Finite Group Databasebeta

← #31908648

Subgroups — #31908648

Order 18176.

274 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 71 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#22598721C211yesmaximal, characteristic
2#22590152C211yesmaximal, characteristic
3#22586374C214yesmaximal, complemented
4C71×#2165C211yesmaximal, characteristic
5#17986556C411yescharacteristic
6C2×#14556128C411yescharacteristic
7#17986368C412yes—
8#17986368C2212yes—
9#17986350—21no—
10#17986350—21no—
11#17986009—21no—
12#17986009—21no—
13#17985820—21no—
14#17985781—21no—
15#17985777C2212yes—
16C4×M16×C71C2211yescharacteristic
17#364×C71—21no—
18#349×C71—21no—
19#347×C71C2212yes—
20C22×Dic142—21no—
21#14556128—21no—
22#14556128—21no—
23#14556128—41no—
24#14556128C2×C412yes—
25#14556120C2×C412yes—
26#14556107—41no—
27C4×Dic142—41no—
28#14556087—21no—
29C2×C4⋊Dic71—41no—
30#14556068—22no—
31#14556053—22no—
32#14556053—22no—
33#14556053—22no—
34#14556053—22no—
35C2×M16×C71D411yescharacteristic
36C2×M16×C71D411yescharacteristic
37C2×C42×C71C2×C411yescharacteristic
38M32×C71—22no—
39#97×C71C2×C411yescharacteristic
40#97×C71C2311yescharacteristic
41C4×C8×C71D411yescharacteristic
42C4×C8×C71D411yescharacteristic
43C2×Dic142—22no—
44C2×Dic142—42no—
45C2×Dic142—42no—
46C2×Dic142—41no—
47C2×Dic142—41no—
48C22×Dic71—41no—
49C4⋊Dic71—22no—
50C4⋊Dic71—42no—
51C4⋊Dic71—41no—
52C4⋊Dic71—42no—
53Dic71⋊C4—42no—
54C22⋊Dic71—41no—
55C4×Dic71—41no—
56C71⋊C16—42no—
57C22×C4×C71—21no—
58C22×C4×C71C22⋊C411yescharacteristic
59M16×C71—41no—
60M16×C71—41no—
61C2×C8×C71C22⋊C411yescharacteristic
62C2×C8×C71C22⋊C411yescharacteristic
63C2×C8×C71C2×D411yescharacteristic
64C2×C8×C71C2×D411yescharacteristic
65C42×C71—21no—
66C42×C71C22⋊C411yescharacteristic
67C42×C71C22×C411yescharacteristic
68C16×C71—42no—
69C23×C71#10011yescharacteristic
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—21no—
73C2×C4×C71—21no—
74C2×C4×C71—42no—
75C2×C4×C71#10011yescharacteristic
76C2×C4×C71C2×C22⋊C411yescharacteristic
77C2×Dic71—42no—
78C2×Dic71—42no—
79C2×Dic71—82no—
80Dic142—42no—
81Dic142—42no—
82Dic142—82no—
83Dic142—82no—
84C8×C71—21no—
85C8×C71—21no—
86C8×C71—21no—
87C8×C71—21no—
88C22×C71—21no—
89C22×C71—41no—
90C22×C71C2×#10011yescharacteristic
91C4×C71—22no—
92C4×C71—21no—
93C4×C71—41no—
94C4×C71#42111yescharacteristic
95C4×C71#42111yescharacteristic
96Dic71—82no—
97Dic71—82no—
98#11916—711nomaximal, Sylow
99C2×C71—21no—
100C2×C71—41no—
101C2×C71#153311yescharacteristic
102#3139—711no—
103#2165—711no—
104#2165D7111yescharacteristic
105#1356—714no—
106C71#1191611yescharacteristic, complemented, Sylow
107#533—711no—
108C2×C4⋊Q8—711no—
109#491—712no—
110#491—712no—
111#487—1421no—
112#487—1421no—
113#419—1421no—
114#419—1421no—
115C4×M16C2×D7111yescharacteristic
116#364—21no—
117#364—1421no—
118#349—21no—
119#349—1421no—
120#347—712no—
121#347C2×D7112yes—
122C22×Q8—1421no—
123C2×M16D28411yescharacteristic
124C2×M16C22⋊D7111yescharacteristic
125C4⋊Q8—2841no—
126C4⋊Q8—712no—
127C4⋊Q8—1421no—
128C4⋊Q8—1421no—
129#125—712no—
130C22⋊Q8—2841no—
131C4×Q8—2841no—
132#117—1421no—
133C2×C4⋊C4—2841no—
134C2×C42C4×D7111yescharacteristic
135M32—22no—
136M32—1422no—
137#103—1422no—
138#103—1422no—
139#103—1422no—
140#103—1422no—
141#97C4×D7111yescharacteristic
142#97C22×D7111yescharacteristic
143C4×C8D28411yescharacteristic
144C4×C8C22⋊D7111yescharacteristic
145C2×Q8—2841no—
146C2×Q8—2842no—
147C2×Q8—2842no—
148C2×Q8—2841no—
149C2×Q8—1422no—
150C22×C4—21no—
151C22×C4—2841no—
152C22×C4(C2×D71)⋊C411yescharacteristic
153M16—41no—
154M16—41no—
155C2×C8(C2×D71)⋊C411yescharacteristic
156C2×C8(C2×D71)⋊C411yescharacteristic
157C2×C8C2×C22⋊D7111yescharacteristic
158C2×C8C2×D28411yescharacteristic
159C4⋊C4—2842no—
160C4⋊C4—2842no—
161C4⋊C4—2842no—
162C4⋊C4—2841no—
163C4⋊C4—1422no—
164C22⋊C4—2841no—
165C42—21no—
166C42—2841no—
167C42(C2×D71)⋊C411yescharacteristic
168C42C2×C4×D7111yescharacteristic
169C16—42no—
170C16—2842no—
171C23#1455604411yescharacteristic
172Q8—2842no—
173Q8—2842no—
174Q8—5682no—
175Q8—5682no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—2842no—
182C2×C4—2842no—
183C2×C4—5682no—
184C2×C4#1455604411yescharacteristic
185C2×C4C2×(C2×D71)⋊C411yescharacteristic
186C8—21no—
187C8—21no—
188C8—21no—
189C8—21no—
190C22—21no—
191C22—41no—
192C22C2×#1455604411yescharacteristic
193C4—22no—
194C4—21no—
195C4—41no—
196C4—5682no—
197C4—5682no—
198C4#1798601511yescharacteristic
199C4#1798601711yescharacteristic
200C2—21no—
201C2—41no—
202C2#2258696311yescharacteristic
subgroupG/H
centerC2#22586963
commutatorC42×C71C22×C4
Frattini#97C22×D71
FittingC71×#2165C2
socleC2×C71#1533
prime radicalC23×C71#100
cosocle#97×C71C23
hypercenter#2165D71
hypocenterC71#11916
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
3. #22586374C2semidirect
106. C71#11916semidirect

Maximal subgroups

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

subgroupindex
#225987212
#225901522
#225863742
C71×#21652
#1191671