Finite Group Databasebeta

← #31796107

Subgroups — #31796107

Order 17536.

234 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 62 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#22578404C211yesmaximal, characteristic, complemented
2C4×#14555520C211yesmaximal, characteristic, complemented
3#22578253C212yesmaximal, complemented
4#22578251C212yesmaximal, complemented
5#22577938C211yesmaximal, characteristic
6C2×#14555520—21no—
7C2×#14555518—21no—
8#17984155—21no—
9C4⋊#11764047C2212yes—
10#17984143—21no—
11C4×#11764047C2212yes—
12#17984083—21no—
13#17984076C411yescharacteristic
14#17984073—21no—
15#17984068C411yescharacteristic
16#17984063C411yescharacteristic
17#17984053C2211yescharacteristic
18#17984050—21no—
19#17984050C2211yescharacteristic
20#17984046—21no—
21C42×D137C2211yescharacteristic
22#125×C137C411yescharacteristic
23#14555520—24no—
24#14555518—42no—
25C2×#11764047—22no—
26C2×#11764047—22no—
27C2×Dic274C2×C411yescharacteristic
28C2×D548—21no—
29C2×D548C2×C411yescharacteristic
30C2×C4×D137—21no—
31C2×C4×D137C2×C411yescharacteristic
32C2×C4×D137D411yescharacteristic
33C2×C4×D137D411yescharacteristic
34C4⋊Dic137—21no—
35#14555489—21no—
36#14555487—21no—
37#14555487—21no—
38#14555487—21no—
39#14555487—21no—
40#14555487—21no—
41C4×Dic137—21no—
42C4×Dic137D411yescharacteristic
43C4×Dic137D411yescharacteristic
44C4×Dic137C2311yescharacteristic
45C4⋊C4×C137—21no—
46C4⋊C4×C137—21no—
47C4⋊C4×C137C2×C411yescharacteristic
48C4⋊C4×C137C2×C411yescharacteristic
49C42×C137C2×C411yescharacteristic
50#11764047—24no—
51#11764047—42no—
52C22×D137—21no—
53C22×D137C22⋊C411yescharacteristic
54Dic274—22no—
55D548—22no—
56D548—42no—
57C2×Dic137—21no—
58C2×Dic137C22⋊C411yescharacteristic
59C2×Dic137C2×D411yescharacteristic
60C2×Dic137C2×D411yescharacteristic
61C4×D137—22no—
62C4×D137—22no—
63C4×D137—21no—
64C4×D137—21no—
65C4×D137—41no—
66C4×D137—41no—
67C2×C4×C137—21no—
68C2×C4×C137—21no—
69C2×C4×C137C22⋊C411yescharacteristic
70C2×C4×C137C22⋊C411yescharacteristic
71C2×C4×C137C22×C411yescharacteristic
72C22×C137C2×C22⋊C411yescharacteristic
73C2×D137—21no—
74C2×D137—21no—
75C2×D137—41no—
76C2×D137—41no—
77C2×D137—41no—
78C2×D137#10412yes—
79C4×C137—21no—
80C4×C137—21no—
81C4×C137—41no—
82C4×C137—41no—
83C4×C137#13112yes—
84Dic137—21no—
85Dic137—21no—
86Dic137—41no—
87Dic137AGL(1,ℤ/8)11yescharacteristic
88Dic137#13711yescharacteristic
89C2×C137#40811yescharacteristic
90C2×C137#41711yescharacteristic
91C2×C137C2×#10411yescharacteristic
92D137—22no—
93D137—81no—
94C137#148311yescharacteristic, complemented, Sylow
95#1483—1371nomaximal, Sylow
96#529—1371no—
97C2×C4⋊C8—1371no—
98C4×M16—1371no—
99Q8⋊C8—1372no—
100D4⋊C8—1372no—
101C2×M16—2741no—
102C22×C8—2741no—
103#126—2741no—
104#125#1058773911yescharacteristic
105#124—1371no—
106#123—2741no—
107C22⋊Q8—1371no—
108#121—1371no—
109C4×Q8—1371no—
110C4×D4—2741no—
111C4×D4—1371no—
112#117—2741no—
113C2×C42—1371no—
114C4⋊C8—2741no—
115C4⋊C8—1372no—
116#97—2741no—
117C4×C8—1372no—
118C2×Q8—1371no—
119C2×D4—2741no—
120C2×D4—1371no—
121C22×C4—2741no—
122C22×C4—1371no—
123C22×C4—1371no—
124C22×C4—1371no—
125M16—2744no—
126C2×C8—2742no—
127C2×C8—2742no—
128C2×C8—5482no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—2741no—
132C4⋊C4—2741no—
133C4⋊C4C2×#1058773911yescharacteristic
134C4⋊C4C2×#1058773911yescharacteristic
135C22⋊C4—2741no—
136C22⋊C4—2741no—
137C22⋊C4—2741no—
138C22⋊C4—2741no—
139C22⋊C4—2741no—
140C42—2741no—
141C42—1371no—
142C42—1371no—
143C42—1371no—
144C42C2×#1058773911yescharacteristic
145C23—2741no—
146C23—1371no—
147Q8—2742no—
148D4—2742no—
149D4—5482no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—2742no—
153C2×C4—2741no—
154C2×C4—2741no—
155C2×C4—2742no—
156C2×C4—2741no—
157C2×C4—5481no—
158C2×C4—5481no—
159C2×C4—1371no—
160C2×C4—1371no—
161C2×C4—1371no—
162C2×C4C22⋊#1058773911yescharacteristic
163C2×C4C22⋊#1058773911yescharacteristic
164C2×C4C22×#1058773911yescharacteristic
165C8—2744no—
166C8—5482no—
167C22—2741no—
168C22—2741no—
169C22—5481no—
170C22—5481no—
171C22—5481no—
172C22—1372no—
173C22C2×C22⋊#1058773911yescharacteristic
174C4—21no—
175C4—21no—
176C4—41no—
177C4—41no—
178C4—2741no—
179C4—2741no—
180C4—5481no—
181C4—1371no—
182C4—1371no—
183C4#1798417612yes—
184C2—2742no—
185C2—10961no—
186C2#2257837811yescharacteristic
187C2#2257839411yescharacteristic
188C2D137⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#125×C137C4
socleC22×C137C2×C22⋊C4
prime radical#17984063C4
cosocleC4×Dic137C23
hypercenter#125#10587739
hypocenterC137#1483
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
1. #22578404C2semidirect
2. C4×#14555520C2semidirect
3. #22578253C2semidirect
4. #22578251C2semidirect
94. C137#1483semidirect

Maximal subgroups

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

subgroupindex
#225784042
C4×#145555202
#225782532
#225782512
#225779382
#1483137