Finite Group Databasebeta

← #31899743

Subgroups — #31899743

Order 18176.

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

normal subgroups only

This group has 466 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#22599484C212yesmaximal
2#22598864C211yesmaximal, characteristic
3#22586903C212yesmaximal
4#22586185C211yesmaximal, characteristic
5C71×#1960C211yesmaximal, characteristic
6#17986556C2212yes—
7#17986408—22no—
8#17986408—22no—
9#17986395—22no—
10#17986351—22no—
11#17986329—24no—
12#17986281—24no—
13#17986052C2212yes—
14#17986037—22no—
15#17985735—24no—
16C2×C4⋊Q8×C71C2211yescharacteristic
17C2×#103×C71—22no—
18C2×#97×C71C2211yescharacteristic
19#382×C71C2211yescharacteristic
20#14556128D412yes—
21#14556128D412yes—
22#14556120—22no—
23#14556107—22no—
24#14556107—22no—
25C4×Dic142—22no—
26C4×Dic142—22no—
27#14556087—22no—
28#14556062—24no—
29#14556060—22no—
30#14556060—22no—
31#14556059—28no—
32C4⋊(C71⋊C8)—22no—
33#14556051—28no—
34C22⋊(C71⋊C8)D414yes—
35C4×C71⋊C8—22no—
36C22×Q8×C71—21no—
37C22×C8×C71D412yes—
38C4⋊Q8×C71—22no—
39C4⋊Q8×C71—21no—
40C4⋊Q8×C71D412yes—
41C2×C4⋊C4×C71—22no—
42C2×C4⋊C4×C71—21no—
43C2×C42×C71C2311yescharacteristic
44#103×C71—44no—
45#97×C71—22no—
46#95×C71—22no—
47C2×Dic142—22no—
48C2×Dic142—22no—
49C2×C71⋊C8—24no—
50C4⋊Dic71—22no—
51C4⋊Dic71—22no—
52C4⋊Dic71—22no—
53C4⋊Dic71—22no—
54Dic71⋊C4—42no—
55Dic71⋊C4—42no—
56C22⋊Dic71—42no—
57C4×Dic71—42no—
58C2×Q8×C71—22no—
59C2×Q8×C71—41no—
60C2×Q8×C71—42no—
61C2×Q8×C71—42no—
62C22×C4×C71—41no—
63C22×C4×C71—41no—
64C22×C4×C71C2×D412yes—
65C22×C4×C71C2×D411yescharacteristic
66C2×C8×C71—22no—
67C2×C8×C71—22no—
68C2×C8×C71SD1614yes—
69C4⋊C4×C71—24no—
70C4⋊C4×C71—42no—
71C4⋊C4×C71—42no—
72C42×C71C2×D412yes—
73C42×C71C2×D411yescharacteristic
74C42×C71#4111yescharacteristic
75C23×C71#12711yescharacteristic
76Q8×C71—44no—
77Q8×C71—82no—
78C2×C4×C71—42no—
79C2×C4×C71—42no—
80C2×C4×C71—81no—
81C2×C4×C71—81no—
82C2×C4×C71—82no—
83C2×C4×C71—81no—
84C2×C4×C71#12011yescharacteristic
85C2×C4×C71#12012yes—
86C2×C4×C71#12112yes—
87C2×C4×C71#12111yescharacteristic
88C2×C4×C71C2×SD1612yes—
89C2×C4×C71#13712yes—
90C2×Dic71—42no—
91C2×Dic71—42no—
92Dic142—44no—
93Dic142—82no—
94C8×C71—24no—
95C71⋊C8—44no—
96C22×C71—22no—
97C22×C71#39211yescharacteristic
98C22×C71#49211yescharacteristic
99C22×C71#49711yescharacteristic
100C4×C71—22no—
101C4×C71—22no—
102C4×C71—82no—
103C4×C71—161no—
104C4×C71#44812yes—
105C4×C71#46112yes—
106Dic71—82no—
107Dic71—82no—
108#10227—711nomaximal, Sylow
109C2×C71—21no—
110C2×C71—21no—
111C2×C71#200912yes—
112C2×C71#201311yescharacteristic
113#3229—712no—
114#3162—711no—
115#1960D7111yescharacteristic
116#1518—712no—
117#1281—711no—
118C71#1022711yescharacteristic, complemented, Sylow
119#533—712no—
120C2×C4⋊Q8C2×D7111yescharacteristic
121#501—1422no—
122#501—1422no—
123#498—1422no—
124#487—1422no—
125#483—1424no—
126#475—1424no—
127#431—712no—
128#427—1422no—
129C2×#103—22no—
130C2×#97C2×D7111yescharacteristic
131#382C2×D7111yescharacteristic
132#328—1424no—
133C22×Q8—21no—
134C22×C8C22⋊D7112yes—
135C4⋊Q8—22no—
136C4⋊Q8—21no—
137C4⋊Q8—712no—
138C4⋊Q8—712no—
139C4⋊Q8D28412yes—
140#125—1422no—
141C22⋊Q8—1422no—
142C22⋊Q8—1422no—
143C4×Q8—1422no—
144C4×Q8—1422no—
145#117—1422no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C42C22×D7111yescharacteristic
149#107—1424no—
150#106—1428no—
151#106—1422no—
152#106—1422no—
153C4⋊C8—1422no—
154#103—44no—
155#103—1428no—
156C22⋊C8—714no—
157#97—22no—
158C4×C8—1422no—
159#95—22no—
160C2×Q8—22no—
161C2×Q8—41no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—1422no—
165C2×Q8—1422no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4D4×D7112yes—
169C22×C4C2×C22⋊D7111yescharacteristic
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—1424no—
173C2×C8#1176585314yes—
174C4⋊C4—24no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—2842no—
178C4⋊C4—2842no—
179C4⋊C4—1422no—
180C4⋊C4—1422no—
181C4⋊C4—1422no—
182C4⋊C4—1422no—
183C22⋊C4—2842no—
184C42—2842no—
185C42D4×D7112yes—
186C42C2×D28411yescharacteristic
187C42#1176586511yescharacteristic
188C23#1455612411yescharacteristic
189Q8—44no—
190Q8—82no—
191Q8—2844no—
192Q8—5682no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—82no—
198C2×C4—81no—
199C2×C4—2842no—
200C2×C4—2842no—
201C2×C4#1455609611yescharacteristic
202C2×C4#1455609812yes—
203C2×C4#1455610112yes—
204C2×C4#1455610211yescharacteristic
205C2×C4C2×#1176585312yes—
206C2×C4#1455615812yes—
207C8—24no—
208C8—2844no—
209C22—22no—
210C22#1798590811yescharacteristic
211C22#1798637111yescharacteristic
212C22#1798639411yescharacteristic
213C4—22no—
214C4—22no—
215C4—82no—
216C4—161no—
217C4—5682no—
218C4—5682no—
219C4#1798613712yes—
220C4#1798620412yes—
221C2—21no—
222C2—21no—
223C2#2258940212yes—
224C2#2258942211yescharacteristic
subgroupG/H
centerC22#17985908
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1960C2
socleC22×C71#392
prime radicalC23×C71#127
hypercenter#1960D71
hypocenterC71#10227
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
118. C71#10227semidirect

Maximal subgroups

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

subgroupindex
#225994842
#225988642
#225869032
#225861852
C71×#19602
#1022771