Finite Group Databasebeta

← #31885035

Subgroups — #31885035

Order 18176.

214 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 37 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#22589507C211yesmaximal, characteristic
2#22586248C211yesmaximal, characteristic
3C71×#1317C211yesmaximal, characteristic
4#17986625C411yescharacteristic
5#17985963—21no—
6#17985862—21no—
7#17985766C411yescharacteristic
8#17985765—21no—
9#381×C71C2211yescharacteristic
10#323×C71—21no—
11C2×#11765849—41no—
12#14556107—41no—
13#14556105—42no—
14C2×C22⋊Dic71—21no—
15C2×C22⋊Dic71—21no—
16C22⋊(C71⋊C8)—22no—
17C22⋊(C71⋊C8)—41no—
18C71⋊#95—42no—
19C23×C4×C71D411yescharacteristic
20C2×C22⋊C4×C71C2×C411yescharacteristic
21C2×C22⋊C4×C71D411yescharacteristic
22C22⋊C8×C71—41no—
23#95×C71—41no—
24C2×Dic142—41no—
25C22×Dic71—22no—
26C22×Dic71—41no—
27#11765849—81no—
28#11765849—81no—
29C2×C71⋊C8—42no—
30C4⋊Dic71—41no—
31Dic71⋊C4—42no—
32C22⋊Dic71—41no—
33C22⋊Dic71—41no—
34C22⋊Dic71—41no—
35C22⋊Dic71—41no—
36C24×C71C22⋊C411yescharacteristic
37C22×C4×C71—21no—
38C22×C4×C71—21no—
39C22×C4×C71—21no—
40C22×C4×C71—21no—
41C22×C4×C71—41no—
42C22×C4×C71—41no—
43C22×C4×C71SD1611yescharacteristic
44C22×C4×C71Q1611yescharacteristic
45C2×C8×C71—81no—
46C22⋊C4×C71—41no—
47C22⋊C4×C71—41no—
48C23×C71—21no—
49C23×C71—21no—
50C23×C71—81no—
51C23×C71#9911yescharacteristic
52C23×C71#10311yescharacteristic
53C23×C71#10411yescharacteristic
54C2×C4×C71—21no—
55C2×C4×C71—21no—
56C2×C4×C71—41no—
57C2×C4×C71—41no—
58C2×C4×C71—41no—
59C2×C4×C71—41no—
60C2×C4×C71—41no—
61C2×C4×C71—41no—
62C2×C4×C71—81no—
63C2×C4×C71—81no—
64C2×C4×C71—81no—
65C2×Dic71—42no—
66C2×Dic71—82no—
67C2×Dic71—81no—
68C2×Dic71—82no—
69Dic142—161no—
70C8×C71—161no—
71C71⋊C8—82no—
72C22×C71—21no—
73C22×C71—21no—
74C22×C71—41no—
75C22×C71—41no—
76C22×C71—41no—
77C22×C71—42no—
78C22×C71—81no—
79C22×C71#32811yescharacteristic
80C22×C71AΣL(1,𝔽16)11yescharacteristic
81C22×C71#35411yescharacteristic
82C4×C71—41no—
83C4×C71—41no—
84C4×C71—81no—
85C4×C71—81no—
86Dic71—82no—
87Dic71—161no—
88#7427—711nomaximal, Sylow
89C2×C71—21no—
90C2×C71—21no—
91C2×C71—21no—
92C2×C71—41no—
93C2×C71—41no—
94C2×C71#135111yescharacteristic
95#2030—711no—
96#1317D7111yescharacteristic
97#1306—711no—
98C71#742711yescharacteristic, complemented, Sylow
99#543—711no—
100#407—1421no—
101#381—1421no—
102#381C2×D7111yescharacteristic
103#342—711no—
104#342—1421no—
105#323—21no—
106C23×C4D28411yescharacteristic
107C2×M16—2841no—
108C22⋊Q8—2842no—
109C22⋊Q8—2841no—
110C2×C22⋊C4—1421no—
111C2×C22⋊C4—1421no—
112C2×C22⋊C4C4×D7111yescharacteristic
113C2×C22⋊C4C22⋊D7111yescharacteristic
114C22⋊C8—41no—
115C22⋊C8—2841no—
116C22⋊C8—1422no—
117#95—41no—
118#95—2842no—
119C24(C2×D71)⋊C411yescharacteristic
120C2×Q8—2841no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—2841no—
128C22×C4—1422no—
129C22×C4#1176585411yescharacteristic
130C22×C4Dic28411yescharacteristic
131M16—5681no—
132M16—5681no—
133C2×C8—81no—
134C2×C8—2842no—
135C4⋊C4—2842no—
136C4⋊C4—2841no—
137C22⋊C4—41no—
138C22⋊C4—41no—
139C22⋊C4—2841no—
140C22⋊C4—2841no—
141C22⋊C4—2841no—
142C22⋊C4—2841no—
143C23—21no—
144C23—21no—
145C23—81no—
146C23#1455604211yescharacteristic
147C23#1455605311yescharacteristic
148C23#1455605411yescharacteristic
149Q8—11361no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—81no—
159C2×C4—81no—
160C2×C4—81no—
161C2×C4—2842no—
162C2×C4—5682no—
163C2×C4—5681no—
164C2×C4—5682no—
165C8—161no—
166C8—5682no—
167C22—21no—
168C22—21no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22—42no—
173C22—81no—
174C22#1798573711yescharacteristic
175C22#1798578711yescharacteristic
176C22#1798579311yescharacteristic
177C4—41no—
178C4—41no—
179C4—81no—
180C4—81no—
181C4—5682no—
182C4—11361no—
183C2—21no—
184C2—21no—
185C2—21no—
186C2—41no—
187C2—41no—
188C2#2258636211yescharacteristic
subgroupG/H
centerC2#22586362
commutatorC2×C22⋊C4×C71C2×C4
Frattini#381C2×D71
FittingC71×#1317C2
socleC2×C71#1351
prime radicalC24×C71C22⋊C4
cosocle#381×C71C22
hypercenter#1317D71
hypocenterC71#7427
LCSG ▷ C2×C22⋊C4×C71 ▷ C24×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ #381 ◁ #1317

Complements

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

NH ≅ G/Ntype
98. C71#7427semidirect

Maximal subgroups

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

subgroupindex
#225895072
#225862482
C71×#13172
#742771