Finite Group Databasebeta

← #18009994

Subgroups — #18009994

Order 4736.

294 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 60 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#14558505C211yesmaximal, characteristic
2C4×C22⋊#3866C211yesmaximal, characteristic
3#14558387C212yesmaximal, complemented
4#14558386C212yesmaximal, complemented
5C22⋊Q8×D37C211yesmaximal, characteristic
6C2×#10596957—21no—
7C2×C22⋊#3866—21no—
8C2×C4×#3866—21no—
9#11924593—21no—
10#11924593C2212yes—
11(C2×#3866)⋊C4C2212yes—
12C2×Q8×D37C411yescharacteristic, complemented
13C22×C4×D37C2211yescharacteristic
14#11924516C2211yescharacteristic
15#11924515C411yescharacteristic, complemented
16#11924514—21no—
17#11924513—21no—
18#11924512C2211yescharacteristic
19C4⋊C4×D37—21no—
20C4⋊C4×D37C411yescharacteristic, complemented
21C22⋊C4×D37—21no—
22C22⋊Q8×C37C411yescharacteristic, complemented
23C23×D37D411yescharacteristic
24C22×#3866—22no—
25C2×Q8×C37C2×C411yescharacteristic
26C22×C4×C37C2×C411yescharacteristic
27C22×Dic37D411yescharacteristic
28Q8×D37—42no—
29C2×C4×D37—21no—
30C2×C4×D37—21no—
31C2×C4×D37—42no—
32C2×C4×D37C2×C411yescharacteristic
33C2×C4×D37D411yescharacteristic
34C2×C4×D37D411yescharacteristic
35C2×C4×D37C2311yescharacteristic
36C2×Dic74—21no—
37C2×Dic74C2×C411yescharacteristic
38C22⋊#3866—24no—
39#10596957—42no—
40C2×#64438—22no—
41C4×#3866—42no—
42C4⋊C4×C37—21no—
43C4⋊C4×C37C2×C411yescharacteristic
44C22⋊C4×C37—21no—
45C22⋊Dic37—21no—
46#10596938—21no—
47#10596938—21no—
48C4⋊Dic37—21no—
49C4⋊Dic37C2×C411yescharacteristic
50#10596936—21no—
51#10596936—21no—
52#10596936—21no—
53C23×C37C22⋊C411yescharacteristic
54C22×D37—22no—
55C22×D37—22no—
56C22×D37—41no—
57C22×D37C22⋊C411yescharacteristic
58C22×D37C2×D411yescharacteristic
59C2×#3866—24no—
60C2×#3866—42no—
61C2×#3866—42no—
62Q8×C37—41no—
63C2×C4×C37—21no—
64C2×C4×C37—21no—
65C2×C4×C37—41no—
66C2×C4×C37C22⋊C411yescharacteristic
67C2×C4×C37C22×C411yescharacteristic
68C2×Dic37—22no—
69C2×Dic37—21no—
70C2×Dic37—21no—
71C2×Dic37C22⋊C411yescharacteristic
72C2×Dic37C2×D411yescharacteristic
73C4×D37—22no—
74C4×D37—22no—
75C4×D37—41no—
76C4×D37—41no—
77C4×D37—42no—
78C4×D37—42no—
79Dic74—42no—
80Dic74—41no—
81#64438—42no—
82C22×C37—21no—
83C22×C37—21no—
84C22×C37—21no—
85C22×C37C2×C22⋊C411yescharacteristic
86C2×D37—21no—
87C2×D37—21no—
88C2×D37—21no—
89C2×D37—21no—
90C2×D37—44no—
91C2×D37#10412yes—
92C2×D37#13712yes—
93#3866—44no—
94C4×C37—21no—
95C4×C37—21no—
96C4×C37—41no—
97C4×C37—41no—
98Dic37—21no—
99Dic37—41no—
100Dic37—41no—
101Dic37#9912yes—
102#1508—371nomaximal, Sylow
103C2×C37—41no—
104C2×C37C2×#9911yescharacteristic
105C2×C37#41911yescharacteristic
106C2×C37C2×#10411yescharacteristic
107D37—22no—
108D37—22no—
109C2×C22⋊Q8—371no—
110#407—371no—
111C4×C22⋊C4—371no—
112#328—372no—
113#328—372no—
114C37#150811yescharacteristic, complemented, Sylow
115C22×Q8—371no—
116C23×C4—371no—
117C2×M16—741no—
118C22⋊Q8—371no—
119C22⋊Q8—371no—
120C22⋊Q8—371no—
121C22⋊Q8—741no—
122C22⋊Q8—741no—
123C22⋊Q8#386611yescharacteristic, complemented
124C2×C4⋊C4—371no—
125C2×C4⋊C4—741no—
126C2×C22⋊C4—741no—
127C2×C22⋊C4—741no—
128C2×C42—741no—
129C22⋊C8—372no—
130C22⋊C8—741no—
131#95—372no—
132C24—371no—
133C2×Q8—371no—
134C2×Q8—741no—
135C2×Q8—1482no—
136C2×Q8C2×#386611yescharacteristic
137C22×C4—371no—
138C22×C4—371no—
139C22×C4—371no—
140C22×C4—371no—
141C22×C4—371no—
142C22×C4—741no—
143C22×C4—741no—
144C22×C4—742no—
145C22×C4—1482no—
146C22×C4C2×#386611yescharacteristic
147M16—1482no—
148C2×C8—742no—
149C4⋊C4—21no—
150C4⋊C4—371no—
151C4⋊C4—741no—
152C4⋊C4—741no—
153C4⋊C4—741no—
154C4⋊C4—741no—
155C4⋊C4C2×#386611yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—741no—
158C22⋊C4—741no—
159C22⋊C4—744no—
160C22⋊C4—741no—
161C42—1482no—
162C23—371no—
163C23—371no—
164C23—742no—
165C23—742no—
166C23—1481no—
167C23C22⋊#386611yescharacteristic
168Q8—41no—
169Q8—1481no—
170Q8—1482no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—41no—
174C2×C4—371no—
175C2×C4—371no—
176C2×C4—742no—
177C2×C4—742no—
178C2×C4—742no—
179C2×C4—744no—
180C2×C4—741no—
181C2×C4—741no—
182C2×C4—1481no—
183C2×C4—1481no—
184C2×C4—1482no—
185C2×C4—1482no—
186C2×C4—1482no—
187C2×C4—1482no—
188C2×C4C22⋊#386611yescharacteristic
189C2×C4C22×#386611yescharacteristic
190C8—1482no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—372no—
195C22—372no—
196C22—741no—
197C22—741no—
198C22—741no—
199C22—741no—
200C22—1484no—
201C22C2×C22⋊#386611yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—372no—
207C4—741no—
208C4—1484no—
209C4—1481no—
210C4—1481no—
211C2—41no—
212C2—742no—
213C2—742no—
214C2D37⋊#9911yescharacteristic
215C2#1455852611yescharacteristic
216C2D37⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#3866
commutatorC2×C4×C37C22×C4
FrattiniC2×C4C22×#3866
FittingC22⋊Q8×C37C4
socleC22×C37C2×C22⋊C4
prime radicalC23×D37D4
cosocleC2×C4×D37C23
hypercenterC22⋊Q8#3866
hypocenterC37#1508
LCSG ▷ C2×C4×C37 ▷ C22×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #14558387C2semidirect
4. #14558386C2semidirect
12. C2×Q8×D37C4semidirect
15. #11924515C4semidirect
20. C4⋊C4×D37C4semidirect
22. C22⋊Q8×C37C4semidirect
114. C37#1508semidirect
123. C22⋊Q8#3866semidirect

Maximal subgroups

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

subgroupindex
#145585052
C4×C22⋊#38662
#145583872
#145583862
C22⋊Q8×D372
#150837