Finite Group Databasebeta

← #34001010

Subgroups — #34001010

Order 18688.

198 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 36 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#22624874C211yesmaximal, characteristic
2#22624809C211yesmaximal, characteristic
3#22611273C211yesmaximal, characteristic
4C2×#14556658—21no—
5#17989432—21no—
6#17988630—21no—
7#17988456C2211yescharacteristic
8#17988356C411yescharacteristic
9#17988346—21no—
10#544×C73C411yescharacteristic
11C22×#10596833—41no—
12#14556658—22no—
13#14556658—41no—
14C22⋊#10596833—41no—
15C2×#11924012—21no—
16C2×C4⋊Dic73D411yescharacteristic
17C2×#11924007—41no—
18C2×C4×Dic73D411yescharacteristic
19#14556524—41no—
20#14556522—41no—
21C4⋊Q8×C73—41no—
22#125×C73—41no—
23C22⋊Q8×C73—41no—
24#117×C73—21no—
25C2×C4⋊C4×C73C2×C411yescharacteristic
26C2×#10596833—42no—
27C2×#10596833—81no—
28C2×#10596833—81no—
29C2×#10596833—81no—
30C22×Dic73—21no—
31C22×Dic73—21no—
32#11924012—22no—
33#11924012—41no—
34C2×#10596831—41no—
35C4⋊Dic73—41no—
36#11924007—81no—
37#11924007—81no—
38C4×Dic73—21no—
39C4×Dic73SD1611yescharacteristic
40C4×Dic73Q1611yescharacteristic
41C2×Q8×C73—41no—
42C22×C4×C73—21no—
43C22×C4×C73C22⋊C411yescharacteristic
44C4⋊C4×C73—21no—
45C4⋊C4×C73—21no—
46C4⋊C4×C73—41no—
47C4⋊C4×C73—41no—
48C4⋊C4×C73—42no—
49C22⋊C4×C73—41no—
50C42×C73—41no—
51C23×C73#9911yescharacteristic
52Q8×C73—161no—
53C2×C4×C73—21no—
54C2×C4×C73—21no—
55C2×C4×C73—41no—
56C2×C4×C73—41no—
57C2×C4×C73—41no—
58C2×C4×C73—81no—
59C2×C4×C73#10311yescharacteristic
60C2×C4×C73#10411yescharacteristic
61C2×Dic73—21no—
62C2×Dic73—21no—
63C2×Dic73—21no—
64C2×Dic73—21no—
65C2×Dic73—21no—
66C2×Dic73—21no—
67C2×Dic73—41no—
68C2×Dic73—81no—
69#10596833—82no—
70#10596833—161no—
71#10596831—42no—
72C22×C73—21no—
73C22×C73—21no—
74C22×C73#32811yescharacteristic
75C22×C73AΣL(1,𝔽16)11yescharacteristic
76C22×C73#35611yescharacteristic
77C4×C73—21no—
78C4×C73—21no—
79C4×C73—81no—
80C4×C73—81no—
81C4×C73—81no—
82Dic73—21no—
83Dic73—21no—
84Dic73—21no—
85Dic73—21no—
86Dic73—81no—
87#7443—731nomaximal, Sylow
88C2×C73—21no—
89C2×C73—21no—
90C2×C73#135411yescharacteristic
91C2×C73#140211yescharacteristic
92C2×C73#140511yescharacteristic
93#2060—731no—
94#1325—731no—
95#1296—731no—
96C73#744311yescharacteristic, complemented, Sylow
97#544#6440811yescharacteristic
98#428—1461no—
99C2×C4⋊C8—1461no—
100#384—731no—
101#343—731no—
102#339—1461no—
103#324—1461no—
104C2×M16—1461no—
105C22×C8—2921no—
106C4⋊Q8—41no—
107#125—41no—
108C22⋊Q8—41no—
109#117—21no—
110C2×C4⋊C4—2921no—
111C2×C4⋊C4—731no—
112C2×C4⋊C4C2×#6440811yescharacteristic
113C2×C42—731no—
114#107—2921no—
115#106—2921no—
116C4⋊C8—2921no—
117C4⋊C8—1462no—
118C22⋊C8—2921no—
119C2×Q8—41no—
120C22×C4—21no—
121C22×C4—1461no—
122C22×C4—1461no—
123C22×C4C22⋊#6440811yescharacteristic
124M16—2921no—
125M16—1462no—
126C2×C8—2921no—
127C2×C8—2922no—
128C2×C8—5841no—
129C2×C8—5841no—
130C2×C8—5841no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—42no—
136C4⋊C4—2921no—
137C4⋊C4—5841no—
138C4⋊C4—5841no—
139C22⋊C4—41no—
140C42—41no—
141C42—731no—
142C42—731no—
143C42—1461no—
144C23#1455664911yescharacteristic
145Q8—161no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—81no—
152C2×C4—2921no—
153C2×C4—5841no—
154C2×C4—1461no—
155C2×C4—1461no—
156C2×C4—1461no—
157C2×C4—1461no—
158C2×C4—1461no—
159C2×C4—1461no—
160C2×C4#1455665411yescharacteristic
161C2×C4#1455665611yescharacteristic
162C8—2922no—
163C8—5842no—
164C8—11681no—
165C22—21no—
166C22—21no—
167C22#1798944111yescharacteristic
168C22#1798948711yescharacteristic
169C22#1798949711yescharacteristic
170C4—21no—
171C4—21no—
172C4—81no—
173C4—81no—
174C4—81no—
175C4—5841no—
176C4—1461no—
177C4—1461no—
178C4—1461no—
179C4—1461no—
180C2—21no—
181C2—21no—
182C2#2262493211yescharacteristic
183C2#2262505011yescharacteristic
184C2#2262505611yescharacteristic
subgroupG/H
centerC22#17989487
commutatorC2×C4⋊C4×C73C2×C4
FrattiniC2×C4⋊C4C2×#64408
Fitting#544×C73C4
socleC22×C73AΣL(1,𝔽16)
prime radicalC23×C73#99
cosocle#17988456C22
hypercenter#544#64408
hypocenterC73#7443
LCSG ▷ C2×C4⋊C4×C73 ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4 ◁ #544

Complements

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

NH ≅ G/Ntype
96. C73#7443semidirect

Maximal subgroups

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

subgroupindex
#226248742
#226248092
#226112732
#744373