Finite Group Databasebeta

← #31790320

Subgroups — #31790320

Order 17536.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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×#17984121C211yesmaximal, characteristic
2C22×Dic548C211yesmaximal, characteristic, complemented
3#22577979C211yesmaximal, characteristic
4#22577880C211yesmaximal, characteristic, complemented
5#22577626C214yesmaximal, complemented
6#22577614C214yesmaximal, complemented
7#22577413C211yesmaximal, characteristic, complemented
8#22577400C211yesmaximal, characteristic
9#424×C137C211yesmaximal, characteristic, complemented
10C2×#14555510C2211yescharacteristic
11C22×Dic274C2211yescharacteristic
12C22×Dic274C2211yescharacteristic
13#17984121—24no—
14C2×Dic548—24no—
15C2×Dic548C2214yes—
16C2×Dic548C2212yes—
17C2×#14555499C2212yes—
18#17984088C2212yes—
19#17984076C2212yes—
20#17984072—22no—
21#17984068—22no—
22C4×Dic274C2214yes—
23#17984053—22no—
24#17984047C2211yescharacteristic
25#17984044—22no—
26#17984013C2214yes—
27#17984013C2212yes—
28#17984010C2212yes—
29C2×M16×C137C2211yescharacteristic
30C22×C8×C137C2211yescharacteristic
31#117×C137C2211yescharacteristic
32C4⋊C8×C137C2214yes—
33#14555510—22no—
34#14555510—22no—
35C2×Dic274—21no—
36C2×Dic274—21no—
37C2×Dic274—22no—
38C2×Dic274—22no—
39C2×Dic274—22no—
40C2×Dic274D414yes—
41C2×Dic274C2312yes—
42C2×Dic274C2311yescharacteristic
43C2×Dic274C2312yes—
44C2×D548C2311yescharacteristic
45C2×C22⋊D137—21no—
46C22×Dic137—21no—
47C22×Dic137—21no—
48C2×C4×D137—21no—
49Dic548—28no—
50Dic548—24no—
51#14555499—24no—
52C4⋊Dic137—24no—
53C4⋊Dic137C2312yes—
54#14555489—24no—
55C22⋊Dic137—21no—
56#14555487—24no—
57C4×Dic137—24no—
58C4×Dic137—21no—
59C22×C4×C137C2311yescharacteristic
60M16×C137—22no—
61C2×C8×C137—22no—
62C2×C8×C137C2312yes—
63C2×C8×C137C2312yes—
64C4⋊C4×C137D412yes—
65C22⋊C4×C137D412yes—
66C42×C137C2312yes—
67C22×D137—21no—
68Dic274—24no—
69Dic274—22no—
70Dic274—22no—
71Dic274—21no—
72Dic274—22no—
73Dic274—21no—
74Dic274—42no—
75Dic274C2×D414yes—
76D548—21no—
77D548—21no—
78C22⋊D137—42no—
79C2×Dic137—24no—
80C2×Dic137—22no—
81C2×Dic137—22no—
82C2×Dic137—22no—
83C2×Dic137—21no—
84C2×Dic137—42no—
85C4×D137—42no—
86C23×C137C2×D411yescharacteristic
87C2×C4×C137C2×D412yes—
88C2×C4×C137C2×D414yes—
89C2×C4×C137C2×D411yescharacteristic
90C2×C4×C137#4112yes—
91C2×C4×C137C2411yescharacteristic
92C8×C137—22no—
93C8×C137—22no—
94C22×C137—22no—
95C22×C137#12112yes—
96C22×C137C22×D411yescharacteristic
97C2×D137—41no—
98C2×D137—42no—
99C4×C137—24no—
100C4×C137#12112yes—
101C4×C137C22×D411yescharacteristic
102C4×C137C2×#4111yescharacteristic
103Dic137—24no—
104Dic137—41no—
105Dic137—42no—
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#519—1371no—
117#464—1374no—
118#462—1374no—
119#424D13711yescharacteristic, complemented
120#419—1371no—
121#416—1371no—
122C2×#41—1371no—
123C22×Q8—1371no—
124C22×Q8—1371no—
125#137—2744no—
126C2×Q16—2744no—
127C2×Q16—1374no—
128C2×Q16—1372no—
129C2×SD16—1372no—
130C2×M16C2×D13711yescharacteristic
131C22×C8C2×D13711yescharacteristic
132C4⋊Q8—1372no—
133#124—1372no—
134#123—2742no—
135C22⋊Q8—2742no—
136C4×Q8—2742no—
137C4×Q8—1374no—
138#117—2742no—
139#117—1371no—
140#117C2×D13711yescharacteristic
141C4⋊C8C2×D13714yes—
142#103—1372no—
143#103—1374no—
144#102—1372no—
145#41—2742no—
146#41—2742no—
147C2×Q8—2741no—
148C2×Q8—2741no—
149C2×Q8—2742no—
150C2×Q8—2742no—
151C2×Q8—2742no—
152C2×Q8—1374no—
153C2×Q8—1372no—
154C2×Q8—1371no—
155C2×Q8—1372no—
156C2×D4—2741no—
157C2×D4—1371no—
158C22×C4—2741no—
159C22×C4—2741no—
160C22×C4—2741no—
161C22×C4C22×D13711yescharacteristic
162Q16—2748no—
163Q16—2744no—
164SD16—2744no—
165M16—22no—
166C2×C8—22no—
167C2×C8C22×D13712yes—
168C2×C8C22×D13712yes—
169C4⋊C4—2744no—
170C4⋊C4—2744no—
171C4⋊C4—1372no—
172C4⋊C4D54812yes—
173C22⋊C4—2741no—
174C22⋊C4—2744no—
175C22⋊C4D54812yes—
176C42—2744no—
177C42—2741no—
178C42C22×D13712yes—
179C23—2741no—
180C23C2×D54811yescharacteristic
181Q8—2744no—
182Q8—2742no—
183Q8—2742no—
184Q8—2741no—
185Q8—2742no—
186Q8—2741no—
187Q8—5482no—
188Q8—1374no—
189D4—2741no—
190D4—2741no—
191D4—5482no—
192C2×C4—2744no—
193C2×C4—2742no—
194C2×C4—2742no—
195C2×C4—2742no—
196C2×C4—2741no—
197C2×C4—5482no—
198C2×C4—5482no—
199C2×C4D4×D13712yes—
200C2×C4C2×D54814yes—
201C2×C4C2×D54811yescharacteristic
202C2×C4#1455551112yes—
203C2×C4C23×D13711yescharacteristic
204C8—22no—
205C8—22no—
206C22—22no—
207C22—5482no—
208C22—5481no—
209C22#1798406312yes—
210C22C22×D54811yescharacteristic
211C4—24no—
212C4—2744no—
213C4—5482no—
214C4—5481no—
215C4—5482no—
216C4#1798406312yes—
217C4C2×D4×D13711yescharacteristic
218C4C2×#1455551111yescharacteristic
219C2—22no—
220C2—10961no—
221C2C2×#1798406311yescharacteristic
222C2#2257822012yes—
subgroupG/H
centerC22C22×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#424×C137C2
socleC22×C137C22×D4
prime radicalC2×#14555510C22
hypercenter#424D137
hypocenterC137#3044
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. C22×Dic548C2semidirect
4. #22577880C2semidirect
5. #22577626C2semidirect
6. #22577614C2semidirect
7. #22577413C2semidirect
9. #424×C137C2semidirect
111. C137#3044semidirect
119. #424D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841212
C22×Dic5482
#225779792
#225778802
#225776262
#225776142
#225774132
#225774002
#424×C1372
#3044137