Finite Group Databasebeta

← #31805669

Subgroups — #31805669

Order 17792.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×Dic278C211yesmaximal, characteristic
2#22581351C211yesmaximal, characteristic, complemented
3#22581239C211yesmaximal, characteristic, complemented
4#22581158C211yesmaximal, characteristic
5#22580700C211yesmaximal, characteristic, complemented
6#22580685C211yesmaximal, characteristic, complemented
7D4⋊C8×C139C211yesmaximal, characteristic
8C22×Dic278—21no—
9C2×#14555773—21no—
10#17984842C2211yescharacteristic
11#17984842C2211yescharacteristic
12#17984841—21no—
13#17984821—21no—
14#17984819—22no—
15C4×Dic278C2211yescharacteristic
16D4×Dic139—21no—
17#17984774—22no—
18#17984774—21no—
19C4⋊(C139⋊C8)C2211yescharacteristic
20#17984767—21no—
21#17984765—21no—
22C4×D4×C139C2211yescharacteristic
23C4⋊C8×C139C2211yescharacteristic
24C4×C8×C139C2211yescharacteristic
25C2×Dic278—22no—
26C2×Dic278—21no—
27C2×Dic278—21no—
28C2×Dic278—42no—
29C2×Dic278—42no—
30C2×Dic278D411yescharacteristic
31C22×Dic139—22no—
32#14555773—42no—
33C2×C139⋊C8—21no—
34C4⋊Dic139—21no—
35C4⋊Dic139—21no—
36C4⋊Dic139—21no—
37C4⋊Dic139D411yescharacteristic
38C4⋊Dic139D412yes—
39Dic139⋊C4—22no—
40Dic139⋊C4—21no—
41C22⋊Dic139—22no—
42C4×Dic139—21no—
43C2×D4×C139D411yescharacteristic
44C22×C4×C139—21no—
45C2×C8×C139—21no—
46C2×C8×C139—21no—
47C4⋊C4×C139D411yescharacteristic
48C22⋊C4×C139—21no—
49C42×C139C2311yescharacteristic
50Dic278—21no—
51Dic278—44no—
52Dic278—41no—
53Dic278—41no—
54Dic278—41no—
55Dic278—41no—
56Dic278SD1612yescomplemented
57C2×Dic139—21no—
58C2×Dic139—21no—
59C2×Dic139—22no—
60C2×Dic139—44no—
61C2×Dic139—41no—
62C139⋊C8—41no—
63C23×C139—21no—
64D4×C139—21no—
65D4×C139SD1612yes—
66C2×C4×C139—21no—
67C2×C4×C139—41no—
68C2×C4×C139—41no—
69C2×C4×C139C2×D411yescharacteristic
70C2×C4×C139C2×D411yescharacteristic
71C2×C4×C139C2×D411yescharacteristic
72C8×C139—22no—
73C8×C139—41no—
74C22×C139—22no—
75C22×C139—41no—
76C22×C139—41no—
77C22×C139#12011yescharacteristic
78C4×C139—21no—
79C4×C139—41no—
80C4×C139C2×SD1611yescharacteristic
81C4×C139C2×SD1611yescharacteristic
82C4×C139AGL(1,ℤ/8)11yescharacteristic
83C4×C139#13711yescharacteristic
84Dic139—22no—
85Dic139—42no—
86Dic139—41no—
87Dic139—81no—
88C2×C139—42no—
89C2×C139#44811yescharacteristic
90C2×C139#45011yescharacteristic
91C2×C139#45611yescharacteristic
92C139#165111yescharacteristic, complemented, Sylow
93#1651—1391nomaximal, Sylow
94D4×Q8—1391no—
95#498—1391no—
96#475—1391no—
97#461—1391no—
98#332—1391no—
99Q8⋊C8—1391no—
100D4⋊C8D13911yescharacteristic
101C22×Q8—2781no—
102C2×SD16—2781no—
103C4⋊Q8—2781no—
104C4⋊Q8—1391no—
105C4⋊Q8—1391no—
106C22⋊Q8—2782no—
107C22⋊Q8—2781no—
108C4×Q8—1391no—
109C4×D4—2781no—
110C4×D4C2×D13911yescharacteristic
111#106—2781no—
112#106—2782no—
113C4⋊C8—1391no—
114C4⋊C8C2×D13911yescharacteristic
115#103—2781no—
116#103—2781no—
117C4×C8C2×D13911yescharacteristic
118C2×Q8—2782no—
119C2×Q8—2781no—
120C2×Q8—2781no—
121C2×Q8—5562no—
122C2×Q8—5562no—
123C2×Q8—1391no—
124C2×D4D55611yescharacteristic
125C22×C4—21no—
126C22×C4—2782no—
127SD16—5562no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—2781no—
131C4⋊C4—2782no—
132C4⋊C4—2781no—
133C4⋊C4—2781no—
134C4⋊C4—2781no—
135C4⋊C4—2781no—
136C4⋊C4—1391no—
137C4⋊C4—1392no—
138C4⋊C4D55611yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—2782no—
141C42—2781no—
142C42C22×D13911yescharacteristic
143C23—21no—
144Q8—2781no—
145Q8—5564no—
146Q8—5561no—
147Q8—5561no—
148Q8—5561no—
149Q8—5561no—
150Q8—1392no—
151D4—21no—
152D4#1455577412yes—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—2781no—
157C2×C4—2781no—
158C2×C4—2782no—
159C2×C4—5564no—
160C2×C4—5561no—
161C2×C4D4×D13911yescharacteristic
162C2×C4D4×D13911yescharacteristic
163C2×C4C2×D55611yescharacteristic
164C8—22no—
165C8—41no—
166C8—5561no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1798481211yescharacteristic
171C4—21no—
172C4—41no—
173C4—2782no—
174C4—5562no—
175C4—5561no—
176C4—11121no—
177C4SD16×D13911yescharacteristic
178C4C2×#1455577411yescharacteristic
179C4#1798486611yescharacteristic
180C4#1798487511yescharacteristic
181C2—42no—
182C2#2258109411yescharacteristic
183C2#2258110511yescharacteristic
184C2#2258113711yescharacteristic
subgroupG/H
centerC22#17984812
commutator, cosocleC42×C139C23
FrattiniC42C22×D139
FittingD4⋊C8×C139C2
socleC22×C139#120
prime radicalC2×D4×C139D4
hypercenterD4⋊C8D139
hypocenterC139#1651
LCSG ▷ C42×C139 ▷ C22×C139 ▷ C139
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #22581351C2semidirect
3. #22581239C2semidirect
5. #22580700C2semidirect
6. #22580685C2semidirect
56. Dic278SD16semidirect
92. C139#1651semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2782
#225813512
#225812392
#225811582
#225807002
#225806852
D4⋊C8×C1392
#1651139