Finite Group Databasebeta

← #1913×C167 #36320273

Subgroups — #1913×C167 #36320273

Order 21376.

380 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 108 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#534×C167C211yesmaximal, characteristic
2#518×C167C211yesmaximal, characteristic, complemented
3C2×#99×C167C211yesmaximal, characteristic, complemented
4C2×#99×C167C212yesmaximal, complemented
5#342×C167C212yesmaximal, complemented
6C22×D4×C167C2211yescharacteristic
7C22×D4×C167C2211yescharacteristic
8#123×C167—21no—
9#123×C167C412yescomplemented
10#121×C167—22no—
11#120×C167—21no—
12#120×C167C412yescomplemented
13C4×D4×C167—24no—
14#117×C167—21no—
15C2×C22⋊C4×C167—21no—
16C2×C22⋊C4×C167C2212yes—
17C2×C22⋊C4×C167C2211yescharacteristic
18C2×C22⋊C4×C167C2212yes—
19#99×C167—22no—
20#99×C167—24no—
21C24×C167C2×C411yescharacteristic, complemented
22C24×C167D411yescharacteristic
23C24×C167C2311yescharacteristic
24C2×D4×C167—22no—
25C2×D4×C167—22no—
26C2×D4×C167—22no—
27C2×D4×C167—21no—
28C2×D4×C167—21no—
29C2×D4×C167—22no—
30C2×D4×C167—22no—
31C2×D4×C167C2×C412yes—
32C2×D4×C167D412yes—
33C22×C4×C167—22no—
34C22×C4×C167—22no—
35C22×C4×C167—21no—
36C22×C4×C167—22no—
37C22×C4×C167C2×C411yescharacteristic, complemented
38C22×C4×C167D411yescharacteristic
39C4⋊C4×C167—22no—
40C4⋊C4×C167—22no—
41C22⋊C4×C167—22no—
42C22⋊C4×C167—22no—
43C22⋊C4×C167—22no—
44C22⋊C4×C167—21no—
45C22⋊C4×C167—22no—
46C22⋊C4×C167—22no—
47C22⋊C4×C167—22no—
48C22⋊C4×C167C2×C412yescomplemented
49C22⋊C4×C167D414yescomplemented
50C42×C167—22no—
51C23×C167—21no—
52C23×C167—21no—
53C23×C167—21no—
54C23×C167—22no—
55C23×C167—81no—
56C23×C167—81no—
57C23×C167—81no—
58C23×C167C22⋊C412yes—
59C23×C167C22×C411yescharacteristic
60C23×C167C2×D411yescharacteristic
61C23×C167C2×D411yescharacteristic
62C23×C167C2×D412yes—
63C23×C167#4112yes—
64D4×C167—24no—
65D4×C167—24no—
66D4×C167—41no—
67D4×C167—42no—
68D4×C167—41no—
69D4×C167—82no—
70C2×C4×C167—21no—
71C2×C4×C167—21no—
72C2×C4×C167—22no—
73C2×C4×C167—21no—
74C2×C4×C167—22no—
75C2×C4×C167—24no—
76C2×C4×C167—42no—
77C2×C4×C167—82no—
78C2×C4×C167—82no—
79C2×C4×C167—81no—
80C2×C4×C167—82no—
81C2×C4×C167C22⋊C412yes—
82C22×C167—22no—
83C22×C167—42no—
84C22×C167—41no—
85C22×C167—42no—
86C22×C167—41no—
87C22×C167—41no—
88C22×C167—42no—
89C22×C167—42no—
90C22×C167—42no—
91C22×C167—41no—
92C22×C167—81no—
93C22×C167—82no—
94C22×C167C2×C22⋊C411yescharacteristic
95C22×C167C4×D412yes—
96C22×C167#12011yescharacteristic
97C22×C167#12112yes—
98C22×C167#12311yescharacteristic
99C4×C167—22no—
100C4×C167—44no—
101C4×C167—42no—
102C4×C167—41no—
103C4×C167—82no—
104C2×C167—21no—
105C2×C167—21no—
106C2×C167—22no—
107C2×C167—22no—
108C2×C167—21no—
109C2×C167—42no—
110C2×C167—81no—
111C2×C167#38611yescharacteristic
112C167#191311yescharacteristic, direct product factor, Sylow
113#1913C16711yesmaximal, characteristic, direct product factor, Sylow
114#534C2×C16711yescharacteristic
115#518C2×C16711yescharacteristic, complemented
116C2×#99C2×C16711yescharacteristic, complemented
117C2×#99C2×C16712yescomplemented
118#342C2×C16712yescomplemented
119C22×D4C22×C16711yescharacteristic
120C22×D4C22×C16711yescharacteristic
121#123—21no—
122#123C4×C16712yescomplemented
123#121—22no—
124#120—21no—
125#120C4×C16712yescomplemented
126C4×D4—24no—
127#117—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4C22×C16712yes—
130C2×C22⋊C4C22×C16711yescharacteristic
131C2×C22⋊C4C22×C16712yes—
132#99—22no—
133#99—24no—
134C24C2×C4×C16711yescharacteristic, complemented
135C24D4×C16711yescharacteristic
136C24C23×C16711yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4C2×C4×C16712yes—
145C2×D4D4×C16712yes—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4C2×C4×C16711yescharacteristic, complemented
151C22×C4D4×C16711yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—22no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—22no—
157C22⋊C4—21no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4C2×C4×C16712yescomplemented
162C22⋊C4D4×C16714yescomplemented
163C42—22no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—22no—
168C23—81no—
169C23—81no—
170C23—81no—
171C23C22⋊C4×C16712yes—
172C23C22×C4×C16711yescharacteristic
173C23C2×D4×C16711yescharacteristic
174C23C2×D4×C16711yescharacteristic
175C23C2×D4×C16712yes—
176C23#41×C16712yes—
177D4—24no—
178D4—24no—
179D4—41no—
180D4—42no—
181D4—41no—
182D4—82no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—42no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—81no—
193C2×C4—82no—
194C2×C4C22⋊C4×C16712yes—
195C22—22no—
196C22—42no—
197C22—41no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—41no—
205C22—81no—
206C22—82no—
207C22C2×C22⋊C4×C16711yescharacteristic
208C22C4×D4×C16712yes—
209C22#120×C16711yescharacteristic
210C22#121×C16712yes—
211C22#123×C16711yescharacteristic
212C4—22no—
213C4—44no—
214C4—42no—
215C4—41no—
216C4—82no—
217C2—21no—
218C2—21no—
219C2—22no—
220C2—22no—
221C2—21no—
222C2—42no—
223C2—81no—
224C2#386×C16711yescharacteristic
subgroupG/H
center, socleC2×C167#386
commutatorC23C22×C4×C167
Frattini, cosocleC24C23×C167
prime radical#534×C167C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C167 ◁ C24×C167 ◁ G

Complements

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

NH ≅ G/Ntype
2. #518×C167C2semidirect
3. C2×#99×C167C2semidirect
4. C2×#99×C167C2semidirect
5. #342×C167C2semidirect
9. #123×C167C4semidirect
12. #120×C167C4semidirect
21. C24×C167C2×C4semidirect
37. C22×C4×C167C2×C4semidirect
48. C22⋊C4×C167C2×C4semidirect
49. C22⋊C4×C167D4semidirect
112. C167#1913direct (N × H)
113. #1913C167direct (N × H)
115. #518C2×C167semidirect
116. C2×#99C2×C167semidirect
117. C2×#99C2×C167semidirect
118. #342C2×C167semidirect
122. #123C4×C167semidirect
125. #120C4×C167semidirect
134. C24C2×C4×C167semidirect
150. C22×C4C2×C4×C167semidirect
161. C22⋊C4C2×C4×C167semidirect
162. C22⋊C4D4×C167semidirect

Maximal subgroups

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

subgroupindex
#534×C1672
#518×C1672
C2×#99×C1672
#342×C1672
#1913167