Finite Group Databasebeta

← C2×#354×C167 #36320485

Subgroups — C2×#354×C167 #36320485

Order 21376.

284 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C2×#124×C167C211yesmaximal, characteristic
2C2×#99×C167C212yesmaximal
3#354×C167C214yesmaximal, direct product factor
4C22×Q8×C167C411yescharacteristic, complemented
5C22×D4×C167C2211yescharacteristic
6#124×C167—22no—
7#124×C167C412yescomplemented
8#124×C167C2212yes—
9C2×C22⋊C4×C167—22no—
10C2×C22⋊C4×C167—22no—
11C2×C42×C167C411yescharacteristic, complemented
12#99×C167—22no—
13#99×C167C2214yes—
14C24×C167D412yes—
15C2×Q8×C167—21no—
16C2×Q8×C167—41no—
17C2×Q8×C167—41no—
18C2×Q8×C167C2×C412yescomplemented
19C2×D4×C167—41no—
20C2×D4×C167—41no—
21C2×D4×C167C2×C411yescharacteristic
22C2×D4×C167D412yes—
23C2×D4×C167C2311yescharacteristic
24C22×C4×C167—21no—
25C22×C4×C167—21no—
26C22×C4×C167—42no—
27C22×C4×C167C2×C411yescharacteristic
28C22⋊C4×C167—24no—
29C22⋊C4×C167—24no—
30C22⋊C4×C167—24no—
31C22⋊C4×C167—42no—
32C42×C167—21no—
33C42×C167C2×C412yescomplemented
34C23×C167—22no—
35C23×C167—22no—
36C23×C167—44no—
37C23×C167C22⋊C412yes—
38C23×C167C22⋊C411yescharacteristic
39C23×C167C2×D412yes—
40Q8×C167—42no—
41Q8×C167—42no—
42D4×C167—41no—
43D4×C167—42no—
44D4×C167—41no—
45C2×C4×C167—21no—
46C2×C4×C167—21no—
47C2×C4×C167—22no—
48C2×C4×C167—22no—
49C2×C4×C167—42no—
50C2×C4×C167—41no—
51C2×C4×C167—41no—
52C2×C4×C167—44no—
53C2×C4×C167—84no—
54C2×C4×C167C22⋊C411yescharacteristic
55C2×C4×C167C22×C411yescharacteristic
56C22×C167—22no—
57C22×C167—22no—
58C22×C167—22no—
59C22×C167—42no—
60C22×C167—42no—
61C22×C167—44no—
62C22×C167—82no—
63C22×C167#9911yescharacteristic
64C22×C167#9911yescharacteristic
65C22×C167C2×C22⋊C411yescharacteristic
66C4×C167—21no—
67C4×C167—21no—
68C4×C167—42no—
69C4×C167—42no—
70C4×C167—84no—
71C2×C167—21no—
72C2×C167—21no—
73C2×C167—42no—
74C2×C167—42no—
75C2×C167#35412yesdirect product factor
76C2×C167C2×#9911yescharacteristic
77C167C2×#35411yescharacteristic, direct product factor, Sylow
78C2×#354C16711yesmaximal, characteristic, direct product factor, Sylow
79C2×#124C2×C16711yescharacteristic
80C2×#99C2×C16712yes—
81#354C2×C16714yesdirect product factor
82C22×Q8C4×C16711yescharacteristic, complemented
83C22×D4C22×C16711yescharacteristic
84#124—22no—
85#124C4×C16712yescomplemented
86#124C22×C16712yes—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C42C4×C16711yescharacteristic, complemented
90#99—22no—
91#99C22×C16714yes—
92C24D4×C16712yes—
93C2×Q8—21no—
94C2×Q8—41no—
95C2×Q8—41no—
96C2×Q8C2×C4×C16712yescomplemented
97C2×D4—41no—
98C2×D4—41no—
99C2×D4C2×C4×C16711yescharacteristic
100C2×D4D4×C16712yes—
101C2×D4C23×C16711yescharacteristic
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—42no—
105C22×C4C2×C4×C16711yescharacteristic
106C22⋊C4—24no—
107C22⋊C4—24no—
108C22⋊C4—24no—
109C22⋊C4—42no—
110C42—21no—
111C42C2×C4×C16712yescomplemented
112C23—22no—
113C23—22no—
114C23—44no—
115C23C22⋊C4×C16711yescharacteristic
116C23C22⋊C4×C16712yes—
117C23C2×D4×C16712yes—
118Q8—42no—
119Q8—42no—
120D4—41no—
121D4—42no—
122D4—41no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—42no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—44no—
131C2×C4—84no—
132C2×C4C22⋊C4×C16711yescharacteristic
133C2×C4C22×C4×C16711yescharacteristic
134C22—22no—
135C22—22no—
136C22—22no—
137C22—42no—
138C22—42no—
139C22—44no—
140C22—82no—
141C22#99×C16711yescharacteristic
142C22#99×C16711yescharacteristic
143C22C2×C22⋊C4×C16711yescharacteristic
144C4—21no—
145C4—21no—
146C4—42no—
147C4—42no—
148C4—84no—
149C2—21no—
150C2—21no—
151C2—42no—
152C2—42no—
153C2#354×C16712yesdirect product factor
154C2C2×#99×C16711yescharacteristic
subgroupG/H
center, socleC22×C167#99
commutatorC2×C4C22×C4×C167
Frattini, cosocleC2×D4C23×C167
prime radicalC22×D4×C167C22
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C167 ◁ C23×C167 ◁ C22×D4×C167 ◁ G

Complements

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

NH ≅ G/Ntype
3. #354×C167C2direct (N × H)
4. C22×Q8×C167C4semidirect
7. #124×C167C4semidirect
11. C2×C42×C167C4semidirect
18. C2×Q8×C167C2×C4semidirect
33. C42×C167C2×C4semidirect
75. C2×C167#354direct (N × H)
77. C167C2×#354direct (N × H)
78. C2×#354C167direct (N × H)
81. #354C2×C167direct (N × H)
82. C22×Q8C4×C167semidirect
85. #124C4×C167semidirect
89. C2×C42C4×C167semidirect
96. C2×Q8C2×C4×C167semidirect
111. C42C2×C4×C167semidirect
153. C2#354×C167direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#124×C1672
C2×#99×C1672
#354×C1672
C2×#354167