Finite Group Databasebeta

← #31780890

Subgroups — #31780890

Order 17536.

262 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 81 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#41×Dic137C211yesmaximal, characteristic
2#22577387C211yesmaximal, characteristic
3#22577162C212yesmaximal, complemented
4#22577159C212yesmaximal, complemented
5#410×C137C211yesmaximal, characteristic
6C2×#14555494C2212yes—
7Q8×Dic137—21no—
8D4×Dic137—22no—
9D4×Dic137—21no—
10#17984047C412yescomplemented
11#17984047C2211yescharacteristic
12C2×C4×Dic137C412yescomplemented
13C2×C4×Dic137C2211yescharacteristic
14#17984007—21no—
15#17984005—21no—
16C2×#41×C137C2211yescharacteristic
17#117×C137C2212yes—
18#99×C137—22no—
19C22×Dic137—22no—
20C22×Dic137—21no—
21#14555494—22no—
22#14555494—22no—
23C2×#11764040—22no—
24C4⋊Dic137—22no—
25C4⋊Dic137—21no—
26C22⋊Dic137—22no—
27C22⋊Dic137—21no—
28C4×Dic137—22no—
29C4×Dic137—21no—
30C4×Dic137C2×C411yescharacteristic
31C4×Dic137C2×C412yes—
32C4×Dic137C2×C411yescharacteristic
33#41×C137—22no—
34#41×C137—22no—
35C2×Q8×C137D411yescharacteristic
36C2×D4×C137D412yes—
37C2×D4×C137D411yescharacteristic
38C22×C4×C137C2×C412yes—
39C22×C4×C137C2311yescharacteristic
40C4⋊C4×C137—22no—
41C22⋊C4×C137—22no—
42C42×C137—22no—
43C2×Dic137—21no—
44C2×Dic137—22no—
45C2×Dic137—21no—
46C2×Dic137—42no—
47C2×Dic137—44no—
48#11764040—42no—
49C23×C137C22⋊C412yes—
50C23×C137C2×D411yescharacteristic
51Q8×C137—22no—
52D4×C137—24no—
53D4×C137—22no—
54C2×C4×C137—22no—
55C2×C4×C137—22no—
56C2×C4×C137—42no—
57C2×C4×C137C22⋊C412yes—
58C2×C4×C137C22×C411yescharacteristic
59C2×C4×C137C2×D411yescharacteristic
60C2×C4×C137#4112yes—
61C2×C4×C137#4112yes—
62C22×C137—22no—
63C22×C137—41no—
64C22×C137—42no—
65C22×C137C2×C22⋊C411yescharacteristic
66C22×C137#12312yes—
67C4×C137—22no—
68C4×C137—82no—
69C4×C137#11711yescharacteristic
70C4×C137#11711yescharacteristic
71C4×C137#12312yes—
72Dic137—22no—
73Dic137—42no—
74Dic137—41no—
75C2×C137—21no—
76C2×C137—22no—
77C2×C137—42no—
78C2×C137#38111yescharacteristic
79C137#182711yescharacteristic, complemented, Sylow
80#1827—1371nomaximal, Sylow
81C4×#41—1371no—
82#413—1371no—
83#410D13711yescharacteristic
84#339—1372no—
85H(3,ℤ/4)—1372no—
86C2×#41C2×D13711yescharacteristic
87C2×M16—1372no—
88C4×Q8—2741no—
89C4×D4—2742no—
90C4×D4—2741no—
91#117—1372no—
92#117—1371no—
93#117C2×D13712yes—
94C2×C42—1372no—
95C2×C42—1371no—
96#101—2741no—
97#100—2741no—
98#99—22no—
99#41—22no—
100#41—22no—
101C2×Q8D54811yescharacteristic
102C2×D4C22⋊D13712yes—
103C2×D4D54811yescharacteristic
104C22×C4—2742no—
105C22×C4—2741no—
106C22×C4C4×D13712yes—
107C22×C4C22×D13711yescharacteristic
108M16—2742no—
109M16—2742no—
110C2×C8—2742no—
111C4⋊C4—22no—
112C4⋊C4—2742no—
113C4⋊C4—2741no—
114C22⋊C4—22no—
115C22⋊C4—2742no—
116C22⋊C4—2741no—
117C42—22no—
118C42—2742no—
119C42—2741no—
120C42—1371no—
121C42—1372no—
122C42—1371no—
123C23#1455548712yes—
124C23C2×C22⋊D13711yescharacteristic
125Q8—22no—
126D4—24no—
127D4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—2741no—
132C2×C4—2742no—
133C2×C4—2741no—
134C2×C4—5482no—
135C2×C4—5484no—
136C2×C4#1455548712yes—
137C2×C4C2×C4×D13711yescharacteristic
138C2×C4C2×D54811yescharacteristic
139C2×C4#1455550912yes—
140C2×C4#1455550912yes—
141C8—5482no—
142C22—22no—
143C22—41no—
144C22—42no—
145C22C2×#1455548711yescharacteristic
146C22#1798406912yes—
147C4—22no—
148C4—82no—
149C4—2742no—
150C4—5482no—
151C4—5481no—
152C4#1798404411yescharacteristic
153C4#1798404411yescharacteristic
154C4#1798407212yes—
155C2—22no—
156C2—21no—
157C2—42no—
158C2#2257726411yescharacteristic
subgroupG/H
centerC4#17984044
commutatorC2×C4×C137C22×C4
FrattiniC22×C4C22×D137
Fitting#410×C137C2
socleC2×C137#381
prime radicalC2×#41×C137C22
cosocleC22×C4×C137C23
hypercenter#410D137
hypocenterC137#1827
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #22577162C2semidirect
4. #22577159C2semidirect
10. #17984047C4semidirect
12. C2×C4×Dic137C4semidirect
79. C137#1827semidirect

Maximal subgroups

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

subgroupindex
#41×Dic1372
#225773872
#225771622
#225771592
#410×C1372
#1827137