Finite Group Databasebeta

← #1705×D71 #31985087

Subgroups — #1705×D71 #31985087

Order 18176.

994 conjugacy classes of subgroups; only the 239 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 239 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#498×D71C213yesmaximal, complemented
2#492×D71C213yesmaximal
3#22587928C213yesmaximal, complemented
4#22587927C211yesmaximal, characteristic, complemented
5#22587926C213yesmaximal, complemented
6C82×D71C211yesmaximal, characteristic, complemented
7C71×#1705C211yesmaximal, characteristic, complemented
8C2×SD16×D71—26no—
9C4⋊Q8×D71C2213yes—
10#127×D71C2211yescharacteristic
11#17986397C2213yescomplemented
12#17986396—26no—
13#17986395C2216yescomplemented
14#17986373C2213yes—
15#17986372C2213yes—
16#17986371C2213yes—
17#17986370—26no—
18#106×D71—26no—
19C4×C8×D71C2213yes—
20C8×C71⋊C8C2213yescomplemented
21#498×C71C2213yescomplemented
22#492×C71C2213yes—
23C82×C71C2211yescharacteristic, complemented
24C2×Q8×D71—23no—
25C2×D4×D71—23no—
26C2×#11765854—26no—
27C2×#11765853—26no—
28C2×#11765852—26no—
29SD16×D71—424no—
30C2×C8×D71D416yes—
31#14556128C2313yes—
32#14556127—26no—
33#14556126—23no—
34#14556125C2311yescharacteristic
35#14556124—23no—
36C4⋊C4×D71—26no—
37C42×D71C2311yescharacteristic
38#14556060—26no—
39#14556059—212no—
40C4×C71⋊C8C2313yes—
41C8×Dic71D416yes—
42C2×SD16×C71—26no—
43C4⋊Q8×C71C2313yes—
44#127×C71C2311yescharacteristic
45#106×C71—26no—
46C4×C8×C71C2313yes—
47C23×D71—41no—
48C2×Dic142—23no—
49C2×Dic142—43no—
50Q8×D71—412no—
51C2×D284—23no—
52C2×C22⋊D71—43no—
53D4×D71—412no—
54C2×C4×D71—43no—
55C2×C4×D71C2×D413yes—
56#11765854—412no—
57#11765853—412no—
58#11765852—412no—
59C2×C71⋊C8C2×D416yes—
60C8×D71SD16124yescomplemented
61C4⋊Dic71—26no—
62Dic71⋊C4—46no—
63C4×Dic71C2×D413yes—
64C2×Q8×C71—23no—
65C2×D4×C71—23no—
66SD16×C71—412no—
67C2×C8×C71C2×D416yes—
68C4⋊C4×C71—26no—
69C42×C71C2411yescharacteristic
70C23×C71—41no—
71C22×D71—41no—
72C22×D71—86no—
73C22×D71#12711yescharacteristic
74Q8×C71—46no—
75D4×C71—46no—
76C2×C4×C71—43no—
77C2×C4×C71C22×D413yes—
78C22⋊D71—86no—
79C2×Dic71—43no—
80C2×Dic71#12713yes—
81D284—46no—
82C4×D71—86no—
83C4×D71C2×SD16112yes—
84Dic142—46no—
85Dic142—86no—
86C8×C71C2×SD16112yescomplemented
87C71⋊C8C2×SD16112yescomplemented
88C22×C71—83no—
89C22×C71C2×#12711yescharacteristic
90C2×D71—83no—
91C2×D71—164no—
92C2×D71#49216yes—
93C4×C71—83no—
94C4×C71C22×SD1616yes—
95Dic71—83no—
96Dic71#49216yes—
97C2×#1705—711nomaximal, Sylow
98C2×C71—161no—
99C2×C71C2×#49213yes—
100D71—161no—
101D71#170514yesdirect product factor
102C2×#498—713no—
103C2×#492—713no—
104#1705—713no—
105#1705—713no—
106#1705—711no—
107#1705D7111yescharacteristic, direct product factor
108C2×C82—711no—
109C71C2×#170511yescharacteristic, complemented, Sylow
110C22×SD16—1426no—
111C2×C4⋊Q8—713no—
112C2×#127—711no—
113#498—716no—
114#498—713no—
115#498—1426no—
116#498C2×D7113yescomplemented
117#492—713no—
118#492—713no—
119#492—713no—
120#492—1426no—
121#492C2×D7113yes—
122C2×#106—1426no—
123C2×C4×C8—713no—
124C82—713no—
125C82C2×D7111yescharacteristic, complemented
126C22×Q8—1423no—
127C22×D4—1423no—
128C2×SD16—26no—
129C2×SD16—28424no—
130C2×SD16—1426no—
131C2×SD16—1426no—
132C2×SD16—1426no—
133C22×C8—716no—
134C4⋊Q8—713no—
135C4⋊Q8—1426no—
136C4⋊Q8—1423no—
137C4⋊Q8C22×D7113yes—
138#127—711no—
139#127—1423no—
140#127C22×D7111yescharacteristic
141C2×C4⋊C4—1426no—
142C2×C42—711no—
143#106—26no—
144#106—14212no—
145#106—1426no—
146C4×C8—716no—
147C4×C8—713no—
148C4×C8C22×D7113yes—
149C24—2841no—
150C2×Q8—23no—
151C2×Q8—2843no—
152C2×Q8—28412no—
153C2×Q8—1423no—
154C2×D4—23no—
155C2×D4—28412no—
156C2×D4—2843no—
157C2×D4—1423no—
158C22×C4—2843no—
159C22×C4—713no—
160SD16—412no—
161SD16—28412no—
162SD16—28412no—
163SD16—28412no—
164C2×C8—716no—
165C2×C8—7124no—
166C2×C8D4×D7116yes—
167C4⋊C4—26no—
168C4⋊C4—2846no—
169C4⋊C4—1426no—
170C42—713no—
171C42C23×D7111yescharacteristic
172C23—41no—
173C23—2841no—
174C23—5686no—
175C23—711no—
176Q8—46no—
177Q8—2846no—
178Q8—5686no—
179D4—46no—
180D4—2846no—
181D4—5686no—
182C2×C4—43no—
183C2×C4—2843no—
184C2×C4—5686no—
185C2×C4—713no—
186C2×C4—7112no—
187C2×C4C2×D4×D7113yes—
188C8—7112no—
189C8SD16×D71112yescomplemented
190C22—83no—
191C22—5683no—
192C22—716no—
193C22—11364no—
194C22#127×D7111yescharacteristic
195C4—83no—
196C4—5683no—
197C4—716no—
198C4C2×SD16×D7116yes—
199C2—161no—
200C2—714no—
201C2—11361no—
202C2#492×D7113yes—
subgroupG/H
centerC22#127×D71
commutator, cosocleC42×C71C24
FrattiniC42C23×D71
FittingC71×#1705C2
socleC22×C71C2×#127
prime radical#127×D71C22
hypercenter#1705D71
hypocenterC71C2×#1705
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #1705

Complements

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

NH ≅ G/Ntype
1. #498×D71C2semidirect
3. #22587928C2semidirect
4. #22587927C2semidirect
5. #22587926C2semidirect
6. C82×D71C2semidirect
7. C71×#1705C2semidirect
11. #17986397C22semidirect
13. #17986395C22semidirect
20. C8×C71⋊C8C22semidirect
21. #498×C71C22semidirect
23. C82×C71C22semidirect
60. C8×D71SD16semidirect
86. C8×C71C2×SD16semidirect
87. C71⋊C8C2×SD16semidirect
101. D71#1705direct (N × H)
107. #1705D71direct (N × H)
109. C71C2×#1705semidirect
116. #498C2×D71semidirect
125. C82C2×D71semidirect
189. C8SD16×D71semidirect

Maximal subgroups

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

subgroupindex
#498×D712
#492×D712
#225879282
#225879272
#225879262
C82×D712
C71×#17052
C2×#170571