Finite Group Databasebeta

← #1711×D71 #31985126

Subgroups — #1711×D71 #31985126

Order 18176.

802 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 179 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#487×D71C213yesmaximal, complemented
2#486×D71C213yesmaximal
3#22587953C213yesmaximal, complemented
4#22587952C211yesmaximal, characteristic, complemented
5#22587951C213yesmaximal, complemented
6C82×D71C211yesmaximal, characteristic, complemented
7C71×#1711C211yesmaximal, characteristic, complemented
8C4⋊Q8×D71C2213yes—
9#127×D71C2211yescharacteristic
10#17986351C2213yescomplemented
11#17986350C2213yescomplemented
12#17986349C2213yescomplemented
13#17986348—26no—
14#17986347C2213yes—
15#17986346C2213yes—
16#17986345C2213yes—
17#17986344—26no—
18#103×D71—26no—
19#102×D71—26no—
20C4×C8×D71C2213yes—
21C8×C71⋊C8C2213yescomplemented
22#487×C71C2213yescomplemented
23#486×C71C2213yes—
24C82×C71C2211yescharacteristic, complemented
25C2×Q8×D71—23no—
26C2×D4×D71—23no—
27C2×C8×D71D416yes—
28#14556128C2313yes—
29#14556127—23no—
30#14556127—23no—
31#14556126—23no—
32#14556125C2311yescharacteristic
33#14556124—23no—
34C4⋊C4×D71—23no—
35C4⋊C4×D71—23no—
36C42×D71C2311yescharacteristic
37#14556053—26no—
38#14556052—26no—
39#14556051—26no—
40#14556050—26no—
41#14556049—26no—
42#14556048—26no—
43C4×C71⋊C8C2313yes—
44C8×Dic71D416yes—
45C4⋊Q8×C71C2313yes—
46#127×C71C2311yescharacteristic
47#103×C71—26no—
48#102×C71—26no—
49C4×C8×C71C2313yes—
50C23×D71—41no—
51C2×Dic142—23no—
52C2×Dic142—43no—
53Q8×D71—412no—
54C2×D284—23no—
55C2×C22⋊D71—43no—
56D4×D71—412no—
57C2×C4×D71—43no—
58C2×C4×D71C2×D413yes—
59C2×C71⋊C8C2×D416yes—
60C8×D71—212no—
61C4⋊Dic71—23no—
62C4⋊Dic71—23no—
63Dic71⋊C4—43no—
64Dic71⋊C4—43no—
65C4×Dic71C2×D413yes—
66C2×Q8×C71—23no—
67C2×D4×C71—23no—
68C2×C8×C71C2×D416yes—
69C4⋊C4×C71—23no—
70C4⋊C4×C71—23no—
71C42×C71C2411yescharacteristic
72C23×C71—41no—
73C22×D71—41no—
74C22×D71—86no—
75C22×D71#12711yescharacteristic
76Q8×C71—46no—
77D4×C71—46no—
78C2×C4×C71—43no—
79C2×C4×C71C22×D413yes—
80C22⋊D71—86no—
81C2×Dic71—43no—
82C2×Dic71#12713yes—
83D284—46no—
84C4×D71—86no—
85C4×D71#135112yes—
86Dic142—46no—
87Dic142—86no—
88C8×C71—26no—
89C71⋊C8—26no—
90C22×C71—83no—
91C22×C71C2×#12711yescharacteristic
92C2×D71—83no—
93C2×D71—164no—
94C2×D71#49516yes—
95C4×C71—83no—
96C4×C71C2×#13516yes—
97Dic71—83no—
98Dic71#49516yes—
99C2×#1711—711nomaximal, Sylow
100C2×C71—161no—
101C2×C71C2×#49513yes—
102D71—161no—
103D71#171114yesdirect product factor
104C2×#487—713no—
105C2×#486—713no—
106#1711—713no—
107#1711—713no—
108#1711—711no—
109#1711D7111yescharacteristic, direct product factor
110C2×C82—711no—
111C71C2×#171111yescharacteristic, complemented, Sylow
112C2×C4⋊Q8—713no—
113C2×#127—711no—
114#487—713no—
115#487—713no—
116#487—713no—
117#487—1426no—
118#487C2×D7113yescomplemented
119#486—713no—
120#486—713no—
121#486—713no—
122#486—1426no—
123#486C2×D7113yes—
124C2×#103—1426no—
125C2×#102—1426no—
126C2×C4×C8—713no—
127C82—713no—
128C82C2×D7111yescharacteristic, complemented
129C22×Q8—1423no—
130C22×D4—1423no—
131C22×C8—716no—
132C4⋊Q8—713no—
133C4⋊Q8—1423no—
134C4⋊Q8—1423no—
135C4⋊Q8—1423no—
136C4⋊Q8C22×D7113yes—
137#127—711no—
138#127—1423no—
139#127C22×D7111yescharacteristic
140C2×C4⋊C4—1423no—
141C2×C4⋊C4—1423no—
142C2×C42—711no—
143#103—26no—
144#103—1426no—
145#103—1426no—
146#103—1426no—
147#102—26no—
148#102—1426no—
149#102—1426no—
150#102—1426no—
151C4×C8—716no—
152C4×C8—713no—
153C4×C8C22×D7113yes—
154C24—2841no—
155C2×Q8—23no—
156C2×Q8—2843no—
157C2×Q8—28412no—
158C2×Q8—1423no—
159C2×D4—23no—
160C2×D4—28412no—
161C2×D4—2843no—
162C2×D4—1423no—
163C22×C4—2843no—
164C22×C4—713no—
165C2×C8—716no—
166C2×C8—14212no—
167C2×C8D4×D7116yes—
168C4⋊C4—23no—
169C4⋊C4—23no—
170C4⋊C4—2843no—
171C4⋊C4—2843no—
172C4⋊C4—1423no—
173C4⋊C4—1423no—
174C42—713no—
175C42C23×D7111yescharacteristic
176C23—41no—
177C23—2841no—
178C23—5686no—
179C23—711no—
180Q8—46no—
181Q8—2846no—
182Q8—5686no—
183D4—46no—
184D4—2846no—
185D4—5686no—
186C2×C4—43no—
187C2×C4—2843no—
188C2×C4—5686no—
189C2×C4—713no—
190C2×C4—7112no—
191C2×C4C2×D4×D7113yes—
192C8—26no—
193C8—1426no—
194C22—83no—
195C22—5683no—
196C22—716no—
197C22—11364no—
198C22#127×D7111yescharacteristic
199C4—83no—
200C4—5683no—
201C4—716no—
202C4#135×D7116yes—
203C2—161no—
204C2—714no—
205C2—11361no—
206C2#495×D7113yes—
subgroupG/H
centerC22#127×D71
commutator, cosocleC42×C71C24
FrattiniC42C23×D71
FittingC71×#1711C2
socleC22×C71C2×#127
prime radical#127×D71C22
hypercenter#1711D71
hypocenterC71C2×#1711
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #1711

Complements

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

NH ≅ G/Ntype
1. #487×D71C2semidirect
3. #22587953C2semidirect
4. #22587952C2semidirect
5. #22587951C2semidirect
6. C82×D71C2semidirect
7. C71×#1711C2semidirect
10. #17986351C22semidirect
11. #17986350C22semidirect
12. #17986349C22semidirect
21. C8×C71⋊C8C22semidirect
22. #487×C71C22semidirect
24. C82×C71C22semidirect
103. D71#1711direct (N × H)
109. #1711D71direct (N × H)
111. C71C2×#1711semidirect
118. #487C2×D71semidirect
128. C82C2×D71semidirect

Maximal subgroups

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

subgroupindex
#487×D712
#486×D712
#225879532
#225879522
#225879512
C82×D712
C71×#17112
C2×#171171