Finite Group Databasebeta

← #31902222

Subgroups — #31902222

Order 18176.

330 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 93 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#22588494C211yesmaximal, characteristic, complemented
3#22586987C212yesmaximal
4#22586236C212yesmaximal, complemented
5C71×#1274C211yesmaximal, characteristic, complemented
6#17986549C2211yescharacteristic
7#17986023C2211yescharacteristic
8C2×C4⋊(C71⋊C8)—21no—
9C2×C4⋊(C71⋊C8)—21no—
10C2×C4⋊(C71⋊C8)C2212yes—
11C2×#14556056—21no—
12C2×#14556056—21no—
13#17985744—22no—
14#17985742—22no—
15#431×C71C2212yes—
16#376×C71C2211yescharacteristic
17C2×#11765865—21no—
18C2×#11765849—21no—
19C22×C71⋊C8—22no—
20#14556128D411yescharacteristic
21#14556125D411yescharacteristic
22#14556116—21no—
23#14556102—22no—
24C4×D284—22no—
25C4⋊(C71⋊C8)—22no—
26C4⋊(C71⋊C8)—22no—
27C4⋊(C71⋊C8)—22no—
28C4⋊(C71⋊C8)—42no—
29C4⋊(C71⋊C8)—42no—
30#14556056—42no—
31#14556056—42no—
32C2×C42×C71D411yescharacteristic
33C2×C42×C71D411yescharacteristic
34C2×C42×C71C2311yescharacteristic
35C4⋊C8×C71—22no—
36C22⋊C8×C71—22no—
37C4×C8×C71—22no—
38#95×C71—22no—
39#11765865—41no—
40#11765865—41no—
41#11765865—41no—
42#11765865—41no—
43C2×Dic142—21no—
44C2×D284—21no—
45C2×D284—22no—
46C2×C22⋊D71—41no—
47C2×C4×D71—41no—
48#11765849—41no—
49#11765849—41no—
50C2×C71⋊C8—42no—
51C2×C71⋊C8—41no—
52C2×C71⋊C8—42no—
53C2×C71⋊C8—42no—
54C4⋊Dic71—22no—
55(C2×D71)⋊C4—42no—
56C22×C4×C71—21no—
57C22×C4×C71—21no—
58C22×C4×C71C2×D411yescharacteristic
59C22×C4×C71#4112yes—
60C2×C8×C71—42no—
61C42×C71—21no—
62C42×C71—21no—
63C42×C71D812yes—
64C42×C71SD1612yes—
65C42×C71C2×D411yescharacteristic
66C42×C71#4112yes—
67C42×C71#4111yescharacteristic
68C23×C71#12311yescharacteristic
69C22×D71—41no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71—42no—
73C2×C4×C71—42no—
74C2×C4×C71—42no—
75C2×C4×C71#12111yescharacteristic
76C2×C4×C71#12312yes—
77C2×C4×C71#12411yescharacteristic
78C2×C4×C71#12612yes—
79C2×C4×C71C2×D811yescharacteristic
80C2×C4×C71C2×SD1611yescharacteristic
81C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
82C2×C4×C71#13711yescharacteristic
83C22⋊D71—81no—
84C22⋊D71—81no—
85C2×Dic71—41no—
86D284—41no—
87D284—41no—
88D284—82no—
89C4×D71—81no—
90C4×D71—81no—
91Dic142—41no—
92Dic142—41no—
93C8×C71—82no—
94C71⋊C8—82no—
95C71⋊C8—81no—
96C22×C71—22no—
97C22×C71#39711yescharacteristic
98C22×C71#48011yescharacteristic
99C22×C71#48111yescharacteristic
100C2×D71—81no—
101C2×D71—82no—
102C4×C71—22no—
103C4×C71—22no—
104C4×C71—42no—
105C4×C71—42no—
106C4×C71#45911yescharacteristic
107C4×C71#46111yescharacteristic
108C4×C71#48611yescharacteristic
109C4×C71#48911yescharacteristic
110Dic71—81no—
111#10627—711nomaximal, Sylow
112C2×C71—21no—
113C2×C71—21no—
114C2×C71#207111yescharacteristic
115C2×C71#208012yes—
116D71—161no—
117#1963—711no—
118#1843—711no—
119#1540—712no—
120#1301—712no—
121#1274D7111yescharacteristic, complemented
122C71#1062711yescharacteristic, complemented, Sylow
123#532—711no—
124#431C2×D7112yes—
125#423—711no—
126C2×C4⋊C8—712no—
127C2×C4⋊C8—1421no—
128C2×C4⋊C8—1421no—
129C2×#104—1421no—
130C2×#104—1421no—
131#376C2×D7111yescharacteristic
132#332—1422no—
133#331—1422no—
134C2×#41—1421no—
135C2×M16—1421no—
136C22×C8—1422no—
137C4⋊Q8—711no—
138#127—711no—
139#124—1421no—
140#121—1422no—
141C4×D4—1422no—
142C2×C42D28411yescharacteristic
143C2×C42D28411yescharacteristic
144C2×C42C22×D7111yescharacteristic
145C4⋊C8—22no—
146C4⋊C8—2842no—
147C4⋊C8—2842no—
148C4⋊C8—1422no—
149C4⋊C8—1422no—
150C4⋊C8—1422no—
151#104—2842no—
152#104—2842no—
153C22⋊C8—22no—
154C4×C8—22no—
155#95—22no—
156#41—2841no—
157#41—2841no—
158#41—2841no—
159#41—2841no—
160C2×Q8—1421no—
161C2×D4—2841no—
162C2×D4—1421no—
163C2×D4—1422no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—2841no—
167C22×C4C2×D28411yescharacteristic
168C22×C4#1176586412yes—
169M16—2841no—
170M16—2841no—
171C2×C8—42no—
172C2×C8—2842no—
173C2×C8—2841no—
174C2×C8—2842no—
175C2×C8—2842no—
176C4⋊C4—1422no—
177C22⋊C4—2842no—
178C42—21no—
179C42—21no—
180C42D56812yes—
181C42#1176585412yes—
182C42D4×D7111yescharacteristic
183C42#1176586512yes—
184C42#1176586611yescharacteristic
185C23—2841no—
186C23#1455611211yescharacteristic
187Q8—2841no—
188Q8—2841no—
189D4—2841no—
190D4—2841no—
191D4—5682no—
192D4—5681no—
193D4—5681no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—2841no—
200C2×C4—5681no—
201C2×C4—5681no—
202C2×C4#1455610311yescharacteristic
203C2×C4#1455611012yes—
204C2×C4#1455611611yescharacteristic
205C2×C4#1455612312yes—
206C2×C4C2×D56811yescharacteristic
207C2×C4C2×#1176585411yescharacteristic
208C2×C4#1455615611yescharacteristic
209C2×C4#1455616111yescharacteristic
210C8—82no—
211C8—5681no—
212C8—5682no—
213C22—22no—
214C22—5681no—
215C22—5682no—
216C22#1798592611yescharacteristic
217C22#1798631311yescharacteristic
218C22#1798631811yescharacteristic
219C4—22no—
220C4—22no—
221C4—42no—
222C4—42no—
223C4—5681no—
224C4#1798619511yescharacteristic
225C4#1798620511yescharacteristic
226C4#1798634611yescharacteristic
227C4#1798635911yescharacteristic
228C2—21no—
229C2—21no—
230C2—11361no—
231C2#2258974511yescharacteristic
232C2#2258979212yes—
subgroupG/H
centerC2×C4#14556116
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1274C2
socleC22×C71#397
prime radical#17986549C22
hypercenter#1274D71
hypocenterC71#10627
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
2. #22588494C2semidirect
4. #22586236C2semidirect
5. C71×#1274C2semidirect
121. #1274D71semidirect
122. C71#10627semidirect

Maximal subgroups

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

subgroupindex
#225891722
#225884942
#225869872
#225862362
C71×#12742
#1062771