Finite Group Databasebeta

← #18010603

Subgroups — #18010603

Order 4736.

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

normal subgroups only

This group has 444 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
1#14558600C211yesmaximal, characteristic, complemented
2#14558525C212yesmaximal, complemented
3D37⋊#100C211yesmaximal, characteristic, complemented
4#14558404C212yesmaximal, complemented
5#127×D37C211yesmaximal, characteristic
6D37⋊M16C2212yes—
7D4×#3866—24no—
8#11924615C2212yes—
9C2×C22⋊#3866—22no—
10#11924598—24no—
11#11924596—22no—
12C2×D4×D37—22no—
13C2×D4×D37C2211yescharacteristic
14C2×D4×D37C2211yescharacteristic
15#11924533C411yescharacteristic, complemented
16#11924532—22no—
17#11924532C411yescharacteristic, complemented
18#11924532C411yescharacteristic, complemented
19C42×D37C2211yescharacteristic
20#127×C37C411yescharacteristic, complemented
21C23×D37—21no—
22C23×D37D412yes—
23C22×#3866—24no—
24C2×D4×C37—22no—
25C2×D4×C37C2×C411yescharacteristic
26C2×D4×C37C2×C411yescharacteristic, complemented
27C2×C22⋊D37—22no—
28C2×C22⋊D37—22no—
29C2×C22⋊D37—21no—
30C2×C22⋊D37D412yes—
31D4×D37—22no—
32D4×D37—22no—
33D4×D37—22no—
34D4×D37—22no—
35D4×D37—48no—
36C2×D148—22no—
37C2×D148C2×C411yescharacteristic
38C2×D148C2×C411yescharacteristic, complemented
39C2×C4×D37—21no—
40C2×C4×D37C2311yescharacteristic
41C22⋊#3866—24no—
42C22⋊#3866—22no—
43#10596957—22no—
44C4⋊#3866D414yescomplemented
45C4×#3866—22no—
46#10596953—22no—
47D37⋊C8—22no—
48C42×C37C2×C411yescharacteristic, complemented
49C4×Dic37—21no—
50C4×Dic37C2×C411yescharacteristic, complemented
51C23×C37—21no—
52C23×C37C22⋊C412yes—
53C22×D37—22no—
54C22×D37—22no—
55C22×D37—21no—
56C22×D37—44no—
57C22×D37—44no—
58C22×D37—41no—
59C22×D37—41no—
60C22×D37C22⋊C412yes—
61C22×D37C2×D411yescharacteristic
62C2×#3866—24no—
63C2×#3866—84no—
64D4×C37—22no—
65D4×C37—22no—
66D4×C37—44no—
67C2×C4×C37—21no—
68C2×C4×C37C22×C411yescharacteristic
69C22⋊D37—24no—
70C22⋊D37—44no—
71C22⋊D37—44no—
72C22⋊D37—42no—
73C2×Dic37—21no—
74C2×Dic37C2×D411yescharacteristic
75D148—22no—
76D148—22no—
77D148—44no—
78C4×D37—24no—
79C4×D37C2×D412yes—
80C4×D37#4112yes—
81#64438—42no—
82C22×C37—22no—
83C22×C37—42no—
84C22×C37—42no—
85C22×C37—41no—
86C22×C37C2×C22⋊C411yescharacteristic
87C2×D37—22no—
88C2×D37—22no—
89C2×D37—44no—
90C2×D37—42no—
91C2×D37—42no—
92C2×D37—41no—
93C2×D37—82no—
94C2×D37—82no—
95C2×D37—82no—
96C2×D37#12011yescharacteristic
97C2×D37#12311yescharacteristic
98#3866—44no—
99C4×C37—22no—
100C4×C37C4×D412yes—
101Dic37—22no—
102Dic37#12112yes—
103#1887—371nomaximal, Sylow
104C2×C37—21no—
105C2×C37—42no—
106C2×C37—81no—
107C2×C37#38611yescharacteristic
108D37—21no—
109D37—42no—
110D37—81no—
111D37#45312yes—
112C2×#127—371no—
113#518—371no—
114#418—372no—
115C2×#100—371no—
116#339—372no—
117C37#188711yescharacteristic, complemented, Sylow
118C22×D4—371no—
119C22×D4—371no—
120C22×D4—742no—
121C2×M16—372no—
122#127—371no—
123#127—371no—
124#127—371no—
125#127—742no—
126#127#386611yescharacteristic, complemented
127C4×D4—744no—
128#117—372no—
129C2×C22⋊C4—742no—
130C2×C42—371no—
131#102—744no—
132#100—742no—
133C24—372no—
134C24—741no—
135C2×D4—22no—
136C2×D4—372no—
137C2×D4—371no—
138C2×D4—371no—
139C2×D4—742no—
140C2×D4—742no—
141C2×D4—742no—
142C2×D4—742no—
143C2×D4—742no—
144C2×D4—741no—
145C2×D4—742no—
146C2×D4—742no—
147C2×D4—1488no—
148C2×D4C2×#386611yescharacteristic
149C2×D4C2×#386611yescharacteristic, complemented
150C22×C4—371no—
151C22×C4—741no—
152C22×C4—744no—
153M16—742no—
154M16—742no—
155C2×C8—742no—
156C4⋊C4—374no—
157C22⋊C4—742no—
158C22⋊C4—744no—
159C42—371no—
160C42—741no—
161C42—742no—
162C42C2×#386611yescharacteristic, complemented
163C23—21no—
164C23—371no—
165C23—372no—
166C23—742no—
167C23—742no—
168C23—741no—
169C23—1481no—
170C23—1481no—
171C23—1484no—
172C23—1484no—
173C23C22⋊#386612yes—
174D4—22no—
175D4—22no—
176D4—44no—
177D4—744no—
178D4—742no—
179D4—742no—
180D4—1484no—
181D4—1482no—
182D4—1484no—
183D4—1484no—
184C2×C4—21no—
185C2×C4—371no—
186C2×C4—372no—
187C2×C4—372no—
188C2×C4—2964no—
189C2×C4—744no—
190C2×C4—741no—
191C2×C4—744no—
192C2×C4C22×#386611yescharacteristic
193C8—1482no—
194C22—22no—
195C22—42no—
196C22—42no—
197C22—41no—
198C22—371no—
199C22—371no—
200C22—2962no—
201C22—2962no—
202C22—2962no—
203C22—742no—
204C22—742no—
205C22—1482no—
206C22—1482no—
207C22—1484no—
208C22—1481no—
209C22C2×C22⋊#386611yescharacteristic
210C4—22no—
211C4—372no—
212C4—742no—
213C4—1484no—
214C4D4×#386612yes—
215C2—21no—
216C2—42no—
217C2—81no—
218C2—372no—
219C2—2961no—
220C2—741no—
221C2—1482no—
222C2#1455848111yescharacteristic
subgroupG/H
centerC2#14558481
commutatorC2×C4×C37C22×C4
FrattiniC2×C4C22×#3866
Fitting#127×C37C4
socleC2×C37#386
prime radical#127×D37C2
cosocleC2×C4×D37C23
hypercenter#127#3866
hypocenterC37#1887
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #14558600C2semidirect
2. #14558525C2semidirect
3. D37⋊#100C2semidirect
4. #14558404C2semidirect
15. #11924533C4semidirect
17. #11924532C4semidirect
18. #11924532C4semidirect
20. #127×C37C4semidirect
26. C2×D4×C37C2×C4semidirect
38. C2×D148C2×C4semidirect
44. C4⋊#3866D4semidirect
48. C42×C37C2×C4semidirect
50. C4×Dic37C2×C4semidirect
117. C37#1887semidirect
126. #127#3866semidirect
149. C2×D4C2×#3866semidirect
162. C42C2×#3866semidirect

Maximal subgroups

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

subgroupindex
#145586002
#145585252
D37⋊#1002
#145584042
#127×D372
#188737