Finite Group Databasebeta

← #31805386

Subgroups — #31805386

Order 17792.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 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
1#22581541C211yesmaximal, characteristic
2#22581111C211yesmaximal, characteristic, complemented
3#22581094C211yesmaximal, characteristic, complemented
4#22580865C211yesmaximal, characteristic, complemented
5#22580691C211yesmaximal, characteristic, complemented
6#22580688C211yesmaximal, characteristic, complemented
7#323×C139C211yesmaximal, characteristic, complemented
8C22×Dic278C2211yescharacteristic
9C2×Dic556—21no—
10C2×#14555774—21no—
11#17984835—21no—
12#17984828—21no—
13#17984824—21no—
14#17984821C2211yescharacteristic
15#17984816C2211yescharacteristic
16#17984815—21no—
17#17984811—21no—
18C4×C22⋊D139—21no—
19C2×C22⋊Dic139—21no—
20#17984767—21no—
21#17984767—21no—
22C139⋊#95—21no—
23C139⋊#95C2211yescharacteristic
24C2×C22⋊C4×C139C2211yescharacteristic
25C22⋊C8×C139C2211yescharacteristic
26C22⋊C8×C139C2211yescharacteristic
27C2×Dic278—21no—
28C2×Dic278—21no—
29C2×Dic278—42no—
30C2×Dic278D411yescharacteristic
31C2×D556D411yescharacteristic
32C2×C22⋊D139—21no—
33C2×C22⋊D139—21no—
34C2×C22⋊D139—21no—
35C22×Dic139—21no—
36C22×Dic139—21no—
37C2×C4×D139—21no—
38Dic556—42no—
39#14555774—42no—
40C4⋊Dic139D411yescharacteristic
41C4⋊Dic139D411yescharacteristic
42Dic139⋊C4—21no—
43Dic139⋊C4—21no—
44C22⋊Dic139—21no—
45C22⋊Dic139—21no—
46C22⋊Dic139—21no—
47C22⋊Dic139—42no—
48(C2×D139)⋊C4—21no—
49(C2×D139)⋊C4—21no—
50C4×Dic139—21no—
51C24×C139D411yescharacteristic
52C22×C4×C139D411yescharacteristic
53C22×C4×C139C2311yescharacteristic
54C2×C8×C139—21no—
55C2×C8×C139—21no—
56C22⋊C4×C139—21no—
57C22⋊C4×C139—21no—
58C22×D139—21no—
59Dic278—22no—
60Dic278—22no—
61Dic278—41no—
62Dic278—41no—
63Dic278—41no—
64D556—41no—
65C22⋊D139—41no—
66C22⋊D139—41no—
67C22⋊D139—81no—
68C22⋊D139—81no—
69C2×Dic139—22no—
70C2×Dic139—21no—
71C2×Dic139—21no—
72C2×Dic139—21no—
73C2×Dic139—41no—
74C2×Dic139—41no—
75C2×Dic139—41no—
76C2×Dic139—41no—
77C2×Dic139—41no—
78C4×D139—81no—
79C23×C139—21no—
80C23×C139—21no—
81C23×C139—21no—
82C23×C139—41no—
83C23×C139—41no—
84C23×C139C2×D411yescharacteristic
85C2×C4×C139—21no—
86C2×C4×C139—41no—
87C2×C4×C139—41no—
88C2×C4×C139C2×D411yescharacteristic
89C2×C4×C139C2×D411yescharacteristic
90C8×C139—41no—
91C8×C139—41no—
92C22×C139—21no—
93C22×C139—21no—
94C22×C139—42no—
95C22×C139—41no—
96C22×C139—41no—
97C22×C139—41no—
98C22×C139—81no—
99C22×C139#12011yescharacteristic
100C22×C139#13511yescharacteristic
101C22×C139#13711yescharacteristic
102C2×D139—41no—
103C2×D139—41no—
104C2×D139—41no—
105C4×C139—21no—
106C4×C139—21no—
107C4×C139—41no—
108Dic139—41no—
109Dic139—41no—
110Dic139—42no—
111Dic139—41no—
112Dic139—81no—
113C2×C139—21no—
114C2×C139—21no—
115C2×C139—81no—
116C2×C139#45211yescharacteristic
117C2×C139#45511yescharacteristic
118C2×C139B(4,𝔽2)11yescharacteristic
119D139—81no—
120C139#160611yescharacteristic, complemented, Sylow
121#1606—1391nomaximal, Sylow
122#538—1391no—
123#451—1391no—
124#448—1391no—
125#393—1391no—
126#328—1391no—
127#327—1391no—
128#323D13911yescharacteristic, complemented
129C22×Q8—1391no—
130C2×Q16—2781no—
131C2×SD16—2781no—
132#126—2781no—
133#124—2781no—
134#123—2781no—
135C22⋊Q8—1391no—
136#121—2781no—
137#121—1391no—
138#120—2781no—
139C4×D4—2781no—
140C2×C22⋊C4—2781no—
141C2×C22⋊C4C2×D13911yescharacteristic
142#103—2781no—
143#103—2781no—
144C22⋊C8C2×D13911yescharacteristic
145C22⋊C8C2×D13911yescharacteristic
146#95—2781no—
147#95—1391no—
148C24D55611yescharacteristic
149C2×Q8—2781no—
150C2×Q8—2781no—
151C2×Q8—5562no—
152C2×Q8—1391no—
153C2×D4—2781no—
154C2×D4—2781no—
155C2×D4—2781no—
156C2×D4—1391no—
157C22×C4—2781no—
158C22×C4—2781no—
159C22×C4—2781no—
160C22×C4D55611yescharacteristic
161C22×C4C22×D13911yescharacteristic
162Q16—5562no—
163SD16—5562no—
164C2×C8—21no—
165C2×C8—21no—
166C4⋊C4—2781no—
167C4⋊C4—2781no—
168C4⋊C4—1391no—
169C4⋊C4—1391no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—2781no—
173C22⋊C4—2781no—
174C22⋊C4—2781no—
175C22⋊C4—2781no—
176C22⋊C4—2781no—
177C22⋊C4—5562no—
178C42—2781no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23—2781no—
185C23C2×D55611yescharacteristic
186Q8—2782no—
187Q8—2782no—
188Q8—5561no—
189Q8—5561no—
190Q8—5561no—
191D4—5561no—
192D4—5561no—
193D4—5561no—
194D4—11121no—
195D4—11121no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—2782no—
200C2×C4—2781no—
201C2×C4—2781no—
202C2×C4—2781no—
203C2×C4—5561no—
204C2×C4—5561no—
205C2×C4—5561no—
206C2×C4—5561no—
207C2×C4—5561no—
208C2×C4—11121no—
209C2×C4D4×D13911yescharacteristic
210C2×C4D4×D13911yescharacteristic
211C8—41no—
212C8—41no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—5561no—
221C22—5561no—
222C22—5561no—
223C22#1798481211yescharacteristic
224C22#1798486411yescharacteristic
225C22#1798487511yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—5561no—
230C4—5561no—
231C4—5562no—
232C4—5561no—
233C4—11121no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—11121no—
238C2#2258111711yescharacteristic
239C2#2258113311yescharacteristic
240C2#2258114111yescharacteristic
subgroupG/H
centerC22#17984812
commutator, cosocleC22×C4×C139C23
FrattiniC22×C4C22×D139
Fitting#323×C139C2
socleC22×C139#120
prime radical#22581541C2
hypercenter#323D139
hypocenterC139#1606
LCSG ▷ C22×C4×C139 ▷ C22×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #22581111C2semidirect
3. #22581094C2semidirect
4. #22580865C2semidirect
5. #22580691C2semidirect
6. #22580688C2semidirect
7. #323×C139C2semidirect
120. C139#1606semidirect
128. #323D139semidirect

Maximal subgroups

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

subgroupindex
#225815412
#225811112
#225810942
#225808652
#225806912
#225806882
#323×C1392
#1606139