Finite Group Databasebeta

← #31779476

Subgroups — #31779476

Order 17536.

216 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 73 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#22577958C211yesmaximal, characteristic
2#22577425C211yesmaximal, characteristic
3C4⋊C8×D137C211yesmaximal, characteristic
4#22577152C211yesmaximal, characteristic, complemented
5#22577150C211yesmaximal, characteristic, complemented
6#22577148C211yesmaximal, characteristic, complemented
7#333×C137C211yesmaximal, characteristic, complemented
8C2×#14555493—21no—
9C2×C8×D137—21no—
10#17984088C2211yescharacteristic
11#17984079—21no—
12#17984068—21no—
13#17984064C412yes—
14Q8×Dic137C412yes—
15#17984053—21no—
16#17984046—21no—
17C42×D137C2211yescharacteristic
18C4⋊#11764040C2211yescharacteristic
19C4⋊#11764040C2211yescharacteristic
20#17984017—21no—
21#17984017—21no—
22C4⋊Q8×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4⋊C8×C137C2211yescharacteristic
25C2×Dic274—21no—
26C2×C4×D137C2×C411yescharacteristic
27C2×C4×D137D411yescharacteristic
28C2×C4×D137D411yescharacteristic
29#14555493—42no—
30C2×#11764040—21no—
31C2×#11764040—21no—
32C8×D137—42no—
33C4⋊Dic137—22no—
34C4⋊Dic137—21no—
35C4⋊Dic137C2×C412yes—
36#14555489—22no—
37#14555487—22no—
38C4×Dic137—22no—
39C4×Dic137C2×C411yescharacteristic
40C4×Dic137D411yescharacteristic
41C4×Dic137D411yescharacteristic
42C2×Q8×C137C2×C412yes—
43C2×C8×C137—21no—
44C2×C8×C137—21no—
45C4⋊C4×C137—21no—
46C4⋊C4×C137—21no—
47C42×C137C2311yescharacteristic
48C22×D137C22⋊C411yescharacteristic
49Dic274—41no—
50Dic274—41no—
51C2×Dic137—22no—
52C2×Dic137C22⋊C411yescharacteristic
53C2×Dic137C22⋊C411yescharacteristic
54C2×Dic137C22⋊C411yescharacteristic
55C4×D137—21no—
56C4×D137—21no—
57C4×D137—41no—
58C4×D137SD1612yes—
59C4×D137Q1612yes—
60#11764040—41no—
61#11764040—41no—
62Q8×C137—42no—
63C2×C4×C137—22no—
64C2×C4×C137C22×C411yescharacteristic
65C2×C4×C137C2×D411yescharacteristic
66C2×C4×C137C2×D411yescharacteristic
67C8×C137—41no—
68C8×C137—41no—
69C22×C137C2×C22⋊C411yescharacteristic
70C2×D137—21no—
71C2×D137—21no—
72C2×D137#10312yes—
73C4×C137—21no—
74C4×C137—42no—
75C4×C137C2×SD1611yescharacteristic
76C4×C137C2×Q1611yescharacteristic
77C4×C137AGL(1,ℤ/8)11yescharacteristic
78C4×C137#13711yescharacteristic
79Dic137—21no—
80Dic137—21no—
81Dic137—42no—
82Dic137#10312yes—
83C2×C137#41311yescharacteristic
84C2×C137C2×#10311yescharacteristic
85C2×C137#41711yescharacteristic
86D137—22no—
87C137#155011yescharacteristic, complemented, Sylow
88#1550—1371nomaximal, Sylow
89#533—1371no—
90#423—1371no—
91C2×C4⋊C8—1371no—
92#333—1371no—
93#333D13711yescharacteristic, complemented
94#332—1371no—
95#332—1371no—
96C2×M16—2741no—
97C22×C8—2741no—
98C4⋊Q8—1371no—
99C4⋊Q8C2×D13711yescharacteristic
100#125—2741no—
101C22⋊Q8—2741no—
102C22⋊Q8—1372no—
103C4×Q8—2741no—
104C4×Q8—1372no—
105#117—2741no—
106C2×C42—1371no—
107C4⋊C8—2741no—
108C4⋊C8—2741no—
109C4⋊C8—1371no—
110C4⋊C8—1371no—
111C4⋊C8C2×D13711yescharacteristic
112C4⋊C8C2×D13711yescharacteristic
113C2×Q8—2741no—
114C2×Q8C4×D13712yes—
115C22×C4—1371no—
116C22×C4—1371no—
117C22×C4—1371no—
118M16—5482no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—2741no—
122C2×C8—2741no—
123C2×C8—5482no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—2742no—
127C4⋊C4—2742no—
128C4⋊C4—2741no—
129C4⋊C4—1372no—
130C22⋊C4—2742no—
131C42—2742no—
132C42—1371no—
133C42—1371no—
134C42—1371no—
135C42C22×D13711yescharacteristic
136C23—1371no—
137Q8—42no—
138Q8—5481no—
139Q8—5481no—
140C2×C4—22no—
141C2×C4—2741no—
142C2×C4—2741no—
143C2×C4—2742no—
144C2×C4—5481no—
145C2×C4—1371no—
146C2×C4—1371no—
147C2×C4—1371no—
148C2×C4—1372no—
149C2×C4—1372no—
150C2×C4C2×C4×D13711yescharacteristic
151C2×C4D4×D13711yescharacteristic
152C2×C4D4×D13711yescharacteristic
153C8—41no—
154C8—41no—
155C8—5481no—
156C8—5481no—
157C22—2741no—
158C22—2741no—
159C22—1372no—
160C22C22⋊C4×D13711yescharacteristic
161C4—21no—
162C4—42no—
163C4—2741no—
164C4—2741no—
165C4—5482no—
166C4—1372no—
167C4SD16×D13711yescharacteristic
168C4Q16×D13711yescharacteristic
169C4#1798411511yescharacteristic
170C4#1798412011yescharacteristic
171C2—2742no—
172C2#2257738411yescharacteristic
173C2#103×D13711yescharacteristic
174C2#2257740311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D137
commutatorC2×C4×C137C22×C4
FrattiniC42C22×D137
Fitting#333×C137C2
socleC22×C137C2×C22⋊C4
prime radicalC22×D137C22⋊C4
cosocleC42×C137C23
hypercenter#333D137
hypocenterC137#1550
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
4. #22577152C2semidirect
5. #22577150C2semidirect
6. #22577148C2semidirect
7. #333×C137C2semidirect
87. C137#1550semidirect
93. #333D137semidirect

Maximal subgroups

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

subgroupindex
#225779582
#225774252
C4⋊C8×D1372
#225771522
#225771502
#225771482
#333×C1372
#1550137