Finite Group Databasebeta

← #31817792

Subgroups — #31817792

Order 17792.

386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 125 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#22581510C211yesmaximal, characteristic, complemented
2#22581503C211yesmaximal, characteristic
3#22581249C212yesmaximal, complemented
4#22581234C212yesmaximal, complemented
5#22581199C214yesmaximal, complemented
6#22581159C212yesmaximal, complemented
7#22581154C212yesmaximal, complemented
8#423×C139C211yesmaximal, characteristic, complemented
9C2×#14555785—21no—
10C2×#14555774C2214yes—
11#17984842C2211yescharacteristic
12#17984842C2211yescharacteristic
13#17984842C2211yescharacteristic
14#17984839C2211yescharacteristic
15#17984834—21no—
16#17984830—21no—
17#17984821—21no—
18#17984821C2212yes—
19#17984816—21no—
20#17984816C2212yes—
21C4×Dic278—21no—
22C4×Dic278C2212yes—
23C4×D556—21no—
24C4×D556C2212yes—
25#17984801—21no—
26#17984776C2214yes—
27#17984767C2214yes—
28#17984764C2214yes—
29C2×M16×C139C2212yes—
30C2×C42×C139C2211yescharacteristic
31C4⋊C8×C139C2212yes—
32C4⋊C8×C139C2212yes—
33#14555785—44no—
34C2×Dic278—21no—
35C2×Dic278—21no—
36C2×Dic278C2312yes—
37C2×D556—21no—
38C2×D556—21no—
39C2×D556C2312yes—
40C2×C22⋊D139—22no—
41C2×C4×D139—22no—
42#14555774—44no—
43C4⋊Dic139—21no—
44C4⋊Dic139—21no—
45C4⋊Dic139—21no—
46C4⋊Dic139—21no—
47C4⋊Dic139C2312yes—
48C4⋊Dic139C2312yes—
49Dic139⋊C4—22no—
50Dic139⋊C4—22no—
51C22⋊Dic139—22no—
52(C2×D139)⋊C4—22no—
53(C2×D139)⋊C4—22no—
54C4×Dic139—22no—
55C22×C4×C139D411yescharacteristic
56C22×C4×C139D411yescharacteristic
57C22×C4×C139C2311yescharacteristic
58M16×C139—42no—
59C2×C8×C139C2314yes—
60C42×C139D411yescharacteristic
61C42×C139D411yescharacteristic
62C42×C139C2311yescharacteristic
63C42×C139C2311yescharacteristic
64C22×D139—22no—
65Dic278—22no—
66Dic278—22no—
67Dic278—42no—
68Dic278—42no—
69D556—22no—
70D556—22no—
71D556—42no—
72D556—42no—
73C22⋊D139—44no—
74C2×Dic139—22no—
75C2×Dic139—22no—
76C2×Dic139—22no—
77C4×D139—44no—
78C23×C139C2×D411yescharacteristic
79C2×C4×C139—22no—
80C2×C4×C139—22no—
81C2×C4×C139—21no—
82C2×C4×C139—21no—
83C2×C4×C139C2×D411yescharacteristic
84C2×C4×C139C2×D411yescharacteristic
85C2×C4×C139C2×D411yescharacteristic
86C2×C4×C139C2×D411yescharacteristic
87C2×C4×C139C2×D411yescharacteristic
88C2×C4×C139C2411yescharacteristic
89C8×C139—24no—
90C22×C139—21no—
91C22×C139—21no—
92C22×C139—21no—
93C22×C139C22×D411yescharacteristic
94C2×D139—42no—
95C2×D139—42no—
96C2×D139—42no—
97C4×C139—21no—
98C4×C139—21no—
99C4×C139—21no—
100C4×C139AGL(1,ℤ/8)12yes—
101C4×C139#13712yes—
102C4×C139H(5,𝔽2)11yescharacteristic
103C4×C1392−1+411yescharacteristic
104Dic139—42no—
105Dic139—42no—
106Dic139—42no—
107C2×C139—41no—
108C2×C139#53711yescharacteristic
109C2×C139C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C139C2×#13711yescharacteristic
111D139—82no—
112C139#325411yescharacteristic, complemented, Sylow
113#3254—1391nomaximal, Sylow
114#533—1391no—
115#532—1391no—
116#477—1392no—
117#474—1392no—
118#468—1394no—
119#461—1392no—
120#460—1392no—
121#423D13911yescharacteristic, complemented
122C2×#41—2781no—
123C2×SD16—1394no—
124C2×M16C2×D13912yes—
125C4⋊Q8—1391no—
126C4⋊Q8—1391no—
127C4⋊Q8—1391no—
128#127—1391no—
129#125—2781no—
130#124—2781no—
131C22⋊Q8—2781no—
132C22⋊Q8—1392no—
133#121—2781no—
134#121—1392no—
135C4×Q8—2781no—
136C4×Q8—1392no—
137C4×D4—2781no—
138C4×D4—1392no—
139#117—2781no—
140C2×C42C2×D13911yescharacteristic
141#107—1394no—
142C4⋊C8C2×D13912yes—
143C4⋊C8C2×D13912yes—
144#103—1394no—
145#102—1394no—
146#41—5564no—
147C2×Q8—2781no—
148C2×Q8—2781no—
149C2×Q8—1392no—
150C2×D4—2782no—
151C2×D4—2781no—
152C2×D4—2781no—
153C2×D4—1392no—
154C22×C4—2782no—
155C22×C4D55611yescharacteristic
156C22×C4D55611yescharacteristic
157C22×C4C22×D13911yescharacteristic
158SD16—5564no—
159M16—42no—
160C2×C8C22×D13914yes—
161C4⋊C4—2782no—
162C4⋊C4—2782no—
163C4⋊C4—2781no—
164C4⋊C4—2781no—
165C4⋊C4—2781no—
166C4⋊C4—2781no—
167C4⋊C4—1392no—
168C4⋊C4—1392no—
169C22⋊C4—2782no—
170C22⋊C4—2782no—
171C22⋊C4—2782no—
172C42—2782no—
173C42D55611yescharacteristic
174C42D55611yescharacteristic
175C42C22×D13911yescharacteristic
176C42C22×D13911yescharacteristic
177C23—2782no—
178C23C2×D55611yescharacteristic
179Q8—2782no—
180Q8—2782no—
181Q8—5562no—
182Q8—5562no—
183D4—2782no—
184D4—2782no—
185D4—5562no—
186D4—5564no—
187D4—5562no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—2782no—
193C2×C4—2782no—
194C2×C4—2782no—
195C2×C4—5564no—
196C2×C4C2×D55611yescharacteristic
197C2×C4C2×D55611yescharacteristic
198C2×C4C2×D55611yescharacteristic
199C2×C4C2×D55611yescharacteristic
200C2×C4C2×D55611yescharacteristic
201C2×C4C23×D13911yescharacteristic
202C8—24no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—5562no—
207C22—5562no—
208C22—5562no—
209C22C22×D55611yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—5562no—
214C4—5562no—
215C4—5562no—
216C4#1798487012yes—
217C4#1798487512yes—
218C4#1798488711yescharacteristic
219C4#1798489011yescharacteristic
220C2—41no—
221C2—11122no—
222C2#2258153411yescharacteristic
223C2C2×#1798487011yescharacteristic
224C2C2×#1798487511yescharacteristic
subgroupG/H
centerC22C22×D556
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#423×C139C2
socleC22×C139C22×D4
prime radical#22581503C2
hypercenter#423D139
hypocenterC139#3254
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #22581510C2semidirect
3. #22581249C2semidirect
4. #22581234C2semidirect
5. #22581199C2semidirect
6. #22581159C2semidirect
7. #22581154C2semidirect
8. #423×C139C2semidirect
112. C139#3254semidirect
121. #423D139semidirect

Maximal subgroups

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

subgroupindex
#225815102
#225815032
#225812492
#225812342
#225811992
#225811592
#225811542
#423×C1392
#3254139