Finite Group Databasebeta

← #2006×C167 #36320366

Subgroups — #2006×C167 #36320366

Order 21376.

406 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 92 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×AGL(1,ℤ/8)×C167C212yesmaximal, complemented
2#537×C167C211yesmaximal, characteristic, complemented
3C2×#104×C167C212yesmaximal, complemented
4#390×C167C211yesmaximal, characteristic, complemented
5#343×C167C211yesmaximal, characteristic, complemented
6C2×#41×C167C2212yes—
7C22×D4×C167C2211yescharacteristic
8AGL(1,ℤ/8)×C167—24no—
9AGL(1,ℤ/8)×C167—24no—
10C2×SD16×C167—22no—
11C2×D8×C167—22no—
12C2×M16×C167C2212yes—
13C22⋊Q8×C167—22no—
14#121×C167—22no—
15#121×C167—22no—
16C2×C4⋊C4×C167C2211yescharacteristic
17C2×C22⋊C4×C167—22no—
18C2×C42×C167C2211yescharacteristic
19#104×C167—24no—
20C24×C167—21no—
21#41×C167—22no—
22#41×C167—22no—
23#41×C167—22no—
24#41×C167—22no—
25C2×Q8×C167D412yes—
26C2×D4×C167—22no—
27C2×D4×C167—22no—
28C2×D4×C167—22no—
29C2×D4×C167—22no—
30C2×D4×C167—21no—
31C2×D4×C167D412yes—
32C2×D4×C167D412yescomplemented
33C22×C4×C167—21no—
34C22×C4×C167—22no—
35C22×C4×C167D412yes—
36C22×C4×C167C2311yescharacteristic
37SD16×C167—28no—
38D8×C167—28no—
39M16×C167—22no—
40M16×C167D414yescomplemented
41C2×C8×C167—22no—
42C4⋊C4×C167—22no—
43C4⋊C4×C167—22no—
44C22⋊C4×C167—24no—
45C22⋊C4×C167—44no—
46C42×C167—22no—
47C23×C167—22no—
48C23×C167—22no—
49C23×C167—42no—
50C23×C167—42no—
51C23×C167—41no—
52C23×C167—41no—
53C23×C167C2×D411yescharacteristic
54Q8×C167—22no—
55Q8×C167—22no—
56D4×C167—22no—
57D4×C167—24no—
58D4×C167—22no—
59D4×C167—42no—
60D4×C167—42no—
61D4×C167—42no—
62D4×C167—82no—
63C2×C4×C167—22no—
64C2×C4×C167—22no—
65C2×C4×C167—22no—
66C2×C4×C167—41no—
67C2×C4×C167—41no—
68C2×C4×C167—42no—
69C2×C4×C167—42no—
70C2×C4×C167—42no—
71C2×C4×C167C2×D411yescharacteristic
72C2×C4×C167C2×D412yes—
73C2×C4×C167C2×D412yes—
74C2×C4×C167#4111yescharacteristic
75C8×C167—24no—
76C22×C167—21no—
77C22×C167—21no—
78C22×C167—44no—
79C22×C167—42no—
80C22×C167—42no—
81C22×C167—42no—
82C22×C167—42no—
83C22×C167—82no—
84C22×C167—81no—
85C22×C167—81no—
86C22×C167#12011yescharacteristic
87C22×C167#12111yescharacteristic
88C22×C167#12711yescharacteristic
89C4×C167—42no—
90C4×C167—42no—
91C4×C167—42no—
92C4×C167#12012yes—
93C4×C167#12112yes—
94C2×C167—21no—
95C2×C167—21no—
96C2×C167—82no—
97C2×C167—82no—
98C2×C167#39211yescharacteristic
99C2×C167#45311yescharacteristic
100C2×C167#45411yescharacteristic
101C167#200611yescharacteristic, direct product factor, Sylow
102#2006C16711yesmaximal, characteristic, direct product factor, Sylow
103C2×AGL(1,ℤ/8)C2×C16712yescomplemented
104#537C2×C16711yescharacteristic, complemented
105C2×#104C2×C16712yescomplemented
106#390C2×C16711yescharacteristic, complemented
107#343C2×C16711yescharacteristic, complemented
108C2×#41C22×C16712yes—
109C22×D4C22×C16711yescharacteristic
110AGL(1,ℤ/8)—24no—
111AGL(1,ℤ/8)—24no—
112C2×SD16—22no—
113C2×D8—22no—
114C2×M16C22×C16712yes—
115C22⋊Q8—22no—
116#121—22no—
117#121—22no—
118C2×C4⋊C4C22×C16711yescharacteristic
119C2×C22⋊C4—22no—
120C2×C42C22×C16711yescharacteristic
121#104—24no—
122C24—21no—
123#41—22no—
124#41—22no—
125#41—22no—
126#41—22no—
127C2×Q8D4×C16712yes—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—21no—
133C2×D4D4×C16712yes—
134C2×D4D4×C16712yescomplemented
135C22×C4—21no—
136C22×C4—22no—
137C22×C4D4×C16712yes—
138C22×C4C23×C16711yescharacteristic
139SD16—28no—
140D8—28no—
141M16—22no—
142M16D4×C16714yescomplemented
143C2×C8—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C22⋊C4—24no—
147C22⋊C4—44no—
148C42—22no—
149C23—22no—
150C23—22no—
151C23—42no—
152C23—42no—
153C23—41no—
154C23—41no—
155C23C2×D4×C16711yescharacteristic
156Q8—22no—
157Q8—22no—
158D4—22no—
159D4—24no—
160D4—22no—
161D4—42no—
162D4—42no—
163D4—42no—
164D4—82no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4C2×D4×C16711yescharacteristic
174C2×C4C2×D4×C16712yes—
175C2×C4C2×D4×C16712yes—
176C2×C4#41×C16711yescharacteristic
177C8—24no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—42no—
185C22—82no—
186C22—81no—
187C22—81no—
188C22#120×C16711yescharacteristic
189C22#121×C16711yescharacteristic
190C22#127×C16711yescharacteristic
191C4—42no—
192C4—42no—
193C4—42no—
194C4#120×C16712yes—
195C4#121×C16712yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—82no—
200C2#392×C16711yescharacteristic
201C2#453×C16711yescharacteristic
202C2#454×C16711yescharacteristic
subgroupG/H
center, socleC22×C167#120
commutator, Frattini, cosocleC22×C4C23×C167
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C167 ◁ C22×C4×C167 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×AGL(1,ℤ/8)×C167C2semidirect
2. #537×C167C2semidirect
3. C2×#104×C167C2semidirect
4. #390×C167C2semidirect
5. #343×C167C2semidirect
32. C2×D4×C167D4semidirect
40. M16×C167D4semidirect
101. C167#2006direct (N × H)
102. #2006C167direct (N × H)
103. C2×AGL(1,ℤ/8)C2×C167semidirect
104. #537C2×C167semidirect
105. C2×#104C2×C167semidirect
106. #390C2×C167semidirect
107. #343C2×C167semidirect
134. C2×D4D4×C167semidirect
142. M16D4×C167semidirect

Maximal subgroups

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

subgroupindex
C2×AGL(1,ℤ/8)×C1672
#537×C1672
C2×#104×C1672
#390×C1672
#343×C1672
#2006167