Finite Group Databasebeta

← #33006338

Subgroups — #33006338

Order 18688.

326 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 79 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#22611596C212yesmaximal
2#22608102C214yesmaximal, complemented
3C73×#3142C211yesmaximal, characteristic
4#17988561—22no—
5#17988382—22no—
6#17988380C2214yes—
7#532×C73C2211yescharacteristic
8C2×#125×C73C411yescharacteristic, complemented
9#491×C73C412yescomplemented
10#490×C73C2212yes—
11#488×C73—22no—
12#417×C73—22no—
13C4×M16×C73C411yescharacteristic, complemented
14#14556550—42no—
15#14556506—24no—
16#14556506—42no—
17C2×#41×C73D412yes—
18C2×M16×C73—21no—
19C4⋊Q8×C73C2×C411yescharacteristic
20#127×C73C2311yescharacteristic
21#125×C73—21no—
22#125×C73—41no—
23#125×C73C2×C412yes—
24#124×C73D412yes—
25#121×C73—41no—
26C4×D4×C73—41no—
27C2×C4⋊C4×C73—21no—
28C2×C4⋊C4×C73—41no—
29C2×C42×C73C2×C411yescharacteristic
30#103×C73—24no—
31#102×C73—24no—
32#97×C73C2×C412yes—
33C4×C8×C73—21no—
34C4⋊Dic73—82no—
35C22⋊Dic73—44no—
36C4×Dic73—82no—
37#41×C73—42no—
38#41×C73—42no—
39C2×Q8×C73C22⋊C412yes—
40C2×D4×C73—22no—
41C2×D4×C73—41no—
42C2×D4×C73C2×D412yes—
43C22×C4×C73—21no—
44C22×C4×C73—22no—
45C22×C4×C73—41no—
46C22×C4×C73C22⋊C411yescharacteristic
47M16×C73—42no—
48C2×C8×C73—22no—
49C4⋊C4×C73—22no—
50C4⋊C4×C73—41no—
51C4⋊C4×C73—42no—
52C4⋊C4×C73—42no—
53C4⋊C4×C73—41no—
54C22⋊C4×C73—42no—
55C42×C73—21no—
56C42×C73C22⋊C411yescharacteristic
57C42×C73C22×C411yescharacteristic
58C23×C73—22no—
59C23×C73#9911yescharacteristic
60Q8×C73—42no—
61D4×C73—42no—
62D4×C73—82no—
63D4×C73—82no—
64C2×C4×C73—22no—
65C2×C4×C73—21no—
66C2×C4×C73—21no—
67C2×C4×C73—22no—
68C2×C4×C73—42no—
69C2×C4×C73—42no—
70C2×C4×C73—42no—
71C2×C4×C73—81no—
72C2×C4×C73—82no—
73C2×C4×C73#9911yescharacteristic
74C2×C4×C73C2×C22⋊C411yescharacteristic
75C2×Dic73—84no—
76C8×C73—42no—
77C22×C73—21no—
78C22×C73—41no—
79C22×C73—42no—
80C22×C73—82no—
81C22×C73C2×#9911yescharacteristic
82C4×C73—21no—
83C4×C73—41no—
84C4×C73—41no—
85C4×C73—42no—
86C4×C73—82no—
87C4×C73#35412yes—
88Dic73—164no—
89#13596—731nomaximal, Sylow
90C2×C73—21no—
91C2×C73—41no—
92C2×C73—82no—
93C2×C73C2×#35411yescharacteristic
94#3142D7311yescharacteristic
95#2126—732no—
96#1412—734no—
97C73#1359611yescharacteristic, complemented, Sylow
98#532C2×D7311yescharacteristic
99C2×#125Dic7311yescharacteristic, complemented
100#491Dic7312yescomplemented
101#490C2×D7312yes—
102#488—22no—
103#417—22no—
104#410—1462no—
105C4×M16Dic7311yescharacteristic, complemented
106#354—1462no—
107#353—734no—
108C2×#41C22⋊D7312yes—
109C2×M16—21no—
110C4⋊Q8C2×Dic7311yescharacteristic
111#127C22×D7311yescharacteristic
112#125—21no—
113#125—41no—
114#125C2×Dic7312yes—
115#124C22⋊D7312yes—
116#121—41no—
117C4×D4—41no—
118#117—2922no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—41no—
121C2×C42C2×Dic7311yescharacteristic
122#103—24no—
123#102—24no—
124#99—2922no—
125#99—1464no—
126#97C2×Dic7312yes—
127C4×C8—21no—
128#41—42no—
129#41—42no—
130C2×Q8C22⋊Dic7312yes—
131C2×D4—22no—
132C2×D4—41no—
133C2×D4C2×C22⋊D7312yes—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—41no—
137C22×C4C22⋊Dic7311yescharacteristic
138M16—42no—
139C2×C8—22no—
140C4⋊C4—22no—
141C4⋊C4—41no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—41no—
145C4⋊C4—5842no—
146C22⋊C4—42no—
147C22⋊C4—2924no—
148C42—21no—
149C42—5842no—
150C42C22⋊Dic7311yescharacteristic
151C42C22×Dic7311yescharacteristic
152C23—22no—
153C23#1455650611yescharacteristic
154Q8—42no—
155D4—42no—
156D4—82no—
157D4—82no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—81no—
166C2×C4—82no—
167C2×C4—5844no—
168C2×C4#1455650611yescharacteristic
169C2×C4C2×C22⋊Dic7311yescharacteristic
170C8—42no—
171C22—21no—
172C22—41no—
173C22—42no—
174C22—82no—
175C22C2×#1455650611yescharacteristic
176C4—21no—
177C4—41no—
178C4—41no—
179C4—42no—
180C4—82no—
181C4—11684no—
182C4#1798838212yes—
183C2—21no—
184C2—41no—
185C2—82no—
186C2C2×#1798838211yescharacteristic
subgroupG/H
centerC2C2×#17988382
commutatorC42×C73C22×C4
Frattini#127C22×D73
FittingC73×#3142C2
socleC2×C73C2×#354
prime radical#532×C73C22
cosocle#127×C73C23
hypercenter#3142D73
hypocenterC73#13596
LCSG ▷ C42×C73 ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3142

Complements

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

NH ≅ G/Ntype
2. #22608102C2semidirect
8. C2×#125×C73C4semidirect
9. #491×C73C4semidirect
13. C4×M16×C73C4semidirect
97. C73#13596semidirect
99. C2×#125Dic73semidirect
100. #491Dic73semidirect
105. C4×M16Dic73semidirect

Maximal subgroups

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

subgroupindex
#226115962
#226081022
C73×#31422
#1359673