Finite Group Databasebeta

← #31885004

Subgroups — #31885004

Order 18176.

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

normal subgroups only

This group has 228 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, complemented
2#22586223C211yesmaximal, characteristic, complemented
3C71×#2039C211yesmaximal, characteristic
4C2×C4⋊(C71⋊C8)—21no—
5#17985725—21no—
6#537×C71C411yescharacteristic
7C2×#104×C71—21no—
8#417×C71—21no—
9#384×C71C2211yescharacteristic
10#343×C71C411yescharacteristic
11C22×C71⋊C8—41no—
12C4⋊(C71⋊C8)—22no—
13C4⋊(C71⋊C8)—41no—
14C22⋊(C71⋊C8)—41no—
15C2×#41×C71—21no—
16C2×M16×C71—21no—
17C22⋊Q8×C71—41no—
18#121×C71—41no—
19#121×C71—41no—
20C2×C4⋊C4×C71—41no—
21C2×C4⋊C4×C71C2×C411yescharacteristic
22C2×C4⋊C4×C71D411yescharacteristic
23C2×C42×C71D411yescharacteristic
24#104×C71—41no—
25#104×C71—41no—
26#103×C71—41no—
27#102×C71—41no—
28C2×C71⋊C8—42no—
29C2×C71⋊C8—81no—
30C2×C71⋊C8—81no—
31C2×C71⋊C8—81no—
32#41×C71—41no—
33#41×C71—41no—
34#41×C71—41no—
35#41×C71—41no—
36C2×Q8×C71—21no—
37C2×D4×C71—21no—
38C2×D4×C71—41no—
39C2×D4×C71—41no—
40C22×C4×C71—21no—
41C22×C4×C71—21no—
42C22×C4×C71—21no—
43C22×C4×C71—41no—
44C22×C4×C71C22⋊C411yescharacteristic
45M16×C71—41no—
46M16×C71—41no—
47C2×C8×C71—41no—
48C4⋊C4×C71—41no—
49C4⋊C4×C71—41no—
50C4⋊C4×C71—41no—
51C4⋊C4×C71—81no—
52C4⋊C4×C71—81no—
53C22⋊C4×C71—42no—
54C42×C71—21no—
55C42×C71D811yescharacteristic
56C42×C71SD1611yescharacteristic
57C23×C71—41no—
58C23×C71#9911yescharacteristic
59Q8×C71—41no—
60Q8×C71—41no—
61D4×C71—41no—
62D4×C71—41no—
63D4×C71—81no—
64D4×C71—81no—
65D4×C71—161no—
66C2×C4×C71—21no—
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—21no—
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—21no—
73C2×C4×C71—21no—
74C2×C4×C71—41no—
75C2×C4×C71—41no—
76C2×C4×C71—41no—
77C2×C4×C71—81no—
78C2×C4×C71—81no—
79C2×C4×C71—81no—
80C2×C4×C71—81no—
81C2×C4×C71#10211yescharacteristic
82C2×C4×C71#10411yescharacteristic
83C8×C71—81no—
84C71⋊C8—82no—
85C71⋊C8—161no—
86C22×C71—21no—
87C22×C71—21no—
88C22×C71—81no—
89C22×C71—81no—
90C22×C71—81no—
91C22×C71#32711yescharacteristic
92C22×C71AΣL(1,𝔽16)11yescharacteristic
93C22×C71#35511yescharacteristic
94C4×C71—21no—
95C4×C71—21no—
96C4×C71—21no—
97C4×C71—21no—
98C4×C71—21no—
99C4×C71—21no—
100C4×C71—81no—
101C4×C71—81no—
102C4×C71—81no—
103#7417—711nomaximal, Sylow
104C2×C71—21no—
105C2×C71—21no—
106C2×C71—161no—
107C2×C71#134511yescharacteristic
108C2×C71#140211yescharacteristic
109C2×C71#140511yescharacteristic
110#2039D7111yescharacteristic
111#1325—711no—
112#1296—711no—
113C71#741711yescharacteristic, complemented, Sylow
114#537Dic7111yescharacteristic
115C2×C4⋊C8—1421no—
116C2×#104—21no—
117#417—21no—
118#384C2×D7111yescharacteristic
119#343Dic7111yescharacteristic
120#324—1421no—
121C2×#41—21no—
122C2×M16—21no—
123C22×C8—2841no—
124C22⋊Q8—41no—
125#121—41no—
126#121—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4C2×Dic7111yescharacteristic
129C2×C4⋊C4C22⋊D7111yescharacteristic
130C2×C42C22⋊D7111yescharacteristic
131C4⋊C8—2841no—
132C4⋊C8—1422no—
133#104—41no—
134#104—41no—
135#103—41no—
136#102—41no—
137C22⋊C8—2841no—
138#41—41no—
139#41—41no—
140#41—41no—
141#41—41no—
142C2×Q8—21no—
143C2×D4—21no—
144C2×D4—41no—
145C2×D4—41no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4C22⋊Dic7111yescharacteristic
151M16—41no—
152M16—41no—
153C2×C8—41no—
154C2×C8—2842no—
155C2×C8—5681no—
156C2×C8—5681no—
157C2×C8—5681no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—81no—
162C4⋊C4—81no—
163C22⋊C4—42no—
164C42—21no—
165C42#1176585011yescharacteristic
166C42#1176585311yescharacteristic
167C23—41no—
168C23#1455604311yescharacteristic
169Q8—41no—
170Q8—41no—
171D4—41no—
172D4—41no—
173D4—81no—
174D4—81no—
175D4—161no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4#1455604911yescharacteristic
192C2×C4#1455605511yescharacteristic
193C8—81no—
194C8—5682no—
195C8—11361no—
196C22—21no—
197C22—21no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22#1798573311yescharacteristic
202C22#1798578611yescharacteristic
203C22#1798579611yescharacteristic
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—21no—
210C4—81no—
211C4—81no—
212C4—81no—
213C2—21no—
214C2—21no—
215C2—161no—
216C2#2258634311yescharacteristic
217C2#2258648511yescharacteristic
218C2#2258649411yescharacteristic
subgroupG/H
centerC22#17985786
commutatorC2×C4⋊C4×C71C2×C4
Frattini#384C2×D71
FittingC71×#2039C2
socleC22×C71AΣL(1,𝔽16)
prime radical#537×C71C4
cosocle#384×C71C22
hypercenter#2039D71
hypocenterC71#7417
LCSG ▷ C2×C4⋊C4×C71 ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #384 ◁ #2039

Complements

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

NH ≅ G/Ntype
1. #22586286C2semidirect
2. #22586223C2semidirect
113. C71#7417semidirect

Maximal subgroups

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

subgroupindex
#225862862
#225862232
C71×#20392
#741771