Finite Group Databasebeta

← #31781732

Subgroups — #31781732

Order 17536.

260 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 93 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#22577960C211yesmaximal, characteristic
2C4×#14555494C211yesmaximal, characteristic
3#22577163C214yesmaximal
4C4×C4⋊C4×C137C211yesmaximal, characteristic
5C2×#14555494C2212yes—
6#17984088C411yescharacteristic, complemented
7#17984088C411yescharacteristic, complemented
8#17984080C412yescomplemented
9#17984067—21no—
10#17984067—21no—
11C4×Dic274—21no—
12C4×Dic274—21no—
13#17984047C2212yes—
14#17983999—21no—
15C4×#11764040—21no—
16C2×C4⋊C4×C137—21no—
17C2×C42×C137C2212yes—
18C2×C42×C137C2211yescharacteristic
19#95×C137—21no—
20C2×Dic274—21no—
21C2×Dic274—21no—
22#14555494—22no—
23#14555494Q814yescomplemented
24C2×#11764040—22no—
25C4⋊Dic137—21no—
26C4⋊Dic137—21no—
27C4⋊Dic137C2×C412yes—
28C4⋊Dic137C2×C412yes—
29#14555489—22no—
30#14555489—22no—
31C22⋊Dic137—22no—
32C4×Dic137—22no—
33C22×C4×C137—22no—
34C22×C4×C137D411yescharacteristic
35C22×C4×C137D411yescharacteristic
36C22×C4×C137C2311yescharacteristic
37C4⋊C4×C137—24no—
38C42×C137—24no—
39C42×C137C2×C411yescharacteristic
40C42×C137C2×C411yescharacteristic
41C42×C137D411yescharacteristic
42C42×C137D411yescharacteristic
43Dic274—42no—
44Dic274—42no—
45C2×Dic137—22no—
46C2×Dic137—22no—
47#11764040—24no—
48C23×C137C2×D411yescharacteristic
49C2×C4×C137—22no—
50C2×C4×C137—24no—
51C2×C4×C137—41no—
52C2×C4×C137—42no—
53C2×C4×C137—42no—
54C2×C4×C137C22⋊C411yescharacteristic
55C2×C4×C137C22⋊C411yescharacteristic
56C2×C4×C137C22⋊C411yescharacteristic
57C2×C4×C137C22⋊C411yescharacteristic
58C2×C4×C137C22×C411yescharacteristic
59C2×C4×C137C2×D411yescharacteristic
60C2×C4×C137C2×Q812yes—
61C2×C4×C137#4112yes—
62C22×C137—21no—
63C22×C137—21no—
64C22×C137C2×C22⋊C411yescharacteristic
65C22×C137#12411yescharacteristic
66C22×C137C4⋊Q811yescharacteristic
67C4×C137—21no—
68C4×C137—21no—
69C4×C137—21no—
70C4×C137—44no—
71C4×C137#10412yes—
72C4×C137C4×Q812yes—
73C4×C137C22⋊Q812yes—
74Dic137—42no—
75Dic137—42no—
76C2×C137—21no—
77C2×C137—21no—
78C2×C137#39111yescharacteristic
79C2×C137C2×#10411yescharacteristic
80C2×C137#42111yescharacteristic
81C137#198611yescharacteristic, complemented, Sylow
82#1986—1371nomaximal, Sylow
83#533—1371no—
84C4×M16—1371no—
85C4×C4⋊C4D13711yescharacteristic
86#339—1374no—
87C2×M16—1372no—
88C4⋊Q8—1371no—
89C4⋊Q8—1371no—
90#125—1372no—
91C22⋊Q8—2741no—
92C22⋊Q8—2741no—
93C4×Q8—2741no—
94C4×Q8—2741no—
95#117—1372no—
96C2×C4⋊C4—21no—
97C2×C42C2×D13712yes—
98C2×C42C2×D13711yescharacteristic
99#97—2741no—
100C4×C8—2741no—
101#95—21no—
102C2×Q8—2741no—
103C2×Q8—2741no—
104C22×C4—22no—
105C22×C4C22⋊D13711yescharacteristic
106C22×C4C22⋊D13711yescharacteristic
107C22×C4C22×D13711yescharacteristic
108M16—2742no—
109M16—1374no—
110C2×C8—2742no—
111C4⋊C4—24no—
112C4⋊C4—2742no—
113C4⋊C4—2742no—
114C4⋊C4—2741no—
115C4⋊C4—2741no—
116C4⋊C4—1372no—
117C4⋊C4—1372no—
118C22⋊C4—2742no—
119C42—24no—
120C42—2742no—
121C42C4×D13711yescharacteristic
122C42C4×D13711yescharacteristic
123C42D54811yescharacteristic
124C42D54811yescharacteristic
125C23C2×C22⋊D13711yescharacteristic
126Q8—5482no—
127Q8—5482no—
128C2×C4—22no—
129C2×C4—24no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—2742no—
134C2×C4—2742no—
135C2×C4#1455548711yescharacteristic
136C2×C4#1455548711yescharacteristic
137C2×C4#1455548711yescharacteristic
138C2×C4#1455548711yescharacteristic
139C2×C4C2×C4×D13711yescharacteristic
140C2×C4C2×D54811yescharacteristic
141C2×C4Q8×D13712yes—
142C2×C4#1455550912yes—
143C8—2744no—
144C22—21no—
145C22—21no—
146C22C2×#1455548711yescharacteristic
147C22#1798407511yescharacteristic
148C22#1798408611yescharacteristic
149C4—21no—
150C4—21no—
151C4—21no—
152C4—44no—
153C4—5482no—
154C4—5482no—
155C4#1798401612yes—
156C4#1798405312yes—
157C4#1798406812yes—
158C2—21no—
159C2—21no—
160C2#2257730811yescharacteristic
161C2C2×#1798401611yescharacteristic
162C2#2257742111yescharacteristic
subgroupG/H
centerC2×C4C2×D548
commutatorC2×C4×C137C22×C4
FrattiniC22×C4C22×D137
FittingC4×C4⋊C4×C137C2
socleC22×C137C2×C22⋊C4
prime radicalC23×C137C2×D4
cosocleC22×C4×C137C23
hypercenterC4×C4⋊C4D137
hypocenterC137#1986
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #17984088C4semidirect
7. #17984088C4semidirect
8. #17984080C4semidirect
23. #14555494Q8semidirect
81. C137#1986semidirect

Maximal subgroups

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

subgroupindex
#225779602
C4×#145554942
#225771632
C4×C4⋊C4×C1372
#1986137