Finite Group Databasebeta

← #31796698

Subgroups — #31796698

Order 17536.

262 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 62 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#22578389C211yesmaximal, characteristic
2#22578382C211yesmaximal, characteristic
3#22578274C212yesmaximal, complemented
4#22578270C212yesmaximal, complemented
5C4×#14555510C211yesmaximal, characteristic
6D137⋊M16C2212yes—
7#17984167C2212yes—
8#17984149—21no—
9#17984148—21no—
10#17984147—21no—
11#17984146—21no—
12C2×#14555510C2211yescharacteristic
13C4×Dic274—21no—
14C4×D548—21no—
15C4×C22⋊D137—21no—
16C4×C22⋊D137—21no—
17#17984047C411yescharacteristic, complemented
18#17984045C411yescharacteristic, complemented
19#17984045C2211yescharacteristic
20C42×D137C411yescharacteristic, complemented
21C42×D137C2211yescharacteristic
22C2×C42×C137C411yescharacteristic, complemented
23#14555520—22no—
24#14555519—22no—
25#14555518—22no—
26C4⋊#10587739—22no—
27C22⋊#10587739—22no—
28C4×#10587739—22no—
29#14555510—22no—
30#14555510—22no—
31C2×Dic274D411yescharacteristic
32C2×D548D411yescharacteristic
33C2×C22⋊D137D411yescharacteristic
34C2×C22⋊D137D411yescharacteristic
35C2×C4×D137—21no—
36C2×C4×D137—21no—
37C2×C4×D137C2×C411yescharacteristic
38C2×C4×D137C2311yescharacteristic
39C4⋊Dic137—21no—
40#14555489—21no—
41#14555489—21no—
42C22⋊Dic137—21no—
43#14555487—21no—
44#14555487—21no—
45C4×Dic137—21no—
46C4×Dic137—21no—
47C4×Dic137C2×C411yescharacteristic
48C4×Dic137C2×C411yescharacteristic
49C22×C4×C137—21no—
50C22×C4×C137C2×C411yescharacteristic
51C42×C137—21no—
52C42×C137C2×C411yescharacteristic
53C42×C137C2×C411yescharacteristic
54C2×#10587739—42no—
55#11764047—42no—
56C22×D137C22⋊C411yescharacteristic
57C22×D137C2×D411yescharacteristic
58Dic274—22no—
59D548—22no—
60C22⋊D137—22no—
61C22⋊D137—22no—
62C2×Dic137—21no—
63C2×Dic137—21no—
64C2×Dic137C22⋊C411yescharacteristic
65C2×Dic137C2×D411yescharacteristic
66C4×D137—21no—
67C4×D137—21no—
68C4×D137—42no—
69C4×D137—42no—
70C4×D137#4112yes—
71C4×D137#4112yes—
72C23×C137C22⋊C411yescharacteristic
73C2×C4×C137—21no—
74C2×C4×C137—21no—
75C2×C4×C137—21no—
76C2×C4×C137—21no—
77C2×C4×C137—42no—
78C2×C4×C137C22⋊C411yescharacteristic
79C2×C4×C137C22×C411yescharacteristic
80C22×C137—21no—
81C22×C137—41no—
82C22×C137C2×C22⋊C411yescharacteristic
83C2×D137—21no—
84C2×D137—41no—
85C2×D137—41no—
86C2×D137#12312yes—
87#10587739—82no—
88C4×C137—22no—
89C4×C137—21no—
90C4×C137—41no—
91C4×C137#11711yescharacteristic
92C4×C137#11711yescharacteristic
93Dic137—21no—
94Dic137—41no—
95Dic137—41no—
96Dic137#12312yes—
97C2×C137—21no—
98C2×C137—41no—
99C2×C137#38111yescharacteristic
100D137—22no—
101D137—41no—
102C137#182711yescharacteristic, complemented, Sylow
103#1827—1371nomaximal, Sylow
104C4×#41—1371no—
105#413—1371no—
106#410—1371no—
107#339—1372no—
108H(3,ℤ/4)—1372no—
109C2×#41—1371no—
110C2×M16—1372no—
111C4×Q8—2741no—
112C4×D4—2741no—
113C4×D4—2741no—
114C4×D4—2741no—
115#117—1372no—
116#117—1371no—
117#117—1371no—
118#117—1371no—
119C2×C42—1371no—
120C2×C42—1371no—
121C2×C42#1058773911yescharacteristic, complemented
122#101—2741no—
123#100—2741no—
124#99—2741no—
125#99—2741no—
126#41—2742no—
127#41—2742no—
128C2×Q8—1371no—
129C2×D4—1371no—
130C2×D4—1371no—
131C2×D4—1371no—
132C22×C4—21no—
133C22×C4—2741no—
134C22×C4—2741no—
135C22×C4—1371no—
136C22×C4—1371no—
137C22×C4C2×#1058773911yescharacteristic
138M16—2742no—
139M16—2742no—
140C2×C8—2742no—
141C4⋊C4—2741no—
142C4⋊C4—2741no—
143C4⋊C4—2741no—
144C4⋊C4—2742no—
145C22⋊C4—2741no—
146C22⋊C4—2741no—
147C22⋊C4—2741no—
148C22⋊C4—2742no—
149C42—21no—
150C42—2741no—
151C42—2741no—
152C42—2742no—
153C42—1371no—
154C42—1371no—
155C42C2×#1058773911yescharacteristic
156C42C2×#1058773911yescharacteristic
157C23—1371no—
158C23—1371no—
159C23C22⋊#1058773911yescharacteristic
160Q8—2742no—
161D4—2742no—
162D4—2742no—
163D4—2742no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—42no—
169C2×C4—2741no—
170C2×C4—2741no—
171C2×C4—2741no—
172C2×C4—2741no—
173C2×C4—5482no—
174C2×C4—5482no—
175C2×C4—5482no—
176C2×C4—1371no—
177C2×C4—1371no—
178C2×C4—1372no—
179C2×C4—1372no—
180C2×C4C22⋊#1058773911yescharacteristic
181C2×C4C22×#1058773911yescharacteristic
182C8—5482no—
183C22—21no—
184C22—41no—
185C22—2741no—
186C22—5481no—
187C22—5481no—
188C22—1372no—
189C22C2×C22⋊#1058773911yescharacteristic
190C4—22no—
191C4—21no—
192C4—41no—
193C4—2741no—
194C4—5481no—
195C4—5481no—
196C4—10962no—
197C4—1372no—
198C4#1798416711yescharacteristic
199C4#1798416711yescharacteristic
200C2—21no—
201C2—41no—
202C2—2742no—
203C2—5481no—
204C2#2257834611yescharacteristic
subgroupG/H
centerC4#17984167
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC2×C42×C137C4
socleC2×C137#381
prime radicalC2×#14555510C22
cosocleC2×C4×D137C23
hypercenterC2×C42#10587739
hypocenterC137#1827
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C4 ◁ C42 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
3. #22578274C2semidirect
4. #22578270C2semidirect
17. #17984047C4semidirect
18. #17984045C4semidirect
20. C42×D137C4semidirect
22. C2×C42×C137C4semidirect
102. C137#1827semidirect
121. C2×C42#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783892
#225783822
#225782742
#225782702
C4×#145555102
#1827137