Finite Group Databasebeta

← #2011×C167 #36320371

Subgroups — #2011×C167 #36320371

Order 21376.

510 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C22×D8×C167C212yesmaximal, complemented
2C2×#127×C167C211yesmaximal, characteristic, complemented
3C2×#102×C167C212yesmaximal, complemented
4#390×C167C211yesmaximal, characteristic, complemented
5#336×C167C211yesmaximal, characteristic, complemented
6C22×D4×C167—22no—
7C22×D4×C167C2212yes—
8C22×D4×C167C2211yescharacteristic
9C2×D8×C167—24no—
10C2×D8×C167—24no—
11C2×D8×C167—24no—
12C22×C8×C167C2212yes—
13#127×C167—24no—
14C2×C4⋊C4×C167C2211yescharacteristic
15C2×C22⋊C4×C167—22no—
16C2×C42×C167C2211yescharacteristic
17#102×C167—24no—
18C24×C167—21no—
19C24×C167—22no—
20C2×D4×C167—28no—
21C2×D4×C167—22no—
22C2×D4×C167—22no—
23C2×D4×C167—22no—
24C2×D4×C167—22no—
25C2×D4×C167—22no—
26C2×D4×C167—22no—
27C2×D4×C167—22no—
28C2×D4×C167—21no—
29C2×D4×C167—44no—
30C2×D4×C167—44no—
31C2×D4×C167D414yes—
32C2×D4×C167D412yescomplemented
33C22×C4×C167—21no—
34C22×C4×C167D412yes—
35C22×C4×C167C2311yescharacteristic
36D8×C167—216no—
37C2×C8×C167—22no—
38C2×C8×C167—22no—
39C2×C8×C167D414yescomplemented
40C4⋊C4×C167—22no—
41C22⋊C4×C167—44no—
42C42×C167—22no—
43C23×C167—24no—
44C23×C167—22no—
45C23×C167—41no—
46C23×C167—42no—
47C23×C167—42no—
48C23×C167—41no—
49C23×C167—42no—
50C23×C167—44no—
51C23×C167—44no—
52C23×C167—42no—
53C23×C167C2×D411yescharacteristic
54D4×C167—24no—
55D4×C167—24no—
56D4×C167—24no—
57D4×C167—416no—
58D4×C167—42no—
59D4×C167—42no—
60D4×C167—42no—
61C2×C4×C167—22no—
62C2×C4×C167—22no—
63C2×C4×C167—41no—
64C2×C4×C167—42no—
65C2×C4×C167D814yescomplemented
66C2×C4×C167C2×D411yescharacteristic
67C2×C4×C167C2×D412yes—
68C2×C4×C167C2×D412yes—
69C2×C4×C167#4111yescharacteristic
70C8×C167—24no—
71C22×C167—44no—
72C22×C167—42no—
73C22×C167—48no—
74C22×C167—44no—
75C22×C167—82no—
76C22×C167—84no—
77C22×C167—81no—
78C22×C167—82no—
79C22×C167—82no—
80C22×C167—81no—
81C22×C167#12011yescharacteristic
82C22×C167#12111yescharacteristic
83C22×C167#12711yescharacteristic
84C22×C167C2×D812yes—
85C22×C167AGL(1,ℤ/8)12yes—
86C4×C167—24no—
87C4×C167—81no—
88C4×C167#12012yes—
89C4×C167#12112yes—
90C2×C167—84no—
91C2×C167—82no—
92C2×C167#39211yescharacteristic
93C2×C167#44712yes—
94C2×C167#45912yes—
95C2×C167#49311yescharacteristic
96C2×C167#49611yescharacteristic
97C167#201111yescharacteristic, direct product factor, Sylow
98#2011C16711yesmaximal, characteristic, direct product factor, Sylow
99C22×D8C2×C16712yescomplemented
100C2×#127C2×C16711yescharacteristic, complemented
101C2×#102C2×C16712yescomplemented
102#390C2×C16711yescharacteristic, complemented
103#336C2×C16711yescharacteristic, complemented
104C22×D4—22no—
105C22×D4C22×C16712yes—
106C22×D4C22×C16711yescharacteristic
107C2×D8—24no—
108C2×D8—24no—
109C2×D8—24no—
110C22×C8C22×C16712yes—
111#127—24no—
112C2×C4⋊C4C22×C16711yescharacteristic
113C2×C22⋊C4—22no—
114C2×C42C22×C16711yescharacteristic
115#102—24no—
116C24—21no—
117C24—22no—
118C2×D4—28no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—44no—
128C2×D4—44no—
129C2×D4D4×C16714yes—
130C2×D4D4×C16712yescomplemented
131C22×C4—21no—
132C22×C4D4×C16712yes—
133C22×C4C23×C16711yescharacteristic
134D8—216no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8D4×C16714yescomplemented
138C4⋊C4—22no—
139C22⋊C4—44no—
140C42—22no—
141C23—24no—
142C23—22no—
143C23—41no—
144C23—42no—
145C23—42no—
146C23—41no—
147C23—42no—
148C23—44no—
149C23—44no—
150C23—42no—
151C23C2×D4×C16711yescharacteristic
152D4—24no—
153D4—24no—
154D4—24no—
155D4—416no—
156D4—42no—
157D4—42no—
158D4—42no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4D8×C16714yescomplemented
164C2×C4C2×D4×C16711yescharacteristic
165C2×C4C2×D4×C16712yes—
166C2×C4C2×D4×C16712yes—
167C2×C4#41×C16711yescharacteristic
168C8—24no—
169C22—44no—
170C22—42no—
171C22—48no—
172C22—44no—
173C22—81no—
174C22—82no—
175C22—82no—
176C22—81no—
177C22—82no—
178C22—84no—
179C22#120×C16711yescharacteristic
180C22#121×C16711yescharacteristic
181C22#127×C16711yescharacteristic
182C22C2×D8×C16712yes—
183C22AGL(1,ℤ/8)×C16712yes—
184C4—24no—
185C4—81no—
186C4#120×C16712yes—
187C4#121×C16712yes—
188C2—84no—
189C2—82no—
190C2#392×C16711yescharacteristic
191C2#447×C16712yes—
192C2#459×C16712yes—
193C2#493×C16711yescharacteristic
194C2#496×C16711yescharacteristic
subgroupG/H
center, socleC23×C167C2×D4
commutator, Frattini, cosocleC22×C4C23×C167
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C167 ◁ C2×C42×C167 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×D8×C167C2semidirect
2. C2×#127×C167C2semidirect
3. C2×#102×C167C2semidirect
4. #390×C167C2semidirect
5. #336×C167C2semidirect
32. C2×D4×C167D4semidirect
39. C2×C8×C167D4semidirect
65. C2×C4×C167D8semidirect
97. C167#2011direct (N × H)
98. #2011C167direct (N × H)
99. C22×D8C2×C167semidirect
100. C2×#127C2×C167semidirect
101. C2×#102C2×C167semidirect
102. #390C2×C167semidirect
103. #336C2×C167semidirect
130. C2×D4D4×C167semidirect
137. C2×C8D4×C167semidirect
163. C2×C4D8×C167semidirect

Maximal subgroups

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

subgroupindex
C22×D8×C1672
C2×#127×C1672
C2×#102×C1672
#390×C1672
#336×C1672
#2011167