Finite Group Databasebeta

← #31779862

Subgroups — #31779862

Order 17536.

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

normal subgroups only

This group has 236 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#22578114C211yesmaximal, characteristic, complemented
2#22577777C211yesmaximal, characteristic, complemented
3#22577604C211yesmaximal, characteristic, complemented
4#22577599C211yesmaximal, characteristic
5#22577133C211yesmaximal, characteristic, complemented
6#22577130C211yesmaximal, characteristic, complemented
7#333×C137C211yesmaximal, characteristic, complemented
8C2×#14555499—21no—
9C2×#14555497—22no—
10C2×D1096—21no—
11#17984085C2211yescharacteristic
12#17984079—21no—
13#17984068—21no—
14#17984064—22no—
15C4×Dic274C2211yescharacteristic
16C4×D548C2211yescharacteristic
17#17984050—21no—
18C4⋊C4×D137—21no—
19#17984010—21no—
20#17984010—21no—
21C4×#11764040C2211yescharacteristic
22C4⋊Q8×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4⋊C8×C137C2211yescharacteristic
25C2×Dic274D411yescharacteristic
26C2×D548—21no—
27C2×D548—21no—
28C2×D548—21no—
29C2×D548D411yescharacteristic
30C2×C4×D137—22no—
31C2×C4×D137—21no—
32#14555499—42no—
33#14555497—44no—
34D1096—42no—
35C2×#11764040—21no—
36C4⋊Dic137—22no—
37C4⋊Dic137D411yescharacteristic
38C4⋊Dic137D411yescharacteristic
39#14555489—22no—
40#14555489—21no—
41#14555487—22no—
42#14555487—21no—
43C4×Dic137—21no—
44C2×Q8×C137D412yes—
45C2×C8×C137—21no—
46C2×C8×C137—21no—
47C4⋊C4×C137—21no—
48C4⋊C4×C137—21no—
49C42×C137C2311yescharacteristic
50C22×D137—21no—
51C22×D137—41no—
52Dic274—21no—
53Dic274—21no—
54D548—21no—
55D548—42no—
56D548—41no—
57D548—41no—
58D548—41no—
59D548—41no—
60D548SD1612yescomplemented
61C2×Dic137—21no—
62C2×Dic137—21no—
63C2×Dic137—21no—
64C4×D137—41no—
65C4×D137—41no—
66C4×D137—82no—
67#11764040—22no—
68Q8×C137—42no—
69C2×C4×C137—22no—
70C2×C4×C137C2×D411yescharacteristic
71C2×C4×C137C2×D411yescharacteristic
72C2×C4×C137C2×D411yescharacteristic
73C8×C137—41no—
74C8×C137—41no—
75C22×C137#12011yescharacteristic
76C2×D137—22no—
77C2×D137—41no—
78C2×D137—41no—
79C2×D137—81no—
80C2×D137—81no—
81C2×D137—81no—
82C4×C137—21no—
83C4×C137—42no—
84C4×C137C2×SD1611yescharacteristic
85C4×C137#13511yescharacteristic
86C4×C137AGL(1,ℤ/8)11yescharacteristic
87C4×C137#13711yescharacteristic
88Dic137—41no—
89Dic137—41no—
90Dic137—41no—
91C2×C137#44811yescharacteristic
92C2×C137#44911yescharacteristic
93C2×C137#45311yescharacteristic
94D137—42no—
95D137—161no—
96C137#162211yescharacteristic, complemented, Sylow
97#1622—1371nomaximal, Sylow
98#554—1371no—
99#492—1371no—
100#460—1371no—
101#459—1371no—
102#333D13711yescharacteristic, complemented
103Q8⋊C8—1371no—
104D4⋊C8—1371no—
105C2×SD16—2741no—
106C2×SD16—2742no—
107C2×D8—2741no—
108C4⋊Q8C2×D13711yescharacteristic
109#127—1371no—
110#125—2741no—
111C22⋊Q8—2741no—
112C22⋊Q8—2742no—
113C4×Q8—1371no—
114C4×D4—2741no—
115C4×D4—1371no—
116C2×C4⋊C4—2741no—
117C4⋊C8C2×D13711yescharacteristic
118C4⋊C8C2×D13711yescharacteristic
119#102—2741no—
120#102—2741no—
121C4×C8—1371no—
122C2×Q8—1371no—
123C2×Q8D54812yes—
124C2×D4—2741no—
125C2×D4—2741no—
126C2×D4—2741no—
127C2×D4—1371no—
128C22×C4—2742no—
129C22×C4—2741no—
130SD16—5482no—
131SD16—5484no—
132D8—5482no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—2741no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—2742no—
139C4⋊C4—2742no—
140C4⋊C4—2741no—
141C4⋊C4—1371no—
142C4⋊C4—1371no—
143C22⋊C4—2742no—
144C22⋊C4—2741no—
145C42—2741no—
146C42C22×D13711yescharacteristic
147C23—2741no—
148C23—5481no—
149Q8—42no—
150Q8—2741no—
151Q8—2741no—
152D4—2741no—
153D4—5482no—
154D4—5481no—
155D4—5481no—
156D4—5481no—
157D4—5481no—
158D4—1372no—
159C2×C4—22no—
160C2×C4—2741no—
161C2×C4—2741no—
162C2×C4—2741no—
163C2×C4—5481no—
164C2×C4—5481no—
165C2×C4—10962no—
166C2×C4D4×D13711yescharacteristic
167C2×C4D4×D13711yescharacteristic
168C2×C4C2×D54811yescharacteristic
169C8—41no—
170C8—41no—
171C8—2742no—
172C22—2742no—
173C22—5481no—
174C22—5481no—
175C22—10961no—
176C22—10961no—
177C22—10961no—
178C22#1798405811yescharacteristic
179C4—21no—
180C4—42no—
181C4—5481no—
182C4—5481no—
183C4—5481no—
184C4SD16×D13711yescharacteristic
185C4#1798411111yescharacteristic
186C4#1798411511yescharacteristic
187C4#1798412011yescharacteristic
188C2—5482no—
189C2—21921no—
190C2#2257754311yescharacteristic
191C2#2257755111yescharacteristic
192C2#2257757311yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#333×C137C2
socleC22×C137#120
prime radical#22577599C2
hypercenter#333D137
hypocenterC137#1622
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
1. #22578114C2semidirect
2. #22577777C2semidirect
3. #22577604C2semidirect
5. #22577133C2semidirect
6. #22577130C2semidirect
7. #333×C137C2semidirect
60. D548SD16semidirect
96. C137#1622semidirect
102. #333D137semidirect

Maximal subgroups

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

subgroupindex
#225781142
#225777772
#225776042
#225775992
#225771332
#225771302
#333×C1372
#1622137