Finite Group Databasebeta

← #31911733

Subgroups — #31911733

Order 18176.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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#22590248C211yesmaximal, characteristic, complemented
2#22587707C211yesmaximal, characteristic, complemented
3#22587697C211yesmaximal, characteristic
4#22586412C211yesmaximal, characteristic, complemented
5#22586389C211yesmaximal, characteristic, complemented
6#22586386C211yesmaximal, characteristic
7C71×#2248C211yesmaximal, characteristic
8#17986281—21no—
9#17986196—21no—
10C8×Dic142C2211yescharacteristic
11C4⋊(C71⋊C16)C2211yescharacteristic
12Dic71⋊C16—21no—
13#17985774—21no—
14#17985752C2211yescharacteristic
15#17985752C2211yescharacteristic
16#17985745—21no—
17#17985743—21no—
18C2×SD32×C71—21no—
19C2×SD32×C71—21no—
20#494×C71C2211yescharacteristic
21#493×C71C2211yescharacteristic
22C4×C16×C71C2211yescharacteristic
23C2×#11765853—41no—
24C4×Dic142D411yescharacteristic
25C2×C71⋊C16—21no—
26#14556059—41no—
27C4⋊(C71⋊C8)—21no—
28C4⋊(C71⋊C8)—21no—
29C4⋊(C71⋊C8)D411yescharacteristic
30Dic71⋊C8—21no—
31#14556051—41no—
32#14556049—41no—
33C8×Dic71—21no—
34C2×Q16×C71—21no—
35C2×Q16×C71—21no—
36C2×D8×C71—21no—
37C2×D8×C71—21no—
38C4⋊Q8×C71—21no—
39#127×C71—21no—
40SD32×C71—42no—
41SD32×C71—42no—
42C2×C16×C71D411yescharacteristic
43C2×C16×C71D411yescharacteristic
44C4×C8×C71C2311yescharacteristic
45C2×Dic142D811yescharacteristic
46#11765853—82no—
47C2×C71⋊C8—21no—
48C2×C71⋊C8—41no—
49C2×C71⋊C8—41no—
50C4⋊Dic71D811yescharacteristic
51Dic71⋊C4—21no—
52C4×Dic71—21no—
53C71⋊C16—41no—
54C2×Q8×C71—41no—
55C2×D4×C71—41no—
56C2×D4×C71—41no—
57C2×D4×C71—41no—
58Q16×C71—41no—
59Q16×C71—41no—
60Q16×C71—81no—
61D8×C71—41no—
62D8×C71—41no—
63D8×C71—81no—
64C2×C8×C71C2×D411yescharacteristic
65C2×C8×C71#4111yescharacteristic
66C4⋊C4×C71—41no—
67C4⋊C4×C71—41no—
68C42×C71C2×D411yescharacteristic
69C16×C71—21no—
70C16×C71SD1612yescomplemented
71C23×C71—81no—
72Q8×C71—81no—
73Q8×C71—81no—
74D4×C71—81no—
75D4×C71—81no—
76D4×C71—81no—
77D4×C71—81no—
78D4×C71—161no—
79C2×C4×C71—81no—
80C2×C4×C71#12111yescharacteristic
81C2×C4×C71C2×D811yescharacteristic
82C2×C4×C71#13511yescharacteristic
83C2×Dic71—21no—
84C2×Dic71—21no—
85Dic142—21no—
86Dic142—21no—
87C8×C71—21no—
88C8×C71C2×SD1611yescharacteristic
89C8×C71AGL(1,ℤ/8)11yescharacteristic
90C71⋊C8—41no—
91C71⋊C8—81no—
92C71⋊C8—81no—
93C22×C71—161no—
94C22×C71—161no—
95C22×C71—161no—
96C22×C71#46611yescharacteristic
97C4×C71—21no—
98C4×C71—161no—
99C4×C71#46011yescharacteristic
100C4×C71#47211yescharacteristic
101C4×C71#50911yescharacteristic
102C4×C71#51011yescharacteristic
103Dic71—41no—
104Dic71—41no—
105#12444—711nomaximal, Sylow
106C2×C71—321no—
107C2×C71#166811yescharacteristic
108C2×C71#222011yescharacteristic
109C2×C71#222311yescharacteristic
110#2248D7111yescharacteristic
111#2183—711no—
112#1670—711no—
113#1668—711no—
114#1371—711no—
115#1362—711no—
116#1361—711no—
117C71#1244411yescharacteristic, complemented, Sylow
118C2×SD32—21no—
119C2×SD32—21no—
120#494C2×D7111yescharacteristic
121#493C2×D7111yescharacteristic
122#475—1421no—
123#460—1421no—
124C8×Q8—711no—
125C4⋊C16—711no—
126C4⋊C16—1421no—
127#346—1421no—
128C4×C16C2×D7111yescharacteristic
129#335—711no—
130#335—711no—
131#332—1421no—
132#331—1421no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×SD16—2841no—
136C2×D8—21no—
137C2×D8—21no—
138C4⋊Q8—21no—
139#127—21no—
140C4×Q8—711no—
141SD32—42no—
142SD32—42no—
143C2×C16—1421no—
144C2×C16C22⋊D7111yescharacteristic
145C2×C16C22⋊D7111yescharacteristic
146#106—2841no—
147C4⋊C8—711no—
148C4⋊C8—1421no—
149C4⋊C8—1421no—
150C4⋊C8—1421no—
151#103—2841no—
152#102—2841no—
153C4×C8—1421no—
154C4×C8C22×D7111yescharacteristic
155C2×Q8—41no—
156C2×Q8—711no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160Q16—41no—
161Q16—41no—
162Q16—81no—
163SD16—5682no—
164D8—41no—
165D8—41no—
166D8—81no—
167C2×C8—2841no—
168C2×C8—2841no—
169C2×C8—1421no—
170C2×C8C2×C22⋊D7111yescharacteristic
171C2×C8#1176586411yescharacteristic
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—711no—
175C4⋊C4—1421no—
176C42—1421no—
177C42D4×D7111yescharacteristic
178C16—21no—
179C16—2841no—
180C16#1176585312yescomplemented
181C23—81no—
182Q8—81no—
183Q8—81no—
184Q8—1421no—
185Q8—1421no—
186D4—81no—
187D4—81no—
188D4—81no—
189D4—81no—
190D4—161no—
191C2×C4—81no—
192C2×C4—1421no—
193C2×C4—1421no—
194C2×C4#1455609911yescharacteristic
195C2×C4D8×D7111yescharacteristic
196C2×C4#1455614711yescharacteristic
197C8—21no—
198C8—2841no—
199C8—5681no—
200C8—5681no—
201C8C2×#1176585311yescharacteristic
202C8#1455615411yescharacteristic
203C22—161no—
204C22—161no—
205C22—161no—
206C22#1798623211yescharacteristic
207C4—21no—
208C4—161no—
209C4—2841no—
210C4—2841no—
211C4#1798619611yescharacteristic
212C4#1798626611yescharacteristic
213C4#1798643611yescharacteristic
214C4#1798643811yescharacteristic
215C2—321no—
216C2#2258769711yescharacteristic
217C2#2259045011yescharacteristic
218C2#2259046711yescharacteristic
subgroupG/H
centerC22#17986232
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#2248C2
socleC22×C71#466
prime radical#493×C71C22
hypercenter#2248D71
hypocenterC71#12444
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2248

Complements

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

NH ≅ G/Ntype
1. #22590248C2semidirect
2. #22587707C2semidirect
4. #22586412C2semidirect
5. #22586389C2semidirect
70. C16×C71SD16semidirect
117. C71#12444semidirect
180. C16#11765853semidirect

Maximal subgroups

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

subgroupindex
#225902482
#225877072
#225876972
#225864122
#225863892
#225863862
C71×#22482
#1244471