Finite Group Databasebeta

← #2122×C167 #36320482

Subgroups — #2122×C167 #36320482

Order 21376.

266 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 84 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#535×C167C211yesmaximal, characteristic
2#413×C167C211yesmaximal, characteristic, complemented
3#410×C167C211yesmaximal, characteristic, complemented
4AΣL(1,𝔽16)×C167C214yesmaximal, complemented
5C2×#41×C167C2211yescharacteristic
6C22×D4×C167C411yescharacteristic, complemented
7C2×M16×C167—21no—
8#127×C167—21no—
9#124×C167—21no—
10#121×C167—21no—
11#121×C167—21no—
12#120×C167C412yescomplemented
13#120×C167C2212yes—
14#117×C167—21no—
15#117×C167C411yescharacteristic, complemented
16#101×C167—21no—
17#100×C167C2212yes—
18#99×C167—21no—
19#99×C167C2212yes—
20C24×C167C2×C412yes—
21#41×C167—41no—
22#41×C167—41no—
23C2×Q8×C167D411yescharacteristic
24C2×D4×C167—22no—
25C2×D4×C167—22no—
26C2×D4×C167—21no—
27C2×D4×C167—21no—
28C2×D4×C167—41no—
29C2×D4×C167—41no—
30C2×D4×C167C2×C411yescharacteristic
31C2×D4×C167D411yescharacteristic
32C2×D4×C167C2311yescharacteristic
33C22×C4×C167C2×C411yescharacteristic
34C22×C4×C167D411yescharacteristic
35C22×C4×C167D411yescharacteristic
36M16×C167—22no—
37M16×C167—41no—
38C2×C8×C167—41no—
39C4⋊C4×C167—21no—
40C4⋊C4×C167—41no—
41C22⋊C4×C167—22no—
42C22⋊C4×C167—22no—
43C22⋊C4×C167—22no—
44C22⋊C4×C167C2×C412yes—
45C42×C167—21no—
46C42×C167—41no—
47C23×C167—22no—
48C23×C167—42no—
49C23×C167—82no—
50C23×C167C22⋊C411yescharacteristic
51C23×C167C22×C411yescharacteristic
52C23×C167C2×D411yescharacteristic
53Q8×C167—41no—
54D4×C167—41no—
55D4×C167—41no—
56D4×C167—41no—
57D4×C167—42no—
58D4×C167—41no—
59D4×C167—41no—
60D4×C167—82no—
61D4×C167—82no—
62C2×C4×C167—21no—
63C2×C4×C167—21no—
64C2×C4×C167—21no—
65C2×C4×C167—21no—
66C2×C4×C167—21no—
67C2×C4×C167—21no—
68C2×C4×C167—22no—
69C2×C4×C167—42no—
70C2×C4×C167C22⋊C411yescharacteristic
71C2×C4×C167C22⋊C411yescharacteristic
72C2×C4×C167C22⋊C411yescharacteristic
73C2×C4×C167C2×D411yescharacteristic
74C8×C167—42no—
75C22×C167—21no—
76C22×C167—21no—
77C22×C167—21no—
78C22×C167—41no—
79C22×C167—41no—
80C22×C167—41no—
81C22×C167—42no—
82C22×C167—42no—
83C22×C167—42no—
84C22×C167—84no—
85C22×C167C2×C22⋊C411yescharacteristic
86C4×C167—21no—
87C4×C167—21no—
88C4×C167—21no—
89C4×C167—42no—
90C4×C167—82no—
91C4×C167#9911yescharacteristic
92C4×C167#9911yescharacteristic
93C2×C167—21no—
94C2×C167—41no—
95C2×C167—41no—
96C2×C167—41no—
97C2×C167—82no—
98C2×C167C2×#9911yescharacteristic
99C167#212211yescharacteristic, direct product factor, Sylow
100#2122C16711yesmaximal, characteristic, direct product factor, Sylow
101#535C2×C16711yescharacteristic
102#413C2×C16711yescharacteristic, complemented
103#410C2×C16711yescharacteristic, complemented
104AΣL(1,𝔽16)C2×C16714yescomplemented
105C2×#41C22×C16711yescharacteristic
106C22×D4C4×C16711yescharacteristic, complemented
107C2×M16—21no—
108#127—21no—
109#124—21no—
110#121—21no—
111#121—21no—
112#120C4×C16712yescomplemented
113#120C22×C16712yes—
114#117—21no—
115#117C4×C16711yescharacteristic, complemented
116#101—21no—
117#100C22×C16712yes—
118#99—21no—
119#99C22×C16712yes—
120C24C2×C4×C16712yes—
121#41—41no—
122#41—41no—
123C2×Q8D4×C16711yescharacteristic
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4C2×C4×C16711yescharacteristic
131C2×D4D4×C16711yescharacteristic
132C2×D4C23×C16711yescharacteristic
133C22×C4C2×C4×C16711yescharacteristic
134C22×C4D4×C16711yescharacteristic
135C22×C4D4×C16711yescharacteristic
136M16—22no—
137M16—41no—
138C2×C8—41no—
139C4⋊C4—21no—
140C4⋊C4—41no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4C2×C4×C16712yes—
145C42—21no—
146C42—41no—
147C23—22no—
148C23—42no—
149C23—82no—
150C23C22⋊C4×C16711yescharacteristic
151C23C22×C4×C16711yescharacteristic
152C23C2×D4×C16711yescharacteristic
153Q8—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—42no—
158D4—41no—
159D4—41no—
160D4—82no—
161D4—82no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—42no—
170C2×C4C22⋊C4×C16711yescharacteristic
171C2×C4C22⋊C4×C16711yescharacteristic
172C2×C4C22⋊C4×C16711yescharacteristic
173C2×C4C2×D4×C16711yescharacteristic
174C8—42no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—84no—
185C22C2×C22⋊C4×C16711yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—42no—
190C4—82no—
191C4#99×C16711yescharacteristic
192C4#99×C16711yescharacteristic
193C2—21no—
194C2—41no—
195C2—41no—
196C2—41no—
197C2—82no—
198C2C2×#99×C16711yescharacteristic
subgroupG/H
center, socleC2×C167C2×#99
commutatorC23C22×C4×C167
Frattini, cosocleC2×D4C23×C167
prime radical#535×C167C2
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C167 ◁ C2×C4×C167 ◁ C2×#41×C167 ◁ G

Complements

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

NH ≅ G/Ntype
2. #413×C167C2semidirect
3. #410×C167C2semidirect
4. AΣL(1,𝔽16)×C167C2semidirect
6. C22×D4×C167C4semidirect
12. #120×C167C4semidirect
15. #117×C167C4semidirect
99. C167#2122direct (N × H)
100. #2122C167direct (N × H)
102. #413C2×C167semidirect
103. #410C2×C167semidirect
104. AΣL(1,𝔽16)C2×C167semidirect
106. C22×D4C4×C167semidirect
112. #120C4×C167semidirect
115. #117C4×C167semidirect

Maximal subgroups

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

subgroupindex
#535×C1672
#413×C1672
#410×C1672
AΣL(1,𝔽16)×C1672
#2122167