Finite Group Databasebeta

← #31781309

Subgroups — #31781309

Order 17536.

312 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 85 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#22578170C211yesmaximal, characteristic
2#22577974C211yesmaximal, characteristic
3#22577411C212yesmaximal, complemented
4#22577159C212yesmaximal, complemented
5#410×C137C211yesmaximal, characteristic
6C22×Dic274C2211yescharacteristic
7#17984095—24no—
8C2×#14555494—22no—
9C2×#14555494C2212yes—
10#17984086C412yescomplemented
11#17984075C412yescomplemented
12#17984069—22no—
13#17984067—22no—
14#17984047C2211yescharacteristic
15#17984011—24no—
16C2×#41×C137C2211yescharacteristic
17#117×C137C2212yes—
18#99×C137—22no—
19C2×Dic274—21no—
20C2×Dic274—22no—
21C2×Dic274—22no—
22C2×Dic274C2×C412yes—
23C22×Dic137—21no—
24#14555494—24no—
25#14555494—22no—
26#14555494—22no—
27C2×#11764040—24no—
28C2×#11764040D414yes—
29C4⋊Dic137—21no—
30#14555489—24no—
31C22⋊Dic137—24no—
32C22⋊Dic137—21no—
33C4×Dic137C2×C412yes—
34#41×C137—22no—
35#41×C137—22no—
36C2×Q8×C137C2×C411yescharacteristic
37C2×D4×C137C2×C411yescharacteristic
38C2×D4×C137D412yes—
39C22×C4×C137D412yes—
40C22×C4×C137C2311yescharacteristic
41C4⋊C4×C137—22no—
42C22⋊C4×C137—22no—
43C42×C137—22no—
44Dic274—24no—
45Dic274—41no—
46Dic274—41no—
47C2×Dic137—22no—
48C2×Dic137—22no—
49C2×Dic137—42no—
50#11764040—24no—
51C23×C137C22⋊C412yes—
52C23×C137C2×D411yescharacteristic
53Q8×C137—22no—
54D4×C137—24no—
55D4×C137—22no—
56C2×C4×C137—22no—
57C2×C4×C137—22no—
58C2×C4×C137—42no—
59C2×C4×C137C22⋊C412yes—
60C2×C4×C137C22×C411yescharacteristic
61C2×C4×C137C2×D412yes—
62C2×C4×C137C2×D411yescharacteristic
63C2×C4×C137#4112yes—
64C22×C137—22no—
65C22×C137—41no—
66C22×C137—42no—
67C22×C137C2×C22⋊C411yescharacteristic
68C22×C137#12112yes—
69C4×C137—22no—
70C4×C137—82no—
71C4×C137C4×D412yes—
72C4×C137#12011yescharacteristic
73C4×C137#12311yescharacteristic
74Dic137—42no—
75Dic137—42no—
76C2×C137—21no—
77C2×C137—22no—
78C2×C137—42no—
79C2×C137#38611yescharacteristic
80C137#190911yescharacteristic, complemented, Sylow
81#1909—1371nomaximal, Sylow
82#568—1371no—
83#536—1371no—
84#419—1372no—
85#410D13711yescharacteristic
86H(3,ℤ/4)—1372no—
87C2×#41C2×D13711yescharacteristic
88C22×Q8—1371no—
89#131—2744no—
90C2×M16—2742no—
91C2×M16—1372no—
92C4⋊Q8—1372no—
93#124—1372no—
94#123—2742no—
95C22⋊Q8—2742no—
96#117—1371no—
97#117C2×D13712yes—
98#103—2744no—
99#99—22no—
100#41—22no—
101#41—22no—
102C2×Q8—2741no—
103C2×Q8—2742no—
104C2×Q8—2742no—
105C2×Q8—1372no—
106C2×Q8C4×D13711yescharacteristic
107C2×D4C4×D13711yescharacteristic
108C2×D4C22⋊D13712yes—
109C22×C4—2741no—
110C22×C4D54812yes—
111C22×C4C22×D13711yescharacteristic
112M16—2744no—
113M16—2742no—
114M16—2742no—
115C2×C8—2744no—
116C2×C8—1374no—
117C4⋊C4—22no—
118C4⋊C4—2744no—
119C4⋊C4—2741no—
120C22⋊C4—22no—
121C22⋊C4—2744no—
122C22⋊C4—2741no—
123C42—22no—
124C42—1372no—
125C23#1455548712yes—
126C23C2×C22⋊D13711yescharacteristic
127Q8—22no—
128Q8—2744no—
129Q8—5481no—
130Q8—5481no—
131D4—24no—
132D4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—2742no—
137C2×C4—2742no—
138C2×C4—5482no—
139C2×C4#1455548712yes—
140C2×C4C2×C4×D13711yescharacteristic
141C2×C4D4×D13712yes—
142C2×C4C2×D54811yescharacteristic
143C2×C4#1455550912yes—
144C8—2744no—
145C22—22no—
146C22—41no—
147C22—42no—
148C22C2×#1455548711yescharacteristic
149C22#1798406012yes—
150C4—22no—
151C4—82no—
152C4—5482no—
153C4—5482no—
154C4#1798404812yes—
155C4#1798405811yescharacteristic
156C4#1798407211yescharacteristic
157C2—22no—
158C2—21no—
159C2—42no—
160C2#2257728011yescharacteristic
subgroupG/H
centerC2#22577280
commutatorC2×C4×C137C22×C4
FrattiniC22×C4C22×D137
Fitting#410×C137C2
socleC2×C137#386
prime radicalC2×#41×C137C22
cosocleC22×C4×C137C23
hypercenter#410D137
hypocenterC137#1909
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #22577411C2semidirect
4. #22577159C2semidirect
10. #17984086C4semidirect
11. #17984075C4semidirect
80. C137#1909semidirect

Maximal subgroups

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

subgroupindex
#225781702
#225779742
#225774112
#225771592
#410×C1372
#1909137