Finite Group Databasebeta

← #31782759

Subgroups — #31782759

Order 17536.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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#22578204C211yesmaximal, characteristic
2#22577844C211yesmaximal, characteristic, complemented
3#22577675C211yesmaximal, characteristic, complemented
4#22577216C211yesmaximal, characteristic, complemented
5#22577211C211yesmaximal, characteristic, complemented
6#22577186C211yesmaximal, characteristic, complemented
7C2×SD32×C137C211yesmaximal, characteristic, complemented
8#17984133—21no—
9#17984119—21no—
10#17984114—21no—
11#17984112—22no—
12#17984110C2211yescharacteristic
13#17984107—21no—
14D8×D137—21no—
15#17984094C2211yescharacteristic
16#17984034—21no—
17#17984031—21no—
18#17984028C2211yescharacteristic, complemented
19#17984024C2211yescharacteristic, complemented
20#17984007—21no—
21C2×Q16×C137C2211yescharacteristic, complemented
22C2×D8×C137C2211yescharacteristic
23SD32×C137—22no—
24C2×C16×C137C2211yescharacteristic, complemented
25#14555510—21no—
26#14555510D411yescharacteristic
27#14555509—22no—
28#14555509—41no—
29C2×C22⋊D137—42no—
30D4×D137—22no—
31D4×D137—41no—
32Dic548D411yescharacteristic
33#14555500—41no—
34#14555499—21no—
35#14555498—22no—
36#14555497—41no—
37D1096D411yescharacteristic
38#14555495—22no—
39#14555494—21no—
40#14555494D411yescharacteristic
41#14555493—21no—
42C8×D137—21no—
43#14555485—21no—
44C2×Q8×C137—21no—
45C2×D4×C137—21no—
46Q16×C137—21no—
47Q16×C137—21no—
48D8×C137—21no—
49D8×C137D412yes—
50C2×C8×C137C2311yescharacteristic
51C16×C137—21no—
52C22×D137—42no—
53Dic274—21no—
54Dic274D811yescharacteristic
55D548—21no—
56D548D811yescharacteristic
57C22⋊D137—21no—
58C22⋊D137—42no—
59C22⋊D137—42no—
60C22⋊D137—41no—
61C2×Dic137—42no—
62C4×D137—21no—
63C4×D137—41no—
64#11764040—21no—
65#11764040—41no—
66C23×C137—41no—
67Q8×C137—41no—
68Q8×C137—41no—
69D4×C137—22no—
70D4×C137—41no—
71C2×C4×C137—41no—
72C2×C4×C137C2×D411yescharacteristic
73C8×C137C2×D411yescharacteristic
74C8×C137C2×D411yescharacteristic
75C22×C137—42no—
76C22×C137—161no—
77C22×C137AGL(1,ℤ/8)11yescharacteristic
78C2×D137—21no—
79C2×D137—41no—
80C2×D137—162no—
81C4×C137—81no—
82C4×C137#12011yescharacteristic
83C4×C137C2×D811yescharacteristic
84Dic137—21no—
85Dic137—41no—
86C2×C137—21no—
87C2×C137—82no—
88C2×C137#44711yescharacteristic
89D137—41no—
90D137—81no—
91C137#219411yescharacteristic, complemented, Sylow
92#2194—1371nomaximal, Sylow
93#576—1371no—
94#510—1371no—
95C2×SD32D13711yescharacteristic, complemented
96#473—1371no—
97#361—1371no—
98#359—1371no—
99#350—1371no—
100H(5,𝔽2)—2741no—
101#137—2741no—
102AGL(1,ℤ/8)—2742no—
103AGL(1,ℤ/8)—2741no—
104#135—2741no—
105#135—1371no—
106C2×Q16C2×D13711yescharacteristic, complemented
107C2×D8—2741no—
108C2×D8C2×D13711yescharacteristic
109#131—1371no—
110Q32—2741no—
111SD32—22no—
112SD32—2741no—
113M32—1371no—
114C2×C16C2×D13711yescharacteristic, complemented
115#108—1371no—
116#101—2741no—
117#41—2742no—
118#41—2741no—
119#41—5481no—
120#41—1371no—
121C2×Q8—21no—
122C2×D4—21no—
123C2×D4—2742no—
124C2×D4—5481no—
125C2×D4—5482no—
126Q16—21no—
127Q16—21no—
128Q16—5481no—
129Q16—1371no—
130SD16—2742no—
131SD16—2741no—
132SD16—5481no—
133D8—21no—
134D8—2742no—
135D8—1371no—
136D8C22⋊D13712yes—
137M16—2741no—
138M16—2741no—
139M16—1371no—
140C2×C8—2741no—
141C2×C8C22×D13711yescharacteristic
142C16—21no—
143C16—2741no—
144C23—41no—
145C23—5482no—
146Q8—41no—
147Q8—41no—
148Q8—2741no—
149Q8—1371no—
150D4—22no—
151D4—41no—
152D4—2741no—
153D4—2741no—
154D4—5482no—
155D4—5482no—
156D4—5481no—
157D4—1371no—
158C2×C4—41no—
159C2×C4—2741no—
160C2×C4—5482no—
161C2×C4—5481no—
162C2×C4D4×D13711yescharacteristic
163C8—2741no—
164C8—5481no—
165C8D4×D13711yescharacteristic
166C8C2×C22⋊D13711yescharacteristic
167C22—42no—
168C22—161no—
169C22—2741no—
170C22—5481no—
171C22—21922no—
172C22#1798411211yescharacteristic
173C4—81no—
174C4—2741no—
175C4—5481no—
176C4#1798405611yescharacteristic
177C4D8×D13711yescharacteristic
178C2—21no—
179C2—82no—
180C2—5481no—
181C2—10961no—
182C2#2257753511yescharacteristic
subgroupG/H
centerC2#22577535
commutator, cosocleC2×C8×C137C23
FrattiniC2×C8C22×D137
FittingC2×SD32×C137C2
socleC2×C137#447
prime radical#22578204C2
hypercenterC2×SD32D137
hypocenterC137#2194
LCSG ▷ C2×C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
2. #22577844C2semidirect
3. #22577675C2semidirect
4. #22577216C2semidirect
5. #22577211C2semidirect
6. #22577186C2semidirect
7. C2×SD32×C137C2semidirect
18. #17984028C22semidirect
19. #17984024C22semidirect
21. C2×Q16×C137C22semidirect
24. C2×C16×C137C22semidirect
91. C137#2194semidirect
95. C2×SD32D137semidirect
106. C2×Q16C2×D137semidirect
114. C2×C16C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782042
#225778442
#225776752
#225772162
#225772112
#225771862
C2×SD32×C1372
#2194137