Finite Group Databasebeta

← #31796229

Subgroups — #31796229

Order 17536.

288 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 70 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#22578371C211yesmaximal, characteristic
2#22578345C211yesmaximal, characteristic
3#22578257C214yesmaximal, complemented
4C2×#17984064C211yesmaximal, characteristic
5C2×#14555518—21no—
6C2×C4⋊#10587739—21no—
7C2×C22⋊#10587739—21no—
8#17984145C2212yes—
9C22⋊#11764047—21no—
10(C2×#10587739)⋊C4C2212yes—
11C22×C4×D137C2211yescharacteristic
12#17984064—21no—
13#17984064—21no—
14#17984064C412yescomplemented
15#17984064C2212yes—
16C2×C4⋊Dic137C411yescharacteristic, complemented
17C2×#14555489—21no—
18C2×#14555487—21no—
19C22×Q8×C137C411yescharacteristic, complemented
20C22×#10587739—22no—
21#14555518—42no—
22C2×#11764047—22no—
23C4⋊#10587739—42no—
24C22⋊#10587739—42no—
25C23×D137D411yescharacteristic
26C22×Dic137—21no—
27C22×Dic137D411yescharacteristic
28C2×C4×D137—42no—
29C2×C4×D137C2×C411yescharacteristic
30C2×C4×D137D411yescharacteristic
31C2×C4×D137D411yescharacteristic
32C2×C4×D137C2311yescharacteristic
33C4⋊Dic137—21no—
34C4⋊Dic137C2×C412yes—
35#14555489—22no—
36#14555489—22no—
37#14555487—22no—
38#14555487—22no—
39C2×Q8×C137—21no—
40C2×Q8×C137—42no—
41C2×Q8×C137C2×C412yes—
42C22×C4×C137—21no—
43C22×C4×C137C2×C411yescharacteristic
44C2×#10587739—42no—
45C2×#10587739—42no—
46C2×#10587739—42no—
47#11764047—42no—
48C22×D137—41no—
49C22×D137—41no—
50C22×D137C22⋊C411yescharacteristic
51C22×D137SD1612yes—
52C22×D137Q1612yes—
53C22×D137C2×D411yescharacteristic
54C2×Dic137—22no—
55C2×Dic137—41no—
56C2×Dic137—42no—
57C2×Dic137C22⋊C411yescharacteristic
58C2×Dic137C2×D411yescharacteristic
59C4×D137—22no—
60C4×D137—22no—
61C4×D137—41no—
62C4×D137—41no—
63C23×C137C22⋊C411yescharacteristic
64Q8×C137—42no—
65Q8×C137—42no—
66C2×C4×C137—22no—
67C2×C4×C137—22no—
68C2×C4×C137—42no—
69C2×C4×C137C22⋊C411yescharacteristic
70C2×C4×C137C22×C411yescharacteristic
71C22×C137—21no—
72C22×C137—21no—
73C22×C137#10312yes—
74C22×C137C2×C22⋊C411yescharacteristic
75C2×D137—21no—
76C2×D137—21no—
77C2×D137—21no—
78C2×D137—21no—
79C2×D137—42no—
80C2×D137—42no—
81C2×D137#10312yes—
82C2×D137C2×SD1611yescharacteristic
83C2×D137C2×Q1611yescharacteristic
84#10587739—82no—
85C4×C137—21no—
86C4×C137—42no—
87C4×C137#9912yes—
88Dic137—21no—
89Dic137—21no—
90Dic137—42no—
91C2×C137—22no—
92C2×C137C2×#9911yescharacteristic
93C2×C137C2×#10311yescharacteristic
94C2×C137#42111yescharacteristic
95D137—21no—
96D137—22no—
97D137—21no—
98C137#152011yescharacteristic, complemented, Sylow
99#1520—1371nomaximal, Sylow
100C2×C22⋊Q8—1371no—
101C2×C22⋊C8—1371no—
102#380—1371no—
103#328—1374no—
104C22×Q8#1058773911yescharacteristic, complemented
105C23×C4—1371no—
106C22×C8—2741no—
107C22⋊Q8—2741no—
108C22⋊Q8—2741no—
109C22⋊Q8—1372no—
110C22⋊Q8—1372no—
111C2×C4⋊C4—2741no—
112C2×C4⋊C4—2741no—
113C2×C4⋊C4—1371no—
114C2×C22⋊C4—2741no—
115C2×C22⋊C4—2741no—
116C22⋊C8—2741no—
117C22⋊C8—1372no—
118#95—1372no—
119C24—1371no—
120C2×Q8—21no—
121C2×Q8—42no—
122C2×Q8C2×#1058773912yes—
123C22×C4—21no—
124C22×C4—2741no—
125C22×C4—2742no—
126C22×C4—5482no—
127C22×C4—1371no—
128C22×C4—1371no—
129C22×C4—1371no—
130C22×C4—1371no—
131C22×C4—1371no—
132C22×C4C2×#1058773911yescharacteristic
133C2×C8—2742no—
134C2×C8—5482no—
135C4⋊C4—2742no—
136C4⋊C4—2742no—
137C4⋊C4—2741no—
138C4⋊C4—5482no—
139C4⋊C4—1372no—
140C22⋊C4—2742no—
141C22⋊C4—2742no—
142C22⋊C4—5482no—
143C23—5481no—
144C23—5481no—
145C23—1371no—
146C23—1371no—
147C23—1372no—
148C23—1372no—
149C23C22⋊#1058773911yescharacteristic
150Q8—42no—
151Q8—42no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—2742no—
156C2×C4—2742no—
157C2×C4—2742no—
158C2×C4—5481no—
159C2×C4—5481no—
160C2×C4—5481no—
161C2×C4—5482no—
162C2×C4—5482no—
163C2×C4—5482no—
164C2×C4—5482no—
165C2×C4—1371no—
166C2×C4—1371no—
167C2×C4C22⋊#1058773911yescharacteristic
168C2×C4C22×#1058773911yescharacteristic
169C8—5482no—
170C22—21no—
171C22—21no—
172C22—2741no—
173C22—2741no—
174C22—2741no—
175C22—2741no—
176C22—5482no—
177C22—5482no—
178C22—1371no—
179C22—1372no—
180C22—1371no—
181C22#1798415112yes—
182C22C2×C22⋊#1058773911yescharacteristic
183C4—21no—
184C4—42no—
185C4—2741no—
186C4—2741no—
187C4—5482no—
188C4—10962no—
189C4#1798414612yes—
190C2—22no—
191C2—2741no—
192C2—2741no—
193C2—2742no—
194C2C2×#1798414611yescharacteristic
195C2C2×#1798415111yescharacteristic
196C2#2257840211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC22×Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC23×D137D4
cosocleC2×C4×D137C23
hypercenterC22×Q8#10587739
hypocenterC137#1520
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #22578257C2semidirect
14. #17984064C4semidirect
16. C2×C4⋊Dic137C4semidirect
19. C22×Q8×C137C4semidirect
98. C137#1520semidirect
104. C22×Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783712
#225783452
#225782572
C2×#179840642
#1520137