Finite Group Databasebeta

← #31780025

Subgroups — #31780025

Order 17536.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#22578056C211yesmaximal, characteristic
2#22577806C211yesmaximal, characteristic, complemented
3#22577684C211yesmaximal, characteristic, complemented
4#22577609C211yesmaximal, characteristic, complemented
5#22577130C211yesmaximal, characteristic, complemented
6#22577130C211yesmaximal, characteristic, complemented
7#332×C137C211yesmaximal, characteristic, complemented
8C2×#14555511—21no—
9C2×#14555499—21no—
10#17984088C2211yescharacteristic
11#17984073—22no—
12#17984064—21no—
13#17984063—21no—
14C4×D548C2211yescharacteristic
15C4×D548C2211yescharacteristic
16C4⋊C4×D137—21no—
17#17984022—21no—
18#17984021—22no—
19#17984013—21no—
20#17984010—21no—
21C4×#11764040C2211yescharacteristic
22C4⋊Q8×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4⋊C8×C137C2211yescharacteristic
25#14555511—42no—
26#14555511—42no—
27C2×Dic274—21no—
28C2×D548—22no—
29C2×D548D411yescharacteristic
30C2×D548D411yescharacteristic
31C2×C4×D137—22no—
32C2×C4×D137—21no—
33C2×C4×D137—21no—
34#14555499—42no—
35C2×#11764040—21no—
36C4⋊Dic137—21no—
37C4⋊Dic137—21no—
38C4⋊Dic137D411yescharacteristic
39C4⋊Dic137D411yescharacteristic
40#14555489—22no—
41#14555487—22no—
42#14555487—21no—
43#14555487—21no—
44C2×Q8×C137—21no—
45C2×C8×C137—21no—
46C2×C8×C137—21no—
47C4⋊C4×C137—21no—
48C4⋊C4×C137D412yes—
49C42×C137C2311yescharacteristic
50C22×D137—21no—
51C22×D137—21no—
52Dic274—41no—
53Dic274—41no—
54D548—21no—
55D548—21no—
56D548—21no—
57D548—44no—
58D548D812yes—
59C2×Dic137—21no—
60C2×Dic137—21no—
61C2×Dic137—41no—
62C4×D137—44no—
63C4×D137—41no—
64C4×D137—41no—
65C4×D137—41no—
66C4×D137—41no—
67#11764040—22no—
68Q8×C137—41no—
69Q8×C137—41no—
70C2×C4×C137—22no—
71C2×C4×C137C2×D411yescharacteristic
72C2×C4×C137C2×D411yescharacteristic
73C2×C4×C137C2×D411yescharacteristic
74C8×C137—41no—
75C8×C137—41no—
76C22×C137#12011yescharacteristic
77C2×D137—22no—
78C2×D137—41no—
79C2×D137—41no—
80C2×D137—41no—
81C2×D137—41no—
82C2×D137—41no—
83C4×C137—21no—
84C4×C137—42no—
85C4×C137C2×D811yescharacteristic
86C4×C137#13511yescharacteristic
87C4×C137AGL(1,ℤ/8)11yescharacteristic
88C4×C137#13711yescharacteristic
89Dic137—41no—
90Dic137—41no—
91Dic137—81no—
92C2×C137#44711yescharacteristic
93C2×C137#45211yescharacteristic
94C2×C137#45611yescharacteristic
95D137—42no—
96D137—81no—
97C137#164711yescharacteristic, complemented, Sylow
98#1647—1371nomaximal, Sylow
99#547—1371no—
100#500—1371no—
101#474—1371no—
102#461—1371no—
103#332D13711yescharacteristic, complemented
104D4⋊C8—1371no—
105D4⋊C8—1371no—
106C2×#41—2741no—
107C2×SD16—2741no—
108C4⋊Q8—1371no—
109C4⋊Q8C2×D13711yescharacteristic
110#123—2742no—
111C22⋊Q8—2741no—
112#121—2741no—
113C4×D4—1371no—
114C4×D4—1371no—
115C2×C4⋊C4—2741no—
116#107—2741no—
117#107—2742no—
118C4⋊C8C2×D13711yescharacteristic
119C4⋊C8C2×D13711yescharacteristic
120#103—2741no—
121#102—2741no—
122C4×C8—1371no—
123#41—5482no—
124#41—5482no—
125C2×Q8—21no—
126C2×Q8—2741no—
127C2×D4—2742no—
128C2×D4—1371no—
129C2×D4—1371no—
130C22×C4—2742no—
131C22×C4—2741no—
132C22×C4—2741no—
133SD16—5482no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—2741no—
137C4⋊C4—21no—
138C4⋊C4—2742no—
139C4⋊C4—2741no—
140C4⋊C4—2741no—
141C4⋊C4—1371no—
142C4⋊C4—1371no—
143C4⋊C4D54812yes—
144C22⋊C4—2742no—
145C22⋊C4—2741no—
146C22⋊C4—2741no—
147C42C22×D13711yescharacteristic
148C23—2741no—
149C23—2741no—
150Q8—41no—
151Q8—41no—
152Q8—5481no—
153Q8—5481no—
154D4—2741no—
155D4—2741no—
156D4—2741no—
157D4—5484no—
158D4—1372no—
159C2×C4—22no—
160C2×C4—2741no—
161C2×C4—2741no—
162C2×C4—5484no—
163C2×C4—5481no—
164C2×C4—5481no—
165C2×C4—5481no—
166C2×C4—5481no—
167C2×C4—5481no—
168C2×C4D4×D13711yescharacteristic
169C2×C4D4×D13711yescharacteristic
170C2×C4C2×D54811yescharacteristic
171C8—41no—
172C8—41no—
173C8—2742no—
174C22—2742no—
175C22—5481no—
176C22—5481no—
177C22—5481no—
178C22—5481no—
179C22—5481no—
180C22#1798405811yescharacteristic
181C4—21no—
182C4—42no—
183C4—5481no—
184C4—5481no—
185C4—10961no—
186C4D8×D13711yescharacteristic
187C4#1798410911yescharacteristic
188C4#1798411511yescharacteristic
189C4#1798412011yescharacteristic
190C2—5482no—
191C2—10961no—
192C2#2257753811yescharacteristic
193C2#2257756811yescharacteristic
194C2#2257758811yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#332×C137C2
socleC22×C137#120
prime radical#22578056C2
hypercenter#332D137
hypocenterC137#1647
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #22577806C2semidirect
3. #22577684C2semidirect
4. #22577609C2semidirect
5. #22577130C2semidirect
6. #22577130C2semidirect
7. #332×C137C2semidirect
97. C137#1647semidirect
103. #332D137semidirect

Maximal subgroups

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

subgroupindex
#225780562
#225778062
#225776842
#225776092
#225771302
#332×C1372
#1647137