Finite Group Databasebeta

← #32983514

Subgroups — #32983514

Order 18688.

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#22620307C211yesmaximal, characteristic
2#22619037C211yesmaximal, characteristic, complemented
3#22608644C212yesmaximal
4#22607804C212yesmaximal, complemented
5C73×#1537C211yesmaximal, characteristic, complemented
6C2×#14556588C2211yescharacteristic
7#17988953—22no—
8#17988935C414yes—
9#17988718—24no—
10#17988648—22no—
11#17988644C2212yes—
12#17988591—22no—
13C4×#11924012C2211yescharacteristic
14#17988330—24no—
15#533×C73C2211yescharacteristic
16#432×C73C2212yes—
17#330×C73—22no—
18C22×D292—21no—
19C22×D292—22no—
20C2×#11924012—21no—
21#14556588—21no—
22#14556588—22no—
23#14556588C2×C412yes—
24C4⋊#10596831—24no—
25C4⋊#10596831D414yes—
26#14556511—28no—
27C22⋊#10596831—22no—
28#14556502—22no—
29#14556502—22no—
30#14556502—21no—
31C4×#10596831—22no—
32C4×#10596831C2×C412yes—
33C4⋊Q8×C73C2×C412yes—
34#125×C73D412yes—
35C22⋊Q8×C73—22no—
36C4×Q8×C73—22no—
37#117×C73D412yes—
38C2×C42×C73C2311yescharacteristic
39C4⋊C8×C73—22no—
40C22⋊C8×C73—22no—
41#97×C73—22no—
42C23×D73—41no—
43C2×D292—22no—
44C2×D292—24no—
45C2×D292—44no—
46C2×D292—41no—
47C2×D292—42no—
48C2×D292—84no—
49#11924012—42no—
50C2×#10596831—22no—
51C2×#10596831—24no—
52C2×Q8×C73—22no—
53C22×C4×C73C2×D411yescharacteristic
54C22×C4×C73#4112yes—
55C2×C8×C73—42no—
56C4⋊C4×C73—22no—
57C4⋊C4×C73—24no—
58C4⋊C4×C73C22⋊C414yes—
59C22⋊C4×C73—22no—
60C42×C73—22no—
61C42×C73C22×C411yescharacteristic
62C42×C73C2×D412yes—
63C42×C73C2×D411yescharacteristic
64C23×C73#12311yescharacteristic
65C22×D73—42no—
66C22×D73—82no—
67C22×D73—84no—
68Q8×C73—82no—
69C2×C4×C73—22no—
70C2×C4×C73—24no—
71C2×C4×C73—42no—
72C2×C4×C73C2×C22⋊C411yescharacteristic
73C2×C4×C73C4×D412yes—
74C2×C4×C73#12011yescharacteristic
75C2×C4×C73#12112yes—
76D292—44no—
77D292—84no—
78D292—84no—
79D292—82no—
80C8×C73—82no—
81#10596831—42no—
82#10596831—44no—
83C22×C73—21no—
84C22×C73—22no—
85C22×C73#38611yescharacteristic
86C2×D73—82no—
87C2×D73—84no—
88C2×D73—164no—
89C4×C73—21no—
90C4×C73—22no—
91C4×C73—22no—
92C4×C73—44no—
93C4×C73#45412yes—
94#9424—731nomaximal, Sylow
95C2×C73—41no—
96C2×C73#188811yescharacteristic
97C2×C73#190312yes—
98D73—162no—
99#3138—731no—
100#2987—731no—
101#1555—732no—
102#1537D7311yescharacteristic, complemented
103#1291—732no—
104C73#942411yescharacteristic, complemented, Sylow
105#533C2×D7311yescharacteristic
106C2×#127—731no—
107#490—1462no—
108#486—734no—
109#446—1464no—
110#433—1462no—
111#432—732no—
112#432C2×D7312yes—
113#418—1462no—
114C4×M16—731no—
115#331—1464no—
116#330—22no—
117C22×D4—1461no—
118C22×D4—1462no—
119C2×M16—1461no—
120C4⋊Q8C4×D7312yes—
121#127—732no—
122#127—1462no—
123#127—1461no—
124#125C22⋊D7312yes—
125C22⋊Q8—22no—
126C4×Q8—22no—
127#117D29212yes—
128C2×C42C22×D7311yescharacteristic
129C4⋊C8—22no—
130C4⋊C8—734no—
131C4⋊C8—1464no—
132#102—1468no—
133C22⋊C8—22no—
134C22⋊C8—1462no—
135#97—22no—
136#97—1462no—
137#97—1462no—
138#97—1461no—
139C4×C8—732no—
140C4×C8—1462no—
141C24—2921no—
142C2×Q8—22no—
143C2×D4—2924no—
144C2×D4—2921no—
145C2×D4—2922no—
146C2×D4—5844no—
147C2×D4—1462no—
148C2×D4—1464no—
149C22×C4C2×D29211yescharacteristic
150C22×C4#1192402712yes—
151M16—2922no—
152C2×C8—42no—
153C2×C8—1462no—
154C2×C8—1464no—
155C4⋊C4—22no—
156C4⋊C4—24no—
157C4⋊C4#1192400514yes—
158C22⋊C4—22no—
159C42—22no—
160C42C2×C4×D7311yescharacteristic
161C42D4×D7312yes—
162C42C2×C22⋊D7311yescharacteristic
163C23—2922no—
164C23—5842no—
165C23—5844no—
166C23#1455657511yescharacteristic
167Q8—82no—
168D4—2924no—
169D4—5844no—
170D4—5844no—
171D4—5842no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—42no—
175C2×C4C2×#1192400511yescharacteristic
176C2×C4#1455655112yes—
177C2×C4#1455656111yescharacteristic
178C2×C4#1455656312yes—
179C8—82no—
180C8—2922no—
181C8—2924no—
182C22—21no—
183C22—22no—
184C22—5842no—
185C22—5844no—
186C22—11684no—
187C22#1798846411yescharacteristic
188C4—21no—
189C4—22no—
190C4—22no—
191C4—44no—
192C4#1798876212yes—
193C2—41no—
194C2—11682no—
195C2#2261033211yescharacteristic
196C2#2261040912yes—
subgroupG/H
centerC22#17988464
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1537C2
socleC22×C73#386
prime radicalC2×#14556588C22
cosocleC2×C42×C73C23
hypercenter#1537D73
hypocenterC73#9424
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1537

Complements

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

NH ≅ G/Ntype
2. #22619037C2semidirect
4. #22607804C2semidirect
5. C73×#1537C2semidirect
102. #1537D73semidirect
104. C73#9424semidirect

Maximal subgroups

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

subgroupindex
#226203072
#226190372
#226086442
#226078042
C73×#15372
#942473