Finite Group Databasebeta

← #33000227

Subgroups — #33000227

Order 18688.

214 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 59 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#22612183C211yesmaximal, characteristic, complemented
2#22612012C211yesmaximal, characteristic
3#22611768C211yesmaximal, characteristic
4#22608006C211yesmaximal, characteristic, complemented
5#22607970C211yesmaximal, characteristic, complemented
6#22607969C211yesmaximal, characteristic, complemented
7C73×#2214C211yesmaximal, characteristic
8#17989027—21no—
9C2×#14556537—21no—
10C2×#14556537—21no—
11#17989004—21no—
12#17988967C2211yescharacteristic
13#17988408C2211yescharacteristic
14#17988408C2211yescharacteristic
15#17988402—21no—
16#17988371—21no—
17C4×#11924003C2211yescharacteristic
18#510×C73—21no—
19#494×C73C2211yescharacteristic
20#443×C73C2211yescharacteristic
21#364×C73C2211yescharacteristic
22#362×C73—21no—
23C2×Dic292—21no—
24C2×Dic292—21no—
25#14556591—21no—
26#14556537—42no—
27#14556537—41no—
28#14556537—41no—
29#14556537—41no—
30#14556535—41no—
31#14556531—21no—
32#14556531—21no—
33#14556531—41no—
34C2×#11924003—41no—
35C2×#11924003D411yescharacteristic
36C2×#11924003D411yescharacteristic
37#135×C73D411yescharacteristic
38C2×Q16×C73—21no—
39C2×Q16×C73—21no—
40#131×C73—21no—
41C4⋊Q8×C73—21no—
42Q32×C73—41no—
43SD32×C73—41no—
44M32×C73—21no—
45#108×C73D411yescharacteristic
46#104×C73—21no—
47C4×C8×C73C2311yescharacteristic
48C2×Dic146—41no—
49Dic292—41no—
50Dic292—41no—
51Dic292—41no—
52Dic292—41no—
53C4⋊Dic73—81no—
54#11924003—21no—
55#11924003—41no—
56#11924003—41no—
57#11924003D811yescharacteristic
58#11924003D811yescharacteristic
59#41×C73—21no—
60C2×Q8×C73—41no—
61Q16×C73—41no—
62Q16×C73—41no—
63Q16×C73—41no—
64Q16×C73—41no—
65Q16×C73D811yescharacteristic
66SD16×C73—21no—
67D8×C73D811yescharacteristic
68M16×C73—21no—
69M16×C73—41no—
70C2×C8×C73—41no—
71C2×C8×C73C2×D411yescharacteristic
72C2×C8×C73C2×D411yescharacteristic
73C4⋊C4×C73—81no—
74C42×C73#4111yescharacteristic
75C16×C73—41no—
76Q8×C73—21no—
77Q8×C73—81no—
78Q8×C73—81no—
79D4×C73—21no—
80D4×C73—41no—
81C2×C4×C73—21no—
82C2×C4×C73—41no—
83C2×C4×C73—81no—
84C2×C4×C73#12111yescharacteristic
85C2×Dic73—81no—
86Dic146—81no—
87Dic146—81no—
88C8×C73—41no—
89C8×C73C2×D811yescharacteristic
90C8×C73C2×D811yescharacteristic
91C8×C73#13511yescharacteristic
92C8×C73AGL(1,ℤ/8)11yescharacteristic
93C22×C73—41no—
94C22×C73#47211yescharacteristic
95C4×C73—22no—
96C4×C73—41no—
97C4×C73—161no—
98C4×C73#45911yescharacteristic
99C4×C73#46611yescharacteristic
100Dic73—161no—
101#12437—731nomaximal, Sylow
102C2×C73—21no—
103C2×C73—81no—
104C2×C73#166711yescharacteristic
105#2247—731no—
106#2214—731no—
107#2214D7311yescharacteristic
108#2170—731no—
109#1373—731no—
110#1358—731no—
111#1358—731no—
112C73#1243711yescharacteristic, complemented, Sylow
113#510—21no—
114#510—1461no—
115C2×Q32—1461no—
116C2×Q32—1461no—
117#504—1461no—
118#494—731no—
119#494C2×D7311yescharacteristic
120#443C2×D7311yescharacteristic
121#364—731no—
122#364—731no—
123#364C2×D7311yescharacteristic
124#362—21no—
125#362—1461no—
126#350—1461no—
127C4×C16—731no—
128#135C22⋊D7311yescharacteristic
129C2×Q16—21no—
130C2×Q16—21no—
131C2×Q16—1461no—
132C2×Q16—1461no—
133#131—21no—
134C4⋊Q8—21no—
135C4⋊Q8—1461no—
136Q32—41no—
137Q32—2921no—
138Q32—2921no—
139Q32—2921no—
140Q32—2922no—
141SD32—41no—
142SD32—2921no—
143M32—21no—
144M32—2921no—
145M32—1461no—
146M32—1461no—
147C2×C16—2921no—
148C2×C16—731no—
149C2×C16—731no—
150#108C22⋊D7311yescharacteristic
151#104—21no—
152C4×C8C22×D7311yescharacteristic
153#41—21no—
154C2×Q8—41no—
155C2×Q8—2921no—
156Q16—41no—
157Q16—41no—
158Q16—41no—
159Q16—41no—
160Q16—2921no—
161Q16—2921no—
162Q16—2921no—
163Q16—2921no—
164Q16#1192401311yescharacteristic
165SD16—21no—
166D8#1192401311yescharacteristic
167M16—21no—
168M16—41no—
169C2×C8—41no—
170C2×C8D4×D7311yescharacteristic
171C2×C8C2×C22⋊D7311yescharacteristic
172C4⋊C4—81no—
173C4⋊C4—5841no—
174C42#1192402711yescharacteristic
175C16—41no—
176C16—2921no—
177C16—2921no—
178C16—731no—
179C16—731no—
180C16—1461no—
181Q8—21no—
182Q8—81no—
183Q8—81no—
184Q8—5841no—
185Q8—5841no—
186D4—21no—
187D4—41no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—81no—
191C2×C4—5841no—
192C2×C4#1455656311yescharacteristic
193C8—41no—
194C8D8×D7311yescharacteristic
195C8C2×#1192401311yescharacteristic
196C8#1455661111yescharacteristic
197C8#1455661511yescharacteristic
198C22—41no—
199C22#1798885511yescharacteristic
200C4—22no—
201C4—41no—
202C4—161no—
203C4—11681no—
204C4#1798878011yescharacteristic
205C4#1798882111yescharacteristic
206C2—21no—
207C2—81no—
208C2#2260928411yescharacteristic
subgroupG/H
centerC2#22609284
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#2214C2
socleC2×C73#1667
prime radical#135×C73D4
hypercenter#2214D73
hypocenterC73#12437
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2214

Complements

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

NH ≅ G/Ntype
1. #22612183C2semidirect
4. #22608006C2semidirect
5. #22607970C2semidirect
6. #22607969C2semidirect
112. C73#12437semidirect

Maximal subgroups

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

subgroupindex
#226121832
#226120122
#226117682
#226080062
#226079702
#226079692
C73×#22142
#1243773