Finite Group Databasebeta

← #31808435

Subgroups — #31808435

Order 17792.

224 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 41 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#22581683C211yesmaximal, characteristic, complemented
2#22581138C212yesmaximal
3#22581135C211yesmaximal, characteristic
4#22580751C212yesmaximal, complemented
5#356×C139C211yesmaximal, characteristic, complemented
6#17984890C2211yescharacteristic
7#17984875—22no—
8#17984873—21no—
9#17984841—22no—
10#17984840—21no—
11#17984840C2212yes—
12C4×Dic278—21no—
13#17984807—22no—
14#17984770—21no—
15#17984768—22no—
16#17984761C2211yescharacteristic
17C4⋊Q8×C139C2211yescharacteristic
18#101×C139C2212yes—
19#14555786—41no—
20#14555785—22no—
21#14555785—21no—
22C2×Dic278—21no—
23C2×Dic278—21no—
24C2×Dic278D412yes—
25Q8×D139—41no—
26Dic556—42no—
27#14555775—41no—
28#14555774—42no—
29#14555772—41no—
30#14555769—21no—
31C4⋊Dic139—21no—
32Dic139⋊C4—22no—
33Dic139⋊C4—22no—
34Dic139⋊C4—22no—
35C4×Dic139—21no—
36C4×Dic139—22no—
37C4×Dic139D412yes—
38C2×Q8×C139D411yescharacteristic
39C2×Q8×C139C2311yescharacteristic
40M16×C139—22no—
41C4⋊C4×C139—22no—
42C42×C139D411yescharacteristic
43Dic278—24no—
44Dic278—21no—
45Dic278—22no—
46Dic278—42no—
47Dic278—81no—
48D556—21no—
49D556—22no—
50C22⋊D139—41no—
51C2×Dic139—22no—
52C2×Dic139—22no—
53C4×D139—42no—
54C4×D139—41no—
55C139⋊C8—41no—
56Q8×C139—41no—
57Q8×C139—41no—
58C2×C4×C139—21no—
59C2×C4×C139—21no—
60C2×C4×C139C2×D412yes—
61C2×C4×C139C2×D411yescharacteristic
62C8×C139—42no—
63C22×C139#12011yescharacteristic
64C2×D139—41no—
65C4×C139—22no—
66C4×C139—22no—
67C4×C139—21no—
68C4×C139—41no—
69Dic139—24no—
70Dic139—41no—
71Dic139—42no—
72C2×C139—21no—
73C2×C139B(4,𝔽2)11yescharacteristic
74D139—81no—
75C139#220511yescharacteristic, complemented, Sylow
76#2205—1391nomaximal, Sylow
77Q82—1391no—
78#456—1391no—
79#456—1392no—
80#356—1392no—
81#356D13911yescharacteristic, complemented
822−1+4—1391no—
83#137—2782no—
84#137—2781no—
85C4⋊Q8—2782no—
86C4⋊Q8—2781no—
87C4⋊Q8—1392no—
88C4⋊Q8C2×D13911yescharacteristic
89C4×Q8—2782no—
90C4×Q8—2781no—
91#104—2782no—
92#104—2781no—
93#101—1391no—
94#101C2×D13912yes—
95#41—2782no—
96#41—2781no—
97#41—5561no—
98C2×Q8—2781no—
99C2×Q8—2781no—
100C2×Q8—5561no—
101C2×Q8—1392no—
102C2×Q8D55611yescharacteristic
103C2×Q8C22×D13911yescharacteristic
104Q16—5562no—
105Q16—5561no—
106SD16—5562no—
107SD16—5561no—
108M16—22no—
109M16—2781no—
110C4⋊C4—22no—
111C4⋊C4—2782no—
112C4⋊C4—2781no—
113C4⋊C4—2782no—
114C4⋊C4—2782no—
115C42—2782no—
116C42—2781no—
117C42—1392no—
118C42D55611yescharacteristic
119Q8—41no—
120Q8—41no—
121Q8—2784no—
122Q8—2782no—
123Q8—2781no—
124Q8—5562no—
125Q8—11121no—
126D4—2782no—
127D4—2781no—
128D4—5561no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—2782no—
132C2×C4—2782no—
133C2×C4—5562no—
134C2×C4—5561no—
135C2×C4D4×D13912yes—
136C2×C4C2×D55611yescharacteristic
137C8—42no—
138C8—5561no—
139C22—5561no—
140C22#1798481211yescharacteristic
141C4—22no—
142C4—22no—
143C4—21no—
144C4—41no—
145C4—2784no—
146C4—5562no—
147C4—5561no—
148C2—21no—
149C2—11121no—
150C2#2258114111yescharacteristic
subgroupG/H
centerC2#22581141
commutator, cosocleC2×Q8×C139C23
FrattiniC2×Q8C22×D139
Fitting#356×C139C2
socleC2×C139B(4,𝔽2)
prime radical#17984890C22
hypercenter#356D139
hypocenterC139#2205
LCSG ▷ C2×Q8×C139 ▷ C22×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #22581683C2semidirect
4. #22580751C2semidirect
5. #356×C139C2semidirect
75. C139#2205semidirect
81. #356D139semidirect

Maximal subgroups

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

subgroupindex
#225816832
#225811382
#225811352
#225807512
#356×C1392
#2205139