Finite Group Databasebeta

← #31796944

Subgroups — #31796944

Order 17536.

292 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C2×#17984158C211yesmaximal, characteristic, complemented
2C2×#17984152C212yesmaximal, complemented
3#22578377C212yesmaximal, complemented
4C2×#17984142C211yesmaximal, characteristic, complemented
5C2×#17984107C211yesmaximal, characteristic
6C2×#14555520C2212yes—
7C2×#14555518C2211yescharacteristic
8C2×C4×#10587739C2211yescharacteristic
9#17984158—22no—
10#17984152—24no—
11C22⋊#11764047—24no—
12#17984142—22no—
13C2×#14555509C2212yes—
14#17984107—22no—
15#17984107—22no—
16C2×Dic548C411yescharacteristic, complemented
17C2×#14555498C412yescomplemented
18C2×C8×D137C2211yescharacteristic
19C2×D8×C137C411yescharacteristic, complemented
20C22×#10587739—21no—
21#14555520—22no—
22#14555520—22no—
23#14555518—21no—
24#14555518—21no—
25C2×#11764047—22no—
26C2×#11764047D412yes—
27C4×#10587739—21no—
28C4×#10587739D412yes—
29#14555509—22no—
30#14555509—22no—
31#14555509—22no—
32#14555509—22no—
33C2×Dic274C2×C412yes—
34C2×C22⋊D137—22no—
35C22×Dic137—22no—
36C2×C4×D137C2311yescharacteristic
37Dic548—22no—
38#14555498—24no—
39C2×#11764040C2×C411yescharacteristic
40C8×D137D412yes—
41C8×D137D412yes—
42C2×D4×C137C2×C412yes—
43D8×C137—22no—
44C2×C8×C137C2×C411yescharacteristic
45C2×#10587739—22no—
46C2×#10587739—42no—
47#11764047—22no—
48#11764047—42no—
49C22×D137#4111yescharacteristic
50Dic274—22no—
51Dic274—22no—
52C22⋊D137—42no—
53C22⋊D137—42no—
54C2×Dic137—22no—
55C2×Dic137—42no—
56C2×Dic137—42no—
57C2×Dic137C2×D411yescharacteristic
58C4×D137C2×D411yescharacteristic
59C4×D137C2×D411yescharacteristic
60C4×D137C2×D411yescharacteristic
61C4×D137#4111yescharacteristic
62#11764040C22⋊C412yes—
63C23×C137—22no—
64D4×C137—22no—
65D4×C137—22no—
66C2×C4×C137C22×C411yescharacteristic
67C8×C137C22⋊C412yes—
68C22×C137—42no—
69C22×C137—44no—
70C22×C137C4×D411yescharacteristic
71C2×D137—22no—
72C2×D137#12111yescharacteristic
73C2×D137#12411yescharacteristic
74#10587739—42no—
75C4×C137C2×C22⋊C411yescharacteristic
76C4×C137C4×D411yescharacteristic
77Dic137—42no—
78Dic137#12111yescharacteristic
79Dic137#12711yescharacteristic
80C2×C137—82no—
81C2×C137#39011yescharacteristic
82C2×C137#44412yes—
83D137—21no—
84D137—21no—
85C137#197511yescharacteristic, complemented, Sylow
86#1975—1371nomaximal, Sylow
87C2×#135—1371no—
88C2×#108—1371no—
89C2×#104—1372no—
90#408—1372no—
91C2×#97—1371no—
92C2×#41—1372no—
93#135—2742no—
94#135—2742no—
95C2×Q16—1371no—
96C2×SD16—1372no—
97C2×D8#1058773911yescharacteristic, complemented
98C2×M16—1372no—
99C22×C8—1371no—
100C22×C8—1371no—
101C2×C42—1371no—
102#108—2742no—
103#104—2744no—
104C22⋊C8—2744no—
105#97—2742no—
106#41—2742no—
107#41—2742no—
108#41—2742no—
109#41—2742no—
110C2×Q8—1372no—
111C2×D4—2742no—
112C2×D4C2×#1058773912yes—
113C22×C4—2741no—
114C22×C4—2742no—
115C22×C4—1371no—
116Q16—2742no—
117SD16—2744no—
118D8—22no—
119M16—2742no—
120M16—2742no—
121C2×C8—2741no—
122C2×C8—2741no—
123C2×C8—2742no—
124C2×C8—1372no—
125C2×C8—1371no—
126C2×C8—1372no—
127C2×C8—1372no—
128C2×C8C2×#1058773911yescharacteristic
129C42—2741no—
130C42—1372no—
131C23—22no—
132C23—1371no—
133Q8—2742no—
134Q8—2742no—
135D4—22no—
136D4—22no—
137D4—5482no—
138D4—5482no—
139C2×C4—2742no—
140C2×C4—2742no—
141C2×C4—5482no—
142C2×C4—5482no—
143C2×C4—5482no—
144C2×C4—1371no—
145C2×C4—1371no—
146C2×C4—1371no—
147C2×C4—1371no—
148C2×C4—1371no—
149C2×C4C22×#1058773911yescharacteristic
150C8—2742no—
151C8—5482no—
152C8—1372no—
153C8C22⋊#1058773912yes—
154C22—44no—
155C22—42no—
156C22—2742no—
157C22—1371no—
158C22—1371no—
159C22D4×#1058773911yescharacteristic
160C4—5482no—
161C4—5482no—
162C4—1371no—
163C4—1371no—
164C4C2×C22⋊#1058773911yescharacteristic
165C4D4×#1058773911yescharacteristic
166C2—82no—
167C2—2741no—
168C2—2741no—
169C2#2257835511yescharacteristic
170C2#2257845012yes—
subgroupG/H
centerC22D4×#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC2×D8×C137C4
socleC22×C137C4×D4
prime radicalC2×#17984107C2
cosocleC2×C4×D137C23
hypercenterC2×D8#10587739
hypocenterC137#1975
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. C2×#17984158C2semidirect
2. C2×#17984152C2semidirect
3. #22578377C2semidirect
4. C2×#17984142C2semidirect
16. C2×Dic548C4semidirect
17. C2×#14555498C4semidirect
19. C2×D8×C137C4semidirect
85. C137#1975semidirect
97. C2×D8#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×#179841582
C2×#179841522
#225783772
C2×#179841422
C2×#179841072
#1975137