Finite Group Databasebeta

← #31891033

Subgroups — #31891033

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—44no—
30C2×C4⋊C4×C71—22no—
31C2×C4⋊C4×C71—22no—
32C2×C4⋊C4×C71—21no—
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71D412yes—
35C2×C4⋊C4×C71D411yescharacteristic
36C2×C42×C71—22no—
37C2×C42×C71—21no—
38C2×C42×C71C2×C412yes—
39C2×C42×C71C2311yescharacteristic
40#95×C71—22no—
41#95×C71—21no—
42#95×C71C2×C411yescharacteristic
43#95×C71C2×C412yes—
44#95×C71C2×C411yescharacteristic
45#11765849—42no—
46#11765849—42no—
47C2×C71⋊C8—42no—
48C2×C71⋊C8—42no—
49C2×Q8×C71—22no—
50C2×Q8×C71—22no—
51C2×Q8×C71SD1612yes—
52C2×Q8×C71Q1612yes—
53C22×C4×C71—21no—
54C22×C4×C71—21no—
55C22×C4×C71—22no—
56C22×C4×C71C22⋊C412yes—
57C22×C4×C71C22⋊C412yes—
58C22×C4×C71C22×C411yescharacteristic
59C22×C4×C71C2×D411yescharacteristic
60C22×C4×C71C2×D411yescharacteristic
61C4⋊C4×C71—24no—
62C4⋊C4×C71—22no—
63C4⋊C4×C71—42no—
64C4⋊C4×C71—44no—
65C4⋊C4×C71—44no—
66C4⋊C4×C71—42no—
67C42×C71—22no—
68C42×C71—22no—
69C42×C71—44no—
70C42×C71—42no—
71C42×C71#4112yes—
72C42×C71#4112yes—
73C23×C71C2×C22⋊C411yescharacteristic
74Q8×C71—24no—
75Q8×C71—24no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—41no—
81C2×C4×C71—41no—
82C2×C4×C71—41no—
83C2×C4×C71—41no—
84C2×C4×C71—42no—
85C2×C4×C71—41no—
86C2×C4×C71—42no—
87C2×C4×C71—41no—
88C2×C4×C71—42no—
89C2×C4×C71—42no—
90C2×C4×C71—41no—
91C2×C4×C71#10314yes—
92C2×C4×C71#11711yescharacteristic
93C2×C4×C71#11711yescharacteristic
94C2×C4×C71#12312yes—
95C2×C4×C71#12312yes—
96C2×C4×C71C2×SD1611yescharacteristic
97C2×C4×C71C2×Q1611yescharacteristic
98C2×C4×C71AGL(1,ℤ/8)12yes—
99C71⋊C8—82no—
100C71⋊C8—82no—
101C22×C71—21no—
102C22×C71—21no—
103C22×C71#38111yescharacteristic
104C22×C71C2×#10311yescharacteristic
105C22×C71#41811yescharacteristic
106C4×C71—24no—
107C4×C71—22no—
108C4×C71—21no—
109C4×C71—21no—
110C4×C71—42no—
111C4×C71—42no—
112C4×C71—82no—
113C4×C71—81no—
114C4×C71#48112yes—
115C4×C71#48412yes—
116#8767—711nomaximal, Sylow
117C2×C71—21no—
118C2×C71—21no—
119C2×C71#182411yescharacteristic
120C2×C71#182711yescharacteristic
121C2×C71#183011yescharacteristic
122#2351D7111yescharacteristic
123#1801—711no—
124#1461—711no—
125#1303—712no—
126#1286—712no—
127C71#876711yescharacteristic, complemented, Sylow
128C2×C4×Q8—21no—
129C2×C4×Q8C2×D7111yescharacteristic
130#431—712no—
131#423—712no—
132C2×#101—1421no—
133#389—21no—
134#389—22no—
135#382Dic7112yes—
136#382C2×D7111yescharacteristic
137C4×C4⋊C4Dic7112yes—
138C4×C4⋊C4C2×D7111yescharacteristic
139#343—1421no—
140#324—1422no—
141C22×Q8C22⋊D7111yescharacteristic
142C2×M16—1422no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—44no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4C22⋊D7112yes—
151C2×C4⋊C4C22⋊D7111yescharacteristic
152C2×C42—22no—
153C2×C42—21no—
154C2×C42C2×Dic7112yes—
155C2×C42C22×D7111yescharacteristic
156C4⋊C8—1422no—
157C4⋊C8—1422no—
158C4⋊C8—1422no—
159#101—2842no—
160C22⋊C8—1422no—
161C4×C8—1422no—
162#95—22no—
163#95—21no—
164#95C2×Dic7111yescharacteristic
165#95C2×Dic7112yes—
166#95C2×Dic7111yescharacteristic
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8#1176585212yes—
170C2×Q8#1176585512yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4C22⋊Dic7112yes—
175C22×C4C22⋊Dic7112yes—
176C22×C4C22×Dic7111yescharacteristic
177C22×C4C2×C22⋊D7111yescharacteristic
178C22×C4C2×C22⋊D7111yescharacteristic
179M16—2842no—
180M16—2842no—
181C2×C8—2842no—
182C2×C8—2842no—
183C4⋊C4—24no—
184C4⋊C4—22no—
185C4⋊C4—42no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—42no—
189C42—22no—
190C42—22no—
191C42—44no—
192C42—42no—
193C42#1176586512yes—
194C42#1176586512yes—
195C23C2×C22⋊Dic7111yescharacteristic
196Q8—24no—
197Q8—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4#1455605214yes—
214C2×C4#1455608711yescharacteristic
215C2×C4#1455608711yescharacteristic
216C2×C4#1455611112yes—
217C2×C4#1455611112yes—
218C2×C4C2×#1176585211yescharacteristic
219C2×C4C2×#1176585511yescharacteristic
220C2×C4#1455615412yes—
221C8—5682no—
222C8—5682no—
223C22—21no—
224C22—21no—
225C22#1798586211yescharacteristic
226C22C2×#1455605211yescharacteristic
227C22#1798600411yescharacteristic
228C4—24no—
229C4—22no—
230C4—21no—
231C4—21no—
232C4—42no—
233C4—42no—
234C4—82no—
235C4—81no—
236C4#1798631712yes—
237C4#1798633412yes—
238C2—21no—
239C2—21no—
240C2#2258843011yescharacteristic
241C2#2258844411yescharacteristic
242C2#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#8767
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
127. C71#8767semidirect

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
#876771