Finite Group Databasebeta

← #31891043

Subgroups — #31891043

Order 18176.

378 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 115 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#22588336C211yesmaximal, characteristic
2#22586664C211yesmaximal, characteristic
3#22586243C212yesmaximal
4#22586198C212yesmaximal
5C71×#2356C211yesmaximal, characteristic
6#17986052C2212yes—
7#17986023C2212yes—
8C2×#14556047—21no—
9#17985768—21no—
10#17985725—22no—
11C2×C4×Q8×C71C2211yescharacteristic
12#391×C71—21no—
13#387×C71—22no—
14#383×C71C412yes—
15#382×C71C2211yescharacteristic
16C4×C4⋊C4×C71—21no—
17C4×C4⋊C4×C71C2211yescharacteristic
18#376×C71C412yes—
19C2×#11765849—22no—
20C4⋊(C71⋊C8)—22no—
21C4⋊(C71⋊C8)—22no—
22C4⋊(C71⋊C8)—22no—
23#14556047—42no—
24C22⋊(C71⋊C8)—22no—
25C4×C71⋊C8—22no—
26C22×Q8×C71D411yescharacteristic
27C4×Q8×C71—22no—
28C4×Q8×C71—22no—
29C2×C4⋊C4×C71—21no—
30C2×C4⋊C4×C71—21no—
31C2×C4⋊C4×C71D412yes—
32C2×C4⋊C4×C71D411yescharacteristic
33C2×C42×C71—22no—
34C2×C42×C71—21no—
35C2×C42×C71C2×C412yes—
36C2×C42×C71C2311yescharacteristic
37#95×C71—22no—
38#95×C71—22no—
39#95×C71—22no—
40#95×C71—21no—
41#95×C71C2×C411yescharacteristic
42#95×C71C2×C412yes—
43#95×C71C2×C411yescharacteristic
44#11765849—42no—
45#11765849—42no—
46C2×C71⋊C8—42no—
47C2×C71⋊C8—42no—
48C2×Q8×C71—22no—
49C2×Q8×C71—22no—
50C2×Q8×C71—21no—
51C2×Q8×C71—21no—
52C22×C4×C71—21no—
53C22×C4×C71—22no—
54C22×C4×C71—21no—
55C22×C4×C71C22⋊C412yes—
56C22×C4×C71C22⋊C412yes—
57C22×C4×C71C22×C411yescharacteristic
58C22×C4×C71C2×D411yescharacteristic
59C22×C4×C71C2×D411yescharacteristic
60C4⋊C4×C71—22no—
61C4⋊C4×C71—24no—
62C4⋊C4×C71—42no—
63C4⋊C4×C71—41no—
64C4⋊C4×C71—41no—
65C42×C71—22no—
66C42×C71—22no—
67C42×C71—44no—
68C42×C71—42no—
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—22no—
79C2×C4×C71—41no—
80C2×C4×C71—41no—
81C2×C4×C71—41no—
82C2×C4×C71—41no—
83C2×C4×C71—42no—
84C2×C4×C71—42no—
85C2×C4×C71—41no—
86C2×C4×C71—41no—
87C2×C4×C71—42no—
88C2×C4×C71—41no—
89C2×C4×C71—42no—
90C2×C4×C71#11711yescharacteristic
91C2×C4×C71#11711yescharacteristic
92C2×C4×C71#12312yes—
93C2×C4×C71#12312yes—
94C2×C4×C71#13512yes—
95C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C71#13711yescharacteristic
97C71⋊C8—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—42no—
110C4×C71—81no—
111C4×C71—82no—
112C4×C71#48212yes—
113C4×C71#48512yes—
114#8769—711nomaximal, Sylow
115C2×C71—21no—
116C2×C71—21no—
117C2×C71#182511yescharacteristic
118C2×C71#182711yescharacteristic
119C2×C71#183011yescharacteristic
120#2356D7111yescharacteristic
121#1801—711no—
122#1461—711no—
123#1303—712no—
124#1286—712no—
125C71#876911yescharacteristic, complemented, Sylow
126C2×C4×Q8C2×D7111yescharacteristic
127#431—712no—
128#423—712no—
129C2×#101—1421no—
130#391—21no—
131#387—22no—
132#383Dic7112yes—
133#382C2×D7111yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2×D7111yescharacteristic
136#376Dic7112yes—
137#343—1421no—
138#324—1422no—
139C22×Q8C22⋊D7111yescharacteristic
140C2×M16—1422no—
141C4×Q8—22no—
142C4×Q8—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4C22⋊D7112yes—
146C2×C4⋊C4C22⋊D7111yescharacteristic
147C2×C42—22no—
148C2×C42—21no—
149C2×C42C2×Dic7112yes—
150C2×C42C22×D7111yescharacteristic
151C4⋊C8—1422no—
152C4⋊C8—1422no—
153C4⋊C8—1422no—
154#101—2842no—
155C22⋊C8—1422no—
156C4×C8—1422no—
157#95—22no—
158#95—22no—
159#95—22no—
160#95—21no—
161#95C2×Dic7111yescharacteristic
162#95C2×Dic7112yes—
163#95C2×Dic7111yescharacteristic
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—21no—
167C2×Q8—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4C22⋊Dic7112yes—
172C22×C4C22⋊Dic7112yes—
173C22×C4C22×Dic7111yescharacteristic
174C22×C4C2×C22⋊D7111yescharacteristic
175C22×C4C2×C22⋊D7111yescharacteristic
176M16—2842no—
177M16—2842no—
178C2×C8—2842no—
179C2×C8—2842no—
180C4⋊C4—22no—
181C4⋊C4—24no—
182C4⋊C4—42no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C42—22no—
186C42—22no—
187C42—44no—
188C42—42no—
189C42#1176586512yes—
190C42#1176586512yes—
191C23C2×C22⋊Dic7111yescharacteristic
192Q8—42no—
193Q8—42no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4#1455608711yescharacteristic
211C2×C4#1455608711yescharacteristic
212C2×C4#1455611112yes—
213C2×C4#1455611112yes—
214C2×C4#1455614812yes—
215C2×C4#1455615311yescharacteristic
216C2×C4#1455615811yescharacteristic
217C8—5682no—
218C8—5682no—
219C22—21no—
220C22—21no—
221C22#1798586211yescharacteristic
222C22#1798599511yescharacteristic
223C22#1798600011yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—42no—
228C4—42no—
229C4—42no—
230C4—81no—
231C4—82no—
232C4#1798632312yes—
233C4#1798634012yes—
234C2—21no—
235C2—21no—
236C2#2258843511yescharacteristic
237C2#2258844411yescharacteristic
238C2#2258845411yescharacteristic
subgroupG/H
centerC22#17985862
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2356C2
socleC22×C71#381
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2356D71
hypocenterC71#8769
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2356

Complements

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

NH ≅ G/Ntype
125. C71#8769semidirect

Maximal subgroups

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

subgroupindex
#225883362
#225866642
#225862432
#225861982
C71×#23562
#876971