Finite Group Databasebeta

← #31986693

Subgroups — #31986693

Order 18176.

994 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 203 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#22599078C213yesmaximal, complemented
2#22598792C213yesmaximal, complemented
3#22588006C213yesmaximal, complemented
4#22587968C213yesmaximal, complemented
5#22587955C211yesmaximal, characteristic, complemented
6#22587927C211yesmaximal, characteristic, complemented
7C71×#1455C211yesmaximal, characteristic, complemented
8C2×#14556156—26no—
9#17986549C2213yes—
10C2×#14556125C2211yescharacteristic
11#17986397C2213yescomplemented
12#17986388—26no—
13#17986376C2213yes—
14#17986376C2213yes—
15#17986376C2213yescomplemented
16#17986372C2213yes—
17#17986372C2213yes—
18#17986346C2216yescomplemented
19#17986252—26no—
20#17986112—26no—
21C4×M16×C71C2213yes—
22#322×C71C2213yescomplemented
23C82×C71C2211yescharacteristic, complemented
24C2×#11765865—23no—
25C22×D284—23no—
26#14556156—424no—
27C2×#11765854—26no—
28C2×#11765854—26no—
29C2×D568—26no—
30C2×D568—26no—
31C2×D568—26no—
32#14556128C2313yes—
33#14556125—23no—
34#14556125C2313yes—
35#14556125C2311yescharacteristic
36#14556125C2311yescharacteristic
37#14556116—23no—
38#14556102—26no—
39C4×D284—26no—
40#14556060—26no—
41#14556050—212no—
42C2×M16×C71D416yes—
43C2×C42×C71C2311yescharacteristic
44#97×C71D416yes—
45C4×C8×C71C2313yes—
46C4×C8×C71C2313yes—
47C23×D71—41no—
48#11765865—412no—
49C2×Dic142—23no—
50C2×D284—23no—
51C2×D284—23no—
52C2×D284—23no—
53C2×D284—26no—
54C2×D284—412no—
55C2×D284—43no—
56C2×C22⋊D71—43no—
57C2×C4×D71—43no—
58#11765854—412no—
59#11765854—412no—
60D568—412no—
61D568—412no—
62D568—412no—
63C4⋊Dic71—26no—
64(C2×D71)⋊C4—46no—
65C22×C4×C71C2×D413yes—
66M16×C71—212no—
67C2×C8×C71C2×D416yes—
68C2×C8×C71C2×D416yes—
69C42×C71C2×D413yes—
70C42×C71C2411yescharacteristic
71C23×C71#12711yescharacteristic
72C22×D71—43no—
73C22×D71—41no—
74C22×D71—41no—
75C22×D71—86no—
76C2×C4×C71—26no—
77C2×C4×C71#12713yes—
78C2×C4×C71C22×D413yes—
79C22⋊D71—86no—
80C2×Dic71—43no—
81D284—46no—
82D284—46no—
83D284—46no—
84D284—412no—
85D284—86no—
86C4×D71—86no—
87Dic142—46no—
88C8×C71—26no—
89C8×C71AGL(1,ℤ/8)112yescomplemented
90C22×C71—23no—
91C22×C71C2×#12711yescharacteristic
92C2×D71—83no—
93C2×D71—83no—
94C2×D71—83no—
95C2×D71—86no—
96C2×D71—164no—
97C4×C71—23no—
98C4×C71C2×AGL(1,ℤ/8)16yes—
99Dic71—83no—
100#19745—711nomaximal, Sylow
101C2×C71—41no—
102C2×C71#315113yes—
103D71—163no—
104D71—161no—
105D71—161no—
106#3184—713no—
107#3151—713no—
108#1721—713no—
109#1715—713no—
110#1712—711no—
111#1705—711no—
112#1455D7111yescharacteristic, complemented
113C71#1974511yescharacteristic, complemented, Sylow
114C2×AGL(1,ℤ/8)—1426no—
115#532—713no—
116C2×#127—711no—
117#498—713no—
118#496—1426no—
119#493—713no—
120#493—713no—
121#493—713no—
122#492—713no—
123#492—713no—
124#486—716no—
125#469—1426no—
126#442—1426no—
127C4×M16C2×D7113yes—
128#322C2×D7113yescomplemented
129C82C2×D7111yescharacteristic, complemented
130C2×#41—1423no—
131C22×D4—1423no—
132AGL(1,ℤ/8)—28424no—
133C2×SD16—1426no—
134C2×SD16—1426no—
135C2×D8—1426no—
136C2×D8—1426no—
137C2×D8—1426no—
138C2×M16D28416yes—
139C4⋊Q8—713no—
140#127—711no—
141#127—711no—
142#127—713no—
143#127—1423no—
144#124—1423no—
145#121—1426no—
146C4×D4—1426no—
147C2×C42C22×D7111yescharacteristic
148#106—1426no—
149#102—14212no—
150#97D28416yes—
151C4×C8C22×D7113yes—
152C4×C8C22×D7113yes—
153C24—2841no—
154#41—28412no—
155C2×Q8—1423no—
156C2×D4—28412no—
157C2×D4—2843no—
158C2×D4—2843no—
159C2×D4—1426no—
160C2×D4—1423no—
161C2×D4—1423no—
162C2×D4—1423no—
163C22×C4—2843no—
164C22×C4C2×D28413yes—
165SD16—28412no—
166SD16—28412no—
167D8—28412no—
168D8—28412no—
169D8—28412no—
170M16—212no—
171C2×C8C2×D28416yes—
172C2×C8C2×D28416yes—
173C4⋊C4—1426no—
174C22⋊C4—2846no—
175C42C2×D28413yes—
176C42C23×D7111yescharacteristic
177C23—2843no—
178C23—2841no—
179C23—2841no—
180C23—5686no—
181C23#1455612511yescharacteristic
182Q8—2846no—
183D4—2846no—
184D4—2846no—
185D4—2846no—
186D4—28412no—
187D4—5686no—
188D4—5686no—
189C2×C4—26no—
190C2×C4—2843no—
191C2×C4—5686no—
192C2×C4#1455612513yes—
193C2×C4C22×D28413yes—
194C8—26no—
195C8#14556156112yescomplemented
196C22—23no—
197C22—5683no—
198C22—5683no—
199C22—5683no—
200C22—5686no—
201C22—11364no—
202C22C2×#1455612511yescharacteristic
203C4—23no—
204C4—5683no—
205C4C2×#1455615616yes—
206C2—41no—
207C2—11361no—
208C2—11361no—
209C2—11363no—
210C2#2259879213yes—
subgroupG/H
centerC22C2×#14556125
commutator, cosocleC42×C71C24
FrattiniC42C23×D71
FittingC71×#1455C2
socleC22×C71C2×#127
hypercenter#1455D71
hypocenterC71#19745
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1455

Complements

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

NH ≅ G/Ntype
1. #22599078C2semidirect
2. #22598792C2semidirect
3. #22588006C2semidirect
4. #22587968C2semidirect
5. #22587955C2semidirect
6. #22587927C2semidirect
7. C71×#1455C2semidirect
11. #17986397C22semidirect
15. #17986376C22semidirect
18. #17986346C22semidirect
22. #322×C71C22semidirect
23. C82×C71C22semidirect
89. C8×C71AGL(1,ℤ/8)semidirect
112. #1455D71semidirect
113. C71#19745semidirect
128. #322C2×D71semidirect
129. C82C2×D71semidirect
195. C8#14556156semidirect

Maximal subgroups

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

subgroupindex
#225990782
#225987922
#225880062
#225879682
#225879552
#225879272
C71×#14552
#1974571