Finite Group Databasebeta

← #34000994

Subgroups — #34000994

Order 18688.

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

normal subgroups only

This group has 224 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#22625025C211yesmaximal, characteristic
2#22624860C211yesmaximal, characteristic
3#22611465C211yesmaximal, characteristic
4C2×#14556650—21no—
5#17989432—21no—
6C2×#14556522—21no—
7C2×#14556511—21no—
8#17988489C2211yescharacteristic
9#17988342C411yescharacteristic, complemented
10#384×C73C411yescharacteristic, complemented
11C2×C22⋊#64408—41no—
12#14556650—22no—
13#14556650—41no—
14C22⋊#10596833—41no—
15C22×D292D411yescharacteristic
16C22×#10596831—21no—
17C2×C4⋊Dic73D411yescharacteristic
18C2×#11924005—41no—
19#14556522—42no—
20#14556511—42no—
21C2×C4⋊C4×C73—21no—
22C2×C4⋊C4×C73—41no—
23C2×C42×C73C2×C411yescharacteristic
24C22×#64408—81no—
25C2×#10596833—81no—
26C22⋊#64408—42no—
27C22⋊#64408—81no—
28C23×D73—21no—
29C2×D292—21no—
30C2×D292—41no—
31C2×D292—41no—
32C2×D292SD1611yescharacteristic
33C2×D292Q1611yescharacteristic
34C22×Dic73—21no—
35C2×#10596831—22no—
36C2×#10596831—41no—
37C2×#10596831—41no—
38C4⋊Dic73—41no—
39#11924005—82no—
40C22×C4×C73—21no—
41C22×C4×C73—41no—
42C22×C4×C73C22⋊C411yescharacteristic
43C4⋊C4×C73—41no—
44C4⋊C4×C73—41no—
45C4⋊C4×C73—82no—
46C42×C73—41no—
47C23×C73#9911yescharacteristic
48C22×D73—21no—
49C22×D73—21no—
50C22×D73—41no—
51C22×D73—41no—
52C22×D73—81no—
53C22×D73—81no—
54C2×#64408—82no—
55C2×#64408—161no—
56C2×#64408—161no—
57C2×C4×C73—21no—
58C2×C4×C73—21no—
59C2×C4×C73—22no—
60C2×C4×C73—41no—
61C2×C4×C73—41no—
62C2×C4×C73—82no—
63C2×C4×C73—81no—
64C2×C4×C73#10311yescharacteristic
65C2×C4×C73#10411yescharacteristic
66C2×Dic73—41no—
67C2×Dic73—81no—
68D292—41no—
69D292—41no—
70D292—81no—
71#10596833—161no—
72#10596831—42no—
73C22×C73—21no—
74C22×C73—21no—
75C22×C73#32811yescharacteristic
76C22×C73#35211yescharacteristic
77C22×C73#35311yescharacteristic
78C2×D73—41no—
79C2×D73—41no—
80C2×D73—81no—
81C2×D73—81no—
82C2×D73—81no—
83C2×D73—81no—
84C2×D73—161no—
85#64408—162no—
86C4×C73—21no—
87C4×C73—21no—
88C4×C73—42no—
89C4×C73—161no—
90Dic73—81no—
91#7435—731nomaximal, Sylow
92C2×C73—21no—
93C2×C73—21no—
94C2×C73#135211yescharacteristic
95C2×C73#140611yescharacteristic
96C2×C73#140911yescharacteristic
97D73—81no—
98D73—81no—
99#2094—731no—
100#1393—731no—
101#1318—731no—
102C73#743511yescharacteristic, complemented, Sylow
103C2×#106—1461no—
104C2×#102—1461no—
105C2×#99—1461no—
106#390—731no—
107#384#6440811yescharacteristic, complemented
108#336—731no—
109#324—1461no—
110C22×D4—731no—
111C22×C8—1461no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—731no—
115C2×C22⋊C4—2921no—
116C2×C22⋊C4—2921no—
117C2×C42C2×#6440811yescharacteristic
118#106—2922no—
119#102—2922no—
120#99—2921no—
121#99—1462no—
122C22⋊C8—2921no—
123C24—1461no—
124C2×D4—2921no—
125C2×D4—2921no—
126C2×D4—731no—
127C2×D4—731no—
128C2×D4—1461no—
129C22×C4—21no—
130C22×C4—41no—
131C22×C4—5841no—
132C22×C4—1461no—
133C22×C4C22⋊#6440811yescharacteristic
134C2×C8—2921no—
135C2×C8—2921no—
136C2×C8—5841no—
137C2×C8—1462no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—82no—
141C4⋊C4—2921no—
142C22⋊C4—2922no—
143C22⋊C4—5842no—
144C22⋊C4—5841no—
145C42—41no—
146C23—2921no—
147C23—2921no—
148C23—5841no—
149C23—5841no—
150C23—1461no—
151C23—1461no—
152C23#1455664911yescharacteristic
153D4—2921no—
154D4—2921no—
155D4—5841no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—82no—
162C2×C4—81no—
163C2×C4—2921no—
164C2×C4—5841no—
165C2×C4—5842no—
166C2×C4—11681no—
167C2×C4—11681no—
168C2×C4#1455665411yescharacteristic
169C2×C4#1455665611yescharacteristic
170C8—2922no—
171C8—11681no—
172C22—21no—
173C22—21no—
174C22—2921no—
175C22—2921no—
176C22—5841no—
177C22—5841no—
178C22—5841no—
179C22—5841no—
180C22—11681no—
181C22#1798944111yescharacteristic
182C22#1798948911yescharacteristic
183C22#1798949111yescharacteristic
184C4—21no—
185C4—21no—
186C4—42no—
187C4—161no—
188C4—5841no—
189C4—11682no—
190C2—21no—
191C2—21no—
192C2—5841no—
193C2—5841no—
194C2#2262492811yescharacteristic
195C2#2262505811yescharacteristic
196C2#2262506411yescharacteristic
subgroupG/H
centerC22#17989491
commutatorC2×C42×C73C2×C4
FrattiniC2×C42C2×#64408
Fitting#384×C73C4
socleC22×C73#353
prime radicalC22×D292D4
cosocle#17988489C22
hypercenter#384#64408
hypocenterC73#7435
LCSG ▷ C2×C42×C73 ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
9. #17988342C4semidirect
10. #384×C73C4semidirect
102. C73#7435semidirect
107. #384#64408semidirect

Maximal subgroups

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

subgroupindex
#226250252
#226248602
#226114652
#743573