Finite Group Databasebeta

← #18010642

Subgroups — #18010642

Order 4736.

362 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 66 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#14558531C212yesmaximal, complemented
3D37⋊#100C211yesmaximal, characteristic, complemented
4#14558400C212yesmaximal, complemented
5#14558096C211yesmaximal, characteristic
6D37⋊M16C2212yes—
7D4×#3866—24no—
8#11924615C2212yes—
9C2×C22⋊#3866—22no—
10#11924600—24no—
11#11924596—22no—
12C2×#10596964C2211yescharacteristic
13C2×D4×D37C2211yescharacteristic
14#11924532C411yescharacteristic, complemented
15#11924524C411yescharacteristic, complemented
16#11924523C411yescharacteristic, complemented
17#11924508—22no—
18#11924504—22no—
19#11924493C2211yescharacteristic
20#127×C37C411yescharacteristic, complemented
21C23×D37D412yes—
22C22×#3866—24no—
23C2×D4×C37—22no—
24C2×D4×C37C2×C411yescharacteristic
25C2×D4×C37C2×C411yescharacteristic, complemented
26C2×C22⋊D37—21no—
27C2×C22⋊D37—22no—
28C2×C22⋊D37D412yes—
29C22×Dic37—21no—
30#10596964—22no—
31#10596964—22no—
32D4×D37—22no—
33D4×D37—22no—
34C2×D148C2×C411yescharacteristic
35C2×C4×D37C2311yescharacteristic
36C2×Dic74C2×C411yescharacteristic, complemented
37C22⋊#3866—24no—
38C22⋊#3866—22no—
39#10596957—22no—
40C4⋊#3866—22no—
41C4×#3866D414yescomplemented
42#10596953—22no—
43D37⋊C8—22no—
44C42×C37C2×C411yescharacteristic, complemented
45C22⋊Dic37—22no—
46#10596938—22no—
47#10596938—21no—
48#10596936—21no—
49C4×Dic37C2×C411yescharacteristic, complemented
50C23×C37—21no—
51C23×C37C22⋊C412yes—
52C22×D37—22no—
53C22×D37—22no—
54C22×D37—82no—
55C22×D37C22⋊C412yes—
56C22×D37C2×D411yescharacteristic
57C2×#3866—24no—
58C2×#3866—84no—
59D4×C37—22no—
60D4×C37—22no—
61D4×C37—82no—
62C2×C4×C37—21no—
63C2×C4×C37C22×C411yescharacteristic
64C22⋊D37—24no—
65C22⋊D37—41no—
66C22⋊D37—41no—
67C22⋊D37—82no—
68C2×Dic37—21no—
69C2×Dic37—21no—
70C2×Dic37—42no—
71C2×Dic37C2×D411yescharacteristic
72D148—22no—
73C4×D37C2×D412yes—
74C4×D37#4112yes—
75Dic74—22no—
76#64438—42no—
77C22×C37—22no—
78C22×C37—42no—
79C22×C37—42no—
80C22×C37—41no—
81C22×C37C2×C22⋊C411yescharacteristic
82C2×D37—22no—
83C2×D37—41no—
84C2×D37—44no—
85C2×D37—42no—
86C2×D37—82no—
87C2×D37#12011yescharacteristic
88C2×D37#12311yescharacteristic
89#3866—44no—
90C4×C37—41no—
91C4×C37C4×D412yes—
92Dic37—41no—
93Dic37—41no—
94Dic37#12112yes—
95#1911—371nomaximal, Sylow
96C2×C37—21no—
97C2×C37—42no—
98C2×C37—81no—
99C2×C37#38611yescharacteristic
100D37—21no—
101D37—21no—
102D37—42no—
103#535—371no—
104#518—371no—
105#421—372no—
106C2×#100—371no—
107H(3,ℤ/4)—372no—
108C37#191111yescharacteristic, complemented, Sylow
109C2×#41—371no—
110C22×D4—371no—
111C2×M16—372no—
112#127—371no—
113#127#386611yescharacteristic, complemented
114#124—371no—
115#124—371no—
116#121—742no—
117#120—742no—
118C4×D4—744no—
119#117—372no—
120#117—371no—
121C2×C22⋊C4—742no—
122#104—744no—
123#100—742no—
124C24—372no—
125#41—742no—
126#41—742no—
127C2×Q8—371no—
128C2×D4—22no—
129C2×D4—372no—
130C2×D4—371no—
131C2×D4—742no—
132C2×D4—742no—
133C2×D4—742no—
134C2×D4—741no—
135C2×D4C2×#386611yescharacteristic
136C2×D4C2×#386611yescharacteristic, complemented
137C22×C4—371no—
138C22×C4—741no—
139C22×C4—744no—
140M16—742no—
141M16—742no—
142C2×C8—742no—
143C4⋊C4—741no—
144C4⋊C4—742no—
145C22⋊C4—741no—
146C22⋊C4—742no—
147C22⋊C4—742no—
148C22⋊C4—744no—
149C22⋊C4—742no—
150C42—374no—
151C42—371no—
152C42C2×#386611yescharacteristic, complemented
153C23—21no—
154C23—371no—
155C23—372no—
156C23—2962no—
157C23—742no—
158C23—742no—
159C23C22⋊#386612yes—
160Q8—742no—
161D4—22no—
162D4—22no—
163D4—82no—
164D4—2962no—
165D4—744no—
166D4—742no—
167D4—1481no—
168D4—1481no—
169C2×C4—21no—
170C2×C4—371no—
171C2×C4—372no—
172C2×C4—372no—
173C2×C4—2964no—
174C2×C4—744no—
175C2×C4—741no—
176C2×C4—741no—
177C2×C4—1482no—
178C2×C4C22×#386611yescharacteristic
179C8—1482no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—41no—
184C22—371no—
185C22—371no—
186C22—2962no—
187C22—742no—
188C22—1481no—
189C22—1482no—
190C22—1484no—
191C22C2×C22⋊#386611yescharacteristic
192C4—41no—
193C4—372no—
194C4—1481no—
195C4—1484no—
196C4—1481no—
197C4D4×#386612yes—
198C2—21no—
199C2—42no—
200C2—81no—
201C2—741no—
202C2—741no—
203C2—1482no—
204C2#1455848111yescharacteristic
subgroupG/H
centerC2#14558481
commutatorC2×C4×C37C22×C4
FrattiniC2×C4C22×#3866
Fitting#127×C37C4
socleC2×C37#386
prime radical#14558096C2
cosocleC2×C4×D37C23
hypercenter#127#3866
hypocenterC37#1911
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. #14558531C2semidirect
3. D37⋊#100C2semidirect
4. #14558400C2semidirect
14. #11924532C4semidirect
15. #11924524C4semidirect
16. #11924523C4semidirect
20. #127×C37C4semidirect
25. C2×D4×C37C2×C4semidirect
36. C2×Dic74C2×C4semidirect
41. C4×#3866D4semidirect
44. C42×C37C2×C4semidirect
49. C4×Dic37C2×C4semidirect
108. C37#1911semidirect
113. #127#3866semidirect
136. C2×D4C2×#3866semidirect
152. C42C2×#3866semidirect

Maximal subgroups

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

subgroupindex
#145586002
#145585312
D37⋊#1002
#145584002
#145580962
#191137