no—
81C2×C4×C71#11711yescharacteristic
82C2×C4×C71#11711yescharacteristic
83C2×C4×C71#12312yes—
84C2×C4×C71#12312yes—
85C2×C4×C71#13512yes—
86C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
87C2×C4×C71#13711yescharacteristic
88C2×Dic71—22no—
89C2×Dic71—41no—
90C2×Dic71—41no—
91C2×Dic71—42no—
92C2×Dic71—41no—
93C2×Dic71—41no—
94C2×Dic71—42no—
95C2×Dic71—42no—
96C2×Dic71—41no—
97C8×C71—82no—
98C71⋊C8—82no—
99C22×C71—21no—
100C22×C71—21no—
101C22×C71#38111yescharacteristic
102C22×C71#41611yescharacteristic
103C22×C71#41711yescharacteristic
104C4×C71—22no—
105C4×C71—21no—
106C4×C71—21no—
107C4×C71—42no—
108C4×C71—42no—
109C4×C71#48212yes—
110C4×C71#48512yes—
111Dic71—42no—
112Dic71—81no—
113Dic71—82no—
114#8768—711nomaximal, Sylow
115C2×C71—21no—
116C2×C71—21no—
117C2×C71#182511yescharacteristic
118C2×C71#182711yescharacteristic
119C2×C71#183011yescharacteristic
120#2356—711no—
121#1801—711no—
122#1461D7111yescharacteristic
123#1303—712no—
124#1286—712no—
125C71#876811yescharacteristic, complemented, Sylow
126C2×C4×Q8C2×D7111yescharacteristic
127#431C2×D7112yes—
128#423—712no—
129C2×#101—1421no—
130#391—1421no—
131#387—1422no—
132#383—712no—
133#382—711no—
134C4×C4⋊C4—711no—
135C4×C4⋊C4—1421no—
136#376—712no—
137#343—1421no—
138#324—22no—
139C22×Q8D28411yescharacteristic
140C2×M16—1422no—
141C4×Q8—22no—
142C4×Q8—22no—
143C2×C4⋊C4—1421no—
144C2×C4⋊C4—1421no—
145C2×C4⋊C4D28411yescharacteristic
146C2×C4⋊C4C22⋊D7112yes—
147C2×C42—1422no—
148C2×C42—1421no—
149C2×C42C4×D7112yes—
150C2×C42C22×D7111yescharacteristic
151C4⋊C8—22no—
152C4⋊C8—1422no—
153C4⋊C8—1422no—
154#101—2842no—
155C22⋊C8—22no—
156C4×C8—22no—
157#95—711no—
158#95—712no—
159#95—711no—
160#95—1422no—
161#95—1422no—
162#95—1422no—
163#95—1421no—
164C2×Q8—21no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—21no—
168C22×C4—1422no—
169C22×C4—1421no—
170C22×C4—1421no—
171C22×C4(C2×D71)⋊C412yes—
172C22×C4(C2×D71)⋊C412yes—
173C22×C4C2×C4×D7111yescharacteristic
174C22×C4C2×C22⋊D7111yescharacteristic
175C22×C4C2×D28411yescharacteristic
176M16—2842no—
177M16—2842no—
178C2×C8—42no—
179C2×C8—2842no—
180C4⋊C4—24no—
181C4⋊C4—22no—
182C4⋊C4—2842no—
183C4⋊C4—2842no—
184C42—22no—
185C42—22no—
186C42—2844no—
187C42—2841no—
188C42—2841no—
189C42#1176586412yes—
190C42#1176586412yes—
191C23C2×(C2×D71)⋊C411yescharacteristic
192Q8—42no—
193Q8—42no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—2841no—
202C2×C4—2841no—
203C2×C4—2842no—
204C2×C4—2841no—
205C2×C4—2841no—
206C2×C4—2842no—
207C2×C4—2842no—
208C2×C4—2841no—
209C2×C4—1422no—
210C2×C4#1455608411yescharacteristic
211C2×C4#1455608411yescharacteristic
212C2×C4#1455610912yes—
213C2×C4#1455611212yes—
214C2×C4#1455615012yes—
215C2×C4#1455615611yescharacteristic
216C2×C4#1455616111yescharacteristic
217C8—82no—
218C8—5682no—
219C22—21no—
220C22—21no—
221C22#1798586011yescharacteristic
222C22#1798599611yescharacteristic
223C22#1798600111yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—42no—
228C4—42no—
229C4—2842no—
230C4—5681no—
231C4—5682no—
232C4#1798632412yes—
233C4#1798634112yes—
234C2—21no—
235C2—21no—
236C2#2258843611yescharacteristic
237C2#2258844111yescharacteristic
238C2#2258845211yescharacteristic
subgroupG/H
centerC22#17985860
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1461C2
socleC22×C71#381
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1461D71
hypocenterC71#8768
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
125. C71#8768semidirect

Maximal subgroups

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

subgroupindex
#225913052
#225883362
#225862422
#225861992
C71×#14612
#876871