Finite Group Databasebeta

← #31779902

Subgroups — #31779902

Order 17536.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#22578126C211yesmaximal, characteristic, complemented
2#22577750C211yesmaximal, characteristic, complemented
3#22577603C211yesmaximal, characteristic, complemented
4#22577599C211yesmaximal, characteristic
5#22577150C211yesmaximal, characteristic, complemented
6#22577132C211yesmaximal, characteristic, complemented
7Q8⋊C8×C137C211yesmaximal, characteristic, complemented
8C2×#14555497—21no—
9C2×D1096—21no—
10#17984088C2211yescharacteristic
11#17984085C2211yescharacteristic
12#17984083—22no—
13#17984076—21no—
14#17984064—21no—
15C4×Dic274—21no—
16C4×D548C2211yescharacteristic
17#17984046—21no—
18C4⋊#11764040C2211yescharacteristic
19#17984010—22no—
20#17984010—21no—
21#17984008—21no—
22C4×Q8×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4×C8×C137C2211yescharacteristic
25C2×Dic274—21no—
26C2×D548—21no—
27C2×D548—21no—
28C2×D548—21no—
29C2×D548D411yescharacteristic
30C2×C4×D137—21no—
31#14555497—42no—
32D1096—42no—
33C2×#11764040—21no—
34C4⋊Dic137—21no—
35C4⋊Dic137D411yescharacteristic
36C4⋊Dic137D412yes—
37#14555489—22no—
38#14555487—22no—
39#14555487—22no—
40#14555487—21no—
41C4×Dic137—22no—
42C2×Q8×C137D411yescharacteristic
43C2×C8×C137—21no—
44C2×C8×C137—21no—
45C4⋊C4×C137—21no—
46C4⋊C4×C137D411yescharacteristic
47C42×C137—21no—
48C42×C137C2311yescharacteristic
49C22×D137—21no—
50C22×D137—41no—
51Dic274—41no—
52Dic274—41no—
53D548—21no—
54D548—21no—
55D548—42no—
56D548—41no—
57D548—41no—
58D548—41no—
59D548—41no—
60C2×Dic137—21no—
61C2×Dic137—22no—
62C4×D137—41no—
63C4×D137—41no—
64#11764040—41no—
65Q8×C137—21no—
66Q8×C137—21no—
67C2×C4×C137—21no—
68C2×C4×C137—21no—
69C2×C4×C137C2×D411yescharacteristic
70C2×C4×C137C2×D411yescharacteristic
71C2×C4×C137C2×D411yescharacteristic
72C8×C137—41no—
73C8×C137—41no—
74C22×C137#12011yescharacteristic
75C2×D137—41no—
76C2×D137—41no—
77C2×D137—41no—
78C2×D137—81no—
79C2×D137—81no—
80C2×D137—81no—
81C4×C137—21no—
82C4×C137—41no—
83C4×C137—41no—
84C4×C137#13511yescharacteristic
85C4×C137#13511yescharacteristic
86C4×C137AGL(1,ℤ/8)11yescharacteristic
87C4×C137AGL(1,ℤ/8)11yescharacteristic
88Dic137—41no—
89Dic137—42no—
90C2×C137#44911yescharacteristic
91C2×C137#44911yescharacteristic
92C2×C137#45411yescharacteristic
93D137—81no—
94D137—161no—
95C137#162811yescharacteristic, complemented, Sylow
96#1628—1371nomaximal, Sylow
97#556—1371no—
98#486—1371no—
99#460—1371no—
100#459—1371no—
101#332—1371no—
102Q8⋊C8D13711yescharacteristic, complemented
103D4⋊C8—1371no—
104C2×SD16—2741no—
105C2×D8—2741no—
106C4⋊Q8—1371no—
107#127—1371no—
108#126—2742no—
109#124—2741no—
110C22⋊Q8—2741no—
111C4×Q8—2741no—
112C4×Q8C2×D13711yescharacteristic
113C4×D4—1371no—
114#117—2741no—
115C4⋊C8—1371no—
116C4⋊C8C2×D13711yescharacteristic
117#102—2741no—
118#102—2742no—
119#102—2741no—
120C4×C8C2×D13711yescharacteristic
121C2×Q8—2741no—
122C2×Q8D54811yescharacteristic
123C2×D4—2741no—
124C2×D4—2741no—
125C2×D4—2741no—
126C2×D4—1371no—
127C22×C4—2741no—
128SD16—5482no—
129D8—5482no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—2741no—
133C4⋊C4—21no—
134C4⋊C4—2742no—
135C4⋊C4—2741no—
136C4⋊C4—1371no—
137C4⋊C4—1372no—
138C4⋊C4D54811yescharacteristic
139C22⋊C4—2742no—
140C22⋊C4—2742no—
141C22⋊C4—2741no—
142C42—21no—
143C42—2742no—
144C42C22×D13711yescharacteristic
145C23—2741no—
146C23—5481no—
147Q8—21no—
148Q8—21no—
149Q8—5481no—
150Q8—5481no—
151D4—2741no—
152D4—2741no—
153D4—5482no—
154D4—5481no—
155D4—5481no—
156D4—5481no—
157D4—5481no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—2741no—
161C2×C4—2742no—
162C2×C4—5481no—
163C2×C4—5481no—
164C2×C4D4×D13711yescharacteristic
165C2×C4D4×D13711yescharacteristic
166C2×C4C2×D54811yescharacteristic
167C8—41no—
168C8—41no—
169C8—5481no—
170C22—5481no—
171C22—5481no—
172C22—5481no—
173C22—10961no—
174C22—10961no—
175C22—10961no—
176C22#1798405811yescharacteristic
177C4—21no—
178C4—41no—
179C4—41no—
180C4—5481no—
181C4—5482no—
182C4#1798411011yescharacteristic
183C4#1798411111yescharacteristic
184C4#1798411511yescharacteristic
185C4#1798411611yescharacteristic
186C2—10961no—
187C2—21921no—
188C2#2257755011yescharacteristic
189C2#2257755111yescharacteristic
190C2#2257757711yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
FittingQ8⋊C8×C137C2
socleC22×C137#120
prime radical#22577599C2
hypercenterQ8⋊C8D137
hypocenterC137#1628
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
1. #22578126C2semidirect
2. #22577750C2semidirect
3. #22577603C2semidirect
5. #22577150C2semidirect
6. #22577132C2semidirect
7. Q8⋊C8×C137C2semidirect
95. C137#1628semidirect
102. Q8⋊C8D137semidirect

Maximal subgroups

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

subgroupindex
#225781262
#225777502
#225776032
#225775992
#225771502
#225771322
Q8⋊C8×C1372
#1628137