Finite Group Databasebeta

← #31807306

Subgroups — #31807306

Order 17792.

296 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 83 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×#14555772C211yesmaximal, characteristic
2C2×#17984777C211yesmaximal, characteristic, complemented
3#22580957C211yesmaximal, characteristic
4C2×#17984761C211yesmaximal, characteristic, complemented
5C2×#17984758C211yesmaximal, characteristic
6#22580909C211yesmaximal, characteristic, complemented
7#419×C139C211yesmaximal, characteristic, complemented
8C22×D556C2211yescharacteristic
9C2×#14555772—22no—
10C2×#14555772—21no—
11C2×#14555772—21no—
12C2×#14555772C414yescomplemented
13C2×#14555769C2211yescharacteristic
14C2×#14555769C2211yescharacteristic
15C22×C139⋊C8C2211yescharacteristic
16#17984777—22no—
17#17984762—22no—
18#17984761—22no—
19#17984758—22no—
20#17984753—21no—
21C4×C139⋊C8—21no—
22C22×Q8×C139C2211yescharacteristic
23C2×M16×C139C2211yescharacteristic
24#117×C139C2211yescharacteristic
25#103×C139—22no—
26C23×D139—21no—
27C2×D556—21no—
28C2×D556—21no—
29C2×D556—21no—
30C2×D556—21no—
31C2×D556—21no—
32C2×D556C2×C412yes—
33#14555772—24no—
34#14555772—24no—
35#14555769—21no—
36#14555769—21no—
37#14555769—21no—
38#14555769D412yes—
39C2×C139⋊C8—21no—
40C2×C139⋊C8—21no—
41C2×C139⋊C8C2×C412yes—
42C2×C139⋊C8D412yes—
43C2×C139⋊C8D411yescharacteristic
44C2×C139⋊C8D411yescharacteristic
45C2×Q8×C139—21no—
46C2×Q8×C139—21no—
47C2×Q8×C139—21no—
48C2×Q8×C139—21no—
49C2×Q8×C139—21no—
50C2×Q8×C139C2×C412yes—
51C22×C4×C139—21no—
52C22×C4×C139C2311yescharacteristic
53M16×C139—21no—
54M16×C139—21no—
55C2×C8×C139—21no—
56C4⋊C4×C139D412yes—
57C22⋊C4×C139—21no—
58C42×C139—21no—
59C22×D139—22no—
60C22×D139—41no—
61C22×D139—41no—
62C22×D139—82no—
63D556—22no—
64D556—22no—
65D556—41no—
66D556—41no—
67C139⋊C8—22no—
68C139⋊C8—41no—
69C139⋊C8C22⋊C412yes—
70C139⋊C8C22⋊C412yes—
71C23×C139#4111yescharacteristic
72Q8×C139—22no—
73Q8×C139—22no—
74Q8×C139—41no—
75Q8×C139—41no—
76C2×C4×C139—22no—
77C2×C4×C139—22no—
78C2×C4×C139—41no—
79C2×C4×C139—41no—
80C2×C4×C139C22×C411yescharacteristic
81C2×C4×C139C2×D411yescharacteristic
82C2×C4×C139C2×D411yescharacteristic
83C2×C4×C139C2×D411yescharacteristic
84C2×C4×C139C2×D411yescharacteristic
85C2×C4×C139#4111yescharacteristic
86C8×C139—41no—
87C22×C139—22no—
88C22×C139C4×D411yescharacteristic
89C22×C139#12111yescharacteristic
90C22×C139#12411yescharacteristic
91C2×D139—42no—
92C2×D139—82no—
93C2×D139—82no—
94C2×D139—82no—
95C4×C139—42no—
96C4×C139—42no—
97C4×C139C2×C22⋊C411yescharacteristic
98C4×C139C4×D411yescharacteristic
99C4×C139#12111yescharacteristic
100C4×C139#12711yescharacteristic
101C2×C139—21no—
102C2×C139—21no—
103C2×C139#39011yescharacteristic
104C2×C139#47112yes—
105D139—82no—
106C139#197711yescharacteristic, complemented, Sylow
107#1977—1391nomaximal, Sylow
108C22×SD16—1391no—
109C2×#108—1391no—
110#419D13911yescharacteristic, complemented
111#418—1391no—
112C2×#101—1391no—
113C2×#100—1391no—
114#405—1391no—
115C22×Q8C2×D13911yescharacteristic
116C22×D4—1391no—
117C2×SD16—2781no—
118C2×SD16—2781no—
119C2×SD16—2782no—
120C2×SD16—1394no—
121C2×M16—1391no—
122C2×M16—1391no—
123C2×M16C2×D13911yescharacteristic
124C22×C8—1391no—
125#117C2×D13911yescharacteristic
126#108—2782no—
127#103—22no—
128#102—2782no—
129#101—2782no—
130#100—2782no—
131#97—2781no—
132C4×C8—2781no—
133C24—2781no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8C4×D13912yes—
140C2×D4—2781no—
141C2×D4—2781no—
142C2×D4—2781no—
143C2×D4—2781no—
144C2×D4—2781no—
145C2×D4—1392no—
146C22×C4—21no—
147C22×C4C22×D13911yescharacteristic
148SD16—2784no—
149SD16—2784no—
150M16—21no—
151M16—21no—
152M16—2781no—
153M16—2781no—
154M16—2781no—
155M16—1392no—
156C2×C8—21no—
157C2×C8—2781no—
158C2×C8—2781no—
159C2×C8—1392no—
160C2×C8—1391no—
161C2×C8—1391no—
162C2×C8—1392no—
163C4⋊C4C22⋊D13912yes—
164C22⋊C4—21no—
165C42—21no—
166C23—2782no—
167C23—5561no—
168C23—5561no—
169C23—11122no—
170C23#1455578411yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—41no—
174Q8—41no—
175D4—2782no—
176D4—2782no—
177D4—5561no—
178D4—5561no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4C2×C4×D13911yescharacteristic
184C2×C4D4×D13911yescharacteristic
185C2×C4D4×D13911yescharacteristic
186C2×C4D4×D13911yescharacteristic
187C2×C4C2×C22⋊D13911yescharacteristic
188C2×C4#1455578511yescharacteristic
189C8—41no—
190C8—2782no—
191C8—5561no—
192C8—1392no—
193C8—1392no—
194C22—22no—
195C22—5562no—
196C22—11122no—
197C22—11122no—
198C22—11122no—
199C22(C22⋊D139)⋊C411yescharacteristic
200C22#1798481411yescharacteristic
201C22#1798482811yescharacteristic
202C4—42no—
203C4—42no—
204C4C22⋊C4×D13911yescharacteristic
205C4C4×C22⋊D13911yescharacteristic
206C4#1798481511yescharacteristic
207C4#1798483811yescharacteristic
208C2—21no—
209C2—21no—
210C2—11122no—
211C2#2258085311yescharacteristic
212C2#2258121812yes—
subgroupG/H
centerC22#17984814
commutatorC2×C4×C139C22×C4
FrattiniC22×C4C22×D139
Fitting#419×C139C2
socleC22×C139#121
prime radicalC22×D556C22
cosocleC22×C4×C139C23
hypercenter#419D139
hypocenterC139#1977
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C2×#17984777C2semidirect
4. C2×#17984761C2semidirect
6. #22580909C2semidirect
7. #419×C139C2semidirect
12. C2×#14555772C4semidirect
106. C139#1977semidirect
110. #419D139semidirect

Maximal subgroups

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

subgroupindex
C22×#145557722
C2×#179847772
#225809572
C2×#179847612
C2×#179847582
#225809092
#419×C1392
#1977139