Finite Group Databasebeta

← #31790322

Subgroups — #31790322

Order 17536.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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×#17984118C211yesmaximal, characteristic
2C22×Dic548C211yesmaximal, characteristic, complemented
3#22577974C211yesmaximal, characteristic
4#22577880C211yesmaximal, characteristic, complemented
5#22577624C214yesmaximal, complemented
6#22577612C214yesmaximal, complemented
7#22577429C211yesmaximal, characteristic, complemented
8#22577411C211yesmaximal, characteristic, complemented
9#416×C137C211yesmaximal, characteristic
10C22×Dic274C2211yescharacteristic
11C22×Dic274C2211yescharacteristic
12#17984118—24no—
13C2×Dic548—24no—
14C2×Dic548C2214yes—
15C2×#14555500C2212yes—
16C2×#14555498C2212yes—
17C2×#14555494C2211yescharacteristic
18#17984086C2212yes—
19#17984075C2212yes—
20#17984069—22no—
21#17984067—22no—
22C4×Dic274—22no—
23#17984053C2214yes—
24#17984047C2211yescharacteristic
25#17984044—22no—
26#17984017C2214yes—
27#17984011C2214yes—
28C2×#41×C137C2211yescharacteristic
29C22×C8×C137C2211yescharacteristic
30#117×C137C2211yescharacteristic
31#103×C137C2212yes—
32#102×C137C2212yes—
33C2×Dic274—21no—
34C2×Dic274—21no—
35C2×Dic274—22no—
36C2×Dic274—22no—
37C2×Dic274—22no—
38C2×Dic274D414yes—
39C2×Dic274C2312yes—
40C2×Dic274C2312yes—
41C22×Dic137—21no—
42C22×Dic137—21no—
43Dic548—28no—
44#14555500—24no—
45#14555498—24no—
46#14555494—22no—
47C2×#11764040C2312yes—
48C4⋊Dic137D412yes—
49#14555489—24no—
50#14555489—24no—
51C22⋊Dic137—24no—
52C22⋊Dic137D412yes—
53C4×Dic137—24no—
54C4×Dic137C2312yes—
55#41×C137—22no—
56#41×C137—22no—
57C2×Q8×C137C2311yescharacteristic
58C2×D4×C137—21no—
59C2×D4×C137C2311yescharacteristic
60C22×C4×C137—21no—
61C22×C4×C137C2311yescharacteristic
62C2×C8×C137—22no—
63C2×C8×C137C2312yes—
64C4⋊C4×C137C2312yes—
65C22⋊C4×C137—21no—
66C42×C137—21no—
67Dic274—24no—
68Dic274—22no—
69Dic274—22no—
70Dic274—22no—
71Dic274—42no—
72Dic274C2×D414yes—
73C2×Dic137—24no—
74C2×Dic137—22no—
75C2×Dic137—22no—
76C2×Dic137—42no—
77C2×Dic137C2×D414yes—
78#11764040—22no—
79C23×C137—21no—
80C23×C137C2×D411yescharacteristic
81Q8×C137—21no—
82Q8×C137—21no—
83D4×C137—21no—
84D4×C137—21no—
85D4×C137—42no—
86C2×C4×C137—22no—
87C2×C4×C137—21no—
88C2×C4×C137—42no—
89C2×C4×C137C2×D411yescharacteristic
90C2×C4×C137C2×D412yes—
91C2×C4×C137#4112yes—
92C2×C4×C137C2411yescharacteristic
93C8×C137—22no—
94C22×C137—22no—
95C22×C137—41no—
96C22×C137—42no—
97C22×C137#12112yes—
98C22×C137C22×D411yescharacteristic
99C4×C137—41no—
100C4×C137—42no—
101C4×C137#12112yes—
102C4×C137C22×D411yescharacteristic
103C4×C137C2×#4111yescharacteristic
104Dic137—24no—
105Dic137—24no—
106Dic137—42no—
107C2×C137—22no—
108C2×C137—81no—
109C2×C137C2×#12111yescharacteristic
110C2×C137#57812yes—
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#424—1371no—
120#419—1371no—
121#416D13711yescharacteristic
122C2×#41C2×D13711yescharacteristic
123C22×Q8—1371no—
124C22×Q8—1371no—
125#137—2744no—
126C2×Q16—2744no—
127C2×Q16—1372no—
128C2×Q16—1374no—
129C2×SD16—1372no—
130C2×M16—1371no—
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⋊C8—1374no—
142#103—1374no—
143#103C2×D13712yes—
144#102C2×D13712yes—
145#41—22no—
146#41—22no—
147C2×Q8—2741no—
148C2×Q8—2741no—
149C2×Q8—2742no—
150C2×Q8—2742no—
151C2×Q8—2742no—
152C2×Q8—1374no—
153C2×Q8—1372no—
154C2×Q8—1372no—
155C2×Q8C22×D13711yescharacteristic
156C2×D4—21no—
157C2×D4C22×D13711yescharacteristic
158C22×C4—21no—
159C22×C4—2741no—
160C22×C4—2741no—
161C22×C4C22×D13711yescharacteristic
162Q16—2748no—
163Q16—2744no—
164SD16—2744no—
165M16—2742no—
166C2×C8—22no—
167C2×C8—1372no—
168C2×C8C22×D13712yes—
169C4⋊C4—2744no—
170C4⋊C4—2744no—
171C4⋊C4—1372no—
172C4⋊C4C22×D13712yes—
173C22⋊C4—21no—
174C22⋊C4—2744no—
175C22⋊C4—1372no—
176C42—21no—
177C42—2744no—
178C42—1372no—
179C23—21no—
180C23D4×D13711yescharacteristic
181Q8—21no—
182Q8—21no—
183Q8—2744no—
184Q8—2742no—
185Q8—2742no—
186Q8—2742no—
187Q8—5482no—
188Q8—1374no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—2744no—
196C2×C4—2742no—
197C2×C4—2742no—
198C2×C4—5482no—
199C2×C4—1374no—
200C2×C4D4×D13711yescharacteristic
201C2×C4D4×D13712yes—
202C2×C4#1455551012yes—
203C2×C4C23×D13711yescharacteristic
204C8—22no—
205C8—2742no—
206C22—22no—
207C22—42no—
208C22—41no—
209C22#1798406112yes—
210C22C2×D4×D13711yescharacteristic
211C4—42no—
212C4—41no—
213C4—2744no—
214C4—2744no—
215C4—5482no—
216C4#1798406112yes—
217C4C2×D4×D13711yescharacteristic
218C4C2×#1455551011yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1798406111yescharacteristic
222C2#2257821712yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#416×C137C2
socleC22×C137C22×D4
prime radicalC2×#41×C137C22
hypercenter#416D137
hypocenterC137#3044
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C22×Dic548C2semidirect
4. #22577880C2semidirect
5. #22577624C2semidirect
6. #22577612C2semidirect
7. #22577429C2semidirect
8. #22577411C2semidirect
111. C137#3044semidirect

Maximal subgroups

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

subgroupindex
C2×#179841182
C22×Dic5482
#225779742
#225778802
#225776242
#225776122
#225774292
#225774112
#416×C1372
#3044137