Finite Group Databasebeta

← #31898447

Subgroups — #31898447

Order 18176.

476 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 103 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#22598864C211yesmaximal, characteristic
2#22598379C212yesmaximal
3#22586742C212yesmaximal
4#22586171C211yesmaximal, characteristic
5C71×#1985C211yesmaximal, characteristic
6#17986556C2212yes—
7#17986408—22no—
8#17986408—22no—
9#17986334—22no—
10#17986294—22no—
11#17986209—22no—
12#17986155—22no—
13C4×#11765855—24no—
14#17986052C2212yes—
15#17986037—22no—
16#17985729—24no—
17C2×C4×Q8×C71C2211yescharacteristic
18C2×C4⋊C8×C71C2211yescharacteristic
19C2×#97×C71C2211yescharacteristic
20#336×C71—22no—
21C2×#11765855—24no—
22#14556128D412yes—
23#14556128D412yes—
24#14556120—22no—
25#14556107—22no—
26#14556107—22no—
27C4×Dic142—22no—
28C4×Dic142—22no—
29#14556087—22no—
30#14556062—22no—
31#14556062—22no—
32#14556061—24no—
33#14556060—24no—
34C4⋊(C71⋊C8)—22no—
35#14556052—24no—
36#14556051—24no—
37C22⋊(C71⋊C8)—22no—
38C4×C71⋊C8D414yes—
39C22×Q8×C71D411yescharacteristic
40C22×C8×C71—21no—
41C22×C8×C71D412yes—
42C4×Q8×C71—22no—
43C4×Q8×C71—22no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71D411yescharacteristic
46C2×C42×C71—21no—
47C2×C42×C71C2311yescharacteristic
48C4⋊C8×C71—22no—
49#97×C71—22no—
50C2×Dic142—22no—
51C2×Dic142—22no—
52#11765855—48no—
53C2×C71⋊C8—24no—
54C4⋊Dic71—22no—
55C4⋊Dic71—22no—
56C4⋊Dic71—22no—
57C4⋊Dic71—22no—
58Dic71⋊C4—42no—
59Dic71⋊C4—42no—
60C22⋊Dic71—42no—
61C4×Dic71—42no—
62C2×Q8×C71—22no—
63C2×Q8×C71—42no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71C2×D411yescharacteristic
67C22×C4×C71C2×D411yescharacteristic
68C22×C4×C71#4111yescharacteristic
69C2×C8×C71—22no—
70C2×C8×C71—22no—
71C2×C8×C71—41no—
72C2×C8×C71—42no—
73C2×C8×C71Q1614yescomplemented
74C4⋊C4×C71—22no—
75C4⋊C4×C71—42no—
76C42×C71—41no—
77C42×C71—41no—
78C42×C71C2×D412yes—
79C42×C71C2×D412yes—
80C23×C71#12111yescharacteristic
81Q8×C71—24no—
82Q8×C71—42no—
83C2×C4×C71—22no—
84C2×C4×C71—41no—
85C2×C4×C71—41no—
86C2×C4×C71—41no—
87C2×C4×C71—41no—
88C2×C4×C71—42no—
89C2×C4×C71—41no—
90C2×C4×C71#12011yescharacteristic
91C2×C4×C71#12012yes—
92C2×C4×C71#12112yes—
93C2×C4×C71#12711yescharacteristic
94C2×C4×C71C2×Q1612yes—
95C2×C4×C71#13712yes—
96C2×Dic71—42no—
97C2×Dic71—42no—
98Dic142—44no—
99Dic142—44no—
100C8×C71—24no—
101C8×C71—81no—
102C71⋊C8—44no—
103C22×C71—21no—
104C22×C71—21no—
105C22×C71#39211yescharacteristic
106C22×C71#46212yes—
107C4×C71—22no—
108C4×C71—22no—
109C4×C71—42no—
110C4×C71—81no—
111C4×C71#45112yes—
112C4×C71#49411yescharacteristic
113C4×C71#49711yescharacteristic
114Dic71—82no—
115Dic71—82no—
116#10037—711nomaximal, Sylow
117C2×C71—22no—
118C2×C71#201011yescharacteristic
119C2×C71#201611yescharacteristic
120C2×C71#201711yescharacteristic
121#3162—711no—
122#3104—712no—
123#1985D7111yescharacteristic
124#1481—712no—
125#1275—711no—
126C71#1003711yescharacteristic, complemented, Sylow
127#533—712no—
128C2×C4×Q8C2×D7111yescharacteristic
129#501—1422no—
130#501—1422no—
131#484—1422no—
132#477—1422no—
133#462—1422no—
134#451—1422no—
135C4×Q16—1424no—
136#431—712no—
137#427—1422no—
138C2×C4⋊C8C2×D7111yescharacteristic
139C2×#97C2×D7111yescharacteristic
140#336—22no—
141Q8⋊C8—1424no—
142C22×Q8D28411yescharacteristic
143C2×Q16—1424no—
144C22×C8—21no—
145C22×C8C22⋊D7112yes—
146C4⋊Q8—712no—
147C4⋊Q8—712no—
148#125—1422no—
149C22⋊Q8—1422no—
150C22⋊Q8—1422no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—1422no—
154C4×Q8—1422no—
155#117—1422no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4D28411yescharacteristic
158C2×C42—21no—
159C2×C42C22×D7111yescharacteristic
160#107—1424no—
161#107—1422no—
162#107—1422no—
163#106—1424no—
164C4⋊C8—22no—
165C4⋊C8—1422no—
166#103—1424no—
167#103—1424no—
168C22⋊C8—1422no—
169#97—22no—
170C4×C8—714no—
171C2×Q8—22no—
172C2×Q8—42no—
173C2×Q8—1422no—
174C2×Q8—1422no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4C2×C22⋊D7111yescharacteristic
178C22×C4C2×D28411yescharacteristic
179C22×C4#1176586511yescharacteristic
180Q16—2848no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—41no—
184C2×C8—42no—
185C2×C8—1424no—
186C2×C8#1176585514yescomplemented
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4—2842no—
190C4⋊C4—2842no—
191C4⋊C4—1422no—
192C4⋊C4—1422no—
193C4⋊C4—1422no—
194C4⋊C4—1422no—
195C22⋊C4—2842no—
196C42—41no—
197C42—41no—
198C42—2842no—
199C42D4×D7112yes—
200C42D4×D7112yes—
201C23#1455610211yescharacteristic
202Q8—24no—
203Q8—42no—
204Q8—2844no—
205Q8—2844no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—2842no—
214C2×C4—2842no—
215C2×C4#1455609611yescharacteristic
216C2×C4#1455609812yes—
217C2×C4#1455610112yes—
218C2×C4#1455612411yescharacteristic
219C2×C4C2×#1176585512yes—
220C2×C4#1455615812yes—
221C8—24no—
222C8—81no—
223C8—2844no—
224C22—21no—
225C22—21no—
226C22#1798590811yescharacteristic
227C22#1798620912yes—
228C4—22no—
229C4—22no—
230C4—42no—
231C4—81no—
232C4—5682no—
233C4—5682no—
234C4#1798615412yes—
235C4#1798637811yescharacteristic
236C4#1798639411yescharacteristic
237C2—22no—
238C2#2258940711yescharacteristic
239C2#2258943911yescharacteristic
240C2#2258944411yescharacteristic
subgroupG/H
centerC2×C4#14556124
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1985C2
socleC22×C71#392
prime radicalC23×C71#121
hypercenter#1985D71
hypocenterC71#10037
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1985

Complements

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

NH ≅ G/Ntype
73. C2×C8×C71Q16semidirect
126. C71#10037semidirect
186. C2×C8#11765855semidirect

Maximal subgroups

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

subgroupindex
#225988642
#225983792
#225867422
#225861712
C71×#19852
#1003771