Finite Group Databasebeta

← #31781191

Subgroups — #31781191

Order 17536.

344 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#17984076C211yesmaximal, characteristic
2#22577880C211yesmaximal, characteristic, complemented
3#22577409C211yesmaximal, characteristic
4#22577407C211yesmaximal, characteristic
5C2×#17984006C211yesmaximal, characteristic, complemented
6#22577163C211yesmaximal, characteristic, complemented
7#339×C137C211yesmaximal, characteristic, complemented
8C22×Dic274C2211yescharacteristic
9C22×D548C2211yescharacteristic
10C2×#14555494C2211yescharacteristic
11#17984076—22no—
12#17984076C414yescomplemented
13C4×Dic274—22no—
14#17984053—22no—
15#17984047C2211yescharacteristic
16#17984044—22no—
17C2×#14555487—22no—
18#17984010—22no—
19#17984008—22no—
20#17984006—22no—
21C2×M16×C137C2211yescharacteristic
22#117×C137C2211yescharacteristic
23C2×C42×C137C2211yescharacteristic
24C23×D137—21no—
25C2×Dic274—21no—
26C2×Dic274—21no—
27C2×Dic274—21no—
28C2×Dic274—21no—
29C2×Dic274C2×C412yes—
30C2×Dic274D412yes—
31C2×D548—21no—
32C2×D548—21no—
33C2×D548—21no—
34C2×D548—21no—
35C2×D548—21no—
36C2×D548C2×C412yes—
37C22×Dic137D412yes—
38#14555494—21no—
39#14555494—21no—
40C2×#11764040—21no—
41C4⋊Dic137D412yes—
42#14555489—24no—
43C22⋊Dic137—21no—
44#14555487—28no—
45C4×Dic137—24no—
46C4×Dic137—21no—
47C22×C4×C137—21no—
48C22×C4×C137C2311yescharacteristic
49M16×C137—21no—
50M16×C137—21no—
51C2×C8×C137—21no—
52C4⋊C4×C137D412yes—
53C22⋊C4×C137—21no—
54C42×C137—21no—
55C42×C137—21no—
56C42×C137C2×C412yes—
57C22×D137—22no—
58C22×D137—44no—
59C22×D137—41no—
60C22×D137—41no—
61Dic274—22no—
62Dic274—22no—
63Dic274—22no—
64Dic274—22no—
65D548—22no—
66D548—22no—
67D548—41no—
68D548—41no—
69C2×Dic137—22no—
70C2×Dic137—22no—
71C2×Dic137—22no—
72C2×Dic137C22⋊C414yes—
73#11764040—41no—
74C23×C137C2×D411yescharacteristic
75C2×C4×C137—22no—
76C2×C4×C137—22no—
77C2×C4×C137—42no—
78C2×C4×C137C22×C411yescharacteristic
79C2×C4×C137C2×D411yescharacteristic
80C2×C4×C137C2×D411yescharacteristic
81C2×C4×C137C2×D411yescharacteristic
82C2×C4×C137#4111yescharacteristic
83C2×C4×C137#4111yescharacteristic
84C8×C137—41no—
85C22×C137—22no—
86C22×C137C2×C22⋊C411yescharacteristic
87C22×C137#12011yescharacteristic
88C22×C137#12311yescharacteristic
89C2×D137—44no—
90C2×D137—42no—
91C2×D137—82no—
92C2×D137—82no—
93C4×C137—42no—
94C4×C137—42no—
95C4×C137C4×D411yescharacteristic
96C4×C137C4×D411yescharacteristic
97C4×C137#12111yescharacteristic
98C4×C137#12111yescharacteristic
99Dic137—24no—
100Dic137—42no—
101C2×C137—21no—
102C2×C137—21no—
103C2×C137#38611yescharacteristic
104C2×C137#45412yes—
105D137—82no—
106C137#188811yescharacteristic, complemented, Sylow
107#1888—1371nomaximal, Sylow
108C2×#124—1371no—
109#519—1371no—
110#418—1371no—
111#418—1371no—
112C2×#101—1371no—
113#339—1371no—
114#339D13711yescharacteristic, complemented
115C22×Q8—1371no—
116C22×D4—1371no—
117C2×M16—1371no—
118C2×M16C2×D13711yescharacteristic
119#124—2742no—
120#124—1374no—
121C4×Q8—2742no—
122C4×Q8—2742no—
123#117—2742no—
124#117—1371no—
125#117C2×D13711yescharacteristic
126C2×C22⋊C4—2742no—
127C2×C42C2×D13711yescharacteristic
128#102—2742no—
129#102—2742no—
130#101—2742no—
131C24—2741no—
132C2×Q8—2741no—
133C2×Q8—2741no—
134C2×Q8—2741no—
135C2×Q8—2741no—
136C2×Q8—1372no—
137C2×Q8—1372no—
138C2×D4—2741no—
139C2×D4—2741no—
140C2×D4—2741no—
141C2×D4—2741no—
142C2×D4—2741no—
143C2×D4—1372no—
144C22×C4—21no—
145C22×C4—1372no—
146C22×C4C22×D13711yescharacteristic
147M16—21no—
148M16—21no—
149M16—2741no—
150M16—2741no—
151C2×C8—21no—
152C2×C8—2741no—
153C4⋊C4—2744no—
154C4⋊C4—1372no—
155C4⋊C4D54812yes—
156C22⋊C4—21no—
157C22⋊C4—2748no—
158C22⋊C4—2741no—
159C42—21no—
160C42—21no—
161C42—2744no—
162C42—2741no—
163C42C4×D13712yes—
164C23—2742no—
165C23—5484no—
166C23—5481no—
167C23—5481no—
168C23D4×D13711yescharacteristic
169Q8—2742no—
170Q8—2742no—
171Q8—2742no—
172Q8—2742no—
173D4—2742no—
174D4—2742no—
175D4—5481no—
176D4—5481no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—2742no—
181C2×C4—2742no—
182C2×C4—2742no—
183C2×C4—1374no—
184C2×C4C2×C4×D13711yescharacteristic
185C2×C4D4×D13711yescharacteristic
186C2×C4D4×D13711yescharacteristic
187C2×C4C2×D54811yescharacteristic
188C2×C4#1455551011yescharacteristic
189C2×C4#1455551111yescharacteristic
190C8—41no—
191C8—5481no—
192C22—22no—
193C22—5484no—
194C22—5482no—
195C22—10962no—
196C22—10962no—
197C22C22⋊C4×D13711yescharacteristic
198C22#1798405811yescharacteristic
199C22#1798407311yescharacteristic
200C4—42no—
201C4—42no—
202C4—2744no—
203C4—5482no—
204C4#1798405011yescharacteristic
205C4C4×D54811yescharacteristic
206C4#1798406111yescharacteristic
207C4#1798406311yescharacteristic
208C2—21no—
209C2—21no—
210C2—10962no—
211C2#2257728111yescharacteristic
212C2#2257757712yes—
subgroupG/H
centerC22#17984058
commutatorC2×C4×C137C22×C4
FrattiniC22×C4C22×D137
Fitting#339×C137C2
socleC22×C137#120
prime radicalC22×D548C22
cosocleC22×C4×C137C23
hypercenter#339D137
hypocenterC137#1888
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #22577880C2semidirect
5. C2×#17984006C2semidirect
6. #22577163C2semidirect
7. #339×C137C2semidirect
12. #17984076C4semidirect
106. C137#1888semidirect
114. #339D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840762
#225778802
#225774092
#225774072
C2×#179840062
#225771632
#339×C1372
#1888137