Finite Group Databasebeta

← #31806358

Subgroups — #31806358

Order 17792.

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

normal subgroups only

This group has 620 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
1C2×#17984769C212yesmaximal, complemented
2#22580954C212yesmaximal, complemented
3C2×#17984759C211yesmaximal, characteristic, complemented
4#22580856C211yesmaximal, characteristic, complemented
5C2×H(5,𝔽2)×C139C211yesmaximal, characteristic
6C2×#14555769C2212yescomplemented
7C2×C4⋊Dic139C2211yescharacteristic, complemented
8C2×C22⋊Dic139—22no—
9C2×C4×Dic139C2211yescharacteristic, complemented
10#17984769—24no—
11#17984766—22no—
12#17984763—22no—
13#17984759—22no—
14H(5,𝔽2)×C139—24no—
15H(5,𝔽2)×C139—22no—
16H(5,𝔽2)×C139C414yescomplemented
17C2×#41×C139—22no—
18C2×#41×C139C2212yes—
19C22×D4×C139—22no—
20C22×D4×C139—22no—
21C22×D4×C139C2211yescharacteristic
22C22×Dic139—21no—
23C22×Dic139—21no—
24#14555769—22no—
25#14555769—22no—
26C2×C139⋊C8—22no—
27C4⋊Dic139—22no—
28C22⋊Dic139—44no—
29C4×Dic139—22no—
30C24×C139—22no—
31C24×C139D412yes—
32#41×C139—28no—
33#41×C139—28no—
34#41×C139—22no—
35#41×C139—22no—
36#41×C139C2×C414yes—
37#41×C139D414yes—
38C2×Q8×C139D412yes—
39C2×D4×C139—24no—
40C2×D4×C139—28no—
41C2×D4×C139—28no—
42C2×D4×C139—28no—
43C2×D4×C139—22no—
44C2×D4×C139—28no—
45C2×D4×C139—22no—
46C2×D4×C139—22no—
47C2×D4×C139C2×C412yes—
48C2×D4×C139D412yes—
49C2×D4×C139D412yes—
50C22×C4×C139—22no—
51C22×C4×C139—22no—
52C22×C4×C139C2311yescharacteristic
53C2×Dic139—22no—
54C2×Dic139—41no—
55C2×Dic139—41no—
56C2×Dic139—42no—
57C2×Dic139—41no—
58C139⋊C8—42no—
59C23×C139—28no—
60C23×C139—22no—
61C23×C139—22no—
62C23×C139—22no—
63C23×C139—42no—
64C23×C139—42no—
65C23×C139—82no—
66C23×C139—82no—
67C23×C139C22⋊C414yes—
68C23×C139C2×D411yescharacteristic
69Q8×C139—22no—
70Q8×C139C22⋊C414yescomplemented
71D4×C139—216no—
72D4×C139—24no—
73D4×C139—24no—
74D4×C139—24no—
75D4×C139—24no—
76D4×C139—22no—
77D4×C139C22⋊C414yescomplemented
78C2×C4×C139—28no—
79C2×C4×C139—22no—
80C2×C4×C139—24no—
81C2×C4×C139—24no—
82C2×C4×C139C22×C411yescharacteristic
83C2×C4×C139C2×D412yes—
84C2×C4×C139C2×D412yes—
85C2×C4×C139C2×D411yescharacteristic
86C22×C139—21no—
87C22×C139—21no—
88C22×C139—24no—
89C22×C139—24no—
90C22×C139—44no—
91C22×C139—42no—
92C22×C139—42no—
93C22×C139—42no—
94C22×C139—42no—
95C22×C139—82no—
96C22×C139—88no—
97C22×C139C2×C22⋊C411yescharacteristic
98C22×C139#12011yescharacteristic
99C22×C139#12011yescharacteristic
100C4×C139—24no—
101C4×C139C2×C22⋊C412yes—
102C4×C139#12012yes—
103Dic139—42no—
104Dic139—81no—
105C2×C139—21no—
106C2×C139—21no—
107C2×C139—44no—
108C2×C139—44no—
109C2×C139#37911yescharacteristic
110C2×C139#45311yescharacteristic
111C2×C139#45511yescharacteristic
112C139#179211yescharacteristic, complemented, Sylow
113#1792—1391nomaximal, Sylow
114C2×H(5,𝔽2)D13911yescharacteristic
115C2×#104—1392no—
116#417—1392no—
117C2×#100—1391no—
118#390—1391no—
119H(5,𝔽2)—24no—
120H(5,𝔽2)—22no—
121H(5,𝔽2)Dic13914yescomplemented
122C2×#41—22no—
123C2×#41C2×D13912yes—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4C2×D13911yescharacteristic
127C2×M16—1392no—
128C2×C4⋊C4—1391no—
129C2×C22⋊C4—2782no—
130C2×C42—1391no—
131#104—2784no—
132#103—2782no—
133#102—2782no—
134#100—2782no—
135C24—22no—
136C24C22⋊D13912yes—
137#41—28no—
138#41—28no—
139#41—22no—
140#41—22no—
141#41C2×Dic13914yes—
142#41C22⋊D13914yes—
143C2×Q8C22⋊D13912yes—
144C2×D4—24no—
145C2×D4—28no—
146C2×D4—28no—
147C2×D4—28no—
148C2×D4—22no—
149C2×D4—28no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4C2×Dic13912yes—
153C2×D4C22⋊D13912yes—
154C2×D4C22⋊D13912yes—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—2781no—
158C22×C4—2781no—
159C22×C4C22×D13911yescharacteristic
160M16—2782no—
161M16—2782no—
162C2×C8—2782no—
163C4⋊C4—2782no—
164C22⋊C4—5564no—
165C42—2782no—
166C23—28no—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—42no—
171C23—42no—
172C23—82no—
173C23—82no—
174C23C22⋊Dic13914yes—
175C23C2×C22⋊D13911yescharacteristic
176Q8—22no—
177Q8C22⋊Dic13914yescomplemented
178D4—216no—
179D4—24no—
180D4—24no—
181D4—24no—
182D4—24no—
183D4—22no—
184D4C22⋊Dic13914yescomplemented
185C2×C4—28no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—2782no—
190C2×C4—5561no—
191C2×C4—5561no—
192C2×C4—5562no—
193C2×C4—5561no—
194C2×C4C22×Dic13911yescharacteristic
195C2×C4C2×C22⋊D13912yes—
196C2×C4C2×C22⋊D13912yes—
197C2×C4C2×C22⋊D13911yescharacteristic
198C8—5562no—
199C22—21no—
200C22—21no—
201C22—24no—
202C22—24no—
203C22—44no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—82no—
209C22—88no—
210C22C2×C22⋊Dic13911yescharacteristic
211C22#1798481111yescharacteristic
212C22#1798481111yescharacteristic
213C4—24no—
214C4—5562no—
215C4—11121no—
216C4C2×C22⋊Dic13912yes—
217C4#1798481112yes—
218C2—21no—
219C2—21no—
220C2—44no—
221C2—44no—
222C2#2258080911yescharacteristic
223C2#2258112111yescharacteristic
224C2#2258113111yescharacteristic
subgroupG/H
centerC22#17984811
commutatorC2×C4×C139C22×C4
FrattiniC22×C4C22×D139
Fitting, prime radicalC2×H(5,𝔽2)×C139C2
socleC22×C139#120
cosocleC22×C4×C139C23
hypercenterC2×H(5,𝔽2)D139
hypocenterC139#1792
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#17984769C2semidirect
2. #22580954C2semidirect
3. C2×#17984759C2semidirect
4. #22580856C2semidirect
6. C2×#14555769C22semidirect
7. C2×C4⋊Dic139C22semidirect
9. C2×C4×Dic139C22semidirect
16. H(5,𝔽2)×C139C4semidirect
70. Q8×C139C22⋊C4semidirect
77. D4×C139C22⋊C4semidirect
112. C139#1792semidirect
121. H(5,𝔽2)Dic139semidirect
177. Q8C22⋊Dic139semidirect
184. D4C22⋊Dic139semidirect

Maximal subgroups

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

subgroupindex
C2×#179847692
#225809542
C2×#179847592
#225808562
C2×H(5,𝔽2)×C1392
#1792139