Finite Group Databasebeta

← #2203×C167 #36320563

Subgroups — #2203×C167 #36320563

Order 21376.

216 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 56 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#563×C167C211yesmaximal, characteristic, complemented
2#456×C167C212yesmaximal, complemented
3#454×C167C211yesmaximal, characteristic
4#356×C167C212yesmaximal, complemented
5#355×C167C211yesmaximal, characteristic, complemented
62−1+4×C167C2211yescharacteristic
7#137×C167—22no—
8AGL(1,ℤ/8)×C167—21no—
9C4⋊Q8×C167C2212yes—
10#126×C167—22no—
11#125×C167—21no—
12#124×C167C2211yescharacteristic
13#123×C167—21no—
14C22⋊Q8×C167—22no—
15#104×C167—21no—
16#104×C167—22no—
17#101×C167C2212yes—
18#101×C167C2211yescharacteristic
19#41×C167—21no—
20#41×C167—22no—
21#41×C167—41no—
22C2×Q8×C167—41no—
23C2×Q8×C167D412yes—
24C2×Q8×C167C2311yescharacteristic
25C2×D4×C167D411yescharacteristic
26C22×C4×C167—21no—
27Q16×C167—42no—
28SD16×C167—42no—
29SD16×C167—41no—
30D8×C167—41no—
31M16×C167—21no—
32M16×C167—22no—
33C4⋊C4×C167—21no—
34C4⋊C4×C167—22no—
35C4⋊C4×C167—21no—
36C4⋊C4×C167—22no—
37C4⋊C4×C167—22no—
38C22⋊C4×C167—22no—
39C22⋊C4×C167—21no—
40C22⋊C4×C167—22no—
41C42×C167D411yescharacteristic
42C42×C167D412yes—
43C23×C167—21no—
44Q8×C167—21no—
45Q8×C167—22no—
46Q8×C167—41no—
47Q8×C167—42no—
48D4×C167—21no—
49D4×C167—22no—
50D4×C167—41no—
51D4×C167—41no—
52C2×C4×C167—21no—
53C2×C4×C167—22no—
54C2×C4×C167—22no—
55C2×C4×C167—41no—
56C2×C4×C167—42no—
57C2×C4×C167—81no—
58C2×C4×C167C2×D412yes—
59C2×C4×C167C2×D411yescharacteristic
60C8×C167—41no—
61C8×C167—42no—
62C22×C167—42no—
63C22×C167—41no—
64C22×C167—41no—
65C22×C167#12011yescharacteristic
66C4×C167—21no—
67C4×C167—22no—
68C4×C167—41no—
69C4×C167—42no—
70C4×C167—42no—
71C4×C167—41no—
72C2×C167—21no—
73C2×C167—81no—
74C2×C167—81no—
75C2×C167B(4,𝔽2)11yescharacteristic
76C167#220311yescharacteristic, direct product factor, Sylow
77#2203C16711yesmaximal, characteristic, direct product factor, Sylow
78#563C2×C16711yescharacteristic, complemented
79#456C2×C16712yescomplemented
80#454C2×C16711yescharacteristic
81#356C2×C16712yescomplemented
82#355C2×C16711yescharacteristic, complemented
832−1+4C22×C16711yescharacteristic
84#137—22no—
85AGL(1,ℤ/8)—21no—
86C4⋊Q8C22×C16712yes—
87#126—22no—
88#125—21no—
89#124C22×C16711yescharacteristic
90#123—21no—
91C22⋊Q8—22no—
92#104—21no—
93#104—22no—
94#101C22×C16712yes—
95#101C22×C16711yescharacteristic
96#41—21no—
97#41—22no—
98#41—41no—
99C2×Q8—41no—
100C2×Q8D4×C16712yes—
101C2×Q8C23×C16711yescharacteristic
102C2×D4D4×C16711yescharacteristic
103C22×C4—21no—
104Q16—42no—
105SD16—42no—
106SD16—41no—
107D8—41no—
108M16—21no—
109M16—22no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42D4×C16711yescharacteristic
119C42D4×C16712yes—
120C23—21no—
121Q8—21no—
122Q8—22no—
123Q8—41no—
124Q8—42no—
125D4—21no—
126D4—22no—
127D4—41no—
128D4—41no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—41no—
133C2×C4—42no—
134C2×C4—81no—
135C2×C4C2×D4×C16712yes—
136C2×C4C2×D4×C16711yescharacteristic
137C8—41no—
138C8—42no—
139C22—42no—
140C22—41no—
141C22—41no—
142C22#120×C16711yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—41no—
149C2—21no—
150C2—81no—
151C2—81no—
152C2B(4,𝔽2)×C16711yescharacteristic
subgroupG/H
center, socleC2×C167B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C23×C167
prime radical#454×C167C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C167 ◁ C22×C167 ◁ C2×Q8×C167 ◁ G

Complements

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

NH ≅ G/Ntype
1. #563×C167C2semidirect
2. #456×C167C2semidirect
4. #356×C167C2semidirect
5. #355×C167C2semidirect
76. C167#2203direct (N × H)
77. #2203C167direct (N × H)
78. #563C2×C167semidirect
79. #456C2×C167semidirect
81. #356C2×C167semidirect
82. #355C2×C167semidirect

Maximal subgroups

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

subgroupindex
#563×C1672
#456×C1672
#454×C1672
#356×C1672
#355×C1672
#2203167