Finite Group Databasebeta

← #31900595

Subgroups — #31900595

Order 18176.

364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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#22591645C211yesmaximal, characteristic
2#22586839C212yesmaximal
3#22586241C212yesmaximal
4#22586169C211yesmaximal, characteristic
5C71×#1995C211yesmaximal, characteristic
6#17986052C2212yes—
7#17985941—22no—
8#17985933—22no—
9C4×Dic71⋊C4—22no—
10C4×C4⋊Dic71C2212yes—
11#17985736—22no—
12#17985736—22no—
13#533×C71C2211yescharacteristic
14#423×C71C2211yescharacteristic
15#376×C71C2211yescharacteristic
16#339×C71—22no—
17C2×C4⋊Dic71—22no—
18C2×Dic71⋊C4—22no—
19C2×Dic71⋊C4—22no—
20C2×C4×Dic71—22no—
21C4⋊(C71⋊C8)—22no—
22C22⋊(C71⋊C8)—22no—
23C4×C71⋊C8—22no—
24C71⋊#95—22no—
25C71⋊#95—22no—
26C71⋊#95D412yes—
27C71⋊#95D412yes—
28C2×M16×C71—21no—
29C4⋊Q8×C71Q812yes—
30#125×C71—21no—
31C22⋊Q8×C71—22no—
32C4×Q8×C71—22no—
33#117×C71—21no—
34C2×C42×C71D412yes—
35C2×C42×C71C2311yescharacteristic
36C4⋊C8×C71—22no—
37#95×C71—22no—
38C22×Dic71—22no—
39C22×Dic71—22no—
40C2×C71⋊C8—42no—
41C4⋊Dic71—44no—
42Dic71⋊C4—44no—
43Dic71⋊C4—44no—
44C4×Dic71—42no—
45C4×Dic71—42no—
46C2×Q8×C71—22no—
47C22×C4×C71—22no—
48C22×C4×C71C2×D412yes—
49C22×C4×C71C2×D411yescharacteristic
50M16×C71—42no—
51C2×C8×C71—41no—
52C4⋊C4×C71—22no—
53C4⋊C4×C71—22no—
54C4⋊C4×C71—42no—
55C22⋊C4×C71—41no—
56C42×C71—22no—
57C42×C71—22no—
58C42×C71—41no—
59C42×C71C2×Q811yescharacteristic
60C42×C71#4112yes—
61C42×C71#4111yescharacteristic
62C23×C71#12011yescharacteristic
63Q8×C71—82no—
64C2×C4×C71—42no—
65C2×C4×C71—42no—
66C2×C4×C71—42no—
67C2×C4×C71—42no—
68C2×C4×C71—42no—
69C2×C4×C71C22⋊Q812yes—
70C2×C4×C71C22⋊Q811yescharacteristic
71C2×C4×C71#12412yes—
72C2×C4×C71#12411yescharacteristic
73C2×C4×C71#13512yes—
74C2×C4×C71#13712yes—
75C2×Dic71—24no—
76C2×Dic71—42no—
77C2×Dic71—42no—
78C2×Dic71—42no—
79C2×Dic71—42no—
80C2×Dic71—42no—
81C8×C71—81no—
82C71⋊C8—82no—
83C22×C71—22no—
84C22×C71#39311yescharacteristic
85C22×C71#45212yes—
86C4×C71—22no—
87C4×C71—22no—
88C4×C71—82no—
89C4×C71—82no—
90C4×C71#47912yes—
91C4×C71#48811yescharacteristic
92C4×C71#49111yescharacteristic
93Dic71—44no—
94Dic71—82no—
95#10373—711nomaximal, Sylow
96C2×C71—22no—
97C2×C71#202611yescharacteristic
98C2×C71#203112yes—
99#2402—711no—
100#1995D7111yescharacteristic
101#1503—712no—
102#1303—712no—
103#1274—711no—
104C71#1037311yescharacteristic, complemented, Sylow
105#533C2×D7111yescharacteristic
106#431—712no—
107#423C2×D7111yescharacteristic
108#400—1422no—
109#398—1422no—
110C4×C4⋊C4—712no—
111C4×C4⋊C4—1422no—
112#376C2×D7111yescharacteristic
113#339—22no—
114#328—1422no—
115#328—1422no—
116C2×M16—21no—
117C4⋊Q8Dic14212yes—
118#125—21no—
119C22⋊Q8—22no—
120C4×Q8—22no—
121#117—21no—
122C2×C4⋊C4—1422no—
123C2×C4⋊C4—1422no—
124C2×C4⋊C4—1422no—
125C2×C42—1422no—
126C2×C42C22⋊D7112yes—
127C2×C42C22×D7111yescharacteristic
128C4⋊C8—22no—
129C4⋊C8—1422no—
130C22⋊C8—1422no—
131C4×C8—1422no—
132#95—22no—
133#95—712no—
134#95—712no—
135#95—1422no—
136#95—1422no—
137C2×Q8—22no—
138C22×C4—22no—
139C22×C4—1422no—
140C22×C4—1422no—
141C22×C4D4×D7112yes—
142C22×C4C2×C22⋊D7111yescharacteristic
143M16—42no—
144C2×C8—41no—
145C2×C8—2842no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—42no—
149C4⋊C4—2844no—
150C4⋊C4—2844no—
151C4⋊C4—2844no—
152C22⋊C4—41no—
153C42—22no—
154C42—22no—
155C42—41no—
156C42—2842no—
157C42—2842no—
158C42C2×Dic14211yescharacteristic
159C42#1176586412yes—
160C42#1176586511yescharacteristic
161C23#1455609611yescharacteristic
162Q8—82no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—2842no—
169C2×C4—2842no—
170C2×C4—2842no—
171C2×C4—2842no—
172C2×C4—2842no—
173C2×C4—1424no—
174C2×C4#1455610512yes—
175C2×C4#1455610711yescharacteristic
176C2×C4#1455611412yes—
177C2×C4#1455611511yescharacteristic
178C2×C4#1455614812yes—
179C2×C4#1455615912yes—
180C8—81no—
181C8—5682no—
182C22—22no—
183C22#1798591111yescharacteristic
184C22#1798615912yes—
185C4—22no—
186C4—22no—
187C4—82no—
188C4—82no—
189C4—2844no—
190C4—5682no—
191C4#1798630612yes—
192C4#1798635311yescharacteristic
193C4#1798636711yescharacteristic
194C2—22no—
195C2#2258949111yescharacteristic
196C2#2258951012yes—
subgroupG/H
centerC2×C4#14556115
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1995C2
socleC22×C71#393
prime radicalC23×C71#120
hypercenter#1995D71
hypocenterC71#10373
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1995

Complements

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

NH ≅ G/Ntype
104. C71#10373semidirect

Maximal subgroups

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

subgroupindex
#225916452
#225868392
#225862412
#225861692
C71×#19952
#1037371