Finite Group Databasebeta

← #31900586

Subgroups — #31900586

Order 18176.

416 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 97 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#22591633C211yesmaximal, characteristic
2#22589321C211yesmaximal, characteristic
3#22586840C212yesmaximal
4#22586242C212yesmaximal
5C71×#1274C211yesmaximal, characteristic
6#17986556C2211yescharacteristic
7C2×C4×Dic142—21no—
8#17986023C2211yescharacteristic
9#17985931—22no—
10#17985920—22no—
11C4×C4⋊Dic71—21no—
12C4×C4⋊Dic71C2212yes—
13#17985759—21no—
14#17985759—21no—
15#17985737—22no—
16#17985737—22no—
17#431×C71C2212yes—
18#376×C71C2211yescharacteristic
19C22×Dic142—21no—
20C2×#11765849—21no—
21#14556128Q812yescomplemented
22#14556120—21no—
23#14556107—22no—
24C4×Dic142—22no—
25C4×Dic142—42no—
26C4×Dic142—42no—
27#14556087—21no—
28C2×C4⋊Dic71—22no—
29C2×C4⋊Dic71—21no—
30C2×C4⋊Dic71—22no—
31C2×Dic71⋊C4—22no—
32C2×C4×Dic71—22no—
33C4⋊(C71⋊C8)—22no—
34C71⋊#95—21no—
35C71⋊#95—21no—
36C71⋊#95D412yes—
37C71⋊#95D412yes—
38C2×C42×C71D411yescharacteristic
39C2×C42×C71D411yescharacteristic
40C2×C42×C71C2311yescharacteristic
41C4⋊C8×C71—22no—
42C22⋊C8×C71—22no—
43C4×C8×C71—22no—
44#95×C71—22no—
45C2×Dic142—22no—
46C2×Dic142—44no—
47C2×Dic142—42no—
48C22×Dic71—22no—
49C22×Dic71—22no—
50#11765849—41no—
51#11765849—41no—
52C2×C71⋊C8—41no—
53C4⋊Dic71—22no—
54C4⋊Dic71—22no—
55C4⋊Dic71—44no—
56C4⋊Dic71—44no—
57C4⋊Dic71—42no—
58Dic71⋊C4—44no—
59Dic71⋊C4—42no—
60C22⋊Dic71—41no—
61C4×Dic71—41no—
62C4×Dic71—42no—
63C4×Dic71—42no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71C2×D412yes—
67C22×C4×C71C2×D411yescharacteristic
68C2×C8×C71—42no—
69C42×C71—21no—
70C42×C71—21no—
71C42×C71SD1612yes—
72C42×C71Q1612yes—
73C42×C71C2×Q811yescharacteristic
74C42×C71#4111yescharacteristic
75C42×C71#4112yes—
76C23×C71#12011yescharacteristic
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—42no—
80C2×C4×C71—42no—
81C2×C4×C71—42no—
82C2×C4×C71C22⋊Q812yes—
83C2×C4×C71C22⋊Q811yescharacteristic
84C2×C4×C71#12412yes—
85C2×C4×C71#12411yescharacteristic
86C2×C4×C71C2×SD1611yescharacteristic
87C2×C4×C71C2×Q1611yescharacteristic
88C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C71#13711yescharacteristic
90C2×Dic71—24no—
91C2×Dic71—42no—
92C2×Dic71—42no—
93C2×Dic71—42no—
94C2×Dic71—42no—
95C2×Dic71—42no—
96C2×Dic71—42no—
97Dic142—48no—
98Dic142—82no—
99C8×C71—82no—
100C71⋊C8—81no—
101C22×C71—22no—
102C22×C71#39311yescharacteristic
103C22×C71#45011yescharacteristic
104C22×C71#45111yescharacteristic
105C4×C71—22no—
106C4×C71—22no—
107C4×C71—42no—
108C4×C71—42no—
109C4×C71#47511yescharacteristic
110C4×C71#47711yescharacteristic
111C4×C71#48711yescharacteristic
112C4×C71#48911yescharacteristic
113Dic71—44no—
114Dic71—82no—
115Dic71—82no—
116#10371—711nomaximal, Sylow
117C2×C71—21no—
118C2×C71—21no—
119C2×C71#202411yescharacteristic
120C2×C71#203112yes—
121#2399—711no—
122#1995—711no—
123#1503—712no—
124#1303—712no—
125#1274D7111yescharacteristic
126C71#1037111yescharacteristic, complemented, Sylow
127#533—711no—
128C2×C4×Q8—1421no—
129#431C2×D7112yes—
130#423—711no—
131#398—1422no—
132#395—1422no—
133C4×C4⋊C4—712no—
134C4×C4⋊C4—1421no—
135#376C2×D7111yescharacteristic
136#339—1421no—
137#339—1421no—
138#328—1422no—
139#328—1422no—
140C22×Q8—1421no—
141C2×M16—1421no—
142C4⋊Q8—712no—
143#125—1421no—
144C22⋊Q8—1422no—
145C4×Q8—2842no—
146C4×Q8—2842no—
147C4×Q8—1422no—
148#117—1421no—
149C2×C4⋊C4—1422no—
150C2×C4⋊C4—1421no—
151C2×C4⋊C4—1422no—
152C2×C4⋊C4—1422no—
153C2×C42—1422no—
154C2×C42D28411yescharacteristic
155C2×C42D28411yescharacteristic
156C2×C42C22×D7111yescharacteristic
157C4⋊C8—22no—
158C4⋊C8—1422no—
159C22⋊C8—22no—
160C4×C8—22no—
161#95—22no—
162#95—712no—
163#95—712no—
164#95—1421no—
165#95—1421no—
166C2×Q8—2842no—
167C2×Q8—2844no—
168C2×Q8—1422no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—1422no—
172C22×C4—1422no—
173C22×C4D4×D7112yes—
174C22×C4C2×D28411yescharacteristic
175M16—2841no—
176M16—2841no—
177C2×C8—42no—
178C2×C8—2841no—
179C4⋊C4—2844no—
180C4⋊C4—2842no—
181C4⋊C4—2844no—
182C4⋊C4—2844no—
183C4⋊C4—2842no—
184C4⋊C4—1422no—
185C4⋊C4—1422no—
186C22⋊C4—2841no—
187C42—21no—
188C42—21no—
189C42—2841no—
190C42—2842no—
191C42—2842no—
192C42#1176585412yes—
193C42Dic28412yes—
194C42Q8×D7111yescharacteristic
195C42#1176586411yescharacteristic
196C42#1176586512yes—
197C23#1455609811yescharacteristic
198Q8—2848no—
199Q8—5682no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—2842no—
206C2×C4—2842no—
207C2×C4—2842no—
208C2×C4—2842no—
209C2×C4—2842no—
210C2×C4—2842no—
211C2×C4—1424no—
212C2×C4#1455610612yes—
213C2×C4#1455610811yescharacteristic
214C2×C4#1455611412yes—
215C2×C4#1455611611yescharacteristic
216C2×C4C2×#1176585411yescharacteristic
217C2×C4C2×Dic28411yescharacteristic
218C2×C4#1455615611yescharacteristic
219C2×C4#1455616111yescharacteristic
220C8—82no—
221C8—5681no—
222C22—22no—
223C22#1798591211yescharacteristic
224C22#1798615211yescharacteristic
225C22#1798615711yescharacteristic
226C4—22no—
227C4—22no—
228C4—42no—
229C4—42no—
230C4—2844no—
231C4—5682no—
232C4—5682no—
233C4#1798628511yescharacteristic
234C4#1798629511yescharacteristic
235C4#1798635011yescharacteristic
236C4#1798635911yescharacteristic
237C2—21no—
238C2—21no—
239C2#2258948211yescharacteristic
240C2#2258951112yes—
subgroupG/H
centerC2×C4#14556116
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1274C2
socleC22×C71#393
prime radicalC23×C71#120
hypercenter#1274D71
hypocenterC71#10371
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
21. #14556128Q8semidirect
126. C71#10371semidirect

Maximal subgroups

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

subgroupindex
#225916332
#225893212
#225868402
#225862422
C71×#12742
#1037171