Finite Group Databasebeta

← #31805409

Subgroups — #31805409

Order 17792.

254 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 53 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#22581534C211yesmaximal, characteristic
2#22581267C212yesmaximal, complemented
3#22580883C211yesmaximal, characteristic, complemented
4#22580688C212yesmaximal, complemented
5#324×C139C211yesmaximal, characteristic, complemented
6C2×#14555785—21no—
7#17984822—22no—
8#17984817—22no—
9#17984816—21no—
10#17984816C2212yes—
11C2×C4⋊Dic139C2211yescharacteristic
12C2×Dic139⋊C4—21no—
13#17984776—22no—
14#17984764—22no—
15C139⋊#95—21no—
16C139⋊#95C2211yescharacteristic
17C2×C4⋊C4×C139C2211yescharacteristic
18C22⋊C8×C139C2212yes—
19#14555785—42no—
20#14555785—42no—
21C2×Dic278—21no—
22C2×D556—21no—
23C2×D556—21no—
24C2×D556D412yes—
25C2×C22⋊D139—22no—
26C22×Dic139—21no—
27C22×Dic139—21no—
28C2×C4×D139—22no—
29C4⋊Dic139—21no—
30C4⋊Dic139—22no—
31C4⋊Dic139D412yes—
32Dic139⋊C4—42no—
33(C2×D139)⋊C4—22no—
34(C2×D139)⋊C4—22no—
35C22×C4×C139D411yescharacteristic
36C22×C4×C139D411yescharacteristic
37C22×C4×C139C2311yescharacteristic
38C2×C8×C139—22no—
39C4⋊C4×C139—22no—
40C22×D139—22no—
41Dic278—42no—
42D556—42no—
43D556—42no—
44D556—81no—
45C22⋊D139—42no—
46C22⋊D139—42no—
47C2×Dic139—22no—
48C2×Dic139—22no—
49C2×Dic139—41no—
50C2×Dic139—41no—
51C2×Dic139—41no—
52C2×Dic139—41no—
53C2×Dic139—41no—
54C4×D139—44no—
55C23×C139C2×D411yescharacteristic
56C2×C4×C139—21no—
57C2×C4×C139—21no—
58C2×C4×C139—21no—
59C2×C4×C139—41no—
60C2×C4×C139—41no—
61C2×C4×C139D812yes—
62C2×C4×C139C2×D412yes—
63C8×C139—42no—
64C22×C139—21no—
65C22×C139—21no—
66C22×C139#12011yescharacteristic
67C22×C139C2×D811yescharacteristic
68C22×C139AGL(1,ℤ/8)11yescharacteristic
69C2×D139—42no—
70C2×D139—42no—
71C2×D139—42no—
72C4×C139—22no—
73C4×C139—22no—
74C4×C139—41no—
75Dic139—42no—
76Dic139—42no—
77Dic139—81no—
78C2×C139—21no—
79C2×C139—21no—
80C2×C139#44711yescharacteristic
81C2×C139#45611yescharacteristic
82C2×C139#45811yescharacteristic
83D139—82no—
84C139#161011yescharacteristic, complemented, Sylow
85#1610—1391nomaximal, Sylow
86#537—1391no—
87#480—1392no—
88#398—1391no—
89#327—1392no—
90#324D13911yescharacteristic, complemented
91C2×#41—2781no—
92C22⋊Q8—2782no—
93#121—2782no—
94#121—2781no—
95#121—1392no—
96C2×C4⋊C4—2781no—
97C2×C4⋊C4—1391no—
98C2×C4⋊C4C2×D13911yescharacteristic
99#107—2782no—
100#102—2782no—
101C22⋊C8C2×D13912yes—
102#95—2781no—
103#95—1391no—
104#41—5562no—
105#41—5562no—
106C2×Q8—2781no—
107C2×D4—2781no—
108C2×D4—2781no—
109C2×D4—2782no—
110C2×D4—1392no—
111C22×C4—2782no—
112C22×C4—2781no—
113C22×C4—2781no—
114C22×C4D55611yescharacteristic
115C22×C4D55611yescharacteristic
116C22×C4C22×D13911yescharacteristic
117C2×C8—22no—
118C4⋊C4—22no—
119C4⋊C4—2781no—
120C4⋊C4—2782no—
121C4⋊C4—5562no—
122C4⋊C4—1392no—
123C22⋊C4—2782no—
124C22⋊C4—2782no—
125C23—2782no—
126C23C2×D55611yescharacteristic
127Q8—5562no—
128D4—5562no—
129D4—5562no—
130D4—5562no—
131D4—5562no—
132D4—11121no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—2782no—
139C2×C4—2782no—
140C2×C4—5564no—
141C2×C4—5561no—
142C2×C4—5561no—
143C2×C4—5561no—
144C2×C4—5561no—
145C2×C4—5561no—
146C2×C4D111212yes—
147C2×C4D4×D13912yes—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—5562no—
152C22—5562no—
153C22—5562no—
154C22#1798481211yescharacteristic
155C22C2×D111211yescharacteristic
156C22#1798487011yescharacteristic
157C4—22no—
158C4—22no—
159C4—41no—
160C4—5562no—
161C4—5562no—
162C4—11121no—
163C2—21no—
164C2—21no—
165C2—11122no—
166C2#2258108911yescharacteristic
167C2#2258113811yescharacteristic
168C2#2258114411yescharacteristic
subgroupG/H
centerC22#17984812
commutator, cosocleC22×C4×C139C23
FrattiniC22×C4C22×D139
Fitting#324×C139C2
socleC22×C139#120
prime radical#22581534C2
hypercenter#324D139
hypocenterC139#1610
LCSG ▷ C22×C4×C139 ▷ C22×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #22581267C2semidirect
3. #22580883C2semidirect
4. #22580688C2semidirect
5. #324×C139C2semidirect
84. C139#1610semidirect
90. #324D139semidirect

Maximal subgroups

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

subgroupindex
#225815342
#225812672
#225808832
#225806882
#324×C1392
#1610139