Finite Group Databasebeta

← #31796953

Subgroups — #31796953

Order 17536.

336 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C2×#17984152C214yesmaximal, complemented
2C4×#14555520C211yesmaximal, characteristic, complemented
3C4×C4⋊#10587739C211yesmaximal, characteristic, complemented
4#22577951C211yesmaximal, characteristic
5C2×#14555520C2212yes—
6C2×C4⋊#10587739—21no—
7C2×C4×#10587739C2212yes—
8#17984152—28no—
9#17984143—21no—
10C4×#11764047—21no—
11(C2×#10587739)⋊C4—21no—
12C2×#14555509C2212yes—
13#17984088C411yescharacteristic, complemented
14#17984075C412yescomplemented
15#17984059—21no—
16#17984059—21no—
17D4×Dic137—21no—
18D4×Dic137—21no—
19C42×D137C2211yescharacteristic
20#127×C137C411yescharacteristic, complemented
21C22×#10587739—22no—
22#14555520—22no—
23#14555520D414yescomplemented
24C2×#11764047—22no—
25C4⋊#10587739—24no—
26C4×#10587739—24no—
27#14555509—24no—
28#14555509—24no—
29C2×Dic274C2×C412yes—
30C2×C22⋊D137—22no—
31C22×Dic137—22no—
32C2×C4×D137D411yescharacteristic
33C2×C4×D137D411yescharacteristic
34C2×C4×D137C2311yescharacteristic
35C4⋊Dic137—21no—
36C4⋊Dic137—21no—
37C22⋊Dic137—22no—
38C22⋊Dic137—22no—
39C4×Dic137C2×C411yescharacteristic
40C4×Dic137D411yescharacteristic
41C4×Dic137D411yescharacteristic
42C2×D4×C137—21no—
43C2×D4×C137—21no—
44C2×D4×C137C2×C412yes—
45C42×C137C2×C411yescharacteristic
46C2×#10587739—24no—
47C2×#10587739—42no—
48C2×#10587739—42no—
49#11764047—24no—
50C22×D137C2×D411yescharacteristic
51Dic274—24no—
52C22⋊D137—42no—
53C22⋊D137—42no—
54C2×Dic137—22no—
55C2×Dic137—44no—
56C2×Dic137C22⋊C411yescharacteristic
57C2×Dic137C22⋊C411yescharacteristic
58C2×Dic137C2×D411yescharacteristic
59C4×D137—22no—
60C4×D137—41no—
61C4×D137C2×D412yes—
62C4×D137#4112yes—
63C23×C137—22no—
64D4×C137—24no—
65D4×C137—42no—
66D4×C137—42no—
67C2×C4×C137C22⋊C411yescharacteristic
68C2×C4×C137C22⋊C411yescharacteristic
69C2×C4×C137C22×C411yescharacteristic
70C22×C137—42no—
71C22×C137—42no—
72C22×C137—42no—
73C22×C137C2×C22⋊C411yescharacteristic
74C2×D137—21no—
75C2×D137—21no—
76C2×D137#12411yescharacteristic
77C2×D137#12711yescharacteristic
78#10587739—44no—
79C4×C137—21no—
80C4×C137#10412yescomplemented
81C4×C137C4×D412yes—
82Dic137—21no—
83Dic137—21no—
84Dic137—42no—
85Dic137#12112yes—
86C2×C137—82no—
87C2×C137#39011yescharacteristic
88C2×C137C2×#10411yescharacteristic
89C2×C137#42111yescharacteristic
90D137—21no—
91D137—21no—
92C137#198011yescharacteristic, complemented, Sylow
93#1980—1371nomaximal, Sylow
94#532—1371no—
95C2×#104—1374no—
96C4×M16—1371no—
97C4×C4⋊C4—1371no—
98C2×#41—1372no—
99C2×M16—1372no—
100C4⋊Q8—1371no—
101#127#1058773911yescharacteristic, complemented
102#124—1372no—
103#121—2741no—
104#121—2741no—
105C4×D4—2741no—
106C4×D4—2741no—
107C2×C4⋊C4—2741no—
108C2×C42—1372no—
109C2×C42—1371no—
110#104—2748no—
111#97—2741no—
112C4×C8—2741no—
113#95—2741no—
114#41—2744no—
115#41—2744no—
116C2×Q8—1372no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—2742no—
120C2×D4C2×#1058773912yes—
121C22×C4—2742no—
122C22×C4—2742no—
123C22×C4—1371no—
124C22×C4—1371no—
125C22×C4—1371no—
126M16—2742no—
127M16—1374no—
128C2×C8—2742no—
129C4⋊C4—2744no—
130C4⋊C4—2741no—
131C4⋊C4—2741no—
132C22⋊C4—2742no—
133C22⋊C4—2742no—
134C42—2744no—
135C42—1371no—
136C42—1371no—
137C42—1371no—
138C42C2×#1058773911yescharacteristic
139C23—22no—
140C23—1371no—
141Q8—2744no—
142D4—24no—
143D4—42no—
144D4—42no—
145D4—5482no—
146D4—5482no—
147C2×C4—2742no—
148C2×C4—2744no—
149C2×C4—2742no—
150C2×C4—5481no—
151C2×C4—5482no—
152C2×C4—5482no—
153C2×C4—5484no—
154C2×C4—1371no—
155C2×C4—1371no—
156C2×C4—1371no—
157C2×C4—1372no—
158C2×C4—1372no—
159C2×C4C22⋊#1058773911yescharacteristic
160C2×C4C22⋊#1058773911yescharacteristic
161C2×C4C22×#1058773911yescharacteristic
162C8—2744no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22—2741no—
167C22—2741no—
168C22—1371no—
169C22—1371no—
170C22C2×C22⋊#1058773911yescharacteristic
171C4—21no—
172C4—2741no—
173C4—2741no—
174C4—5484no—
175C4—5482no—
176C4—1372no—
177C4#1798415212yescomplemented
178C4D4×#1058773912yes—
179C2—82no—
180C2—2741no—
181C2—2741no—
182C2#2257835511yescharacteristic
183C2C2×#1798415211yescharacteristic
184C2#2257840111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#127×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577951C2
cosocleC2×C4×D137C23
hypercenter#127#10587739
hypocenterC137#1980
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C2×#17984152C2semidirect
2. C4×#14555520C2semidirect
3. C4×C4⋊#10587739C2semidirect
13. #17984088C4semidirect
14. #17984075C4semidirect
20. #127×C137C4semidirect
23. #14555520D4semidirect
80. C4×C137#104semidirect
92. C137#1980semidirect
101. #127#10587739semidirect
177. C4#17984152semidirect

Maximal subgroups

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

subgroupindex
C2×#179841522
C4×#145555202
C4×C4⋊#105877392
#225779512
#1980137