Finite Group Databasebeta

← #31796835

Subgroups — #31796835

Order 17536.

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#22578470C211yesmaximal, characteristic, complemented
2#22578401C212yesmaximal, complemented
3D137⋊#100C211yesmaximal, characteristic, complemented
4#22578270C212yesmaximal, complemented
5#22577966C211yesmaximal, characteristic
6D137⋊M16C2212yes—
7D4×#10587739—24no—
8#17984167C2212yes—
9C2×C22⋊#10587739—22no—
10#17984152—24no—
11#17984148—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
21C22×#10587739—24no—
22#14555520—22no—
23#14555519—22no—
24#14555518—22no—
25C4⋊#10587739—22no—
26C22⋊#10587739—24no—
27C22⋊#10587739—22no—
28C4×#10587739D414yescomplemented
29C23×D137D412yes—
30#14555509—22no—
31#14555509—22no—
32C2×Dic274C2×C411yescharacteristic, complemented
33C2×D548C2×C411yescharacteristic
34C2×C22⋊D137—21no—
35C2×C22⋊D137—22no—
36C2×C22⋊D137D412yes—
37D4×D137—22no—
38D4×D137—22no—
39C22×Dic137—21no—
40C2×C4×D137C2311yescharacteristic
41#14555489—21no—
42C22⋊Dic137—22no—
43#14555487—22no—
44#14555487—21no—
45C4×Dic137C2×C411yescharacteristic, complemented
46C2×D4×C137—22no—
47C2×D4×C137C2×C411yescharacteristic
48C2×D4×C137C2×C411yescharacteristic, complemented
49C42×C137C2×C411yescharacteristic, complemented
50C2×#10587739—24no—
51C2×#10587739—84no—
52#11764047—42no—
53C22×D137—22no—
54C22×D137—22no—
55C22×D137—82no—
56C22×D137C22⋊C412yes—
57C22×D137C2×D411yescharacteristic
58Dic274—22no—
59D548—22no—
60C22⋊D137—24no—
61C22⋊D137—41no—
62C22⋊D137—41no—
63C22⋊D137—82no—
64C2×Dic137—21no—
65C2×Dic137—21no—
66C2×Dic137—42no—
67C2×Dic137C2×D411yescharacteristic
68C4×D137C2×D412yes—
69C4×D137#4112yes—
70C23×C137—21no—
71C23×C137C22⋊C412yes—
72D4×C137—22no—
73D4×C137—22no—
74D4×C137—82no—
75C2×C4×C137—21no—
76C2×C4×C137C22×C411yescharacteristic
77C22×C137—22no—
78C22×C137—42no—
79C22×C137—42no—
80C22×C137—41no—
81C22×C137C2×C22⋊C411yescharacteristic
82C2×D137—22no—
83C2×D137—41no—
84C2×D137—44no—
85C2×D137—42no—
86C2×D137—82no—
87C2×D137#12011yescharacteristic
88C2×D137#12311yescharacteristic
89#10587739—44no—
90C4×C137—41no—
91C4×C137C4×D412yes—
92Dic137—41no—
93Dic137—41no—
94Dic137#12112yes—
95C2×C137—21no—
96C2×C137—42no—
97C2×C137—81no—
98C2×C137#38611yescharacteristic
99D137—21no—
100D137—21no—
101D137—42no—
102C137#191111yescharacteristic, complemented, Sylow
103#1911—1371nomaximal, Sylow
104#535—1371no—
105#518—1371no—
106#421—1372no—
107C2×#100—1371no—
108H(3,ℤ/4)—1372no—
109C2×#41—1371no—
110C22×D4—1371no—
111C2×M16—1372no—
112#127—1371no—
113#127#1058773911yescharacteristic, complemented
114#124—1371no—
115#124—1371no—
116#121—2742no—
117#120—2742no—
118C4×D4—2744no—
119#117—1372no—
120#117—1371no—
121C2×C22⋊C4—2742no—
122#104—2744no—
123#100—2742no—
124C24—1372no—
125#41—2742no—
126#41—2742no—
127C2×Q8—1371no—
128C2×D4—22no—
129C2×D4—2741no—
130C2×D4—2742no—
131C2×D4—2742no—
132C2×D4—2742no—
133C2×D4—1372no—
134C2×D4—1371no—
135C2×D4C2×#1058773911yescharacteristic
136C2×D4C2×#1058773911yescharacteristic, complemented
137C22×C4—2744no—
138C22×C4—2741no—
139C22×C4—1371no—
140M16—2742no—
141M16—2742no—
142C2×C8—2742no—
143C4⋊C4—2741no—
144C4⋊C4—2742no—
145C22⋊C4—2741no—
146C22⋊C4—2742no—
147C22⋊C4—2744no—
148C22⋊C4—2742no—
149C22⋊C4—2742no—
150C42—1374no—
151C42—1371no—
152C42C2×#1058773911yescharacteristic, complemented
153C23—21no—
154C23—2742no—
155C23—2742no—
156C23—10962no—
157C23—1372no—
158C23—1371no—
159C23C22⋊#1058773912yes—
160Q8—2742no—
161D4—22no—
162D4—22no—
163D4—82no—
164D4—2744no—
165D4—2742no—
166D4—5481no—
167D4—5481no—
168D4—10962no—
169C2×C4—21no—
170C2×C4—2744no—
171C2×C4—2741no—
172C2×C4—2741no—
173C2×C4—5482no—
174C2×C4—10964no—
175C2×C4—1371no—
176C2×C4—1372no—
177C2×C4—1372no—
178C2×C4C22×#1058773911yescharacteristic
179C8—5482no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—41no—
184C22—2742no—
185C22—5481no—
186C22—5482no—
187C22—5484no—
188C22—10962no—
189C22—1371no—
190C22—1371no—
191C22C2×C22⋊#1058773911yescharacteristic
192C4—41no—
193C4—5481no—
194C4—5484no—
195C4—5481no—
196C4—1372no—
197C4D4×#1058773912yes—
198C2—21no—
199C2—42no—
200C2—81no—
201C2—2741no—
202C2—2741no—
203C2—5482no—
204C2#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#1911
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. #22578401C2semidirect
3. D137⋊#100C2semidirect
4. #22578270C2semidirect
14. #17984084C4semidirect
15. #17984076C4semidirect
16. #17984075C4semidirect
20. #127×C137C4semidirect
28. C4×#10587739D4semidirect
32. C2×Dic274C2×C4semidirect
45. C4×Dic137C2×C4semidirect
48. C2×D4×C137C2×C4semidirect
49. C42×C137C2×C4semidirect
102. C137#1911semidirect
113. #127#10587739semidirect
136. C2×D4C2×#10587739semidirect
152. C42C2×#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784702
#225784012
D137⋊#1002
#225782702
#225779662
#1911137