Finite Group Databasebeta

← #31891028

Subgroups — #31891028

Order 18176.

422 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 131 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×#2351C211yesmaximal, characteristic
6#17986052C2212yes—
7#17986023C2212yes—
8C2×#14556047—21no—
9#17985768—21no—
10#17985725—22no—
11C2×C4×Q8×C71—21no—
12C2×C4×Q8×C71C2211yescharacteristic
13#389×C71—21no—
14#389×C71—22no—
15#382×C71C412yes—
16#382×C71C2211yescharacteristic
17C4×C4⋊C4×C71C412yes—
18C4×C4⋊C4×C71C2211yescharacteristic
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—
29C4×Q8×C71—42no—
30C4×Q8×C71—42no—
31C2×C4⋊C4×C71—22no—
32C2×C4⋊C4×C71—22no—
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71D412yes—
36C2×C4⋊C4×C71D411yescharacteristic
37C2×C42×C71—22no—
38C2×C42×C71—21no—
39C2×C42×C71C2×C412yes—
40C2×C42×C71C2311yescharacteristic
41#95×C71—22no—
42#95×C71—21no—
43#95×C71C2×C411yescharacteristic
44#95×C71C2×C412yes—
45#95×C71C2×C411yescharacteristic
46#11765849—42no—
47#11765849—42no—
48C2×C71⋊C8—42no—
49C2×C71⋊C8—42no—
50C2×Q8×C71—22no—
51C2×Q8×C71—22no—
52C2×Q8×C71D812yes—
53C2×Q8×C71SD1612yes—
54C22×C4×C71—21no—
55C22×C4×C71—21no—
56C22×C4×C71—22no—
57C22×C4×C71C22⋊C412yes—
58C22×C4×C71C22⋊C412yes—
59C22×C4×C71C22×C411yescharacteristic
60C22×C4×C71C2×D411yescharacteristic
61C22×C4×C71C2×D411yescharacteristic
62C4⋊C4×C71—24no—
63C4⋊C4×C71—22no—
64C4⋊C4×C71—42no—
65C4⋊C4×C71—44no—
66C4⋊C4×C71—44no—
67C4⋊C4×C71—41no—
68C4⋊C4×C71—41no—
69C42×C71—22no—
70C42×C71—22no—
71C42×C71—44no—
72C42×C71—41no—
73C42×C71—41no—
74C42×C71#4112yes—
75C42×C71#4112yes—
76C23×C71C2×C22⋊C411yescharacteristic
77Q8×C71—24no—
78Q8×C71—24no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—41no—
84C2×C4×C71—41no—
85C2×C4×C71—41no—
86C2×C4×C71—41no—
87C2×C4×C71—42no—
88C2×C4×C71—41no—
89C2×C4×C71—42no—
90C2×C4×C71—41no—
91C2×C4×C71—42no—
92C2×C4×C71—42no—
93C2×C4×C71—41no—
94C2×C4×C71#10214yes—
95C2×C4×C71#11711yescharacteristic
96C2×C4×C71#11711yescharacteristic
97C2×C4×C71#12312yes—
98C2×C4×C71#12312yes—
99C2×C4×C71C2×D811yescharacteristic
100C2×C4×C71C2×SD1611yescharacteristic
101C2×C4×C71#13712yes—
102C71⋊C8—82no—
103C71⋊C8—82no—
104C22×C71—21no—
105C22×C71—21no—
106C22×C71#38111yescharacteristic
107C22×C71C2×#10211yescharacteristic
108C22×C71#41911yescharacteristic
109C4×C71—24no—
110C4×C71—22no—
111C4×C71—21no—
112C4×C71—21no—
113C4×C71—42no—
114C4×C71—42no—
115C4×C71—82no—
116C4×C71—81no—
117C4×C71#48012yes—
118C4×C71#48312yes—
119#8766—711nomaximal, Sylow
120C2×C71—21no—
121C2×C71—21no—
122C2×C71#182311yescharacteristic
123C2×C71#182711yescharacteristic
124C2×C71#183011yescharacteristic
125#2351D7111yescharacteristic
126#1801—711no—
127#1461—711no—
128#1303—712no—
129#1286—712no—
130C71#876611yescharacteristic, complemented, Sylow
131C2×C4×Q8—21no—
132C2×C4×Q8C2×D7111yescharacteristic
133#431—712no—
134#423—712no—
135C2×#101—1421no—
136#389—21no—
137#389—22no—
138#382Dic7112yes—
139#382C2×D7111yescharacteristic
140C4×C4⋊C4Dic7112yes—
141C4×C4⋊C4C2×D7111yescharacteristic
142#343—1421no—
143#324—1422no—
144C22×Q8C22⋊D7111yescharacteristic
145C2×M16—1422no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—42no—
149C4×Q8—42no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4C22⋊D7112yes—
155C2×C4⋊C4C22⋊D7111yescharacteristic
156C2×C42—22no—
157C2×C42—21no—
158C2×C42C2×Dic7112yes—
159C2×C42C22×D7111yescharacteristic
160C4⋊C8—1422no—
161C4⋊C8—1422no—
162C4⋊C8—1422no—
163#101—2842no—
164C22⋊C8—1422no—
165C4×C8—1422no—
166#95—22no—
167#95—21no—
168#95C2×Dic7111yescharacteristic
169#95C2×Dic7112yes—
170#95C2×Dic7111yescharacteristic
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8#1176585012yes—
174C2×Q8#1176585312yes—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4C22⋊Dic7112yes—
179C22×C4C22⋊Dic7112yes—
180C22×C4C22×Dic7111yescharacteristic
181C22×C4C2×C22⋊D7111yescharacteristic
182C22×C4C2×C22⋊D7111yescharacteristic
183M16—2842no—
184M16—2842no—
185C2×C8—2842no—
186C2×C8—2842no—
187C4⋊C4—24no—
188C4⋊C4—22no—
189C4⋊C4—42no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C42—22no—
195C42—22no—
196C42—44no—
197C42—41no—
198C42—41no—
199C42#1176586512yes—
200C42#1176586512yes—
201C23C2×C22⋊Dic7111yescharacteristic
202Q8—24no—
203Q8—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4#1455604914yes—
220C2×C4#1455608711yescharacteristic
221C2×C4#1455608711yescharacteristic
222C2×C4#1455611112yes—
223C2×C4#1455611112yes—
224C2×C4C2×#1176585011yescharacteristic
225C2×C4C2×#1176585311yescharacteristic
226C2×C4#1455615912yes—
227C8—5682no—
228C8—5682no—
229C22—21no—
230C22—21no—
231C22#1798586211yescharacteristic
232C22C2×#1455604911yescharacteristic
233C22#1798600811yescharacteristic
234C4—24no—
235C4—22no—
236C4—21no—
237C4—21no—
238C4—42no—
239C4—42no—
240C4—82no—
241C4—81no—
242C4#1798631212yes—
243C4#1798632912yes—
244C2—21no—
245C2—21no—
246C2#2258842611yescharacteristic
247C2#2258844411yescharacteristic
248C2#2258845411yescharacteristic
subgroupG/H
centerC22#17985862
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2351C2
socleC22×C71#381
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2351D71
hypocenterC71#8766
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2351

Complements

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

NH ≅ G/Ntype
130. C71#8766semidirect

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×#23512
#876671