Finite Group Databasebeta

← #31779893

Subgroups — #31779893

Order 17536.

250 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 57 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#22578068C211yesmaximal, characteristic, complemented
2#22577751C211yesmaximal, characteristic, complemented
3#22577604C211yesmaximal, characteristic, complemented
4#22577599C211yesmaximal, characteristic
5#22577133C211yesmaximal, characteristic, complemented
6#22577130C211yesmaximal, characteristic, complemented
7#332×C137C211yesmaximal, characteristic, complemented
8C2×Q8×D137—21no—
9C2×#14555499—21no—
10C2×D1096—21no—
11#17984087—21no—
12#17984085C2211yescharacteristic
13#17984066—22no—
14#17984064—21no—
15C4×Dic274C2211yescharacteristic
16C4×D548C2211yescharacteristic
17#17984050—21no—
18#17984010—21no—
19#17984010—21no—
20#17984008—22no—
21C4×#11764040C2211yescharacteristic
22C4⋊Q8×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4⋊C8×C137C2211yescharacteristic
25C2×Dic274—22no—
26C2×Dic274D411yescharacteristic
27Q8×D137—42no—
28Q8×D137—42no—
29C2×D548—21no—
30C2×D548—21no—
31C2×D548—21no—
32C2×D548D411yescharacteristic
33C2×C4×D137—22no—
34C2×C4×D137—21no—
35#14555499—42no—
36D1096—42no—
37C2×#11764040—21no—
38C4⋊Dic137D411yescharacteristic
39C4⋊Dic137D411yescharacteristic
40#14555489—22no—
41#14555489—21no—
42#14555487—22no—
43#14555487—21no—
44C4×Dic137—21no—
45C2×Q8×C137—21no—
46C2×C8×C137—21no—
47C2×C8×C137—21no—
48C4⋊C4×C137—21no—
49C4⋊C4×C137D412yes—
50C42×C137C2311yescharacteristic
51C22×D137—21no—
52C22×D137—41no—
53Dic274—21no—
54Dic274—44no—
55Dic274D812yescomplemented
56D548—21no—
57D548—42no—
58D548—41no—
59D548—41no—
60D548—41no—
61D548—41no—
62D548SD1612yescomplemented
63C2×Dic137—21no—
64C2×Dic137—21no—
65C2×Dic137—21no—
66C4×D137—44no—
67C4×D137—41no—
68C4×D137—41no—
69#11764040—41no—
70Q8×C137—41no—
71Q8×C137—41no—
72C2×C4×C137—22no—
73C2×C4×C137C2×D411yescharacteristic
74C2×C4×C137C2×D411yescharacteristic
75C2×C4×C137C2×D411yescharacteristic
76C8×C137—41no—
77C8×C137—41no—
78C22×C137#12011yescharacteristic
79C2×D137—22no—
80C2×D137—41no—
81C2×D137—41no—
82C2×D137—81no—
83C2×D137—81no—
84C2×D137—81no—
85C4×C137—21no—
86C4×C137—42no—
87C4×C137C2×D811yescharacteristic
88C4×C137C2×SD1611yescharacteristic
89C4×C137AGL(1,ℤ/8)11yescharacteristic
90C4×C137#13711yescharacteristic
91Dic137—22no—
92Dic137—41no—
93Dic137—41no—
94C2×C137#44711yescharacteristic
95C2×C137#44811yescharacteristic
96C2×C137#45411yescharacteristic
97D137—42no—
98D137—161no—
99C137#162711yescharacteristic, complemented, Sylow
100#1627—1371nomaximal, Sylow
101D4×Q8—1371no—
102#486—1371no—
103#460—1371no—
104#459—1371no—
105#332D13711yescharacteristic, complemented
106Q8⋊C8—1371no—
107D4⋊C8—1371no—
108C22×Q8—2741no—
109C2×SD16—2741no—
110C2×D8—2741no—
111C4⋊Q8—2741no—
112C4⋊Q8C2×D13711yescharacteristic
113#127—1371no—
114C22⋊Q8—2741no—
115C22⋊Q8—2742no—
116C4×Q8—1371no—
117C4×D4—2741no—
118C4×D4—1371no—
119C4⋊C8C2×D13711yescharacteristic
120C4⋊C8C2×D13711yescharacteristic
121#102—2741no—
122#102—2741no—
123#102—2742no—
124C4×C8—1371no—
125C2×Q8—21no—
126C2×Q8—2742no—
127C2×Q8—5482no—
128C2×Q8—5482no—
129C2×Q8—1371no—
130C2×D4—2741no—
131C2×D4—2741no—
132C2×D4—2741no—
133C2×D4—1371no—
134C22×C4—2742no—
135C22×C4—2741no—
136SD16—5482no—
137D8—5482no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—2741no—
141C4⋊C4—21no—
142C4⋊C4—2742no—
143C4⋊C4—2741no—
144C4⋊C4—1371no—
145C4⋊C4—1371no—
146C4⋊C4D54812yes—
147C22⋊C4—2742no—
148C22⋊C4—2741no—
149C42—2741no—
150C42C22×D13711yescharacteristic
151C23—2741no—
152C23—5481no—
153Q8—41no—
154Q8—41no—
155Q8—2741no—
156Q8—5484no—
157Q8—1372no—
158D4—2741no—
159D4—5482no—
160D4—5481no—
161D4—5481no—
162D4—5481no—
163D4—5481no—
164D4—1372no—
165C2×C4—22no—
166C2×C4—2741no—
167C2×C4—2741no—
168C2×C4—2741no—
169C2×C4—5484no—
170C2×C4—5481no—
171C2×C4—5481no—
172C2×C4D4×D13711yescharacteristic
173C2×C4D4×D13711yescharacteristic
174C2×C4C2×D54811yescharacteristic
175C8—41no—
176C8—41no—
177C8—5481no—
178C22—2742no—
179C22—5481no—
180C22—5481no—
181C22—10961no—
182C22—10961no—
183C22—10961no—
184C22#1798405811yescharacteristic
185C4—21no—
186C4—42no—
187C4—2742no—
188C4—5481no—
189C4—5481no—
190C4D8×D13711yescharacteristic
191C4SD16×D13711yescharacteristic
192C4#1798411511yescharacteristic
193C4#1798412011yescharacteristic
194C2—5482no—
195C2—21921no—
196C2#2257753811yescharacteristic
197C2#2257754311yescharacteristic
198C2#2257757811yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#332×C137C2
socleC22×C137#120
prime radical#22577599C2
hypercenter#332D137
hypocenterC137#1627
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #22578068C2semidirect
2. #22577751C2semidirect
3. #22577604C2semidirect
5. #22577133C2semidirect
6. #22577130C2semidirect
7. #332×C137C2semidirect
55. Dic274D8semidirect
62. D548SD16semidirect
99. C137#1627semidirect
105. #332D137semidirect

Maximal subgroups

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

subgroupindex
#225780682
#225777512
#225776042
#225775992
#225771332
#225771302
#332×C1372
#1627137