Finite Group Databasebeta

← #31790443

Subgroups — #31790443

Order 17536.

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

normal subgroups only

This group has 460 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
1C22×#14555500C211yesmaximal, characteristic, complemented
2C2×#17984096C211yesmaximal, characteristic
3#22577979C211yesmaximal, characteristic
4#22577977C211yesmaximal, characteristic
5#22577660C214yesmaximal, complemented
6#22577643C214yesmaximal, complemented
7#22577439C211yesmaximal, characteristic, complemented
8#22577413C211yesmaximal, characteristic, complemented
9#419×C137C211yesmaximal, characteristic, complemented
10C2×#14555510C2211yescharacteristic
11C22×Dic274C2211yescharacteristic
12C2×#14555500—24no—
13C2×#14555500C2212yes—
14C2×#14555500C2212yes—
15#17984096—24no—
16C2×#14555493C2212yes—
17C22×#11764040C2211yescharacteristic
18C2×C8×D137C2212yes—
19#17984088—21no—
20#17984086—21no—
21#17984077—21no—
22#17984076—21no—
23#17984072—22no—
24#17984071—22no—
25#17984068C2214yes—
26#17984064C2214yes—
27#17984047C2211yescharacteristic
28#17984021C2212yes—
29#17984019C2212yes—
30#17984013C2214yes—
31C22×Q8×C137C2211yescharacteristic
32C2×M16×C137C2211yescharacteristic
33#117×C137C2211yescharacteristic
34#103×C137C2214yes—
35#14555510—22no—
36#14555510—22no—
37C2×Dic274—21no—
38C2×Dic274—22no—
39C2×Dic274—22no—
40C2×Dic274D411yescharacteristic
41C2×Dic274C2312yes—
42C2×D548D411yescharacteristic
43C2×C22⋊D137D412yes—
44C22×Dic137—21no—
45C2×C4×D137C2312yes—
46#14555500—24no—
47#14555500—24no—
48#14555493—24no—
49C2×#11764040D414yes—
50C2×#11764040C2311yescharacteristic
51C2×#11764040C2311yescharacteristic
52C8×D137—24no—
53C4⋊Dic137—24no—
54C4⋊Dic137C2312yes—
55#14555489—24no—
56C22⋊Dic137—21no—
57#14555487—24no—
58#14555487—24no—
59C4×Dic137—21no—
60C2×Q8×C137—21no—
61C2×Q8×C137—22no—
62C2×Q8×C137—22no—
63C2×Q8×C137C2312yes—
64C22×C4×C137—21no—
65C22×C4×C137C2311yescharacteristic
66M16×C137—22no—
67C2×C8×C137C2312yes—
68C4⋊C4×C137C2312yes—
69C22⋊C4×C137—21no—
70C42×C137—21no—
71C22×D137C2×D412yes—
72Dic274—22no—
73Dic274—22no—
74Dic274—22no—
75Dic274—42no—
76D548—22no—
77C22⋊D137—24no—
78C2×Dic137—22no—
79C2×Dic137—22no—
80C2×Dic137—42no—
81C2×Dic137C2×D412yes—
82C4×D137—22no—
83C4×D137—22no—
84#11764040C2×D412yes—
85#11764040C2×D412yes—
86C23×C137C2×D411yescharacteristic
87Q8×C137—22no—
88Q8×C137—22no—
89Q8×C137—42no—
90C2×C4×C137—22no—
91C2×C4×C137—22no—
92C2×C4×C137—42no—
93C2×C4×C137C2×D411yescharacteristic
94C2×C4×C137C2×D412yes—
95C2×C4×C137#4112yes—
96C2×C4×C137C2411yescharacteristic
97C8×C137—22no—
98C22×C137—22no—
99C22×C137#12112yes—
100C22×C137C22×D411yescharacteristic
101C2×D137—22no—
102C2×D137—24no—
103C4×C137—42no—
104C4×C137—42no—
105C4×C137#12112yes—
106C4×C137C22×D411yescharacteristic
107C4×C137C2×#4111yescharacteristic
108Dic137—22no—
109Dic137—42no—
110Dic137—42no—
111C2×C137—22no—
112C2×C137C2×#12111yescharacteristic
113C2×C137#57812yes—
114D137—42no—
115C137#305911yescharacteristic, complemented, Sylow
116#3059—1371nomaximal, Sylow
117C22×Q16—1371no—
118C2×#131—1371no—
119#536—1371no—
120#536—1371no—
121#470—1374no—
122#467—1374no—
123#427—1371no—
124#419—1371no—
125#419D13711yescharacteristic, complemented
126C2×#41—1371no—
127C22×Q8—1371no—
128C22×Q8C2×D13711yescharacteristic
129C2×Q16—2744no—
130C2×Q16—1372no—
131C2×Q16—1372no—
132#131—2744no—
133C2×M16—1372no—
134C2×M16C2×D13711yescharacteristic
135C22×C8—1372no—
136C22×C8—1371no—
137C4⋊Q8—2741no—
138C4⋊Q8—2741no—
139#124—2741no—
140#124—2741no—
141#123—2742no—
142#123—2742no—
143C22⋊Q8—1374no—
144C22⋊Q8—1374no—
145#117—1371no—
146#117C2×D13711yescharacteristic
147#107—1372no—
148#106—1372no—
149#103—1374no—
150#103C2×D13714yes—
151#41—2742no—
152#41—2742no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—2741no—
157C2×Q8—2742no—
158C2×Q8—2742no—
159C2×Q8—1372no—
160C2×Q8—1371no—
161C2×Q8C22×D13712yes—
162C2×D4—1372no—
163C2×D4—1371no—
164C22×C4—21no—
165C22×C4—2741no—
166C22×C4—1372no—
167C22×C4C22×D13711yescharacteristic
168Q16—2744no—
169Q16—2744no—
170M16—22no—
171M16—2744no—
172C2×C8—2744no—
173C2×C8—1374no—
174C2×C8—1371no—
175C2×C8—1371no—
176C2×C8C22×D13712yes—
177C4⋊C4—2744no—
178C4⋊C4—2744no—
179C4⋊C4—1372no—
180C4⋊C4C22×D13712yes—
181C22⋊C4—21no—
182C22⋊C4—2744no—
183C22⋊C4—2744no—
184C22⋊C4—2741no—
185C42—21no—
186C42—2741no—
187C23—1372no—
188C23D4×D13711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—2742no—
193Q8—2742no—
194Q8—2742no—
195Q8—5482no—
196D4—2744no—
197D4—2742no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—2742no—
202C2×C4—2742no—
203C2×C4—2742no—
204C2×C4—2742no—
205C2×C4—5482no—
206C2×C4—1372no—
207C2×C4D4×D13711yescharacteristic
208C2×C4D4×D13712yes—
209C2×C4#1455551012yes—
210C2×C4C23×D13711yescharacteristic
211C8—22no—
212C8—1372no—
213C8—1372no—
214C22—22no—
215C22—2744no—
216C22—2742no—
217C22#1798406112yes—
218C22C2×D4×D13711yescharacteristic
219C4—42no—
220C4—42no—
221C4—2742no—
222C4—5482no—
223C4—5482no—
224C4#1798406112yes—
225C4C2×D4×D13711yescharacteristic
226C4C2×#1455551011yescharacteristic
227C2—22no—
228C2—5482no—
229C2C2×#1798406111yescharacteristic
230C2#2257821912yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#419×C137C2
socleC22×C137C22×D4
prime radicalC2×#14555510C22
hypercenter#419D137
hypocenterC137#3059
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#14555500C2semidirect
5. #22577660C2semidirect
6. #22577643C2semidirect
7. #22577439C2semidirect
8. #22577413C2semidirect
9. #419×C137C2semidirect
115. C137#3059semidirect
125. #419D137semidirect

Maximal subgroups

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

subgroupindex
C22×#145555002
C2×#179840962
#225779792
#225779772
#225776602
#225776432
#225774392
#225774132
#419×C1372
#3059137