Finite Group Databasebeta

← #31796830

Subgroups — #31796830

Order 17536.

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

normal subgroups only

This group has 312 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
2#22578396C212yesmaximal, complemented
3#22578383C211yesmaximal, characteristic
4#22578270C212yesmaximal, complemented
5#22577973C211yesmaximal, characteristic
6#17984175—24no—
7D137⋊M16C2212yes—
8C2×#14555519—22no—
9#17984167C2212yes—
10#17984151—24no—
11#17984146—22no—
12C2×#14555509C2211yescharacteristic
13C2×Q8×D137C2211yescharacteristic
14#17984088C411yescharacteristic, complemented
15#17984086C411yescharacteristic, complemented
16#17984075C411yescharacteristic, complemented
17#17984070—22no—
18#17984065—22no—
19#17984045C2211yescharacteristic
20#124×C137C411yescharacteristic, complemented
21#14555520—22no—
22#14555519—24no—
23#14555519—22no—
24#14555518D414yes—
25C2×#11764047—24no—
26C4⋊#10587739—22no—
27C22⋊#10587739—22no—
28C4×#10587739—22no—
29#14555509—22no—
30#14555509—22no—
31C2×Dic274—21no—
32C2×Dic274C2×C411yescharacteristic
33C2×Dic274C2×C411yescharacteristic
34Q8×D137—22no—
35Q8×D137—22no—
36C2×C22⋊D137D412yes—
37C22×Dic137D412yes—
38C2×C4×D137—21no—
39C2×C4×D137C2311yescharacteristic
40C4⋊Dic137—22no—
41#14555489—22no—
42#14555489—21no—
43C22⋊Dic137—22no—
44#14555487—21no—
45C4×Dic137C2×C411yescharacteristic
46C2×Q8×C137C2×C411yescharacteristic
47C2×D4×C137C2×C411yescharacteristic
48C22⋊C4×C137—22no—
49C42×C137C2×C411yescharacteristic
50C2×#10587739—42no—
51#11764047—24no—
52C22×D137C2×D411yescharacteristic
53Dic274—22no—
54Dic274—22no—
55Dic274—41no—
56Dic274—41no—
57C22⋊D137—24no—
58C2×Dic137—22no—
59C2×Dic137—22no—
60C2×Dic137—21no—
61C2×Dic137—21no—
62C2×Dic137C22⋊C412yes—
63C2×Dic137C2×D411yescharacteristic
64C4×D137—42no—
65C4×D137C2×D412yes—
66C4×D137#4112yes—
67C23×C137C22⋊C412yes—
68Q8×C137—22no—
69D4×C137—22no—
70C2×C4×C137—21no—
71C2×C4×C137—21no—
72C2×C4×C137C22×C411yescharacteristic
73C22×C137—22no—
74C22×C137—42no—
75C22×C137C2×C22⋊C411yescharacteristic
76C2×D137—41no—
77C2×D137#12112yes—
78#10587739—82no—
79C4×C137—41no—
80C4×C137—41no—
81C4×C137C4×D412yes—
82Dic137—22no—
83Dic137—41no—
84Dic137—41no—
85Dic137#12011yescharacteristic
86Dic137#12311yescharacteristic
87C2×C137—21no—
88C2×C137—42no—
89C2×C137#38611yescharacteristic
90D137—22no—
91C137#190911yescharacteristic, complemented, Sylow
92#1909—1371nomaximal, Sylow
93#568—1371no—
94#536—1371no—
95#419—1372no—
96#410—1371no—
97H(3,ℤ/4)—1372no—
98C2×#41—1371no—
99C22×Q8—1371no—
100#131—2744no—
101C2×M16—2742no—
102C2×M16—1372no—
103C4⋊Q8—1371no—
104C4⋊Q8—1371no—
105#124—1371no—
106#124#1058773911yescharacteristic, complemented
107#123—2742no—
108C22⋊Q8—2742no—
109#117—1372no—
110#117—1371no—
111#103—2744no—
112#99—2742no—
113#41—2742no—
114#41—2742no—
115C2×Q8—2741no—
116C2×Q8—2742no—
117C2×Q8—2742no—
118C2×Q8—1371no—
119C2×Q8—1371no—
120C2×Q8C2×#1058773911yescharacteristic
121C2×D4—1372no—
122C2×D4C2×#1058773911yescharacteristic
123C22×C4—2741no—
124C22×C4—1372no—
125C22×C4—1371no—
126M16—2744no—
127M16—2742no—
128M16—2742no—
129C2×C8—2744no—
130C2×C8—1374no—
131C4⋊C4—2741no—
132C4⋊C4—2742no—
133C4⋊C4—2742no—
134C4⋊C4—2742no—
135C22⋊C4—22no—
136C22⋊C4—2741no—
137C22⋊C4—2742no—
138C22⋊C4—2742no—
139C42—2742no—
140C42—1371no—
141C42C2×#1058773911yescharacteristic
142C23—1371no—
143C23C22⋊#1058773912yes—
144Q8—22no—
145Q8—2742no—
146Q8—2742no—
147Q8—5481no—
148Q8—5481no—
149D4—22no—
150D4—2744no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—2742no—
154C2×C4—2742no—
155C2×C4—2741no—
156C2×C4—2741no—
157C2×C4—5482no—
158C2×C4—5482no—
159C2×C4—1372no—
160C2×C4—1371no—
161C2×C4—1372no—
162C2×C4—1372no—
163C2×C4C22×#1058773911yescharacteristic
164C8—2744no—
165C22—22no—
166C22—42no—
167C22—5481no—
168C22—1372no—
169C22C2×C22⋊#1058773911yescharacteristic
170C4—41no—
171C4—41no—
172C4—2742no—
173C4—5481no—
174C4—5481no—
175C4—10962no—
176C4—1371no—
177C4—1371no—
178C4D4×#1058773912yes—
179C2—21no—
180C2—42no—
181C2—2742no—
182C2#2257835111yescharacteristic
subgroupG/H
centerC2#22578351
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#124×C137C4
socleC2×C137#386
prime radicalC2×#14555509C22
cosocleC2×C4×D137C23
hypercenter#124#10587739
hypocenterC137#1909
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #22578396C2semidirect
4. #22578270C2semidirect
14. #17984088C4semidirect
15. #17984086C4semidirect
16. #17984075C4semidirect
20. #124×C137C4semidirect
91. C137#1909semidirect
106. #124#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784812
#225783962
#225783832
#225782702
#225779732
#1909137