Finite Group Databasebeta

← #31805426

Subgroups — #31805426

Order 17792.

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

normal subgroups only

This group has 288 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#22581515C211yesmaximal, characteristic
2#22581272C212yesmaximal, complemented
3#22580889C211yesmaximal, characteristic, complemented
4#22580688C212yesmaximal, complemented
5#323×C139C211yesmaximal, characteristic, complemented
6C22×C22⋊D139—21no—
7#17984826—22no—
8#17984816—21no—
9#17984816C2212yes—
10#17984812—22no—
11C2×C4⋊Dic139C2211yescharacteristic, complemented
12C2×Dic139⋊C4—21no—
13C2×C22⋊Dic139—21no—
14#17984774—22no—
15#17984764—22no—
16C139⋊#95C2211yescharacteristic, complemented
17C2×C22⋊C4×C139C2211yescharacteristic
18C22⋊C8×C139C2212yescomplemented
19C23×D139—21no—
20C2×D556—21no—
21C2×D556D412yes—
22C2×C22⋊D139—22no—
23C2×C22⋊D139—22no—
24C2×C22⋊D139—42no—
25C2×C22⋊D139—42no—
26C22×Dic139—21no—
27C22×Dic139—21no—
28C22×Dic139—21no—
29C4⋊Dic139—21no—
30C4⋊Dic139—22no—
31C4⋊Dic139D412yes—
32Dic139⋊C4—42no—
33C22⋊Dic139—42no—
34(C2×D139)⋊C4—22no—
35(C2×D139)⋊C4—22no—
36C24×C139D411yescharacteristic
37C22×C4×C139D411yescharacteristic
38C22×C4×C139C2311yescharacteristic
39C2×C8×C139—22no—
40C22⋊C4×C139—22no—
41C22×D139—22no—
42C22×D139—42no—
43C22×D139—81no—
44C22×D139—81no—
45D556—42no—
46D556—81no—
47C22⋊D139—44no—
48C22⋊D139—42no—
49C22⋊D139—42no—
50C2×Dic139—22no—
51C2×Dic139—22no—
52C2×Dic139—41no—
53C2×Dic139—42no—
54C2×Dic139—41no—
55C2×Dic139—41no—
56C2×Dic139—41no—
57C2×Dic139—41no—
58C23×C139—21no—
59C23×C139—21no—
60C23×C139—41no—
61C23×C139—41no—
62C23×C139SD1612yes—
63C23×C139C2×D411yescharacteristic
64C2×C4×C139—21no—
65C2×C4×C139—41no—
66C2×C4×C139—41no—
67C2×C4×C139C2×D412yes—
68C8×C139—42no—
69C22×C139—21no—
70C22×C139—21no—
71C22×C139—22no—
72C22×C139—42no—
73C22×C139—41no—
74C22×C139—41no—
75C22×C139—81no—
76C22×C139#12011yescharacteristic
77C22×C139C2×SD1611yescharacteristic
78C22×C139AGL(1,ℤ/8)11yescharacteristic
79C2×D139—42no—
80C2×D139—42no—
81C2×D139—42no—
82C2×D139—84no—
83C4×C139—22no—
84C4×C139—41no—
85Dic139—42no—
86Dic139—42no—
87Dic139—81no—
88C2×C139—21no—
89C2×C139—21no—
90C2×C139—42no—
91C2×C139#45011yescharacteristic
92C2×C139#45311yescharacteristic
93C2×C139#45811yescharacteristic
94D139—82no—
95C139#161311yescharacteristic, complemented, Sylow
96#1613—1391nomaximal, Sylow
97#534—1391no—
98#481—1392no—
99#399—1391no—
100#327—1392no—
101#323D13911yescharacteristic, complemented
102C22×D4—2781no—
103#123—2782no—
104#121—2781no—
105#121—1392no—
106#120—2782no—
107C2×C4⋊C4—2781no—
108C2×C4⋊C4—1391no—
109C2×C22⋊C4—2781no—
110C2×C22⋊C4C2×D13911yescharacteristic
111#106—2782no—
112#102—2782no—
113C22⋊C8C2×D13912yescomplemented
114#95—1391no—
115C24—2781no—
116C24D55611yescharacteristic
117C2×D4—2781no—
118C2×D4—2782no—
119C2×D4—2782no—
120C2×D4—5562no—
121C2×D4—5562no—
122C2×D4—1392no—
123C22×C4—2781no—
124C22×C4—2781no—
125C22×C4—2781no—
126C22×C4D55611yescharacteristic
127C22×C4C22×D13911yescharacteristic
128C2×C8—22no—
129C4⋊C4—2781no—
130C4⋊C4—2782no—
131C4⋊C4—5562no—
132C4⋊C4—1392no—
133C22⋊C4—22no—
134C22⋊C4—2782no—
135C22⋊C4—2782no—
136C22⋊C4—5562no—
137C23—21no—
138C23—21no—
139C23—41no—
140C23—41no—
141C23—2782no—
142C23—5562no—
143C23—11121no—
144C23—11121no—
145C23#1455577412yes—
146C23C2×D55611yescharacteristic
147D4—5562no—
148D4—5564no—
149D4—5562no—
150D4—5562no—
151D4—11121no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—2782no—
156C2×C4—2782no—
157C2×C4—5562no—
158C2×C4—5561no—
159C2×C4—5561no—
160C2×C4—5561no—
161C2×C4—5561no—
162C2×C4—5561no—
163C2×C4D4×D13912yes—
164C8—42no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—81no—
172C22—5562no—
173C22—5562no—
174C22—5562no—
175C22—11124no—
176C22#1798481211yescharacteristic
177C22C2×#1455577411yescharacteristic
178C22#1798487011yescharacteristic
179C4—22no—
180C4—41no—
181C4—5562no—
182C4—5562no—
183C4—11121no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—11122no—
188C2#2258110611yescharacteristic
189C2#2258112311yescharacteristic
190C2#2258114411yescharacteristic
subgroupG/H
centerC22#17984812
commutator, cosocleC22×C4×C139C23
FrattiniC22×C4C22×D139
Fitting#323×C139C2
socleC22×C139#120
prime radical#22581515C2
hypercenter#323D139
hypocenterC139#1613
LCSG ▷ C22×C4×C139 ▷ C22×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #22581272C2semidirect
3. #22580889C2semidirect
4. #22580688C2semidirect
5. #323×C139C2semidirect
11. C2×C4⋊Dic139C22semidirect
16. C139⋊#95C22semidirect
18. C22⋊C8×C139C22semidirect
95. C139#1613semidirect
101. #323D139semidirect
113. C22⋊C8C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225815152
#225812722
#225808892
#225806882
#323×C1392
#1613139