Finite Group Databasebeta

← #31901519

Subgroups — #31901519

Order 18176.

408 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 81 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#22588493C211yesmaximal, characteristic, complemented
2#22587035C212yesmaximal, complemented
3#22586848C212yesmaximal, complemented
4#22586210C211yesmaximal, characteristic, complemented
5C71×#3245C211yesmaximal, characteristic
6#17986056C2212yes—
7C2×C4⋊(C71⋊C8)C2211yescharacteristic
8C2×Dic71⋊C8—22no—
9#17985845C2211yescharacteristic
10#17985748—22no—
11#17985743—22no—
12#17985733—24no—
13#532×C71C2212yes—
14#474×C71—22no—
15#466×C71—22no—
16#459×C71—22no—
17C2×C4⋊C8×C71C2211yescharacteristic
18C22×C71⋊C8—21no—
19C2×C4×Dic71—21no—
20C4⋊(C71⋊C8)—22no—
21C4⋊(C71⋊C8)—22no—
22Dic71⋊C8—44no—
23C22⋊(C71⋊C8)—22no—
24#14556039—22no—
25C71⋊#95—21no—
26C71⋊#95D412yes—
27C2×#41×C71—22no—
28C2×D8×C71—22no—
29C22×C8×C71D412yes—
30C4⋊Q8×C71D412yes—
31#127×C71D412yes—
32#124×C71—22no—
33#121×C71—22no—
34#121×C71—22no—
35C4×D4×C71—22no—
36C4×D4×C71—22no—
37C2×C42×C71C2311yescharacteristic
38#107×C71—22no—
39C4⋊C8×C71—22no—
40#102×C71—24no—
41C22×Dic71—22no—
42C2×C71⋊C8—42no—
43C2×C71⋊C8—41no—
44C2×C71⋊C8—42no—
45C4×Dic71—22no—
46C4×Dic71—22no—
47#41×C71—84no—
48C2×Q8×C71—22no—
49C2×D4×C71—22no—
50C2×D4×C71—22no—
51C2×D4×C71—22no—
52C2×D4×C71—42no—
53C22×C4×C71—42no—
54C22×C4×C71C2×D411yescharacteristic
55C22×C4×C71C2×D411yescharacteristic
56C22×C4×C71#4111yescharacteristic
57D8×C71—82no—
58C2×C8×C71—22no—
59C2×C8×C71—42no—
60C4⋊C4×C71—22no—
61C4⋊C4×C71—22no—
62C22⋊C4×C71—42no—
63C22⋊C4×C71—42no—
64C42×C71C2×D412yes—
65C42×C71#4112yes—
66C23×C71—42no—
67C23×C71#12111yescharacteristic
68Q8×C71—82no—
69D4×C71—42no—
70D4×C71—42no—
71D4×C71—82no—
72D4×C71—82no—
73D4×C71—84no—
74C2×C4×C71—22no—
75C2×C4×C71—21no—
76C2×C4×C71—21no—
77C2×C4×C71—41no—
78C2×C4×C71—42no—
79C2×C4×C71—84no—
80C2×C4×C71#12011yescharacteristic
81C2×C4×C71#12112yes—
82C2×C4×C71#12312yes—
83C2×C4×C71#12411yescharacteristic
84C2×Dic71—24no—
85C2×Dic71—42no—
86C2×Dic71—42no—
87C8×C71—42no—
88C71⋊C8—82no—
89C71⋊C8—81no—
90C22×C71—21no—
91C22×C71—21no—
92C22×C71—82no—
93C22×C71—82no—
94C22×C71—82no—
95C22×C71#39411yescharacteristic
96C22×C71#47212yes—
97C4×C71—22no—
98C4×C71—21no—
99C4×C71—22no—
100C4×C71—82no—
101C4×C71#45311yescharacteristic
102C4×C71#45611yescharacteristic
103Dic71—44no—
104#10512—711nomaximal, Sylow
105C2×C71—22no—
106C2×C71—162no—
107C2×C71#203911yescharacteristic
108C2×C71#204511yescharacteristic
109C2×C71#205311yescharacteristic
110#3245D7111yescharacteristic
111#1843—711no—
112#1551—712no—
113#1505—712no—
114#1290—711no—
115C71#1051211yescharacteristic, complemented, Sylow
116#532C2×D7112yes—
117#474—22no—
118#466—22no—
119#459—22no—
120#432—712no—
121C2×C4⋊C8—711no—
122C2×C4⋊C8—1422no—
123C2×C4⋊C8C2×D7111yescharacteristic
124#376—711no—
125#333—1422no—
126#331—1422no—
127#327—1424no—
128C2×#41—22no—
129C2×D8—22no—
130C22×C8—1421no—
131C22×C8C22⋊D7112yes—
132C4⋊Q8C22⋊D7112yes—
133#127C22⋊D7112yes—
134#124—22no—
135#121—22no—
136#121—22no—
137C4×D4—22no—
138C4×D4—22no—
139C2×C42—1421no—
140C2×C42C22×D7111yescharacteristic
141#107—22no—
142C4⋊C8—22no—
143C4⋊C8—2844no—
144C4⋊C8—1422no—
145C4⋊C8—1422no—
146#102—24no—
147C22⋊C8—1422no—
148#97—1422no—
149#95—712no—
150#95—1421no—
151#41—84no—
152C2×Q8—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—42no—
157C22×C4—42no—
158C22×C4—1422no—
159C22×C4D4×D7111yescharacteristic
160C22×C4C2×C22⋊D7111yescharacteristic
161C22×C4#1176586411yescharacteristic
162D8—82no—
163C2×C8—22no—
164C2×C8—42no—
165C2×C8—2842no—
166C2×C8—2841no—
167C2×C8—2842no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C22⋊C4—42no—
171C22⋊C4—42no—
172C42—1422no—
173C42—1422no—
174C42C2×C22⋊D7112yes—
175C42#1176586412yes—
176C23—42no—
177C23#1455609911yescharacteristic
178Q8—82no—
179D4—42no—
180D4—42no—
181D4—82no—
182D4—82no—
183D4—84no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—84no—
190C2×C4—2842no—
191C2×C4—2842no—
192C2×C4—1424no—
193C2×C4#1455609711yescharacteristic
194C2×C4#1455610012yes—
195C2×C4#1455610912yes—
196C2×C4#1455611511yescharacteristic
197C8—42no—
198C8—5682no—
199C8—5681no—
200C22—21no—
201C22—21no—
202C22—82no—
203C22—82no—
204C22—82no—
205C22#1798591311yescharacteristic
206C22#1798626612yes—
207C4—21no—
208C4—22no—
209C4—22no—
210C4—82no—
211C4—2844no—
212C4#1798616711yescharacteristic
213C4#1798618211yescharacteristic
214C2—22no—
215C2—162no—
216C2#2258956111yescharacteristic
217C2#2258959311yescharacteristic
218C2#2258964311yescharacteristic
subgroupG/H
centerC22#17985913
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
Fitting, prime radicalC71×#3245C2
socleC22×C71#394
hypercenter#3245D71
hypocenterC71#10512
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #22588493C2semidirect
2. #22587035C2semidirect
3. #22586848C2semidirect
4. #22586210C2semidirect
115. C71#10512semidirect

Maximal subgroups

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

subgroupindex
#225884932
#225870352
#225868482
#225862102
C71×#32452
#1051271