Finite Group Databasebeta

← #31796796

Subgroups — #31796796

Order 17536.

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#22578470C211yesmaximal, characteristic, complemented
2#22578395C212yesmaximal, complemented
3D137⋊#100C211yesmaximal, characteristic, complemented
4#22578274C212yesmaximal, complemented
5#127×D137C211yesmaximal, characteristic
6D137⋊M16C2212yes—
7D4×#10587739—24no—
8#17984167C2212yes—
9C2×C22⋊#10587739—22no—
10#17984150—24no—
11#17984148—22no—
12C2×D4×D137—22no—
13C2×D4×D137C2211yescharacteristic
14C2×D4×D137C2211yescharacteristic
15#17984085C411yescharacteristic, complemented
16#17984084—22no—
17#17984084C411yescharacteristic, complemented
18#17984084C411yescharacteristic, complemented
19C42×D137C2211yescharacteristic
20#127×C137C411yescharacteristic, complemented
21C22×#10587739—24no—
22#14555520—22no—
23#14555519—22no—
24#14555518—22no—
25C4⋊#10587739D414yescomplemented
26C22⋊#10587739—24no—
27C22⋊#10587739—22no—
28C4×#10587739—22no—
29C23×D137—21no—
30C23×D137D412yes—
31C2×D548—22no—
32C2×D548C2×C411yescharacteristic
33C2×D548C2×C411yescharacteristic, complemented
34C2×C22⋊D137—22no—
35C2×C22⋊D137—21no—
36C2×C22⋊D137—22no—
37C2×C22⋊D137D412yes—
38D4×D137—22no—
39D4×D137—22no—
40D4×D137—22no—
41D4×D137—22no—
42D4×D137—48no—
43C2×C4×D137—21no—
44C2×C4×D137C2311yescharacteristic
45C4×Dic137—21no—
46C4×Dic137C2×C411yescharacteristic, complemented
47C2×D4×C137—22no—
48C2×D4×C137C2×C411yescharacteristic
49C2×D4×C137C2×C411yescharacteristic, complemented
50C42×C137C2×C411yescharacteristic, complemented
51C2×#10587739—24no—
52C2×#10587739—84no—
53#11764047—42no—
54C22×D137—22no—
55C22×D137—22no—
56C22×D137—21no—
57C22×D137—44no—
58C22×D137—44no—
59C22×D137—41no—
60C22×D137—41no—
61C22×D137C22⋊C412yes—
62C22×D137C2×D411yescharacteristic
63D548—22no—
64D548—22no—
65D548—44no—
66C22⋊D137—24no—
67C22⋊D137—44no—
68C22⋊D137—44no—
69C22⋊D137—42no—
70C2×Dic137—21no—
71C2×Dic137C2×D411yescharacteristic
72C4×D137—24no—
73C4×D137C2×D412yes—
74C4×D137#4112yes—
75C23×C137—21no—
76C23×C137C22⋊C412yes—
77D4×C137—22no—
78D4×C137—22no—
79D4×C137—44no—
80C2×C4×C137—21no—
81C2×C4×C137C22×C411yescharacteristic
82C22×C137—22no—
83C22×C137—42no—
84C22×C137—42no—
85C22×C137—41no—
86C22×C137C2×C22⋊C411yescharacteristic
87C2×D137—22no—
88C2×D137—22no—
89C2×D137—44no—
90C2×D137—42no—
91C2×D137—42no—
92C2×D137—41no—
93C2×D137—82no—
94C2×D137—82no—
95C2×D137—82no—
96C2×D137#12011yescharacteristic
97C2×D137#12311yescharacteristic
98#10587739—44no—
99C4×C137—22no—
100C4×C137C4×D412yes—
101Dic137—22no—
102Dic137#12112yes—
103C2×C137—21no—
104C2×C137—42no—
105C2×C137—81no—
106C2×C137#38611yescharacteristic
107D137—21no—
108D137—42no—
109D137—81no—
110D137#45312yes—
111C137#188711yescharacteristic, complemented, Sylow
112#1887—1371nomaximal, Sylow
113C2×#127—1371no—
114#518—1371no—
115#418—1372no—
116C2×#100—1371no—
117#339—1372no—
118C22×D4—2742no—
119C22×D4—1371no—
120C22×D4—1371no—
121C2×M16—1372no—
122#127—2742no—
123#127—1371no—
124#127—1371no—
125#127—1371no—
126#127#1058773911yescharacteristic, complemented
127C4×D4—2744no—
128#117—1372no—
129C2×C22⋊C4—2742no—
130C2×C42—1371no—
131#102—2744no—
132#100—2742no—
133C24—2741no—
134C24—1372no—
135C2×D4—22no—
136C2×D4—2742no—
137C2×D4—2742no—
138C2×D4—2742no—
139C2×D4—2742no—
140C2×D4—2742no—
141C2×D4—2742no—
142C2×D4—2742no—
143C2×D4—2741no—
144C2×D4—5488no—
145C2×D4—1372no—
146C2×D4—1371no—
147C2×D4—1371no—
148C2×D4C2×#1058773911yescharacteristic
149C2×D4C2×#1058773911yescharacteristic, complemented
150C22×C4—2741no—
151C22×C4—2744no—
152C22×C4—1371no—
153M16—2742no—
154M16—2742no—
155C2×C8—2742no—
156C4⋊C4—1374no—
157C22⋊C4—2744no—
158C22⋊C4—2742no—
159C42—2741no—
160C42—2742no—
161C42—1371no—
162C42C2×#1058773911yescharacteristic, complemented
163C23—21no—
164C23—2742no—
165C23—2742no—
166C23—2741no—
167C23—5484no—
168C23—5484no—
169C23—5481no—
170C23—5481no—
171C23—1371no—
172C23—1372no—
173C23C22⋊#1058773912yes—
174D4—22no—
175D4—22no—
176D4—44no—
177D4—2744no—
178D4—2742no—
179D4—2742no—
180D4—5484no—
181D4—5484no—
182D4—5484no—
183D4—5482no—
184C2×C4—21no—
185C2×C4—2744no—
186C2×C4—2741no—
187C2×C4—2744no—
188C2×C4—10964no—
189C2×C4—1371no—
190C2×C4—1372no—
191C2×C4—1372no—
192C2×C4C22×#1058773911yescharacteristic
193C8—5482no—
194C22—22no—
195C22—42no—
196C22—42no—
197C22—41no—
198C22—2742no—
199C22—2742no—
200C22—5482no—
201C22—5482no—
202C22—5484no—
203C22—5481no—
204C22—10962no—
205C22—10962no—
206C22—10962no—
207C22—1371no—
208C22—1371no—
209C22C2×C22⋊#1058773911yescharacteristic
210C4—22no—
211C4—2742no—
212C4—5484no—
213C4—1372no—
214C4D4×#1058773912yes—
215C2—21no—
216C2—42no—
217C2—81no—
218C2—2741no—
219C2—5482no—
220C2—10961no—
221C2—1372no—
222C2#2257835111yescharacteristic
subgroupG/H
centerC2#22578351
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#127×C137C4
socleC2×C137#386
prime radical#127×D137C2
cosocleC2×C4×D137C23
hypercenter#127#10587739
hypocenterC137#1887
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #22578470C2semidirect
2. #22578395C2semidirect
3. D137⋊#100C2semidirect
4. #22578274C2semidirect
15. #17984085C4semidirect
17. #17984084C4semidirect
18. #17984084C4semidirect
20. #127×C137C4semidirect
25. C4⋊#10587739D4semidirect
33. C2×D548C2×C4semidirect
46. C4×Dic137C2×C4semidirect
49. C2×D4×C137C2×C4semidirect
50. C42×C137C2×C4semidirect
111. C137#1887semidirect
126. #127#10587739semidirect
149. C2×D4C2×#10587739semidirect
162. C42C2×#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784702
#225783952
D137⋊#1002
#225782742
#127×D1372
#1887137