Finite Group Databasebeta

← #31790319

Subgroups — #31790319

Order 17536.

458 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 149 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×#17984119C211yesmaximal, characteristic
2C22×#14555500C211yesmaximal, characteristic, complemented
3#22577979C211yesmaximal, characteristic
4#22577625C214yesmaximal, complemented
5#22577613C214yesmaximal, complemented
6#22577430C211yesmaximal, characteristic, complemented
7#22577411C211yesmaximal, characteristic, complemented
8#22577397C211yesmaximal, characteristic
9#519×C137C211yesmaximal, characteristic, complemented
10C2×#14555510C2211yescharacteristic
11C22×Dic274C2211yescharacteristic
12#17984119—24no—
13C2×#14555500—24no—
14C2×#14555500C2214yes—
15C2×#14555500C2212yes—
16C2×#14555497C2212yes—
17C2×#14555494C2211yescharacteristic
18C22×#11764040C2211yescharacteristic
19#17984088C2212yes—
20#17984076C2212yes—
21#17984072—22no—
22#17984068—22no—
23C4⋊#11764040C2214yes—
24#17984011C2214yes—
25#17984011C2212yes—
26#17984008C2212yes—
27C22×Q8×C137C2211yescharacteristic
28C4×Q8×C137—22no—
29C4×Q8×C137C2214yes—
30#117×C137—22no—
31#117×C137C2211yescharacteristic
32#117×C137C2211yescharacteristic
33#14555510—22no—
34#14555510—22no—
35C2×Dic274—21no—
36C2×Dic274—22no—
37C2×Dic274—22no—
38C2×Dic274C2312yes—
39C2×Dic274C2311yescharacteristic
40C2×D548C2311yescharacteristic
41C2×C22⋊D137—21no—
42C22×Dic137—21no—
43C2×C4×D137—21no—
44#14555500—28no—
45#14555500—24no—
46#14555497—24no—
47#14555494—22no—
48C2×#11764040—22no—
49C2×#11764040C2312yes—
50C2×#11764040C2312yes—
51C4⋊Dic137—24no—
52#14555487—24no—
53C2×Q8×C137—21no—
54C2×Q8×C137—22no—
55C2×Q8×C137D414yes—
56C2×Q8×C137C2312yes—
57C22×C4×C137—21no—
58C22×C4×C137C2311yescharacteristic
59C4⋊C4×C137—24no—
60C4⋊C4×C137D412yes—
61C4⋊C4×C137C2312yes—
62C22⋊C4×C137—21no—
63C22⋊C4×C137D412yes—
64C42×C137—24no—
65C42×C137—21no—
66C42×C137C2312yes—
67C22×D137—21no—
68Dic274—22no—
69Dic274—21no—
70Dic274—22no—
71Dic274—21no—
72Dic274—42no—
73D548—21no—
74D548—21no—
75C22⋊D137—42no—
76C2×Dic137—22no—
77C2×Dic137—21no—
78C2×Dic137—42no—
79C4×D137—42no—
80#11764040—22no—
81#11764040—22no—
82C23×C137C2×D411yescharacteristic
83Q8×C137—24no—
84Q8×C137—22no—
85Q8×C137C2×D414yes—
86C2×C4×C137—24no—
87C2×C4×C137—22no—
88C2×C4×C137—22no—
89C2×C4×C137C2×D414yes—
90C2×C4×C137C2×D411yescharacteristic
91C2×C4×C137C2×D412yes—
92C2×C4×C137#4112yes—
93C2×C4×C137C2411yescharacteristic
94C22×C137—22no—
95C22×C137#12112yes—
96C22×C137C22×D411yescharacteristic
97C2×D137—41no—
98C2×D137—42no—
99C4×C137—24no—
100C4×C137—24no—
101C4×C137—42no—
102C4×C137#12112yes—
103C4×C137C22×D411yescharacteristic
104C4×C137C2×#4111yescharacteristic
105Dic137—41no—
106Dic137—42no—
107C2×C137—22no—
108C2×C137C2×#12111yescharacteristic
109C2×C137#57812yes—
110D137—81no—
111C137#304411yescharacteristic, complemented, Sylow
112#3044—1371nomaximal, Sylow
113C2×#137—1371no—
114C22×Q16—1371no—
115#536—1371no—
116#519D13711yescharacteristic, complemented
117#464—1374no—
118#462—1374no—
119#424—1371no—
120#419—1371no—
121#416—1371no—
122C2×#41—1371no—
123C22×Q8—1371no—
124C22×Q8C2×D13711yescharacteristic
125#137—2744no—
126C2×Q16—2744no—
127C2×Q16—1374no—
128C2×Q16—1372no—
129C2×SD16—1372no—
130C2×M16—1371no—
131C22×C8—1371no—
132C4⋊Q8—1372no—
133#124—1372no—
134#123—2742no—
135C22⋊Q8—2742no—
136C4×Q8—22no—
137C4×Q8C2×D13714yes—
138#117—22no—
139#117C2×D13711yescharacteristic
140#117C2×D13711yescharacteristic
141C4⋊C8—1374no—
142#103—1372no—
143#103—1374no—
144#102—1372no—
145#41—2742no—
146#41—2742no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—2741no—
150C2×Q8—2742no—
151C2×Q8—2742no—
152C2×Q8—1372no—
153C2×Q8—1371no—
154C2×Q8D54814yes—
155C2×Q8C22×D13712yes—
156C2×D4—2741no—
157C2×D4—1371no—
158C22×C4—21no—
159C22×C4—2741no—
160C22×C4—2741no—
161C22×C4C22×D13711yescharacteristic
162Q16—2748no—
163Q16—2744no—
164SD16—2744no—
165M16—2742no—
166C2×C8—2742no—
167C2×C8—1372no—
168C2×C8—1372no—
169C4⋊C4—24no—
170C4⋊C4—2744no—
171C4⋊C4C22⋊D13712yes—
172C4⋊C4C22×D13712yes—
173C22⋊C4—21no—
174C22⋊C4—2744no—
175C22⋊C4C22⋊D13712yes—
176C42—24no—
177C42—21no—
178C42C22×D13712yes—
179C23—2741no—
180C23C2×C22⋊D13711yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—2742no—
184Q8—2741no—
185Q8—2742no—
186Q8—2741no—
187Q8—5482no—
188Q8C2×D54814yes—
189D4—2741no—
190D4—2741no—
191D4—5482no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—2742no—
196C2×C4—2741no—
197C2×C4—5482no—
198C2×C4—5482no—
199C2×C4C2×C22⋊D13714yes—
200C2×C4C2×C22⋊D13711yescharacteristic
201C2×C4C2×D54812yes—
202C2×C4#1455551012yes—
203C2×C4C23×D13711yescharacteristic
204C8—2742no—
205C8—2742no—
206C22—22no—
207C22—5482no—
208C22—5481no—
209C22#1798406212yes—
210C22C22×C22⋊D13711yescharacteristic
211C4—24no—
212C4—24no—
213C4—42no—
214C4—5481no—
215C4—5482no—
216C4#1798406212yes—
217C4C22×D54811yescharacteristic
218C4C2×#1455551011yescharacteristic
219C2—22no—
220C2—10961no—
221C2C2×#1798406211yescharacteristic
222C2#2257821812yes—
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#519×C137C2
socleC22×C137C22×D4
prime radicalC2×#14555510C22
hypercenter#519D137
hypocenterC137#3044
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. C22×#14555500C2semidirect
4. #22577625C2semidirect
5. #22577613C2semidirect
6. #22577430C2semidirect
7. #22577411C2semidirect
9. #519×C137C2semidirect
111. C137#3044semidirect
116. #519D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841192
C22×#145555002
#225779792
#225776252
#225776132
#225774302
#225774112
#225773972
#519×C1372
#3044137