Finite Group Databasebeta

← #31780752

Subgroups — #31780752

Order 17536.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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#22578239C211yesmaximal, characteristic
2#22577419C212yesmaximal, complemented
3#22577400C212yesmaximal, complemented
4#22577377C211yesmaximal, characteristic, complemented
5#408×C137C211yesmaximal, characteristic, complemented
6#17984136C412yescomplemented
7#17984135—22no—
8#17984134—22no—
9#17984133—22no—
10#17984133C412yescomplemented
11C2×#14555510—22no—
12C2×#14555510C2212yes—
13#41×D137—24no—
14#17984047C2212yescomplemented
15#17984014—24no—
16#17984013—22no—
17#17984010—22no—
18#17984003—22no—
19C2×#41×C137C2211yescharacteristic
20C2×M16×C137C2211yescharacteristic, complemented
21C22×C8×C137C2211yescharacteristic, complemented
22C22⋊C8×C137—22no—
23#14555511—24no—
24#14555510—24no—
25#14555510—24no—
26#14555510—24no—
27#14555510—22no—
28#14555510—22no—
29#14555510C2×C414yes—
30#14555510D414yes—
31#14555509—28no—
32#14555509—24no—
33C2×Dic274—22no—
34C2×Dic274D412yes—
35Q8×D137—24no—
36C2×D548—22no—
37C2×D548D412yes—
38C2×C22⋊D137—24no—
39C2×C22⋊D137—22no—
40D4×D137—28no—
41D4×D137—24no—
42C2×C4×D137—24no—
43C2×C4×D137—22no—
44C4⋊Dic137—22no—
45C22⋊Dic137—22no—
46C4×Dic137—22no—
47#41×C137—22no—
48#41×C137—22no—
49C2×Q8×C137C2×C411yescharacteristic
50C2×D4×C137C2×C411yescharacteristic
51C2×D4×C137D412yes—
52C22×C4×C137D412yes—
53C22×C4×C137C2311yescharacteristic
54M16×C137—22no—
55C2×C8×C137—22no—
56C2×C8×C137—21no—
57C2×C8×C137—21no—
58C22×D137—24no—
59C22×D137—22no—
60Dic274—24no—
61Dic274—22no—
62Dic274C22⋊C414yes—
63D548—24no—
64D548—22no—
65D548C22⋊C414yes—
66C22⋊D137—28no—
67C22⋊D137—24no—
68C22⋊D137—24no—
69C2×Dic137—24no—
70C2×Dic137—22no—
71C2×Dic137—42no—
72C4×D137—28no—
73C4×D137—24no—
74C4×D137—24no—
75C23×C137C22⋊C412yes—
76C23×C137C2×D411yescharacteristic
77Q8×C137—22no—
78D4×C137—24no—
79D4×C137—22no—
80C2×C4×C137—22no—
81C2×C4×C137—22no—
82C2×C4×C137C22⋊C412yes—
83C2×C4×C137C22×C411yescharacteristic
84C2×C4×C137C2×D412yes—
85C2×C4×C137C2×D412yes—
86C2×C4×C137C2×D411yescharacteristic
87C8×C137—41no—
88C8×C137—41no—
89C22×C137—22no—
90C22×C137—41no—
91C22×C137—42no—
92C22×C137C2×C22⋊C411yescharacteristic
93C22×C137#12012yes—
94C2×D137—24no—
95C2×D137—82no—
96C2×D137—84no—
97C4×C137—22no—
98C4×C137C2×C22⋊C412yes—
99C4×C137#12011yescharacteristic
100C4×C137#12011yescharacteristic
101Dic137—24no—
102Dic137—82no—
103C2×C137—22no—
104C2×C137—21no—
105C2×C137—42no—
106C2×C137#37911yescharacteristic
107D137—44no—
108C137#179411yescharacteristic, complemented, Sylow
109#1794—1371nomaximal, Sylow
110#585—1371no—
111#421—1372no—
112#416—1372no—
113#410—1371no—
114#408D13711yescharacteristic, complemented
1152−1+4—2742no—
1162−1+4—1372no—
117H(5,𝔽2)—2742no—
118H(5,𝔽2)—2742no—
119H(5,𝔽2)—1372no—
120C2×#41—2744no—
121C2×#41—2742no—
122C2×#41—1372no—
123C2×#41C2×D13711yescharacteristic
124C2×M16C2×D13711yescharacteristic, complemented
125C22×C8C2×D13711yescharacteristic, complemented
126#117—1372no—
127#104—2744no—
128#103—2742no—
129#102—2742no—
130#99—2742no—
131C22⋊C8—22no—
132#41—22no—
133#41—22no—
134#41—2744no—
135#41—2744no—
136#41—2748no—
137#41—2744no—
138#41—2744no—
139#41—2744no—
140#41—2742no—
141#41—2742no—
142#41—1374no—
143#41—1374no—
144C2×Q8—2742no—
145C2×Q8—2744no—
146C2×Q8—1372no—
147C2×Q8C4×D13711yescharacteristic
148C2×D4—2742no—
149C2×D4—2748no—
150C2×D4—2744no—
151C2×D4—2744no—
152C2×D4—2742no—
153C2×D4—1372no—
154C2×D4C4×D13711yescharacteristic
155C2×D4D54812yes—
156C22×C4—2744no—
157C22×C4—2742no—
158C22×C4C22⋊D13712yes—
159C22×C4C22×D13711yescharacteristic
160M16—22no—
161C2×C8—22no—
162C2×C8—21no—
163C2×C8—21no—
164C4⋊C4—2742no—
165C22⋊C4—2742no—
166C42—2742no—
167C23—2744no—
168C23—2742no—
169C23#1455548712yes—
170C23C2×D54811yescharacteristic
171Q8—22no—
172Q8—2744no—
173Q8—2742no—
174Q8—1374no—
175D4—24no—
176D4—22no—
177D4—2744no—
178D4—2748no—
179D4—2744no—
180D4—2744no—
181D4—2742no—
182D4—1374no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—2748no—
186C2×C4—2744no—
187C2×C4—2742no—
188C2×C4—2744no—
189C2×C4—2744no—
190C2×C4—5482no—
191C2×C4#1455548712yes—
192C2×C4C2×C4×D13711yescharacteristic
193C2×C4D4×D13712yes—
194C2×C4D4×D13712yes—
195C2×C4C2×C22⋊D13711yescharacteristic
196C8—41no—
197C8—41no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—2744no—
202C22—10962no—
203C22—10964no—
204C22C2×#1455548711yescharacteristic
205C22#1798405812yes—
206C4—22no—
207C4—2744no—
208C4—10962no—
209C4C22⋊C4×D13712yes—
210C4#1798405611yescharacteristic
211C4#1798405611yescharacteristic
212C2—22no—
213C2—21no—
214C2—42no—
215C2—5484no—
216C2#2257725811yescharacteristic
subgroupG/H
centerC4#17984056
commutatorC2×C4×C137C22×C4
FrattiniC22×C4C22×D137
Fitting#408×C137C2
socleC2×C137#379
prime radical#22578239C2
cosocleC22×C4×C137C23
hypercenter#408D137
hypocenterC137#1794
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #22577419C2semidirect
3. #22577400C2semidirect
4. #22577377C2semidirect
5. #408×C137C2semidirect
6. #17984136C4semidirect
10. #17984133C4semidirect
14. #17984047C22semidirect
20. C2×M16×C137C22semidirect
21. C22×C8×C137C22semidirect
108. C137#1794semidirect
114. #408D137semidirect
124. C2×M16C2×D137semidirect
125. C22×C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782392
#225774192
#225774002
#225773772
#408×C1372
#1794137