Finite Group Databasebeta

← #31899744

Subgroups — #31899744

Order 18176.

466 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 105 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#22599481C212yesmaximal
2#22589160C211yesmaximal, characteristic
3#22586904C212yesmaximal
4#22586185C211yesmaximal, characteristic
5C71×#3162C211yesmaximal, characteristic
6C2×#14556128C2211yescharacteristic
7#17986395—22no—
8#17986349—22no—
9#17986327—24no—
10#17986284—24no—
11#17986052C2212yes—
12C2×#14556053—22no—
13#17985865C2211yescharacteristic
14#17985736—24no—
15#533×C71C2212yes—
16#501×C71—22no—
17#501×C71—22no—
18#427×C71—22no—
19C2×#97×C71C2211yescharacteristic
20C22×Dic142—21no—
21#14556128—22no—
22#14556128—21no—
23#14556128D412yes—
24C2×C4⋊Dic71—22no—
25C2×C4⋊Dic71—21no—
26#14556059—28no—
27C4⋊(C71⋊C8)—22no—
28#14556053—44no—
29#14556052—28no—
30C22⋊(C71⋊C8)D414yes—
31C4×C71⋊C8—22no—
32C71⋊#95—22no—
33C22×C8×C71D412yes—
34C4⋊Q8×C71D412yes—
35C4⋊Q8×C71D412yes—
36#125×C71—22no—
37C22⋊Q8×C71—22no—
38C22⋊Q8×C71—22no—
39C4×Q8×C71—22no—
40C4×Q8×C71—22no—
41#117×C71—22no—
42C2×C42×C71C2311yescharacteristic
43#107×C71—24no—
44#106×C71—22no—
45#106×C71—22no—
46#97×C71—22no—
47C2×Dic142—22no—
48C2×Dic142—42no—
49C2×Dic142—42no—
50C2×Dic142—41no—
51C22×Dic71—41no—
52C22×Dic71—41no—
53C2×C71⋊C8—24no—
54C4⋊Dic71—24no—
55C4⋊Dic71—42no—
56C4⋊Dic71—42no—
57C2×Q8×C71—22no—
58C2×Q8×C71—22no—
59C22×C4×C71C2×D412yes—
60C22×C4×C71C2×D411yescharacteristic
61C2×C8×C71—22no—
62C2×C8×C71—22no—
63C2×C8×C71SD1614yes—
64C4⋊C4×C71—22no—
65C4⋊C4×C71—22no—
66C4⋊C4×C71—22no—
67C4⋊C4×C71—22no—
68C4⋊C4×C71—42no—
69C4⋊C4×C71—42no—
70C22⋊C4×C71—42no—
71C42×C71—42no—
72C42×C71C2×D411yescharacteristic
73C42×C71C2×D412yes—
74C42×C71#4111yescharacteristic
75C23×C71#12711yescharacteristic
76Q8×C71—44no—
77Q8×C71—82no—
78C2×C4×C71—42no—
79C2×C4×C71—42no—
80C2×C4×C71—42no—
81C2×C4×C71#12012yes—
82C2×C4×C71#12011yescharacteristic
83C2×C4×C71#12111yescharacteristic
84C2×C4×C71#12112yes—
85C2×C4×C71C2×SD1612yes—
86C2×C4×C71#13712yes—
87C2×Dic71—42no—
88C2×Dic71—81no—
89C2×Dic71—81no—
90C2×Dic71—82no—
91C2×Dic71—81no—
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—82no—
104C4×C71#44812yes—
105C4×C71#46112yes—
106Dic71—82no—
107Dic71—161no—
108#10227—711nomaximal, Sylow
109C2×C71—21no—
110C2×C71—21no—
111C2×C71#200912yes—
112C2×C71#201311yescharacteristic
113#3229—712no—
114#3162D7111yescharacteristic
115#1960—711no—
116#1518—712no—
117#1281—711no—
118C71#1022711yescharacteristic, complemented, Sylow
119#533C2×D7112yes—
120C2×C4⋊Q8—711no—
121#501—22no—
122#501—22no—
123#498—1422no—
124#487—1422no—
125#483—1424no—
126#475—1424no—
127#431—712no—
128#427—22no—
129C2×#103—1422no—
130C2×#97C2×D7111yescharacteristic
131#382—711no—
132#328—1424no—
133C22×Q8—1421no—
134C22×C8C22⋊D7112yes—
135C4⋊Q8—712no—
136C4⋊Q8—1422no—
137C4⋊Q8—1421no—
138C4⋊Q8C22⋊D7112yes—
139C4⋊Q8C22⋊D7112yes—
140#125—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145#117—22no—
146C2×C4⋊C4—1421no—
147C2×C4⋊C4—1422no—
148C2×C42C22×D7111yescharacteristic
149#107—24no—
150#106—22no—
151#106—22no—
152#106—1428no—
153C4⋊C8—1422no—
154#103—2844no—
155#103—1428no—
156C22⋊C8—714no—
157#97—22no—
158C4×C8—1422no—
159#95—1422no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—2841no—
163C2×Q8—2842no—
164C2×Q8—2842no—
165C2×Q8—1422no—
166C22×C4—2841no—
167C22×C4—2841no—
168C22×C4D4×D7112yes—
169C22×C4C2×C22⋊D7111yescharacteristic
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—1424no—
173C2×C8#1176585214yes—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—2842no—
181C4⋊C4—2842no—
182C4⋊C4—1424no—
183C22⋊C4—42no—
184C42—42no—
185C42D4×D7111yescharacteristic
186C42C2×C22⋊D7112yes—
187C42#1176586411yescharacteristic
188C23#1455612411yescharacteristic
189Q8—44no—
190Q8—82no—
191Q8—2844no—
192Q8—5682no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—2842no—
197C2×C4—5681no—
198C2×C4—5681no—
199C2×C4—5682no—
200C2×C4—5681no—
201C2×C4#1455609612yes—
202C2×C4#1455609711yescharacteristic
203C2×C4#1455609911yescharacteristic
204C2×C4#1455610012yes—
205C2×C4C2×#1176585212yes—
206C2×C4#1455615912yes—
207C8—24no—
208C8—2844no—
209C22—22no—
210C22#1798590711yescharacteristic
211C22#1798637311yescharacteristic
212C22#1798639111yescharacteristic
213C4—22no—
214C4—22no—
215C4—82no—
216C4—82no—
217C4—5682no—
218C4—11361no—
219C4#1798613812yes—
220C4#1798620112yes—
221C2—21no—
222C2—21no—
223C2#2258939712yes—
224C2#2258942311yescharacteristic
subgroupG/H
centerC22#17985907
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#3162C2
socleC22×C71#392
prime radicalC23×C71#127
hypercenter#3162D71
hypocenterC71#10227
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3162

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
#225994812
#225891602
#225869042
#225861852
C71×#31622
#1022771