Finite Group Databasebeta

← #31900596

Subgroups — #31900596

Order 18176.

364 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 89 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#22589321C211yesmaximal, characteristic
3#22586840C212yesmaximal
4#22586242C212yesmaximal
5C71×#1274C211yesmaximal, characteristic
6#17986556C2211yescharacteristic
7#17986023C2211yescharacteristic
8#17985941—22no—
9#17985933—22no—
10C4×Dic71⋊C4—22no—
11C4×C4⋊Dic71C2212yes—
12#17985759—22no—
13#17985737—22no—
14#17985737—22no—
15#431×C71C2212yes—
16#376×C71C2211yescharacteristic
17C2×#11765849—21no—
18#14556128Q812yes—
19#14556120—21no—
20#14556107—22no—
21C4×Dic142—22no—
22#14556087—21no—
23C2×C4⋊Dic71—22no—
24C2×Dic71⋊C4—22no—
25C2×Dic71⋊C4—22no—
26C2×C4×Dic71—22no—
27C4⋊(C71⋊C8)—22no—
28C71⋊#95—22no—
29C71⋊#95—22no—
30C71⋊#95D412yes—
31C71⋊#95D412yes—
32C2×C42×C71D412yes—
33C2×C42×C71C2311yescharacteristic
34C4⋊C8×C71—22no—
35C22⋊C8×C71—22no—
36C4×C8×C71—22no—
37#95×C71—22no—
38C2×Dic142—22no—
39C22×Dic71—22no—
40C22×Dic71—22no—
41#11765849—42no—
42C2×C71⋊C8—41no—
43C4⋊Dic71—22no—
44C4⋊Dic71—22no—
45C4⋊Dic71—44no—
46Dic71⋊C4—44no—
47Dic71⋊C4—44no—
48Dic71⋊C4—42no—
49C22⋊Dic71—41no—
50C4×Dic71—41no—
51C4×Dic71—42no—
52C4×Dic71—42no—
53C22×C4×C71—22no—
54C22×C4×C71C2×D412yes—
55C22×C4×C71C2×D411yescharacteristic
56C2×C8×C71—42no—
57C42×C71—22no—
58C42×C71—22no—
59C42×C71C2×Q811yescharacteristic
60C42×C71#4111yescharacteristic
61C42×C71#4112yes—
62C23×C71#12011yescharacteristic
63C2×C4×C71—42no—
64C2×C4×C71—42no—
65C2×C4×C71—42no—
66C2×C4×C71—42no—
67C2×C4×C71C22⋊Q812yes—
68C2×C4×C71C22⋊Q811yescharacteristic
69C2×C4×C71#12412yes—
70C2×C4×C71#12411yescharacteristic
71C2×C4×C71#13512yes—
72C2×C4×C71#13712yes—
73C2×Dic71—24no—
74C2×Dic71—42no—
75C2×Dic71—42no—
76C2×Dic71—42no—
77C2×Dic71—42no—
78C2×Dic71—42no—
79C2×Dic71—42no—
80Dic142—82no—
81C8×C71—82no—
82C71⋊C8—81no—
83C22×C71—22no—
84C22×C71#39311yescharacteristic
85C22×C71#45212yes—
86C4×C71—22no—
87C4×C71—22no—
88C4×C71—82no—
89C4×C71#47912yes—
90C4×C71#48811yescharacteristic
91C4×C71#49111yescharacteristic
92Dic71—44no—
93Dic71—82no—
94Dic71—82no—
95#10373—711nomaximal, Sylow
96C2×C71—22no—
97C2×C71#202611yescharacteristic
98C2×C71#203112yes—
99#2402—711no—
100#1995—711no—
101#1503—712no—
102#1303—712no—
103#1274D7111yescharacteristic
104C71#1037311yescharacteristic, complemented, Sylow
105#533—711no—
106#431C2×D7112yes—
107#423—711no—
108#400—1422no—
109#398—1422no—
110C4×C4⋊C4—712no—
111C4×C4⋊C4—1422no—
112#376C2×D7111yescharacteristic
113#339—1422no—
114#328—1422no—
115#328—1422no—
116C2×M16—1421no—
117C4⋊Q8—712no—
118#125—1421no—
119C22⋊Q8—1422no—
120C4×Q8—1422no—
121#117—1421no—
122C2×C4⋊C4—1422no—
123C2×C4⋊C4—1422no—
124C2×C4⋊C4—1422no—
125C2×C42—1422no—
126C2×C42D28412yes—
127C2×C42C22×D7111yescharacteristic
128C4⋊C8—22no—
129C4⋊C8—1422no—
130C22⋊C8—22no—
131C4×C8—22no—
132#95—22no—
133#95—712no—
134#95—712no—
135#95—1422no—
136#95—1422no—
137C2×Q8—1422no—
138C22×C4—22no—
139C22×C4—1422no—
140C22×C4—1422no—
141C22×C4D4×D7112yes—
142C22×C4C2×D28411yescharacteristic
143M16—2842no—
144C2×C8—42no—
145C2×C8—2841no—
146C4⋊C4—2844no—
147C4⋊C4—2844no—
148C4⋊C4—2844no—
149C4⋊C4—2842no—
150C4⋊C4—1422no—
151C4⋊C4—1422no—
152C22⋊C4—2841no—
153C42—22no—
154C42—22no—
155C42—2842no—
156C42—2841no—
157C42—2842no—
158C42Q8×D7111yescharacteristic
159C42#1176586411yescharacteristic
160C42#1176586512yes—
161C23#1455609811yescharacteristic
162Q8—5682no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—2842no—
168C2×C4—2842no—
169C2×C4—2842no—
170C2×C4—2842no—
171C2×C4—2842no—
172C2×C4—2842no—
173C2×C4—1424no—
174C2×C4#1455610612yes—
175C2×C4#1455610811yescharacteristic
176C2×C4#1455611412yes—
177C2×C4#1455611611yescharacteristic
178C2×C4#1455615012yes—
179C2×C4#1455616112yes—
180C8—82no—
181C8—5681no—
182C22—22no—
183C22#1798591211yescharacteristic
184C22#1798616312yes—
185C4—22no—
186C4—22no—
187C4—82no—
188C4—2844no—
189C4—5682no—
190C4—5682no—
191C4#1798630712yes—
192C4#1798635411yescharacteristic
193C4#1798636811yescharacteristic
194C2—22no—
195C2#2258949211yescharacteristic
196C2#2258951112yes—
subgroupG/H
centerC2×C4#14556116
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1274C2
socleC22×C71#393
prime radicalC23×C71#120
hypercenter#1274D71
hypocenterC71#10373
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

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
#225893212
#225868402
#225862422
C71×#12742
#1037371