Finite Group Databasebeta

← #2435×C167 #36320795

Subgroups — #2435×C167 #36320795

Order 21376.

1088 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 248 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
1C23×D4×C167C211yesmaximal, characteristic
2C2×#124×C167C212yesmaximal, complemented
3C2×C22⋊Q8×C167C211yesmaximal, characteristic, complemented
4#394×C167C214yesmaximal, complemented
5#390×C167C212yesmaximal, complemented
6#379×C167C214yesmaximal, complemented
7C4×C22⋊C4×C167C211yesmaximal, characteristic, complemented
8C25×C167C2212yes—
9C22×Q8×C167C2211yescharacteristic, complemented
10C22×D4×C167—28no—
11C22×D4×C167—24no—
12C22×D4×C167C2212yes—
13C22×D4×C167C2212yes—
14C23×C4×C167C2211yescharacteristic
15#124×C167—28no—
16C22⋊Q8×C167—24no—
17C2×C4⋊C4×C167C2211yescharacteristic, complemented
18C2×C4⋊C4×C167C2212yescomplemented
19C2×C22⋊C4×C167C2212yes—
20C2×C22⋊C4×C167C2212yes—
21C2×C22⋊C4×C167C2218yescomplemented
22C2×C22⋊C4×C167C2218yescomplemented
23C2×C42×C167C2212yescomplemented
24#95×C167C2212yescomplemented
25C24×C167—28no—
26C24×C167—24no—
27C24×C167—28no—
28C24×C167—44no—
29C24×C167C2314yes—
30C24×C167C2311yescharacteristic
31C2×Q8×C167—22no—
32C2×Q8×C167—24no—
33C2×D4×C167—232no—
34C2×D4×C167—28no—
35C2×D4×C167—24no—
36C2×D4×C167—24no—
37C2×D4×C167—24no—
38C2×D4×C167D418yes—
39C22×C4×C167—24no—
40C22×C4×C167D414yes—
41C22×C4×C167C2314yes—
42C22×C4×C167C2312yes—
43C22×C4×C167C2312yes—
44C22×C4×C167C2311yescharacteristic
45C22×C4×C167C2311yescharacteristic
46C4⋊C4×C167—22no—
47C4⋊C4×C167—24no—
48C22⋊C4×C167—24no—
49C22⋊C4×C167—216no—
50C22⋊C4×C167—24no—
51C22⋊C4×C167—48no—
52C42×C167—24no—
53C23×C167—22no—
54C23×C167—22no—
55C23×C167—216no—
56C23×C167—24no—
57C23×C167—24no—
58C23×C167—28no—
59C23×C167—28no—
60C23×C167—432no—
61C23×C167—48no—
62C23×C167—48no—
63C23×C167C2×D418yes—
64C23×C167C2×D412yes—
65C23×C167#4114yes—
66C23×C167C2411yescharacteristic
67Q8×C167—44no—
68D4×C167—216no—
69D4×C167—216no—
70C2×C4×C167—22no—
71C2×C4×C167—28no—
72C2×C4×C167—24no—
73C2×C4×C167—24no—
74C2×C4×C167—22no—
75C2×C4×C167—22no—
76C2×C4×C167—24no—
77C2×C4×C167—24no—
78C2×C4×C167—24no—
79C2×C4×C167C2×D412yes—
80C2×C4×C167C2×D414yes—
81C2×C4×C167C2×D412yes—
82C22×C167—22no—
83C22×C167—22no—
84C22×C167—28no—
85C22×C167—28no—
86C22×C167—216no—
87C22×C167—432no—
88C22×C167—44no—
89C22×C167—44no—
90C22×C167—48no—
91C22×C167#12414yescomplemented
92C22×C167C22×D411yescharacteristic
93C22×C167C22×D412yes—
94C22×C167C2×#4112yes—
95C22×C167H(5,𝔽2)12yes—
96C4×C167—22no—
97C4×C167—42no—
98C4×C167—42no—
99C4×C167—44no—
100C4×C167#12014yes—
101C2×C167—24no—
102C2×C167—48no—
103C2×C167C2×#12011yescharacteristic
104C2×C167C2×#12411yescharacteristic
105C2×C167#53511yescharacteristic
106C2×C167#54614yes—
107C167#243511yescharacteristic, direct product factor, Sylow
108#2435C16711yesmaximal, characteristic, direct product factor, Sylow
109C23×D4C2×C16711yescharacteristic
110C2×#124C2×C16712yescomplemented
111C2×C22⋊Q8C2×C16711yescharacteristic, complemented
112#394C2×C16714yescomplemented
113#390C2×C16712yescomplemented
114#379C2×C16714yescomplemented
115C4×C22⋊C4C2×C16711yescharacteristic, complemented
116C25C22×C16712yes—
117C22×Q8C22×C16711yescharacteristic, complemented
118C22×D4—28no—
119C22×D4—24no—
120C22×D4C22×C16712yes—
121C22×D4C22×C16712yes—
122C23×C4C22×C16711yescharacteristic
123#124—28no—
124C22⋊Q8—24no—
125C2×C4⋊C4C22×C16711yescharacteristic, complemented
126C2×C4⋊C4C22×C16712yescomplemented
127C2×C22⋊C4C22×C16712yes—
128C2×C22⋊C4C22×C16712yes—
129C2×C22⋊C4C22×C16718yescomplemented
130C2×C22⋊C4C22×C16718yescomplemented
131C2×C42C22×C16712yescomplemented
132#95C22×C16712yescomplemented
133C24—28no—
134C24—24no—
135C24—28no—
136C24—44no—
137C24C23×C16714yes—
138C24C23×C16711yescharacteristic
139C2×Q8—22no—
140C2×Q8—24no—
141C2×D4—232no—
142C2×D4—28no—
143C2×D4—24no—
144C2×D4—24no—
145C2×D4—24no—
146C2×D4D4×C16718yes—
147C22×C4—24no—
148C22×C4D4×C16714yes—
149C22×C4C23×C16714yes—
150C22×C4C23×C16712yes—
151C22×C4C23×C16712yes—
152C22×C4C23×C16711yescharacteristic
153C22×C4C23×C16711yescharacteristic
154C4⋊C4—22no—
155C4⋊C4—24no—
156C22⋊C4—24no—
157C22⋊C4—216no—
158C22⋊C4—24no—
159C22⋊C4—48no—
160C42—24no—
161C23—22no—
162C23—22no—
163C23—216no—
164C23—24no—
165C23—24no—
166C23—28no—
167C23—28no—
168C23—432no—
169C23—48no—
170C23—48no—
171C23C2×D4×C16718yes—
172C23C2×D4×C16712yes—
173C23#41×C16714yes—
174C23C24×C16711yescharacteristic
175Q8—44no—
176D4—216no—
177D4—216no—
178C2×C4—22no—
179C2×C4—28no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—24no—
187C2×C4C2×D4×C16712yes—
188C2×C4C2×D4×C16714yes—
189C2×C4C2×D4×C16712yes—
190C22—22no—
191C22—22no—
192C22—28no—
193C22—28no—
194C22—216no—
195C22—432no—
196C22—44no—
197C22—44no—
198C22—48no—
199C22#124×C16714yescomplemented
200C22C22×D4×C16711yescharacteristic
201C22C22×D4×C16712yes—
202C22C2×#41×C16712yes—
203C22H(5,𝔽2)×C16712yes—
204C4—22no—
205C4—42no—
206C4—42no—
207C4—44no—
208C4#120×C16714yes—
209C2—24no—
210C2—48no—
211C2C2×#120×C16711yescharacteristic
212C2C2×#124×C16711yescharacteristic
213C2#535×C16711yescharacteristic
214C2#546×C16714yes—
subgroupG/H
center, socleC23×C167C24
commutator, Frattini, cosocleC23C24×C167
prime radicalC23×D4×C167C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C167 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#124×C167C2semidirect
3. C2×C22⋊Q8×C167C2semidirect
4. #394×C167C2semidirect
5. #390×C167C2semidirect
6. #379×C167C2semidirect
7. C4×C22⋊C4×C167C2semidirect
9. C22×Q8×C167C22semidirect
17. C2×C4⋊C4×C167C22semidirect
18. C2×C4⋊C4×C167C22semidirect
21. C2×C22⋊C4×C167C22semidirect
22. C2×C22⋊C4×C167C22semidirect
23. C2×C42×C167C22semidirect
24. #95×C167C22semidirect
91. C22×C167#124semidirect
107. C167#2435direct (N × H)
108. #2435C167direct (N × H)
110. C2×#124C2×C167semidirect
111. C2×C22⋊Q8C2×C167semidirect
112. #394C2×C167semidirect
113. #390C2×C167semidirect
114. #379C2×C167semidirect
115. C4×C22⋊C4C2×C167semidirect
117. C22×Q8C22×C167semidirect
125. C2×C4⋊C4C22×C167semidirect
126. C2×C4⋊C4C22×C167semidirect
129. C2×C22⋊C4C22×C167semidirect
130. C2×C22⋊C4C22×C167semidirect
131. C2×C42C22×C167semidirect
132. #95C22×C167semidirect
199. C22#124×C167semidirect

Maximal subgroups

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

subgroupindex
C23×D4×C1672
C2×#124×C1672
C2×C22⋊Q8×C1672
#394×C1672
#390×C1672
#379×C1672
C4×C22⋊C4×C1672
#2435167