Finite Group Databasebeta

← #2037×C167 #36320397

Subgroups — #2037×C167 #36320397

Order 21376.

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

normal subgroups only

This group has 326 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#428×C167C211yesmaximal, characteristic, complemented
3C2×#104×C167C211yesmaximal, characteristic, complemented
4#417×C167C211yesmaximal, characteristic, complemented
5C2×#100×C167C211yesmaximal, characteristic, complemented
6#390×C167C211yesmaximal, characteristic, complemented
7#339×C167C211yesmaximal, characteristic, complemented
8C2×#41×C167C2211yescharacteristic
9C22×D4×C167C2211yescharacteristic
10C2×M16×C167C2211yescharacteristic
11C2×M16×C167C2211yescharacteristic
12#127×C167—21no—
13#124×C167—21no—
14#121×C167—21no—
15#121×C167—21no—
16#120×C167—22no—
17#117×C167C2211yescharacteristic
18C2×C4⋊C4×C167C2211yescharacteristic
19C2×C22⋊C4×C167—22no—
20C2×C42×C167C2211yescharacteristic
21#107×C167—21no—
22#106×C167—21no—
23#104×C167—22no—
24#103×C167—21no—
25#102×C167—21no—
26#100×C167—22no—
27C24×C167D412yes—
28#41×C167—21no—
29#41×C167—21no—
30#41×C167—21no—
31#41×C167—21no—
32C2×Q8×C167D411yescharacteristic
33C2×D4×C167—21no—
34C2×D4×C167—22no—
35C2×D4×C167—22no—
36C2×D4×C167—21no—
37C2×D4×C167—21no—
38C2×D4×C167—21no—
39C2×D4×C167—21no—
40C2×D4×C167—21no—
41C2×D4×C167—21no—
42C2×D4×C167D411yescharacteristic
43C22×C4×C167—21no—
44C22×C4×C167—21no—
45C22×C4×C167—21no—
46C22×C4×C167C2311yescharacteristic
47M16×C167—21no—
48M16×C167—21no—
49M16×C167—21no—
50M16×C167D412yescomplemented
51C2×C8×C167—21no—
52C2×C8×C167—21no—
53C4⋊C4×C167—21no—
54C4⋊C4×C167—21no—
55C4⋊C4×C167D411yescharacteristic
56C4⋊C4×C167D411yescharacteristic
57C22⋊C4×C167—22no—
58C22⋊C4×C167—22no—
59C22⋊C4×C167—21no—
60C22⋊C4×C167—44no—
61C42×C167—21no—
62C42×C167—21no—
63C42×C167—21no—
64C23×C167—22no—
65C23×C167—21no—
66C23×C167—22no—
67C23×C167—22no—
68C23×C167—42no—
69C23×C167—42no—
70C23×C167C2×D411yescharacteristic
71Q8×C167—21no—
72Q8×C167—21no—
73D4×C167—21no—
74D4×C167—21no—
75D4×C167—41no—
76D4×C167—41no—
77D4×C167—41no—
78D4×C167—41no—
79D4×C167—41no—
80D4×C167—41no—
81D4×C167—82no—
82D4×C167—82no—
83C2×C4×C167—22no—
84C2×C4×C167—21no—
85C2×C4×C167—21no—
86C2×C4×C167—21no—
87C2×C4×C167—41no—
88C2×C4×C167—41no—
89C2×C4×C167—41no—
90C2×C4×C167—41no—
91C2×C4×C167—42no—
92C2×C4×C167—41no—
93C2×C4×C167C2×D411yescharacteristic
94C2×C4×C167C2×D411yescharacteristic
95C2×C4×C167C2×D411yescharacteristic
96C2×C4×C167#4111yescharacteristic
97C2×C4×C167#4111yescharacteristic
98C2×C4×C167#4111yescharacteristic
99C8×C167—22no—
100C8×C167—41no—
101C22×C167—21no—
102C22×C167—21no—
103C22×C167—44no—
104C22×C167—41no—
105C22×C167—42no—
106C22×C167—41no—
107C22×C167—42no—
108C22×C167—41no—
109C22×C167—42no—
110C22×C167—82no—
111C22×C167#12011yescharacteristic
112C22×C167#12111yescharacteristic
113C22×C167#12311yescharacteristic
114C4×C167—42no—
115C4×C167—41no—
116C4×C167—41no—
117C4×C167—41no—
118C4×C167—81no—
119C4×C167#12111yescharacteristic
120C4×C167#12111yescharacteristic
121C4×C167#12311yescharacteristic
122C4×C167#12411yescharacteristic
123C2×C167—21no—
124C2×C167—21no—
125C2×C167—81no—
126C2×C167—82no—
127C2×C167#39411yescharacteristic
128C2×C167#45311yescharacteristic
129C2×C167#45511yescharacteristic
130C167#203711yescharacteristic, direct product factor, Sylow
131#2037C16711yesmaximal, characteristic, direct product factor, Sylow
132#535C2×C16711yescharacteristic
133#428C2×C16711yescharacteristic, complemented
134C2×#104C2×C16711yescharacteristic, complemented
135#417C2×C16711yescharacteristic, complemented
136C2×#100C2×C16711yescharacteristic, complemented
137#390C2×C16711yescharacteristic, complemented
138#339C2×C16711yescharacteristic, complemented
139C2×#41C22×C16711yescharacteristic
140C22×D4C22×C16711yescharacteristic
141C2×M16C22×C16711yescharacteristic
142C2×M16C22×C16711yescharacteristic
143#127—21no—
144#124—21no—
145#121—21no—
146#121—21no—
147#120—22no—
148#117C22×C16711yescharacteristic
149C2×C4⋊C4C22×C16711yescharacteristic
150C2×C22⋊C4—22no—
151C2×C42C22×C16711yescharacteristic
152#107—21no—
153#106—21no—
154#104—22no—
155#103—21no—
156#102—21no—
157#100—22no—
158C24D4×C16712yes—
159#41—21no—
160#41—21no—
161#41—21no—
162#41—21no—
163C2×Q8D4×C16711yescharacteristic
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4D4×C16711yescharacteristic
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4C23×C16711yescharacteristic
178M16—21no—
179M16—21no—
180M16—21no—
181M16D4×C16712yescomplemented
182C2×C8—21no—
183C2×C8—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4D4×C16711yescharacteristic
187C4⋊C4D4×C16711yescharacteristic
188C22⋊C4—22no—
189C22⋊C4—22no—
190C22⋊C4—21no—
191C22⋊C4—44no—
192C42—21no—
193C42—21no—
194C42—21no—
195C23—22no—
196C23—21no—
197C23—22no—
198C23—22no—
199C23—42no—
200C23—42no—
201C23C2×D4×C16711yescharacteristic
202Q8—21no—
203Q8—21no—
204D4—21no—
205D4—21no—
206D4—41no—
207D4—41no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—41no—
212D4—82no—
213D4—82no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4C2×D4×C16711yescharacteristic
225C2×C4C2×D4×C16711yescharacteristic
226C2×C4C2×D4×C16711yescharacteristic
227C2×C4#41×C16711yescharacteristic
228C2×C4#41×C16711yescharacteristic
229C2×C4#41×C16711yescharacteristic
230C8—22no—
231C8—41no—
232C22—21no—
233C22—21no—
234C22—44no—
235C22—41no—
236C22—42no—
237C22—41no—
238C22—42no—
239C22—41no—
240C22—42no—
241C22—82no—
242C22#120×C16711yescharacteristic
243C22#121×C16711yescharacteristic
244C22#123×C16711yescharacteristic
245C4—42no—
246C4—41no—
247C4—41no—
248C4—41no—
249C4—81no—
250C4#121×C16711yescharacteristic
251C4#121×C16711yescharacteristic
252C4#123×C16711yescharacteristic
253C4#124×C16711yescharacteristic
254C2—21no—
255C2—21no—
256C2—82no—
257C2—81no—
258C2#394×C16711yescharacteristic
259C2#453×C16711yescharacteristic
260C2#455×C16711yescharacteristic
subgroupG/H
center, socleC22×C167#120
commutator, Frattini, cosocleC22×C4C23×C167
prime radical#535×C167C2
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
2. #428×C167C2semidirect
3. C2×#104×C167C2semidirect
4. #417×C167C2semidirect
5. C2×#100×C167C2semidirect
6. #390×C167C2semidirect
7. #339×C167C2semidirect
50. M16×C167D4semidirect
130. C167#2037direct (N × H)
131. #2037C167direct (N × H)
133. #428C2×C167semidirect
134. C2×#104C2×C167semidirect
135. #417C2×C167semidirect
136. C2×#100C2×C167semidirect
137. #390C2×C167semidirect
138. #339C2×C167semidirect
181. M16D4×C167semidirect

Maximal subgroups

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

subgroupindex
#535×C1672
#428×C1672
C2×#104×C1672
#417×C1672
C2×#100×C1672
#390×C1672
#339×C1672
#2037167