Finite Group Databasebeta

← #31779983

Subgroups — #31779983

Order 17536.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#22578073C211yesmaximal, characteristic
2#22577749C211yesmaximal, characteristic, complemented
3#22577684C211yesmaximal, characteristic, complemented
4#22577614C211yesmaximal, characteristic, complemented
5#22577133C211yesmaximal, characteristic, complemented
6#22577130C211yesmaximal, characteristic, complemented
7#331×C137C211yesmaximal, characteristic, complemented
8C2×#14555509—21no—
9C2×Dic548—21no—
10#17984088C2211yescharacteristic
11#17984084—21no—
12#17984060—22no—
13#17984059—21no—
14C4×Dic274C2211yescharacteristic
15C4×D548C2211yescharacteristic
16D4×Dic137—21no—
17#17984022—21no—
18#17984013—21no—
19#17984010—21no—
20#17984009—22no—
21C4×#11764040C2211yescharacteristic
22#127×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4⋊C8×C137C2211yescharacteristic
25#14555509—42no—
26#14555509—42no—
27C2×Dic274—21no—
28C2×Dic274D411yescharacteristic
29C2×D548D411yescharacteristic
30C2×C22⋊D137—22no—
31C2×C22⋊D137—22no—
32C22×Dic137—22no—
33C2×C4×D137—21no—
34Dic548—42no—
35C2×#11764040—21no—
36C4⋊Dic137—21no—
37C4⋊Dic137—21no—
38C4⋊Dic137D411yescharacteristic
39C4⋊Dic137D411yescharacteristic
40#14555489—21no—
41C22⋊Dic137—22no—
42#14555487—21no—
43C4×Dic137—21no—
44C2×D4×C137—21no—
45C2×D4×C137—21no—
46C2×D4×C137D412yes—
47C2×C8×C137—21no—
48C2×C8×C137—21no—
49C42×C137C2311yescharacteristic
50C22×D137—21no—
51Dic274—21no—
52Dic274—41no—
53Dic274—41no—
54Dic274D812yes—
55D548—21no—
56D548—21no—
57C22⋊D137—44no—
58C22⋊D137—82no—
59C2×Dic137—21no—
60C2×Dic137—21no—
61C2×Dic137—21no—
62C2×Dic137—44no—
63C2×Dic137—41no—
64C4×D137—41no—
65C4×D137—41no—
66#11764040—41no—
67C23×C137—22no—
68D4×C137—42no—
69D4×C137—41no—
70D4×C137—41no—
71D4×C137—41no—
72D4×C137—41no—
73C2×C4×C137C2×D411yescharacteristic
74C2×C4×C137C2×D411yescharacteristic
75C2×C4×C137C2×D411yescharacteristic
76C8×C137—41no—
77C8×C137—41no—
78C22×C137—42no—
79C22×C137—42no—
80C22×C137—42no—
81C22×C137#12011yescharacteristic
82C2×D137—41no—
83C2×D137—41no—
84C2×D137—41no—
85C4×C137—21no—
86C4×C137C2×D811yescharacteristic
87C4×C137#13511yescharacteristic
88C4×C137AGL(1,ℤ/8)11yescharacteristic
89C4×C137#13711yescharacteristic
90Dic137—22no—
91Dic137—41no—
92Dic137—41no—
93Dic137—81no—
94C2×C137—82no—
95C2×C137#44711yescharacteristic
96C2×C137#45211yescharacteristic
97C2×C137#45511yescharacteristic
98D137—81no—
99C137#164111yescharacteristic, complemented, Sylow
100#1641—1371nomaximal, Sylow
101#550—1371no—
102#486—1371no—
103#474—1371no—
104#462—1371no—
105#331D13711yescharacteristic, complemented
106Q8⋊C8—1371no—
107D4⋊C8—1371no—
108C2×#41—2741no—
109C2×Q16—2741no—
110C4⋊Q8—1371no—
111#127—2741no—
112#127C2×D13711yescharacteristic
113#121—2742no—
114#121—2741no—
115C4×Q8—1371no—
116C4×D4—2741no—
117C4×D4—1371no—
118#107—2741no—
119C4⋊C8C2×D13711yescharacteristic
120C4⋊C8C2×D13711yescharacteristic
121#103—2741no—
122#102—2741no—
123#102—2742no—
124C4×C8—1371no—
125#41—5482no—
126#41—5482no—
127C2×Q8—2741no—
128C2×Q8—1371no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—2742no—
132C2×D4—2742no—
133C2×D4—1371no—
134C2×D4D54812yes—
135C22×C4—2742no—
136C22×C4—2741no—
137Q16—5482no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—2741no—
141C4⋊C4—2741no—
142C4⋊C4—2741no—
143C4⋊C4—2741no—
144C4⋊C4—1371no—
145C4⋊C4—1371no—
146C22⋊C4—2742no—
147C22⋊C4—2741no—
148C42—2741no—
149C42C22×D13711yescharacteristic
150C23—22no—
151C23—2741no—
152Q8—2741no—
153Q8—5481no—
154Q8—5481no—
155Q8—1372no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—2741no—
162D4—2741no—
163D4—5484no—
164D4—10962no—
165C2×C4—2741no—
166C2×C4—2741no—
167C2×C4—2741no—
168C2×C4—5484no—
169C2×C4—5481no—
170C2×C4—5481no—
171C2×C4—5481no—
172C2×C4D4×D13711yescharacteristic
173C2×C4D4×D13711yescharacteristic
174C2×C4C2×D54811yescharacteristic
175C8—41no—
176C8—41no—
177C8—5481no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—5481no—
182C22—5481no—
183C22—5481no—
184C22#1798405811yescharacteristic
185C4—21no—
186C4—2742no—
187C4—5481no—
188C4—5481no—
189C4—10961no—
190C4D8×D13711yescharacteristic
191C4#1798410711yescharacteristic
192C4#1798411511yescharacteristic
193C4#1798411711yescharacteristic
194C2—82no—
195C2—10961no—
196C2#2257753811yescharacteristic
197C2#2257756611yescharacteristic
198C2#2257758311yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#331×C137C2
socleC22×C137#120
prime radical#22578073C2
hypercenter#331D137
hypocenterC137#1641
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #22577749C2semidirect
3. #22577684C2semidirect
4. #22577614C2semidirect
5. #22577133C2semidirect
6. #22577130C2semidirect
7. #331×C137C2semidirect
99. C137#1641semidirect
105. #331D137semidirect

Maximal subgroups

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

subgroupindex
#225780732
#225777492
#225776842
#225776142
#225771332
#225771302
#331×C1372
#1641137