Finite Group Databasebeta

← #31884848

Subgroups — #31884848

Order 18176.

184 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 39 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#22589316C211yesmaximal, characteristic
2#22586291C211yesmaximal, characteristic
3C71×#1327C211yesmaximal, characteristic
4#17986556C411yescharacteristic
5C42×Dic71C411yescharacteristic
6#17985781—21no—
7#17985760—21no—
8#17985760—21no—
9#423×C71C2211yescharacteristic
10#349×C71—21no—
11#14556128—21no—
12#14556120C812yes—
13#14556107—41no—
14C4×Dic142—41no—
15#14556087—21no—
16C2×C4×Dic71—21no—
17C2×C4×Dic71—41no—
18#14556068—41no—
19C2×M16×C71D411yescharacteristic
20C2×M16×C71D411yescharacteristic
21C2×C42×C71C2×C411yescharacteristic
22M32×C71—41no—
23C4⋊C8×C71—41no—
24C2×Dic142—41no—
25C22×Dic71—41no—
26C4⋊Dic71—22no—
27C4⋊Dic71—41no—
28Dic71⋊C4—42no—
29C22⋊Dic71—41no—
30C4×Dic71—22no—
31C4×Dic71—41no—
32C4×Dic71—44no—
33C4×Dic71—41no—
34C71⋊C16—81no—
35C22×C4×C71—21no—
36C22×C4×C71C22⋊C411yescharacteristic
37M16×C71—41no—
38M16×C71—41no—
39C2×C8×C71—21no—
40C2×C8×C71—21no—
41C42×C71—21no—
42C42×C71C2×C811yescharacteristic
43C42×C71M1611yescharacteristic
44C16×C71—81no—
45C23×C71#10011yescharacteristic
46C2×C4×C71—21no—
47C2×C4×C71—21no—
48C2×C4×C71—21no—
49C2×C4×C71—21no—
50C2×C4×C71—42no—
51C2×C4×C71C22⋊C811yescharacteristic
52C2×C4×C71#9911yescharacteristic
53C2×Dic71—42no—
54C2×Dic71—44no—
55C2×Dic71—42no—
56Dic142—82no—
57C8×C71—41no—
58C8×C71—41no—
59C22×C71—21no—
60C22×C71—41no—
61C22×C71#32311yescharacteristic
62C4×C71—22no—
63C4×C71—21no—
64C4×C71—41no—
65C4×C71#35411yescharacteristic
66C4×C71#35611yescharacteristic
67Dic71—44no—
68Dic71—82no—
69#7361—711nomaximal, Sylow
70C2×C71—21no—
71C2×C71—41no—
72C2×C71#131911yescharacteristic
73#1994—711no—
74#1327—711no—
75#1327D7111yescharacteristic
76C71#736111yescharacteristic, complemented, Sylow
77#533—711no—
78#423C2×D7111yescharacteristic
79C43—711no—
80#349—21no—
81#349—1421no—
82#339—1421no—
83#339—1421no—
84C2×M16D28411yescharacteristic
85C2×M16C22⋊D7111yescharacteristic
86C4⋊Q8—1421no—
87#125—712no—
88C22⋊Q8—2841no—
89C4×Q8—2841no—
90#117—1421no—
91C2×C42—2841no—
92C2×C42—1421no—
93C2×C42C4×D7111yescharacteristic
94M32—41no—
95M32—2841no—
96C4⋊C8—41no—
97C2×Q8—2841no—
98C22×C4—21no—
99C22×C4—2841no—
100C22×C4(C2×D71)⋊C411yescharacteristic
101M16—41no—
102M16—41no—
103C2×C8—21no—
104C2×C8—21no—
105C4⋊C4—2842no—
106C4⋊C4—2841no—
107C4⋊C4—1422no—
108C22⋊C4—2841no—
109C42—21no—
110C42—2841no—
111C42—2844no—
112C42—2841no—
113C42—1422no—
114C42C8×D7111yescharacteristic
115C42#1176584811yescharacteristic
116C16—81no—
117C16—5681no—
118C23#1455604411yescharacteristic
119Q8—5682no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—42no—
125C2×C4—2842no—
126C2×C4—2844no—
127C2×C4—2842no—
128C2×C4(C2×D71)⋊C811yescharacteristic
129C2×C4#1455604211yescharacteristic
130C8—41no—
131C8—41no—
132C22—21no—
133C22—41no—
134C22#1798572211yescharacteristic
135C4—22no—
136C4—21no—
137C4—41no—
138C4—2844no—
139C4—5682no—
140C4#1798579311yescharacteristic
141C4#1798579711yescharacteristic
142C2—21no—
143C2—41no—
144C2#2258627311yescharacteristic
subgroupG/H
centerC4#17985793
commutatorC42×C71C2×C8
Frattini#423C2×D71
FittingC71×#1327C2
socleC2×C71#1319
prime radicalC23×C71#100
cosocle#423×C71C22
hypercenter#1327D71
hypocenterC71#7361
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #423 ◁ #1327

Complements

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

NH ≅ G/Ntype
76. C71#7361semidirect

Maximal subgroups

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

subgroupindex
#225893162
#225862912
C71×#13272
#736171