Finite Group Databasebeta

← #31894982

Subgroups — #31894982

Order 18176.

436 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 95 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#22598716C211yesmaximal, characteristic
2#22597446C211yesmaximal, characteristic, complemented
3#22587053C212yesmaximal
4#22586213C212yesmaximal, complemented
5C71×#1537C211yesmaximal, characteristic, complemented
6C2×#14556125C2211yescharacteristic
7#17986365—22no—
8#17986347C414yes—
9#17986130—24no—
10#17986060—22no—
11#17986056C2212yes—
12#17986003—22no—
13C4×#11765849C2211yescharacteristic
14#17985742—24no—
15#533×C71C2211yescharacteristic
16#432×C71C2212yes—
17#330×C71—22no—
18C22×D284—21no—
19C22×D284—22no—
20C2×#11765849—21no—
21#14556125—21no—
22#14556125—22no—
23#14556125C2×C412yes—
24C4⋊(C71⋊C8)—24no—
25C4⋊(C71⋊C8)D414yes—
26#14556048—28no—
27C22⋊(C71⋊C8)—22no—
28#14556039—22no—
29#14556039—22no—
30#14556039—21no—
31C4×C71⋊C8—22no—
32C4×C71⋊C8C2×C412yes—
33C4⋊Q8×C71C2×C412yes—
34#125×C71D412yes—
35C22⋊Q8×C71—22no—
36C4×Q8×C71—22no—
37#117×C71D412yes—
38C2×C42×C71C2311yescharacteristic
39C4⋊C8×C71—22no—
40C22⋊C8×C71—22no—
41#97×C71—22no—
42C23×D71—41no—
43C2×D284—22no—
44C2×D284—24no—
45C2×D284—44no—
46C2×D284—41no—
47C2×D284—42no—
48C2×D284—84no—
49#11765849—42no—
50C2×C71⋊C8—22no—
51C2×C71⋊C8—24no—
52C2×Q8×C71—22no—
53C22×C4×C71C2×D411yescharacteristic
54C22×C4×C71#4112yes—
55C2×C8×C71—42no—
56C4⋊C4×C71—22no—
57C4⋊C4×C71—24no—
58C4⋊C4×C71C22⋊C414yes—
59C22⋊C4×C71—22no—
60C42×C71—22no—
61C42×C71C22×C411yescharacteristic
62C42×C71C2×D412yes—
63C42×C71C2×D411yescharacteristic
64C23×C71#12311yescharacteristic
65C22×D71—42no—
66C22×D71—82no—
67C22×D71—84no—
68Q8×C71—82no—
69C2×C4×C71—22no—
70C2×C4×C71—24no—
71C2×C4×C71—42no—
72C2×C4×C71C2×C22⋊C411yescharacteristic
73C2×C4×C71C4×D412yes—
74C2×C4×C71#12011yescharacteristic
75C2×C4×C71#12112yes—
76D284—44no—
77D284—84no—
78D284—84no—
79D284—82no—
80C8×C71—82no—
81C71⋊C8—42no—
82C71⋊C8—44no—
83C22×C71—21no—
84C22×C71—22no—
85C22×C71#38611yescharacteristic
86C2×D71—82no—
87C2×D71—84no—
88C2×D71—164no—
89C4×C71—21no—
90C4×C71—22no—
91C4×C71—22no—
92C4×C71—44no—
93C4×C71#45412yes—
94#9424—711nomaximal, Sylow
95C2×C71—41no—
96C2×C71#188811yescharacteristic
97C2×C71#190312yes—
98D71—162no—
99#3138—711no—
100#2987—711no—
101#1555—712no—
102#1537D7111yescharacteristic, complemented
103#1291—712no—
104C71#942411yescharacteristic, complemented, Sylow
105#533C2×D7111yescharacteristic
106C2×#127—711no—
107#490—1422no—
108#486—714no—
109#446—1424no—
110#433—1422no—
111#432—712no—
112#432C2×D7112yes—
113#418—1422no—
114C4×M16—711no—
115#331—1424no—
116#330—22no—
117C22×D4—1421no—
118C22×D4—1422no—
119C2×M16—1421no—
120C4⋊Q8C4×D7112yes—
121#127—712no—
122#127—1422no—
123#127—1421no—
124#125C22⋊D7112yes—
125C22⋊Q8—22no—
126C4×Q8—22no—
127#117D28412yes—
128C2×C42C22×D7111yescharacteristic
129C4⋊C8—22no—
130C4⋊C8—714no—
131C4⋊C8—1424no—
132#102—1428no—
133C22⋊C8—22no—
134C22⋊C8—1422no—
135#97—22no—
136#97—1422no—
137#97—1422no—
138#97—1421no—
139C4×C8—712no—
140C4×C8—1422no—
141C24—2841no—
142C2×Q8—22no—
143C2×D4—2844no—
144C2×D4—2841no—
145C2×D4—2842no—
146C2×D4—5684no—
147C2×D4—1422no—
148C2×D4—1424no—
149C22×C4C2×D28411yescharacteristic
150C22×C4#1176586412yes—
151M16—2842no—
152C2×C8—42no—
153C2×C8—1422no—
154C2×C8—1424no—
155C4⋊C4—22no—
156C4⋊C4—24no—
157C4⋊C4(C2×D71)⋊C414yes—
158C22⋊C4—22no—
159C42—22no—
160C42C2×C4×D7111yescharacteristic
161C42D4×D7112yes—
162C42C2×C22⋊D7111yescharacteristic
163C23—2842no—
164C23—5682no—
165C23—5684no—
166C23#1455611211yescharacteristic
167Q8—82no—
168D4—2844no—
169D4—5684no—
170D4—5684no—
171D4—5682no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—42no—
175C2×C4C2×(C2×D71)⋊C411yescharacteristic
176C2×C4(C22⋊D71)⋊C412yes—
177C2×C4#1455609811yescharacteristic
178C2×C4#1455610012yes—
179C8—82no—
180C8—2842no—
181C8—2844no—
182C22—21no—
183C22—22no—
184C22—5682no—
185C22—5684no—
186C22—11364no—
187C22#1798587611yescharacteristic
188C4—21no—
189C4—22no—
190C4—22no—
191C4—44no—
192C4#1798617412yes—
193C2—41no—
194C2—11362no—
195C2#2258874111yescharacteristic
196C2#2258881812yes—
subgroupG/H
centerC22#17985876
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1537C2
socleC22×C71#386
prime radicalC2×#14556125C22
cosocleC2×C42×C71C23
hypercenter#1537D71
hypocenterC71#9424
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1537

Complements

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

NH ≅ G/Ntype
2. #22597446C2semidirect
4. #22586213C2semidirect
5. C71×#1537C2semidirect
102. #1537D71semidirect
104. C71#9424semidirect

Maximal subgroups

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

subgroupindex
#225987162
#225974462
#225870532
#225862132
C71×#15372
#942471