Finite Group Databasebeta

← #31885038

Subgroups — #31885038

Order 18176.

226 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 41 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#22589486C211yesmaximal, characteristic
2#22586465C211yesmaximal, characteristic
3C71×#1318C211yesmaximal, characteristic
4C2×#14556120C411yescharacteristic, complemented
5C2×#14556049—21no—
6C2×#14556043—21no—
7#17985761—21no—
8#17985754C411yescharacteristic, complemented
9#390×C71C2211yescharacteristic
10#324×C71—21no—
11C22×C71⋊C8—21no—
12#14556120—42no—
13C2×C4⋊Dic71—21no—
14C2×Dic71⋊C4—41no—
15C2×C22⋊Dic71—41no—
16#14556049—42no—
17#14556043—22no—
18#14556043—41no—
19C22×D4×C71D411yescharacteristic
20C2×C4⋊C4×C71D411yescharacteristic
21C2×C22⋊C4×C71—41no—
22C2×C42×C71C2×C411yescharacteristic
23C22⋊C8×C71—41no—
24C22×Dic71—41no—
25C22×Dic71—81no—
26C2×C71⋊C8—41no—
27C2×C71⋊C8—41no—
28C2×C71⋊C8—41no—
29C4⋊Dic71—22no—
30C4⋊Dic71—41no—
31Dic71⋊C4—42no—
32Dic71⋊C4—42no—
33C22⋊Dic71—42no—
34C22⋊Dic71—81no—
35C24×C71—21no—
36C2×D4×C71—21no—
37C2×D4×C71—41no—
38C2×D4×C71—41no—
39C2×D4×C71SD1611yescharacteristic
40C2×D4×C71Q1611yescharacteristic
41C22×C4×C71—21no—
42C22×C4×C71—21no—
43C22×C4×C71C22⋊C411yescharacteristic
44C2×C8×C71—81no—
45C4⋊C4×C71—41no—
46C22⋊C4×C71—82no—
47C42×C71—41no—
48C23×C71—21no—
49C23×C71—21no—
50C23×C71—41no—
51C23×C71—41no—
52C23×C71—81no—
53C23×C71—81no—
54C23×C71#9911yescharacteristic
55D4×C71—41no—
56D4×C71—41no—
57D4×C71—81no—
58C2×C4×C71—21no—
59C2×C4×C71—21no—
60C2×C4×C71—41no—
61C2×C4×C71—41no—
62C2×C4×C71—41no—
63C2×C4×C71—41no—
64C2×C4×C71—81no—
65C2×C4×C71#10311yescharacteristic
66C2×C4×C71#10411yescharacteristic
67C2×Dic71—42no—
68C2×Dic71—81no—
69C2×Dic71—81no—
70C2×Dic71—82no—
71C2×Dic71—161no—
72C2×Dic71—161no—
73C8×C71—161no—
74C71⋊C8—81no—
75C22×C71—21no—
76C22×C71—21no—
77C22×C71—41no—
78C22×C71—41no—
79C22×C71—81no—
80C22×C71—81no—
81C22×C71—81no—
82C22×C71—81no—
83C22×C71—161no—
84C22×C71#32811yescharacteristic
85C22×C71AΣL(1,𝔽16)11yescharacteristic
86C22×C71#35411yescharacteristic
87C4×C71—21no—
88C4×C71—21no—
89C4×C71—81no—
90C4×C71—81no—
91Dic71—82no—
92Dic71—162no—
93#7428—711nomaximal, Sylow
94C2×C71—21no—
95C2×C71—21no—
96C2×C71—81no—
97C2×C71—81no—
98C2×C71#135111yescharacteristic
99C2×C71#141211yescharacteristic
100C2×C71#141411yescharacteristic
101#2025—711no—
102#1393—711no—
103#1318D7111yescharacteristic
104C71#742811yescharacteristic, complemented, Sylow
105C2×#125—711no—
106C2×#102—1421no—
107C2×#99—1421no—
108#390C2×D7111yescharacteristic
109#340—1421no—
110#336—711no—
111#324—21no—
112C22×D4D28411yescharacteristic
113C22×C8—1421no—
114#125—2842no—
115C2×C4⋊C4—2841no—
116C2×C4⋊C4—1421no—
117C2×C4⋊C4C22⋊D7111yescharacteristic
118C2×C22⋊C4—41no—
119C2×C22⋊C4—2841no—
120C2×C42C4×D7111yescharacteristic
121#102—2842no—
122#99—2841no—
123#99—1422no—
124C22⋊C8—41no—
125C24—21no—
126C2×D4—21no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4#1176585411yescharacteristic
130C2×D4Dic28411yescharacteristic
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—2841no—
134C22×C4—5681no—
135C22×C4(C2×D71)⋊C411yescharacteristic
136C2×C8—81no—
137C2×C8—2841no—
138C2×C8—2841no—
139C2×C8—2841no—
140C4⋊C4—41no—
141C4⋊C4—2841no—
142C4⋊C4—2842no—
143C4⋊C4—2842no—
144C4⋊C4—1422no—
145C22⋊C4—82no—
146C22⋊C4—2842no—
147C22⋊C4—5681no—
148C42—41no—
149C23—21no—
150C23—21no—
151C23—41no—
152C23—41no—
153C23—81no—
154C23—81no—
155C23#1455604211yescharacteristic
156D4—41no—
157D4—41no—
158D4—81no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—81no—
166C2×C4—2842no—
167C2×C4—5681no—
168C2×C4—5681no—
169C2×C4—5682no—
170C2×C4—11361no—
171C2×C4—11361no—
172C2×C4#1455605311yescharacteristic
173C2×C4#1455605411yescharacteristic
174C8—161no—
175C8—5681no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22—161no—
185C22#1798573711yescharacteristic
186C22#1798578711yescharacteristic
187C22#1798579311yescharacteristic
188C4—21no—
189C4—21no—
190C4—81no—
191C4—81no—
192C4—5682no—
193C4—11362no—
194C2—21no—
195C2—21no—
196C2—81no—
197C2—81no—
198C2#2258636211yescharacteristic
199C2#2258651011yescharacteristic
200C2#2258651611yescharacteristic
subgroupG/H
centerC22#17985793
commutatorC2×C42×C71C2×C4
Frattini#390C2×D71
FittingC71×#1318C2
socleC22×C71#354
prime radicalC22×D4×C71D4
cosocle#390×C71C22
hypercenter#1318D71
hypocenterC71#7428
LCSG ▷ C2×C42×C71 ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #1318

Complements

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

NH ≅ G/Ntype
4. C2×#14556120C4semidirect
8. #17985754C4semidirect
104. C71#7428semidirect

Maximal subgroups

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

subgroupindex
#225894862
#225864652
C71×#13182
#742871