Finite Group Databasebeta

← #31885055

Subgroups — #31885055

Order 18176.

224 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 43 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#22586465C211yesmaximal, characteristic
2#22586272C211yesmaximal, characteristic
3C71×#2093C211yesmaximal, characteristic
4C2×#14556043—21no—
5#17985725—21no—
6C2×#107×C71—21no—
7C2×#102×C71—21no—
8#390×C71C2211yescharacteristic
9#384×C71C411yescharacteristic, complemented
10#336×C71C411yescharacteristic, complemented
11C2×C22⋊Dic71—41no—
12#14556043—22no—
13#14556043—41no—
14C22⋊(C71⋊C8)—41no—
15C22×D4×C71D411yescharacteristic
16C22×C8×C71—21no—
17C2×C4⋊C4×C71—21no—
18C2×C4⋊C4×C71—41no—
19C2×C4⋊C4×C71D411yescharacteristic
20C2×C22⋊C4×C71—41no—
21C2×C42×C71C2×C411yescharacteristic
22#107×C71—42no—
23#102×C71—42no—
24C22×Dic71—81no—
25C2×C71⋊C8—81no—
26C22⋊Dic71—42no—
27C22⋊Dic71—81no—
28C24×C71—21no—
29C2×D4×C71—21no—
30C2×D4×C71—41no—
31C2×D4×C71—41no—
32C2×D4×C71SD1611yescharacteristic
33C2×D4×C71Q1611yescharacteristic
34C22×C4×C71—21no—
35C22×C4×C71—21no—
36C22×C4×C71—41no—
37C22×C4×C71C22⋊C411yescharacteristic
38C2×C8×C71—22no—
39C2×C8×C71—41no—
40C2×C8×C71—41no—
41C4⋊C4×C71—41no—
42C4⋊C4×C71—41no—
43C4⋊C4×C71—41no—
44C4⋊C4×C71—82no—
45C22⋊C4×C71—82no—
46C42×C71—41no—
47C23×C71—21no—
48C23×C71—21no—
49C23×C71—41no—
50C23×C71—41no—
51C23×C71—81no—
52C23×C71—81no—
53C23×C71#9911yescharacteristic
54D4×C71—41no—
55D4×C71—41no—
56D4×C71—81no—
57C2×C4×C71—21no—
58C2×C4×C71—21no—
59C2×C4×C71—22no—
60C2×C4×C71—41no—
61C2×C4×C71—41no—
62C2×C4×C71—41no—
63C2×C4×C71—81no—
64C2×C4×C71—82no—
65C2×C4×C71—81no—
66C2×C4×C71#10311yescharacteristic
67C2×C4×C71#10411yescharacteristic
68C2×Dic71—82no—
69C2×Dic71—161no—
70C2×Dic71—161no—
71C8×C71—42no—
72C71⋊C8—161no—
73C22×C71—21no—
74C22×C71—21no—
75C22×C71—41no—
76C22×C71—41no—
77C22×C71—81no—
78C22×C71—81no—
79C22×C71—81no—
80C22×C71—81no—
81C22×C71—161no—
82C22×C71#32811yescharacteristic
83C22×C71#35211yescharacteristic
84C22×C71#35311yescharacteristic
85C4×C71—21no—
86C4×C71—21no—
87C4×C71—42no—
88C4×C71—81no—
89C4×C71—161no—
90Dic71—162no—
91#7434—711nomaximal, Sylow
92C2×C71—21no—
93C2×C71—21no—
94C2×C71—81no—
95C2×C71—81no—
96C2×C71#135211yescharacteristic
97C2×C71#140711yescharacteristic
98C2×C71#140811yescharacteristic
99#2093D7111yescharacteristic
100#1393—711no—
101#1318—711no—
102C71#743411yescharacteristic, complemented, Sylow
103C2×#107—21no—
104C2×#102—21no—
105C2×#99—1421no—
106#390C2×D7111yescharacteristic
107#384Dic7111yescharacteristic, complemented
108#336Dic7111yescharacteristic, complemented
109#324—1421no—
110C22×D4C22⋊D7111yescharacteristic
111C22×C8—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4C22⋊D7111yescharacteristic
115C2×C22⋊C4—41no—
116C2×C22⋊C4—2841no—
117C2×C42C2×Dic7111yescharacteristic
118#107—42no—
119#102—42no—
120#99—2841no—
121#99—1422no—
122C22⋊C8—2841no—
123C24—21no—
124C2×D4—21no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4#1176585211yescharacteristic
128C2×D4#1176585511yescharacteristic
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—41no—
132C22×C4—5681no—
133C22×C4C22⋊Dic7111yescharacteristic
134C2×C8—22no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—5681no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—82no—
142C22⋊C4—82no—
143C22⋊C4—2842no—
144C22⋊C4—5681no—
145C42—41no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—81no—
151C23—81no—
152C23#1455604311yescharacteristic
153D4—41no—
154D4—41no—
155D4—81no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—81no—
163C2×C4—82no—
164C2×C4—81no—
165C2×C4—5682no—
166C2×C4—11361no—
167C2×C4—11361no—
168C2×C4#1455605211yescharacteristic
169C2×C4#1455605511yescharacteristic
170C8—42no—
171C8—11361no—
172C22—21no—
173C22—21no—
174C22—41no—
175C22—41no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—81no—
180C22—161no—
181C22#1798573611yescharacteristic
182C22#1798578911yescharacteristic
183C22#1798579211yescharacteristic
184C4—21no—
185C4—21no—
186C4—42no—
187C4—81no—
188C4—161no—
189C4—11362no—
190C2—21no—
191C2—21no—
192C2—81no—
193C2—81no—
194C2#2258636411yescharacteristic
195C2#2258650011yescharacteristic
196C2#2258650311yescharacteristic
subgroupG/H
centerC22#17985792
commutatorC2×C42×C71C2×C4
Frattini#390C2×D71
FittingC71×#2093C2
socleC22×C71#353
prime radicalC22×D4×C71D4
cosocle#390×C71C22
hypercenter#2093D71
hypocenterC71#7434
LCSG ▷ C2×C42×C71 ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #2093

Complements

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

NH ≅ G/Ntype
9. #384×C71C4semidirect
10. #336×C71C4semidirect
102. C71#7434semidirect
107. #384Dic71semidirect
108. #336Dic71semidirect

Maximal subgroups

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

subgroupindex
#225864652
#225862722
C71×#20932
#743471