Finite Group Databasebeta

← C83×#12180 #35240004

Subgroups — C83×#12180 #35240004

Order 21248.

472 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C83×#2551C211yesmaximal, characteristic, complemented
2C83×#2039C211yesmaximal, characteristic
3C83×#1604C212yesmaximal
4C83×#1354C212yesmaximal, complemented
5C83×#1325C211yesmaximal, characteristic, complemented
6#537×C83C2211yescharacteristic
7C2×C4⋊Q8×C83—22no—
8#448×C83—22no—
9C2×#104×C83—21no—
10#417×C83—21no—
11#395×C83—21no—
12#395×C83C2212yes—
13#389×C83—22no—
14#384×C83C2211yescharacteristic
15C4×C4⋊C4×C83—21no—
16#343×C83C2211yescharacteristic
17#327×C83—22no—
18#324×C83C2212yes—
19C2×#41×C83—21no—
20C22×Q8×C83—21no—
21C22×Q8×C83D412yes—
22C2×SD16×C83—42no—
23C2×M16×C83—21no—
24C4⋊Q8×C83—48no—
25C22⋊Q8×C83—41no—
26#121×C83—22no—
27#121×C83—41no—
28C2×C4⋊C4×C83—21no—
29C2×C4⋊C4×C83—21no—
30C2×C4⋊C4×C83—22no—
31C2×C4⋊C4×C83—22no—
32C2×C4⋊C4×C83—22no—
33C2×C4⋊C4×C83D411yescharacteristic
34C2×C4⋊C4×C83C2311yescharacteristic
35C2×C42×C83—22no—
36C2×C42×C83D411yescharacteristic
37#104×C83—42no—
38#103×C83—42no—
39#103×C83—41no—
40#102×C83—41no—
41C22⋊C8×C83—22no—
42#95×C83—21no—
43#95×C83—22no—
44#95×C83D412yes—
45#41×C83—41no—
46#41×C83—41no—
47#41×C83—41no—
48#41×C83—41no—
49C2×Q8×C83—21no—
50C2×Q8×C83—44no—
51C2×Q8×C83—44no—
52C2×Q8×C83—42no—
53C2×Q8×C83—81no—
54C2×D4×C83—21no—
55C2×D4×C83—22no—
56C2×D4×C83—41no—
57C22×C4×C83—21no—
58C22×C4×C83—22no—
59C22×C4×C83—22no—
60C22×C4×C83—21no—
61C22×C4×C83—41no—
62C22×C4×C83C2×D411yescharacteristic
63C22×C4×C83C2×D412yes—
64SD16×C83—84no—
65M16×C83—41no—
66M16×C83—41no—
67C2×C8×C83—42no—
68C2×C8×C83—41no—
69C4⋊C4×C83—24no—
70C4⋊C4×C83—24no—
71C4⋊C4×C83—22no—
72C4⋊C4×C83—41no—
73C4⋊C4×C83—41no—
74C4⋊C4×C83—44no—
75C4⋊C4×C83—44no—
76C4⋊C4×C83—44no—
77C22⋊C4×C83—42no—
78C42×C83—21no—
79C42×C83—44no—
80C42×C83SD1612yescomplemented
81C23×C83—41no—
82C23×C83#12011yescharacteristic
83Q8×C83—41no—
84Q8×C83—41no—
85Q8×C83—44no—
86Q8×C83—82no—
87Q8×C83—84no—
88D4×C83—41no—
89D4×C83—41no—
90D4×C83—82no—
91D4×C83—81no—
92D4×C83—81no—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83—21no—
97C2×C4×C83—21no—
98C2×C4×C83—24no—
99C2×C4×C83—24no—
100C2×C4×C83—22no—
101C2×C4×C83—42no—
102C2×C4×C83—42no—
103C2×C4×C83—42no—
104C2×C4×C83—42no—
105C2×C4×C83—42no—
106C2×C4×C83—42no—
107C2×C4×C83—41no—
108C2×C4×C83—41no—
109C2×C4×C83—81no—
110C2×C4×C83—81no—
111C2×C4×C83C2×SD1611yescharacteristic
112C2×C4×C83#13711yescharacteristic
113C8×C83—81no—
114C8×C83—82no—
115C22×C83—21no—
116C22×C83—21no—
117C22×C83—81no—
118C22×C83—81no—
119C22×C83—81no—
120C22×C83#44811yescharacteristic
121C22×C83#45411yescharacteristic
122C22×C83B(4,𝔽2)11yescharacteristic
123C4×C83—24no—
124C4×C83—21no—
125C4×C83—21no—
126C4×C83—44no—
127C4×C83—42no—
128C4×C83—44no—
129C4×C83—81no—
130C4×C83—81no—
131#12180C8311yesmaximal, characteristic, direct product factor, Sylow
132C2×C83—21no—
133C2×C83—21no—
134C2×C83—161no—
135C2×C83#159911yescharacteristic
136C2×C83#219611yescharacteristic
137C2×C83#220511yescharacteristic
138#2551C2×C8311yescharacteristic, complemented
139#2039C2×C8311yescharacteristic
140#1604C2×C8312yes—
141#1354C2×C8312yescomplemented
142#1325C2×C8311yescharacteristic, complemented
143C83#1218011yescharacteristic, direct product factor, Sylow
144#537C22×C8311yescharacteristic
145C2×C4⋊Q8—22no—
146#448—22no—
147C2×#104—21no—
148#417—21no—
149#395—21no—
150#395C22×C8312yes—
151#389—22no—
152#384C22×C8311yescharacteristic
153C4×C4⋊C4—21no—
154#343C22×C8311yescharacteristic
155#327—22no—
156#324C22×C8312yes—
157C2×#41—21no—
158C22×Q8—21no—
159C22×Q8D4×C8312yes—
160C2×SD16—42no—
161C2×M16—21no—
162C4⋊Q8—48no—
163C22⋊Q8—41no—
164#121—22no—
165#121—41no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4D4×C8311yescharacteristic
172C2×C4⋊C4C23×C8311yescharacteristic
173C2×C42—22no—
174C2×C42D4×C8311yescharacteristic
175#104—42no—
176#103—42no—
177#103—41no—
178#102—41no—
179C22⋊C8—22no—
180#95—21no—
181#95—22no—
182#95D4×C8312yes—
183#41—41no—
184#41—41no—
185#41—41no—
186#41—41no—
187C2×Q8—21no—
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8—42no—
191C2×Q8—81no—
192C2×D4—21no—
193C2×D4—22no—
194C2×D4—41no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—41no—
200C22×C4C2×D4×C8311yescharacteristic
201C22×C4C2×D4×C8312yes—
202SD16—84no—
203M16—41no—
204M16—41no—
205C2×C8—42no—
206C2×C8—41no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—22no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C22⋊C4—42no—
216C42—21no—
217C42—44no—
218C42SD16×C8312yescomplemented
219C23—41no—
220C23#120×C8311yescharacteristic
221Q8—41no—
222Q8—41no—
223Q8—44no—
224Q8—82no—
225Q8—84no—
226D4—41no—
227D4—41no—
228D4—82no—
229D4—81no—
230D4—81no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4C2×SD16×C8311yescharacteristic
250C2×C4#137×C8311yescharacteristic
251C8—81no—
252C8—82no—
253C22—21no—
254C22—21no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22#448×C8311yescharacteristic
259C22#454×C8311yescharacteristic
260C22B(4,𝔽2)×C8311yescharacteristic
261C4—24no—
262C4—21no—
263C4—21no—
264C4—44no—
265C4—42no—
266C4—44no—
267C4—81no—
268C4—81no—
269C2—21no—
270C2—21no—
271C2—161no—
272C2C83×#159911yescharacteristic
273C2C83×#219611yescharacteristic
274C2C83×#220511yescharacteristic
subgroupG/H
center, socleC22×C83B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C83
prime radical#537×C83C22
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C23×C83 ◁ C2×C4⋊C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#2551C2semidirect
4. C83×#1354C2semidirect
5. C83×#1325C2semidirect
80. C42×C83SD16semidirect
131. #12180C83direct (N × H)
138. #2551C2×C83semidirect
141. #1354C2×C83semidirect
142. #1325C2×C83semidirect
143. C83#12180direct (N × H)
218. C42SD16×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#25512
C83×#20392
C83×#16042
C83×#13542
C83×#13252
#1218083