Finite Group Databasebeta

← #31796832

Subgroups — #31796832

Order 17536.

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

normal subgroups only

This group has 344 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#22578481C211yesmaximal, characteristic, complemented
2#22578395C212yesmaximal, complemented
3#22578383C211yesmaximal, characteristic, complemented
4#22578270C212yesmaximal, complemented
5#22577966C211yesmaximal, characteristic
6#17984175—24no—
7D137⋊M16C2212yes—
8C2×#14555519—22no—
9#17984167C2212yes—
10#17984150—24no—
11#17984146—22no—
12C2×#14555509C2211yescharacteristic
13C2×D4×D137C2211yescharacteristic
14#17984084C411yescharacteristic, complemented
15#17984076C411yescharacteristic, complemented
16#17984075C411yescharacteristic, complemented
17#17984060—22no—
18#17984056—22no—
19#17984045C2211yescharacteristic
20#127×C137C411yescharacteristic, complemented
21#14555520—22no—
22#14555519—24no—
23#14555519—22no—
24#14555518D414yescomplemented
25C2×#11764047—24no—
26C4⋊#10587739—22no—
27C22⋊#10587739—22no—
28C4×#10587739—22no—
29C23×D137—21no—
30#14555509—22no—
31#14555509—22no—
32C2×Dic274C2×C411yescharacteristic
33C2×D548C2×C411yescharacteristic
34C2×C22⋊D137—21no—
35C2×C22⋊D137—22no—
36C2×C22⋊D137D412yes—
37D4×D137—22no—
38D4×D137—22no—
39C22×Dic137D412yes—
40C2×C4×D137C2311yescharacteristic
41#14555489—21no—
42C22⋊Dic137—22no—
43#14555487—21no—
44#14555487—22no—
45C4×Dic137C2×C411yescharacteristic
46C2×D4×C137—22no—
47C2×D4×C137C2×C411yescharacteristic
48C2×D4×C137C2×C411yescharacteristic
49C42×C137C2×C411yescharacteristic
50C2×#10587739—42no—
51#11764047—24no—
52C22×D137—21no—
53C22×D137—42no—
54C22×D137—82no—
55C22×D137C2×D411yescharacteristic
56Dic274—22no—
57D548—22no—
58C22⋊D137—24no—
59C22⋊D137—41no—
60C22⋊D137—41no—
61C22⋊D137—82no—
62C2×Dic137—22no—
63C2×Dic137—22no—
64C2×Dic137—21no—
65C2×Dic137C22⋊C412yes—
66C2×Dic137C2×D411yescharacteristic
67C4×D137C2×D412yes—
68C4×D137#4112yes—
69C23×C137—21no—
70C23×C137C22⋊C412yes—
71D4×C137—22no—
72D4×C137—22no—
73D4×C137—82no—
74C2×C4×C137—21no—
75C2×C4×C137C22×C411yescharacteristic
76C22×C137—22no—
77C22×C137—42no—
78C22×C137—42no—
79C22×C137—41no—
80C22×C137C2×C22⋊C411yescharacteristic
81C2×D137—41no—
82C2×D137—42no—
83C2×D137—41no—
84C2×D137—84no—
85C2×D137#12112yes—
86#10587739—82no—
87C4×C137—41no—
88C4×C137C4×D412yes—
89Dic137—22no—
90Dic137—41no—
91Dic137#12011yescharacteristic
92Dic137#12311yescharacteristic
93C2×C137—21no—
94C2×C137—42no—
95C2×C137—81no—
96C2×C137#38611yescharacteristic
97D137—22no—
98D137—81no—
99C137#191011yescharacteristic, complemented, Sylow
100#1910—1371nomaximal, Sylow
101#568—1371no—
102#535—1371no—
103#418—1372no—
104#410—1371no—
105H(3,ℤ/4)—1372no—
106C2×#41—1371no—
107C22×D4—1371no—
108#131—2744no—
109C2×M16—2742no—
110C2×M16—1372no—
111#127—1371no—
112#127#1058773911yescharacteristic, complemented
113#124—1371no—
114#124—1371no—
115#121—2742no—
116#120—2742no—
117#117—1372no—
118#117—1371no—
119#102—2744no—
120#99—2742no—
121C24—2741no—
122#41—2742no—
123#41—2742no—
124C2×Q8—1371no—
125C2×D4—22no—
126C2×D4—2742no—
127C2×D4—2742no—
128C2×D4—2742no—
129C2×D4—2741no—
130C2×D4—1372no—
131C2×D4—1371no—
132C2×D4C2×#1058773911yescharacteristic
133C2×D4C2×#1058773911yescharacteristic
134C22×C4—1372no—
135C22×C4—1371no—
136M16—2744no—
137M16—2742no—
138M16—2742no—
139C2×C8—2744no—
140C2×C8—1374no—
141C4⋊C4—2741no—
142C4⋊C4—2742no—
143C22⋊C4—2741no—
144C22⋊C4—2742no—
145C22⋊C4—2742no—
146C22⋊C4—2742no—
147C42—2742no—
148C42—1371no—
149C42C2×#1058773911yescharacteristic
150C23—21no—
151C23—2741no—
152C23—5482no—
153C23—10962no—
154C23—1371no—
155C23C22⋊#1058773912yes—
156Q8—2742no—
157D4—22no—
158D4—22no—
159D4—82no—
160D4—2744no—
161D4—2742no—
162D4—5481no—
163D4—5481no—
164D4—10962no—
165C2×C4—21no—
166C2×C4—2742no—
167C2×C4—2742no—
168C2×C4—2741no—
169C2×C4—5482no—
170C2×C4—1372no—
171C2×C4—1371no—
172C2×C4—1372no—
173C2×C4—1372no—
174C2×C4C22×#1058773911yescharacteristic
175C8—2744no—
176C22—22no—
177C22—42no—
178C22—42no—
179C22—41no—
180C22—5481no—
181C22—5482no—
182C22—5481no—
183C22—10964no—
184C22—1372no—
185C22C2×C22⋊#1058773911yescharacteristic
186C4—41no—
187C4—2742no—
188C4—5481no—
189C4—10962no—
190C4—1371no—
191C4—1371no—
192C4D4×#1058773912yes—
193C2—21no—
194C2—42no—
195C2—81no—
196C2—2742no—
197C2—10961no—
198C2#2257835111yescharacteristic
subgroupG/H
centerC2#22578351
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#127×C137C4
socleC2×C137#386
prime radical#22577966C2
cosocleC2×C4×D137C23
hypercenter#127#10587739
hypocenterC137#1910
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. #22578481C2semidirect
2. #22578395C2semidirect
3. #22578383C2semidirect
4. #22578270C2semidirect
14. #17984084C4semidirect
15. #17984076C4semidirect
16. #17984075C4semidirect
20. #127×C137C4semidirect
24. #14555518D4semidirect
99. C137#1910semidirect
112. #127#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784812
#225783952
#225783832
#225782702
#225779662
#1910137