Finite Group Databasebeta

← #31885082

Subgroups — #31885082

Order 18176.

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

normal subgroups only

This group has 198 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#22586286C211yesmaximal, characteristic
2#22586223C211yesmaximal, characteristic
3C71×#2060C211yesmaximal, characteristic
4C2×C4⋊(C71⋊C8)—21no—
5#17985725—21no—
6#544×C71C411yescharacteristic
7#428×C71—21no—
8#384×C71C2211yescharacteristic
9#343×C71C411yescharacteristic
10#339×C71—21no—
11C22×C71⋊C8—41no—
12C4⋊(C71⋊C8)—22no—
13C4⋊(C71⋊C8)—41no—
14C22⋊(C71⋊C8)—41no—
15C2×M16×C71—21no—
16C4⋊Q8×C71—41no—
17#125×C71—41no—
18C22⋊Q8×C71—41no—
19#117×C71—21no—
20C2×C4⋊C4×C71—41no—
21C2×C4⋊C4×C71C2×C411yescharacteristic
22C2×C4⋊C4×C71D411yescharacteristic
23C2×C42×C71D411yescharacteristic
24#107×C71—41no—
25#106×C71—41no—
26C2×C71⋊C8—42no—
27C2×C71⋊C8—81no—
28C2×C71⋊C8—81no—
29C2×C71⋊C8—81no—
30C2×Q8×C71—41no—
31C22×C4×C71—21no—
32C22×C4×C71—21no—
33C22×C4×C71—21no—
34C22×C4×C71C22⋊C411yescharacteristic
35M16×C71—22no—
36M16×C71—41no—
37C2×C8×C71—41no—
38C4⋊C4×C71—21no—
39C4⋊C4×C71—21no—
40C4⋊C4×C71—41no—
41C4⋊C4×C71—41no—
42C4⋊C4×C71—41no—
43C4⋊C4×C71—42no—
44C4⋊C4×C71—81no—
45C4⋊C4×C71—81no—
46C22⋊C4×C71—41no—
47C42×C71—21no—
48C42×C71—41no—
49C42×C71SD1611yescharacteristic
50C42×C71Q1611yescharacteristic
51C23×C71#9911yescharacteristic
52Q8×C71—161no—
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—21no—
61C2×C4×C71—41no—
62C2×C4×C71—41no—
63C2×C4×C71—41no—
64C2×C4×C71—41no—
65C2×C4×C71—81no—
66C2×C4×C71—81no—
67C2×C4×C71#10311yescharacteristic
68C2×C4×C71#10411yescharacteristic
69C8×C71—42no—
70C71⋊C8—82no—
71C71⋊C8—161no—
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—
85C4×C71—81no—
86C4×C71—81no—
87#7443—711nomaximal, Sylow
88C2×C71—21no—
89C2×C71—21no—
90C2×C71#135411yescharacteristic
91C2×C71#140211yescharacteristic
92C2×C71#140511yescharacteristic
93#2060D7111yescharacteristic
94#1325—711no—
95#1296—711no—
96C71#744311yescharacteristic, complemented, Sylow
97#544Dic7111yescharacteristic
98#428—21no—
99C2×C4⋊C8—1421no—
100#384C2×D7111yescharacteristic
101#343Dic7111yescharacteristic
102#339—21no—
103#324—1421no—
104C2×M16—21no—
105C22×C8—2841no—
106C4⋊Q8—41no—
107#125—41no—
108C22⋊Q8—41no—
109#117—21no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4C2×Dic7111yescharacteristic
112C2×C4⋊C4C22⋊D7111yescharacteristic
113C2×C42C22⋊D7111yescharacteristic
114#107—41no—
115#106—41no—
116C4⋊C8—2841no—
117C4⋊C8—1422no—
118C22⋊C8—2841no—
119C2×Q8—41no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4C22⋊Dic7111yescharacteristic
124M16—22no—
125M16—41no—
126C2×C8—41no—
127C2×C8—2842no—
128C2×C8—5681no—
129C2×C8—5681no—
130C2×C8—5681no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—41no—
136C4⋊C4—42no—
137C4⋊C4—81no—
138C4⋊C4—81no—
139C22⋊C4—41no—
140C42—21no—
141C42—41no—
142C42#1176585211yescharacteristic
143C42#1176585511yescharacteristic
144C23#1455604311yescharacteristic
145Q8—161no—
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—41no—
157C2×C4—41no—
158C2×C4—81no—
159C2×C4—81no—
160C2×C4#1455605211yescharacteristic
161C2×C4#1455605511yescharacteristic
162C8—42no—
163C8—5682no—
164C8—11361no—
165C22—21no—
166C22—21no—
167C22#1798573611yescharacteristic
168C22#1798578611yescharacteristic
169C22#1798579811yescharacteristic
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—81no—
177C4—81no—
178C4—81no—
179C4—81no—
180C2—21no—
181C2—21no—
182C2#2258637011yescharacteristic
183C2#2258648511yescharacteristic
184C2#2258649411yescharacteristic
subgroupG/H
centerC22#17985786
commutatorC2×C4⋊C4×C71C2×C4
Frattini#384C2×D71
FittingC71×#2060C2
socleC22×C71AΣL(1,𝔽16)
prime radicalC23×C71#99
cosocle#384×C71C22
hypercenter#2060D71
hypocenterC71#7443
LCSG ▷ C2×C4⋊C4×C71 ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #384 ◁ #2060

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
#225862862
#225862232
C71×#20602
#744371