Finite Group Databasebeta

← #31884842

Subgroups — #31884842

Order 18176.

180 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 43 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#22589308C211yesmaximal, characteristic
2#22586286C211yesmaximal, characteristic
3C71×#1325C211yesmaximal, characteristic
4C2×#14556120C411yescharacteristic
5C2×#14556039C411yescharacteristic
6#17985761—21no—
7#17985761—21no—
8#17985725—21no—
9#384×C71C2211yescharacteristic
10#324×C71—21no—
11C22×C71⋊C8—21no—
12#14556120—21no—
13#14556120—41no—
14#14556120C812yescomplemented
15C2×C4⋊Dic71—21no—
16C2×Dic71⋊C4—41no—
17C22⋊(C71⋊C8)—41no—
18#14556039—41no—
19C2×C4⋊C4×C71—41no—
20C2×C4⋊C4×C71D411yescharacteristic
21C2×C4⋊C4×C71D411yescharacteristic
22C2×C42×C71C2×C411yescharacteristic
23C22⋊C8×C71—41no—
24C22×Dic71—41no—
25C2×C71⋊C8—22no—
26C2×C71⋊C8—41no—
27C2×C71⋊C8—41no—
28C2×C71⋊C8—81no—
29C4⋊Dic71—22no—
30C4⋊Dic71—41no—
31Dic71⋊C4—42no—
32Dic71⋊C4—42no—
33C22×C4×C71—21no—
34C22×C4×C71—21no—
35C22×C4×C71—21no—
36C22×C4×C71C22⋊C411yescharacteristic
37C2×C8×C71—81no—
38C4⋊C4×C71—41no—
39C4⋊C4×C71—41no—
40C4⋊C4×C71—82no—
41C42×C71—21no—
42C42×C71C2×C811yescharacteristic
43C42×C71M1611yescharacteristic
44C23×C71#9911yescharacteristic
45C2×C4×C71—21no—
46C2×C4×C71—21no—
47C2×C4×C71—21no—
48C2×C4×C71—21no—
49C2×C4×C71—41no—
50C2×C4×C71—41no—
51C2×C4×C71—41no—
52C2×C4×C71—41no—
53C2×C4×C71—81no—
54C2×C4×C71—81no—
55C2×C4×C71C22⋊C811yescharacteristic
56C2×C4×C71#10011yescharacteristic
57C2×Dic71—42no—
58C2×Dic71—81no—
59C2×Dic71—81no—
60C8×C71—161no—
61C71⋊C8—42no—
62C71⋊C8—161no—
63C22×C71—21no—
64C22×C71—21no—
65C22×C71#32311yescharacteristic
66C22×C71#35411yescharacteristic
67C22×C71#35611yescharacteristic
68C4×C71—21no—
69C4×C71—21no—
70C4×C71—41no—
71C4×C71—41no—
72C4×C71—81no—
73C4×C71—81no—
74Dic71—82no—
75#7358—711nomaximal, Sylow
76C2×C71—21no—
77C2×C71—21no—
78C2×C71#131911yescharacteristic
79C2×C71#141211yescharacteristic
80C2×C71#141311yescharacteristic
81#1992—711no—
82#1325—711no—
83#1325D7111yescharacteristic
84C71#735811yescharacteristic, complemented, Sylow
85C2×#125—711no—
86C2×#97—711no—
87#384C2×D7111yescharacteristic
88#340—1421no—
89#340—1421no—
90#324—21no—
91#324—1421no—
92C22×C8—1421no—
93#125—2841no—
94#125—712no—
95#125—1421no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—2841no—
98C2×C4⋊C4—1421no—
99C2×C4⋊C4D28411yescharacteristic
100C2×C4⋊C4C22⋊D7111yescharacteristic
101C2×C42C4×D7111yescharacteristic
102C22⋊C8—41no—
103C22⋊C8—2841no—
104#97—2841no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—2841no—
109C22×C4(C2×D71)⋊C411yescharacteristic
110C2×C8—81no—
111C2×C8—2841no—
112C2×C8—2841no—
113C2×C8—5681no—
114C2×C8—1422no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—82no—
118C4⋊C4—2842no—
119C4⋊C4—2842no—
120C4⋊C4—2841no—
121C4⋊C4—1422no—
122C42—21no—
123C42C8×D7111yescharacteristic
124C42#1176584811yescharacteristic
125C23#1455604211yescharacteristic
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—81no—
135C2×C4—81no—
136C2×C4—2842no—
137C2×C4—5681no—
138C2×C4—5681no—
139C2×C4(C2×D71)⋊C811yescharacteristic
140C2×C4#1455604411yescharacteristic
141C8—161no—
142C8—2842no—
143C8—11361no—
144C22—21no—
145C22—21no—
146C22#1798572211yescharacteristic
147C22#1798579311yescharacteristic
148C22#1798579711yescharacteristic
149C4—21no—
150C4—21no—
151C4—41no—
152C4—41no—
153C4—81no—
154C4—81no—
155C4—5682no—
156C2—21no—
157C2—21no—
158C2#2258627311yescharacteristic
159C2#2258651011yescharacteristic
160C2#2258651211yescharacteristic
subgroupG/H
centerC22#17985793
commutatorC42×C71C2×C8
Frattini#384C2×D71
FittingC71×#1325C2
socleC22×C71#354
prime radicalC23×C71#99
cosocle#384×C71C22
hypercenter#1325D71
hypocenterC71#7358
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #384 ◁ #1325

Complements

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

NH ≅ G/Ntype
14. #14556120C8semidirect
84. C71#7358semidirect

Maximal subgroups

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

subgroupindex
#225893082
#225862862
C71×#13252
#735871