Finite Group Databasebeta

← #31805675

Subgroups — #31805675

Order 17792.

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

normal subgroups only

This group has 200 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#22581677C211yesmaximal, characteristic
2#22581349C211yesmaximal, characteristic, complemented
3#22581234C211yesmaximal, characteristic
4#22581164C211yesmaximal, characteristic, complemented
5#22580683C211yesmaximal, characteristic, complemented
6#22580680C211yesmaximal, characteristic
7#332×C139C211yesmaximal, characteristic, complemented
8C2×Dic556—21no—
9#17984842C2211yescharacteristic
10#17984837—22no—
11#17984831—21no—
12#17984822—21no—
13Q8×Dic139—21no—
14C4×Dic278C2211yescharacteristic
15C4×D556C2211yescharacteristic
16#17984800—21no—
17#17984776—21no—
18#17984773—22no—
19#17984767—21no—
20#17984764—21no—
21C4×C139⋊C8C2211yescharacteristic
22C4⋊Q8×C139C2211yescharacteristic
23C4⋊C8×C139C2211yescharacteristic
24C4⋊C8×C139C2211yescharacteristic
25C2×Dic278—21no—
26C2×Dic278D411yescharacteristic
27C2×D556D411yescharacteristic
28C2×C4×D139—21no—
29Dic556—42no—
30C2×C139⋊C8—21no—
31C4⋊Dic139—22no—
32C4⋊Dic139—21no—
33C4⋊Dic139—21no—
34C4⋊Dic139D411yescharacteristic
35C4⋊Dic139D411yescharacteristic
36Dic139⋊C4—21no—
37(C2×D139)⋊C4—22no—
38(C2×D139)⋊C4—22no—
39(C2×D139)⋊C4—21no—
40C4×Dic139—22no—
41C4×Dic139—21no—
42C2×Q8×C139—21no—
43C2×C8×C139—21no—
44C2×C8×C139—21no—
45C4⋊C4×C139—21no—
46C4⋊C4×C139D412yes—
47C42×C139C2311yescharacteristic
48C22×D139—21no—
49Dic278—21no—
50Dic278—21no—
51Dic278—41no—
52Dic278—41no—
53D556—21no—
54D556—21no—
55C2×Dic139—21no—
56C2×Dic139—21no—
57C2×Dic139—21no—
58C2×Dic139—41no—
59C4×D139—41no—
60C4×D139—41no—
61C139⋊C8—22no—
62Q8×C139—41no—
63Q8×C139—41no—
64C2×C4×C139—22no—
65C2×C4×C139C2×D411yescharacteristic
66C2×C4×C139C2×D411yescharacteristic
67C2×C4×C139C2×D411yescharacteristic
68C8×C139—41no—
69C8×C139—41no—
70C22×C139#12011yescharacteristic
71C2×D139—41no—
72C2×D139—41no—
73C2×D139—41no—
74C4×C139—21no—
75C4×C139—42no—
76C4×C139#13511yescharacteristic
77C4×C139#13511yescharacteristic
78C4×C139AGL(1,ℤ/8)11yescharacteristic
79C4×C139#13711yescharacteristic
80Dic139—41no—
81Dic139—41no—
82Dic139—41no—
83Dic139—81no—
84C2×C139#44911yescharacteristic
85C2×C139#45211yescharacteristic
86C2×C139#45611yescharacteristic
87D139—81no—
88C139#165211yescharacteristic, complemented, Sylow
89#1652—1391nomaximal, Sylow
90#556—1391no—
91#498—1391no—
92#474—1391no—
93#462—1391no—
94#332D13911yescharacteristic, complemented
95Q8⋊C8—1391no—
96D4⋊C8—1391no—
97C2×Q16—2781no—
98C4⋊Q8—1391no—
99C4⋊Q8C2×D13911yescharacteristic
100#126—2782no—
101#124—2781no—
102C22⋊Q8—2781no—
103C4×Q8—2781no—
104C4×Q8—1391no—
105C4×D4—1391no—
106#117—2781no—
107#107—2781no—
108#106—2782no—
109C4⋊C8C2×D13911yescharacteristic
110C4⋊C8C2×D13911yescharacteristic
111#103—2781no—
112#102—2781no—
113C4×C8—1391no—
114C2×Q8—21no—
115C2×Q8—2781no—
116C2×Q8—1391no—
117C2×D4—1391no—
118C22×C4—2781no—
119Q16—5562no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—2781no—
123C4⋊C4—21no—
124C4⋊C4—2782no—
125C4⋊C4—2781no—
126C4⋊C4—2781no—
127C4⋊C4—2781no—
128C4⋊C4—1391no—
129C4⋊C4—1391no—
130C4⋊C4D55612yes—
131C22⋊C4—2782no—
132C22⋊C4—2782no—
133C22⋊C4—2781no—
134C42—2782no—
135C42—2781no—
136C42C22×D13911yescharacteristic
137C23—2781no—
138Q8—41no—
139Q8—41no—
140Q8—2781no—
141Q8—2781no—
142Q8—5561no—
143Q8—5561no—
144D4—2781no—
145D4—2781no—
146C2×C4—22no—
147C2×C4—2781no—
148C2×C4—2781no—
149C2×C4—2781no—
150C2×C4—5561no—
151C2×C4—5561no—
152C2×C4—5561no—
153C2×C4D4×D13911yescharacteristic
154C2×C4D4×D13911yescharacteristic
155C2×C4C2×D55611yescharacteristic
156C8—41no—
157C8—41no—
158C8—2782no—
159C22—5561no—
160C22—5561no—
161C22—5561no—
162C22#1798481211yescharacteristic
163C4—21no—
164C4—42no—
165C4—5561no—
166C4—5561no—
167C4—5561no—
168C4—11121no—
169C4#1798486111yescharacteristic
170C4#1798486511yescharacteristic
171C4#1798486911yescharacteristic
172C4#1798487111yescharacteristic
173C2—11121no—
174C2#2258110111yescharacteristic
175C2#2258111611yescharacteristic
176C2#2258113811yescharacteristic
subgroupG/H
centerC22#17984812
commutator, cosocleC42×C139C23
FrattiniC42C22×D139
Fitting#332×C139C2
socleC22×C139#120
prime radicalC2×D556D4
hypercenter#332D139
hypocenterC139#1652
LCSG ▷ C42×C139 ▷ C22×C139 ▷ C139
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #22581349C2semidirect
4. #22581164C2semidirect
5. #22580683C2semidirect
7. #332×C139C2semidirect
88. C139#1652semidirect
94. #332D139semidirect

Maximal subgroups

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

subgroupindex
#225816772
#225813492
#225812342
#225811642
#225806832
#225806802
#332×C1392
#1652139