Finite Group Databasebeta

← #31884822

Subgroups — #31884822

Order 18176.

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

normal subgroups only

This group has 200 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#22588415C211yesmaximal, characteristic
2#22586270C211yesmaximal, characteristic
3C71×#1317C211yesmaximal, characteristic
4C2×#14556087C411yescharacteristic
5#17985862C411yescharacteristic
6#17985765—21no—
7#17985765—21no—
8#17985723—21no—
9#381×C71C2211yescharacteristic
10#323×C71—21no—
11#14556087—41no—
12#14556087—41no—
13C2×C4⋊Dic71—21no—
14C2×C22⋊Dic71—21no—
15C2×C22⋊Dic71—21no—
16C2×C4×Dic71C812yescomplemented
17C22⋊(C71⋊C8)—41no—
18C71⋊#95—41no—
19C23×C4×C71C2×C411yescharacteristic
20C2×C22⋊C4×C71D411yescharacteristic
21C2×C22⋊C4×C71D411yescharacteristic
22C22⋊C8×C71—41no—
23#95×C71—41no—
24C22×Dic71—22no—
25C22×Dic71—41no—
26C2×C71⋊C8—81no—
27C4⋊Dic71—42no—
28C22⋊Dic71—41no—
29C22⋊Dic71—41no—
30C22⋊Dic71—41no—
31C22⋊Dic71—41no—
32C4×Dic71—42no—
33C24×C71C22⋊C411yescharacteristic
34C22×C4×C71—21no—
35C22×C4×C71—21no—
36C22×C4×C71—21no—
37C22×C4×C71—21no—
38C22×C4×C71—41no—
39C22×C4×C71—41no—
40C22×C4×C71C2×C811yescharacteristic
41C22×C4×C71M1611yescharacteristic
42C2×C8×C71—81no—
43C22⋊C4×C71—41no—
44C22⋊C4×C71—41no—
45C23×C71—21no—
46C23×C71—21no—
47C23×C71—81no—
48C23×C71C22⋊C811yescharacteristic
49C23×C71#9911yescharacteristic
50C23×C71#10011yescharacteristic
51C2×C4×C71—21no—
52C2×C4×C71—21no—
53C2×C4×C71—42no—
54C2×C4×C71—41no—
55C2×C4×C71—41no—
56C2×C4×C71—41no—
57C2×C4×C71—41no—
58C2×C4×C71—41no—
59C2×C4×C71—41no—
60C2×C4×C71—81no—
61C2×C4×C71—81no—
62C2×Dic71—42no—
63C2×Dic71—82no—
64C2×Dic71—81no—
65C2×Dic71—82no—
66C8×C71—161no—
67C71⋊C8—161no—
68C22×C71—21no—
69C22×C71—21no—
70C22×C71—41no—
71C22×C71—41no—
72C22×C71—41no—
73C22×C71—81no—
74C22×C71—81no—
75C22×C71#32311yescharacteristic
76C22×C71#35211yescharacteristic
77C22×C71#35211yescharacteristic
78C4×C71—41no—
79C4×C71—41no—
80C4×C71—81no—
81C4×C71—81no—
82Dic71—82no—
83Dic71—161no—
84#7348—711nomaximal, Sylow
85C2×C71—21no—
86C2×C71—21no—
87C2×C71—21no—
88C2×C71—41no—
89C2×C71—41no—
90C2×C71#131711yescharacteristic
91#1820—711no—
92#1317—711no—
93#1317D7111yescharacteristic
94C71#734811yescharacteristic, complemented, Sylow
95C2×#117—711no—
96#381—711no—
97#381C2×D7111yescharacteristic
98#342—1421no—
99#342—1421no—
100#323—21no—
101#323—1421no—
102C23×C4C4×D7111yescharacteristic
103#117—2841no—
104#117—2841no—
105C2×C4⋊C4—1421no—
106C2×C22⋊C4—1421no—
107C2×C22⋊C4—1421no—
108C2×C22⋊C4D28411yescharacteristic
109C2×C22⋊C4C22⋊D7111yescharacteristic
110C2×C42—712no—
111C22⋊C8—41no—
112C22⋊C8—2841no—
113#95—41no—
114#95—2841no—
115C24(C2×D71)⋊C411yescharacteristic
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—2841no—
123C22×C4—1422no—
124C22×C4C8×D7111yescharacteristic
125C22×C4#1176584811yescharacteristic
126C2×C8—81no—
127C2×C8—5681no—
128C4⋊C4—2842no—
129C22⋊C4—41no—
130C22⋊C4—41no—
131C22⋊C4—2841no—
132C22⋊C4—2841no—
133C22⋊C4—2841no—
134C22⋊C4—2841no—
135C42—2842no—
136C23—21no—
137C23—21no—
138C23—81no—
139C23(C2×D71)⋊C811yescharacteristic
140C23#1455604211yescharacteristic
141C23#1455604411yescharacteristic
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—81no—
152C2×C4—81no—
153C2×C4—2842no—
154C2×C4—5682no—
155C2×C4—5681no—
156C2×C4—5682no—
157C8—161no—
158C8—11361no—
159C22—21no—
160C22—21no—
161C22—41no—
162C22—41no—
163C22—41no—
164C22—81no—
165C22—81no—
166C22#1798572211yescharacteristic
167C22#1798578811yescharacteristic
168C22#1798579011yescharacteristic
169C4—41no—
170C4—41no—
171C4—81no—
172C4—81no—
173C4—5682no—
174C4—11361no—
175C2—21no—
176C2—21no—
177C2—21no—
178C2—41no—
179C2—41no—
180C2#2258626911yescharacteristic
subgroupG/H
centerC2#22586269
commutatorC22×C4×C71C2×C8
Frattini#381C2×D71
FittingC71×#1317C2
socleC2×C71#1317
prime radicalC24×C71C22⋊C4
cosocle#381×C71C22
hypercenter#1317D71
hypocenterC71#7348
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ #381 ◁ #1317

Complements

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

NH ≅ G/Ntype
16. C2×C4×Dic71C8semidirect
94. C71#7348semidirect

Maximal subgroups

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

subgroupindex
#225884152
#225862702
C71×#13172
#734871