Finite Group Databasebeta

← #31902231

Subgroups — #31902231

Order 18176.

310 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 85 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#22589172C211yesmaximal, characteristic
2#22586990C212yesmaximal
3#22586237C212yesmaximal, complemented
4#22586169C211yesmaximal, characteristic, complemented
5C71×#1832C211yesmaximal, characteristic, complemented
6#17986549C2211yescharacteristic
7#17986052C2212yes—
8#17986023C2211yescharacteristic
9C2×#14556056—22no—
10#17985746—22no—
11#17985742—22no—
12C2×C4⋊C8×C71C2212yes—
13#376×C71C2211yescharacteristic
14#336×C71—22no—
15C2×#11765865—21no—
16C2×#11765849—21no—
17#14556128D411yescharacteristic
18#14556125D411yescharacteristic
19#14556116—21no—
20#14556102—22no—
21C4×D284—22no—
22C4⋊(C71⋊C8)—22no—
23C4⋊(C71⋊C8)—22no—
24#14556056—44no—
25C22⋊(C71⋊C8)—22no—
26C4×C71⋊C8—22no—
27C22×C8×C71—22no—
28C2×C42×C71D412yes—
29C2×C42×C71C2311yescharacteristic
30C4⋊C8×C71—22no—
31C4⋊C8×C71—22no—
32#95×C71—22no—
33#11765865—42no—
34#11765865—42no—
35C2×Dic142—21no—
36C2×D284—21no—
37C2×D284—22no—
38C2×C22⋊D71—41no—
39C2×C4×D71—41no—
40#11765849—42no—
41C2×C71⋊C8—42no—
42C2×C71⋊C8—41no—
43C4⋊Dic71—22no—
44(C2×D71)⋊C4—42no—
45C22×C4×C71—22no—
46C22×C4×C71C2×D411yescharacteristic
47C22×C4×C71#4112yes—
48C2×C8×C71—42no—
49C2×C8×C71—42no—
50C2×C8×C71—42no—
51C42×C71—22no—
52C42×C71—22no—
53C42×C71C2×D411yescharacteristic
54C42×C71#4111yescharacteristic
55C42×C71#4112yes—
56C23×C71#12311yescharacteristic
57C22×D71—41no—
58C2×C4×C71—42no—
59C2×C4×C71—44no—
60C2×C4×C71—42no—
61C2×C4×C71#12111yescharacteristic
62C2×C4×C71#12312yes—
63C2×C4×C71#12411yescharacteristic
64C2×C4×C71#12612yes—
65C2×C4×C71#13512yes—
66C2×C4×C71AGL(1,ℤ/8)12yes—
67C22⋊D71—82no—
68C2×Dic71—41no—
69D284—42no—
70D284—82no—
71C4×D71—81no—
72C4×D71—81no—
73Dic142—42no—
74C8×C71—82no—
75C71⋊C8—82no—
76C71⋊C8—81no—
77C22×C71—22no—
78C22×C71#39711yescharacteristic
79C22×C71#48212yes—
80C2×D71—81no—
81C2×D71—82no—
82C4×C71—22no—
83C4×C71—22no—
84C4×C71—82no—
85C4×C71#46312yes—
86C4×C71#48811yescharacteristic
87C4×C71#49011yescharacteristic
88Dic71—81no—
89#10629—711nomaximal, Sylow
90C2×C71—22no—
91C2×C71#207311yescharacteristic
92C2×C71#208012yes—
93D71—161no—
94#1963—711no—
95#1832D7111yescharacteristic, complemented
96#1540—712no—
97#1301—712no—
98#1274—711no—
99C71#1062911yescharacteristic, complemented, Sylow
100#532—711no—
101#431—712no—
102#423—711no—
103C2×C4⋊C8C2×D7112yes—
104C2×#104—1422no—
105#376C2×D7111yescharacteristic
106#336—22no—
107#332—1422no—
108#331—1422no—
109C2×#41—1421no—
110C2×M16—1421no—
111C22×C8—22no—
112C4⋊Q8—711no—
113#127—711no—
114#124—1421no—
115#121—1422no—
116C4×D4—1422no—
117C2×C42C22⋊D7112yes—
118C2×C42C22×D7111yescharacteristic
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—1422no—
122C4⋊C8—1422no—
123#104—2844no—
124C22⋊C8—1422no—
125C4×C8—1422no—
126#95—22no—
127#41—2842no—
128#41—2842no—
129C2×Q8—1421no—
130C2×D4—2841no—
131C2×D4—1421no—
132C2×D4—1422no—
133C22×C4—22no—
134C22×C4—2841no—
135C22×C4C2×C22⋊D7111yescharacteristic
136C22×C4#1176586512yes—
137M16—2842no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—2842no—
142C2×C8—2841no—
143C4⋊C4—1422no—
144C22⋊C4—2842no—
145C42—22no—
146C42—22no—
147C42D4×D7111yescharacteristic
148C42#1176586411yescharacteristic
149C42#1176586612yes—
150C23—2841no—
151C23#1455611111yescharacteristic
152Q8—2842no—
153D4—2842no—
154D4—5682no—
155D4—5682no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—42no—
159C2×C4—2841no—
160C2×C4—5681no—
161C2×C4—5681no—
162C2×C4#1455609911yescharacteristic
163C2×C4#1455611312yes—
164C2×C4#1455611711yescharacteristic
165C2×C4#1455612312yes—
166C2×C4#1455614812yes—
167C2×C4#1455615312yes—
168C8—82no—
169C8—5682no—
170C8—5681no—
171C22—22no—
172C22—5681no—
173C22—5682no—
174C22#1798592511yescharacteristic
175C22#1798632312yes—
176C4—22no—
177C4—22no—
178C4—82no—
179C4—5681no—
180C4#1798621112yes—
181C4#1798635611yescharacteristic
182C4#1798636511yescharacteristic
183C2—22no—
184C2—11361no—
185C2#2258975411yescharacteristic
186C2#2258979112yes—
subgroupG/H
centerC2×C4#14556117
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1832C2
socleC22×C71#397
prime radical#17986549C22
hypercenter#1832D71
hypocenterC71#10629
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1832

Complements

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

NH ≅ G/Ntype
3. #22586237C2semidirect
4. #22586169C2semidirect
5. C71×#1832C2semidirect
95. #1832D71semidirect
99. C71#10629semidirect

Maximal subgroups

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

subgroupindex
#225891722
#225869902
#225862372
#225861692
C71×#18322
#1062971