Finite Group Databasebeta

← #31885083

Subgroups — #31885083

Order 18176.

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

normal subgroups only

This group has 198 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#22589685C211yesmaximal, characteristic
2#22586223C211yesmaximal, characteristic
3C71×#1325C211yesmaximal, characteristic
4#17986630C411yescharacteristic
5#17986039—21no—
6C2×C4⋊(C71⋊C8)—21no—
7#17985768C411yescharacteristic
8#17985759—21no—
9#384×C71C2211yescharacteristic
10#324×C71—21no—
11C2×#11765849—21no—
12C22×C71⋊C8—41no—
13#14556127—41no—
14#14556119—41no—
15#14556107—41no—
16#14556087—21no—
17#14556061—41no—
18#14556059—41no—
19C4⋊(C71⋊C8)—22no—
20C4⋊(C71⋊C8)—41no—
21C2×C4⋊C4×C71—41no—
22C2×C4⋊C4×C71C2×C411yescharacteristic
23C2×C4⋊C4×C71D411yescharacteristic
24C2×C42×C71D411yescharacteristic
25C22⋊C8×C71—41no—
26C2×Dic142—41no—
27#11765849—22no—
28#11765849—41no—
29C2×C71⋊C8—41no—
30C2×C71⋊C8—42no—
31C2×C71⋊C8—81no—
32C2×C71⋊C8—81no—
33C4⋊Dic71—21no—
34C4⋊Dic71—21no—
35C4⋊Dic71—41no—
36Dic71⋊C4—42no—
37C22⋊Dic71—41no—
38C4×Dic71—41no—
39C22×C4×C71—21no—
40C22×C4×C71—21no—
41C22×C4×C71—21no—
42C22×C4×C71C22⋊C411yescharacteristic
43C2×C8×C71—81no—
44C4⋊C4×C71—41no—
45C4⋊C4×C71—41no—
46C4⋊C4×C71—81no—
47C4⋊C4×C71—81no—
48C42×C71—21no—
49C42×C71SD1611yescharacteristic
50C42×C71Q1611yescharacteristic
51C23×C71#9911yescharacteristic
52C2×C4×C71—21no—
53C2×C4×C71—21no—
54C2×C4×C71—21no—
55C2×C4×C71—21no—
56C2×C4×C71—21no—
57C2×C4×C71—21no—
58C2×C4×C71—21no—
59C2×C4×C71—21no—
60C2×C4×C71—41no—
61C2×C4×C71—41no—
62C2×C4×C71—81no—
63C2×C4×C71—81no—
64C2×C4×C71#10311yescharacteristic
65C2×C4×C71#10411yescharacteristic
66C2×Dic71—41no—
67C2×Dic71—41no—
68Dic142—161no—
69C8×C71—161no—
70C71⋊C8—42no—
71C71⋊C8—82no—
72C22×C71—21no—
73C22×C71—21no—
74C22×C71#32811yescharacteristic
75C22×C71AΣL(1,𝔽16)11yescharacteristic
76C22×C71#35611yescharacteristic
77C4×C71—21no—
78C4×C71—21no—
79C4×C71—21no—
80C4×C71—21no—
81C4×C71—21no—
82C4×C71—21no—
83C4×C71—81no—
84C4×C71—81no—
85Dic71—81no—
86Dic71—81no—
87#7443—711nomaximal, Sylow
88C2×C71—21no—
89C2×C71—21no—
90C2×C71#135411yescharacteristic
91C2×C71#140211yescharacteristic
92C2×C71#140511yescharacteristic
93#2060—711no—
94#1325D7111yescharacteristic
95#1296—711no—
96C71#744311yescharacteristic, complemented, Sylow
97#544—711no—
98#428—1421no—
99C2×C4⋊C8—1421no—
100#384C2×D7111yescharacteristic
101#343—711no—
102#339—1421no—
103#324—21no—
104C2×M16—1421no—
105C22×C8—2841no—
106C4⋊Q8—2841no—
107#125—2841no—
108C22⋊Q8—2841no—
109#117—1421no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4C4×D7111yescharacteristic
112C2×C4⋊C4C22⋊D7111yescharacteristic
113C2×C42D28411yescharacteristic
114#107—2841no—
115#106—2841no—
116C4⋊C8—2841no—
117C4⋊C8—1422no—
118C22⋊C8—41no—
119C2×Q8—2841no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4(C2×D71)⋊C411yescharacteristic
124M16—2841no—
125M16—1422no—
126C2×C8—81no—
127C2×C8—2841no—
128C2×C8—2842no—
129C2×C8—5681no—
130C2×C8—5681no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—81no—
134C4⋊C4—81no—
135C4⋊C4—2841no—
136C4⋊C4—2842no—
137C4⋊C4—1421no—
138C4⋊C4—1421no—
139C22⋊C4—2841no—
140C42—21no—
141C42—2841no—
142C42#1176585411yescharacteristic
143C42Dic28411yescharacteristic
144C23#1455604211yescharacteristic
145Q8—11361no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—81no—
157C2×C4—81no—
158C2×C4—2841no—
159C2×C4—2841no—
160C2×C4#1455605311yescharacteristic
161C2×C4#1455605411yescharacteristic
162C8—161no—
163C8—2842no—
164C8—5682no—
165C22—21no—
166C22—21no—
167C22#1798573711yescharacteristic
168C22#1798578511yescharacteristic
169C22#1798579711yescharacteristic
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—81no—
177C4—81no—
178C4—5681no—
179C4—5681no—
180C2—21no—
181C2—21no—
182C2#2258637111yescharacteristic
183C2#2258648411yescharacteristic
184C2#2258649311yescharacteristic
subgroupG/H
centerC22#17985785
commutatorC2×C4⋊C4×C71C2×C4
Frattini#384C2×D71
FittingC71×#1325C2
socleC22×C71AΣL(1,𝔽16)
prime radicalC23×C71#99
cosocle#384×C71C22
hypercenter#1325D71
hypocenterC71#7443
LCSG ▷ C2×C4⋊C4×C71 ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #384 ◁ #1325

Complements

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

NH ≅ G/Ntype
96. C71#7443semidirect

Maximal subgroups

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

subgroupindex
#225896852
#225862232
C71×#13252
#744371