Finite Group Databasebeta

← #18010034

Subgroups — #18010034

Order 4736.

288 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×#11924593C211yesmaximal, characteristic
2#14558475C211yesmaximal, characteristic
3#14558387C212yesmaximal, complemented
4#14558386C212yesmaximal, complemented
5C22⋊Q8×D37C211yesmaximal, characteristic
6C22×#64438—21no—
7C2×C4⋊#3866—21no—
8C2×C22⋊#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—42no—
39C2×#64438—22no—
40C2×#64438—42no—
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—41no—
55C22×D37—41no—
56C22×D37C22⋊C411yescharacteristic
57C22×D37SD1612yes—
58C22×D37Q1612yes—
59C22×D37C2×D411yescharacteristic
60C2×#3866—42no—
61C2×#3866—42no—
62C2×#3866—42no—
63Q8×C37—41no—
64C2×C4×C37—21no—
65C2×C4×C37—21no—
66C2×C4×C37—41no—
67C2×C4×C37C22⋊C411yescharacteristic
68C2×C4×C37C22×C411yescharacteristic
69C2×Dic37—22no—
70C2×Dic37—21no—
71C2×Dic37—21no—
72C2×Dic37C22⋊C411yescharacteristic
73C2×Dic37C2×D411yescharacteristic
74C4×D37—22no—
75C4×D37—22no—
76C4×D37—41no—
77C4×D37—41no—
78C4×D37—42no—
79C4×D37—42no—
80Dic74—42no—
81Dic74—41no—
82#64438—42no—
83C22×C37—21no—
84C22×C37—21no—
85C22×C37#10312yes—
86C22×C37C2×C22⋊C411yescharacteristic
87C2×D37—21no—
88C2×D37—21no—
89C2×D37—21no—
90C2×D37—21no—
91C2×D37—42no—
92C2×D37—42no—
93C2×D37#10312yes—
94C2×D37C2×SD1611yescharacteristic
95C2×D37C2×Q1611yescharacteristic
96#3866—82no—
97C4×C37—21no—
98C4×C37—21no—
99C4×C37—41no—
100C4×C37—41no—
101Dic37—21no—
102Dic37—41no—
103Dic37—41no—
104Dic37#9912yes—
105#1520—371nomaximal, Sylow
106C2×C37—22no—
107C2×C37C2×#9911yescharacteristic
108C2×C37C2×#10311yescharacteristic
109C2×C37#42111yescharacteristic
110D37—21no—
111D37—22no—
112D37—21no—
113C2×C22⋊Q8—371no—
114C2×C22⋊C8—371no—
115#380—371no—
116#328—372no—
117#328—372no—
118C37#152011yescharacteristic, complemented, Sylow
119C22×Q8—371no—
120C23×C4—371no—
121C22×C8—741no—
122C22⋊Q8—371no—
123C22⋊Q8—371no—
124C22⋊Q8—371no—
125C22⋊Q8—741no—
126C22⋊Q8—741no—
127C22⋊Q8#386611yescharacteristic, complemented
128C2×C4⋊C4—371no—
129C2×C4⋊C4—741no—
130C2×C4⋊C4—741no—
131C2×C22⋊C4—741no—
132C2×C22⋊C4—741no—
133C22⋊C8—372no—
134C22⋊C8—741no—
135#95—372no—
136C24—371no—
137C2×Q8—371no—
138C2×Q8—741no—
139C2×Q8—1482no—
140C2×Q8C2×#386611yescharacteristic
141C22×C4—371no—
142C22×C4—371no—
143C22×C4—371no—
144C22×C4—371no—
145C22×C4—371no—
146C22×C4—741no—
147C22×C4—741no—
148C22×C4—742no—
149C22×C4—1482no—
150C22×C4C2×#386611yescharacteristic
151C2×C8—742no—
152C2×C8—1482no—
153C4⋊C4—21no—
154C4⋊C4—371no—
155C4⋊C4—741no—
156C4⋊C4—741no—
157C4⋊C4—741no—
158C4⋊C4—741no—
159C4⋊C4—1482no—
160C4⋊C4C2×#386611yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—741no—
163C22⋊C4—741no—
164C22⋊C4—741no—
165C22⋊C4—1482no—
166C23—371no—
167C23—371no—
168C23—372no—
169C23—372no—
170C23—1481no—
171C23—1481no—
172C23C22⋊#386611yescharacteristic
173Q8—41no—
174Q8—1481no—
175Q8—1482no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—371no—
180C2×C4—371no—
181C2×C4—742no—
182C2×C4—742no—
183C2×C4—742no—
184C2×C4—741no—
185C2×C4—741no—
186C2×C4—1481no—
187C2×C4—1481no—
188C2×C4—1482no—
189C2×C4—1482no—
190C2×C4—1482no—
191C2×C4—1482no—
192C2×C4—1482no—
193C2×C4C22⋊#386611yescharacteristic
194C2×C4C22×#386611yescharacteristic
195C8—1482no—
196C22—21no—
197C22—21no—
198C22—371no—
199C22—372no—
200C22—371no—
201C22—741no—
202C22—741no—
203C22—741no—
204C22—741no—
205C22—1482no—
206C22—1482no—
207C22#1192459912yes—
208C22C2×C22⋊#386611yescharacteristic
209C4—21no—
210C4—21no—
211C4—41no—
212C4—41no—
213C4—372no—
214C4—2962no—
215C4—741no—
216C4—1481no—
217C4—1481no—
218C2—22no—
219C2—741no—
220C2—742no—
221C2—741no—
222C2D37⋊#9911yescharacteristic
223C2C2×#1192459911yescharacteristic
224C2#1455853011yescharacteristic
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#1520
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
118. C37#1520semidirect
127. C22⋊Q8#3866semidirect

Maximal subgroups

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

subgroupindex
C2×#119245932
#145584752
#145583872
#145583862
C22⋊Q8×D372
#152037