Finite Group Databasebeta

← C83×#11475 #35239299

Subgroups — C83×#11475 #35239299

Order 21248.

616 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 134 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
1C4×#127×C83C211yesmaximal, characteristic
2C83×#1459C212yesmaximal, complemented
3C83×#1325C212yesmaximal, complemented
4C83×#1318C212yesmaximal, complemented
5C2×#127×C83C411yescharacteristic
6C2×C4×D4×C83—22no—
7C2×C4×D4×C83C411yescharacteristic
8C2×C4×D4×C83C2211yescharacteristic
9C2×C22⋊C8×C83—22no—
10#390×C83—22no—
11#390×C83C411yescharacteristic
12#390×C83C2211yescharacteristic
13#384×C83C2211yescharacteristic
14C43×C83C411yescharacteristic
15#324×C83C2214yes—
16#323×C83—22no—
17C22×D4×C83—22no—
18C22×D4×C83C2×C411yescharacteristic
19C22×D4×C83C2×C411yescharacteristic
20C23×C4×C83—21no—
21C23×C4×C83D412yes—
22C22×C8×C83—42no—
23#127×C83—22no—
24#127×C83C814yescomplemented
25C4×D4×C83—41no—
26C4×D4×C83—41no—
27C4×D4×C83—48no—
28C4×D4×C83—41no—
29C4×D4×C83—41no—
30C2×C4⋊C4×C83—22no—
31C2×C4⋊C4×C83C2×C411yescharacteristic
32C2×C4⋊C4×C83C2311yescharacteristic
33C2×C22⋊C4×C83—22no—
34C2×C22⋊C4×C83—21no—
35C2×C22⋊C4×C83—22no—
36C2×C22⋊C4×C83D412yes—
37C2×C42×C83—21no—
38C2×C42×C83—21no—
39C2×C42×C83C2×C411yescharacteristic
40C2×C42×C83C2×C411yescharacteristic
41C2×C42×C83C2×C411yescharacteristic
42C22⋊C8×C83—24no—
43C22⋊C8×C83—42no—
44C24×C83—21no—
45C24×C83C22⋊C412yes—
46C2×D4×C83—28no—
47C2×D4×C83—21no—
48C2×D4×C83—41no—
49C2×D4×C83—41no—
50C2×D4×C83—41no—
51C2×D4×C83—41no—
52C2×D4×C83—44no—
53C2×D4×C83—44no—
54C2×D4×C83C2×C812yes—
55C2×D4×C83C2×C812yescomplemented
56C2×D4×C83M1612yes—
57C22×C4×C83—21no—
58C22×C4×C83—21no—
59C22×C4×C83—22no—
60C22×C4×C83—22no—
61C22×C4×C83—21no—
62C22×C4×C83—44no—
63C22×C4×C83—44no—
64C22×C4×C83—41no—
65C22×C4×C83—41no—
66C22×C4×C83C22⋊C411yescharacteristic
67C22×C4×C83C22⋊C411yescharacteristic
68C22×C4×C83C22×C411yescharacteristic
69C22×C4×C83C2×D412yes—
70C2×C8×C83—44no—
71C2×C8×C83—84no—
72C4⋊C4×C83—41no—
73C4⋊C4×C83—41no—
74C4⋊C4×C83—44no—
75C22⋊C4×C83—42no—
76C22⋊C4×C83—44no—
77C22⋊C4×C83—44no—
78C22⋊C4×C83—42no—
79C42×C83—24no—
80C42×C83—21no—
81C42×C83—21no—
82C42×C83—21no—
83C42×C83—41no—
84C42×C83—42no—
85C42×C83C2×C812yescomplemented
86C23×C83—22no—
87C23×C83—22no—
88C23×C83—22no—
89C23×C83—42no—
90C23×C83—42no—
91C23×C83—42no—
92C23×C83—42no—
93C23×C83—41no—
94C23×C83—41no—
95C23×C83C22⋊C814yes—
96C23×C83C2×C22⋊C411yescharacteristic
97D4×C83—42no—
98D4×C83—42no—
99D4×C83—42no—
100D4×C83—416no—
101D4×C83—42no—
102C2×C4×C83—22no—
103C2×C4×C83—22no—
104C2×C4×C83—22no—
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—22no—
108C2×C4×C83—22no—
109C2×C4×C83—21no—
110C2×C4×C83—22no—
111C2×C4×C83—41no—
112C2×C4×C83—41no—
113C2×C4×C83—44no—
114C2×C4×C83—41no—
115C2×C4×C83—42no—
116C2×C4×C83—42no—
117C2×C4×C83—41no—
118C2×C4×C83—82no—
119C2×C4×C83—82no—
120C2×C4×C83—82no—
121C2×C4×C83#9911yescharacteristic
122C2×C4×C83#9911yescharacteristic
123C2×C4×C83C22×C811yescharacteristic
124C2×C4×C83C2×M1611yescharacteristic
125C8×C83—84no—
126C22×C83—21no—
127C22×C83—21no—
128C22×C83—24no—
129C22×C83—44no—
130C22×C83—44no—
131C22×C83—42no—
132C22×C83—42no—
133C22×C83—42no—
134C22×C83—82no—
135C22×C83—82no—
136C22×C83—81no—
137C22×C83—81no—
138C22×C83C2×C22⋊C811yescharacteristic
139C22×C83C2×#9911yescharacteristic
140C22×C83#41311yescharacteristic
141C4×C83—24no—
142C4×C83—21no—
143C4×C83—21no—
144C4×C83—21no—
145C4×C83—42no—
146C4×C83—42no—
147C4×C83—41no—
148C4×C83—81no—
149C4×C83#35312yes—
150#11475C8311yesmaximal, characteristic, direct product factor, Sylow
151C2×C83—21no—
152C2×C83—21no—
153C2×C83—44no—
154C2×C83—82no—
155C2×C83#145911yescharacteristic
156C2×C83C2×#35311yescharacteristic
157C2×C83#213211yescharacteristic
158C4×#127C2×C8311yescharacteristic
159#1459C2×C8312yescomplemented
160#1325C2×C8312yescomplemented
161#1318C2×C8312yescomplemented
162C83#1147511yescharacteristic, direct product factor, Sylow
163C2×#127C4×C8311yescharacteristic
164C2×C4×D4—22no—
165C2×C4×D4C4×C8311yescharacteristic
166C2×C4×D4C22×C8311yescharacteristic
167C2×C22⋊C8—22no—
168#390—22no—
169#390C4×C8311yescharacteristic
170#390C22×C8311yescharacteristic
171#384C22×C8311yescharacteristic
172C43C4×C8311yescharacteristic
173#324C22×C8314yes—
174#323—22no—
175C22×D4—22no—
176C22×D4C2×C4×C8311yescharacteristic
177C22×D4C2×C4×C8311yescharacteristic
178C23×C4—21no—
179C23×C4D4×C8312yes—
180C22×C8—42no—
181#127—22no—
182#127C8×C8314yescomplemented
183C4×D4—41no—
184C4×D4—41no—
185C4×D4—48no—
186C4×D4—41no—
187C4×D4—41no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C2×C4×C8311yescharacteristic
190C2×C4⋊C4C23×C8311yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4D4×C8312yes—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42C2×C4×C8311yescharacteristic
198C2×C42C2×C4×C8311yescharacteristic
199C2×C42C2×C4×C8311yescharacteristic
200C22⋊C8—24no—
201C22⋊C8—42no—
202C24—21no—
203C24C22⋊C4×C8312yes—
204C2×D4—28no—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4C2×C8×C8312yes—
213C2×D4C2×C8×C8312yescomplemented
214C2×D4M16×C8312yes—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4C22⋊C4×C8311yescharacteristic
225C22×C4C22⋊C4×C8311yescharacteristic
226C22×C4C22×C4×C8311yescharacteristic
227C22×C4C2×D4×C8312yes—
228C2×C8—44no—
229C2×C8—84no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—44no—
233C22⋊C4—42no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—42no—
237C42—24no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—41no—
242C42—42no—
243C42C2×C8×C8312yescomplemented
244C23—22no—
245C23—22no—
246C23—22no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—41no—
252C23—41no—
253C23C22⋊C8×C8314yes—
254C23C2×C22⋊C4×C8311yescharacteristic
255D4—42no—
256D4—42no—
257D4—42no—
258D4—416no—
259D4—42no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4#99×C8311yescharacteristic
280C2×C4#99×C8311yescharacteristic
281C2×C4C22×C8×C8311yescharacteristic
282C2×C4C2×M16×C8311yescharacteristic
283C8—84no—
284C22—21no—
285C22—21no—
286C22—24no—
287C22—44no—
288C22—44no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—82no—
293C22—82no—
294C22—81no—
295C22—81no—
296C22C2×C22⋊C8×C8311yescharacteristic
297C22C2×#99×C8311yescharacteristic
298C22#413×C8311yescharacteristic
299C4—24no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—42no—
304C4—42no—
305C4—41no—
306C4—81no—
307C4#353×C8312yes—
308C2—21no—
309C2—21no—
310C2—44no—
311C2—82no—
312C2C83×#145911yescharacteristic
313C2C2×#353×C8311yescharacteristic
314C2C83×#213211yescharacteristic
subgroupG/H
centerC2×C4×C83#99
commutatorC2×C4C22×C8×C83
Frattini, cosocleC2×C4⋊C4C23×C83
socleC22×C83C2×#99
prime radicalC2×#127×C83C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C83 ◁ C22×C4×C83 ◁ C2×C4×D4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#1459C2semidirect
3. C83×#1325C2semidirect
4. C83×#1318C2semidirect
24. #127×C83C8semidirect
55. C2×D4×C83C2×C8semidirect
85. C42×C83C2×C8semidirect
150. #11475C83direct (N × H)
159. #1459C2×C83semidirect
160. #1325C2×C83semidirect
161. #1318C2×C83semidirect
162. C83#11475direct (N × H)
182. #127C8×C83semidirect
213. C2×D4C2×C8×C83semidirect
243. C42C2×C8×C83semidirect

Maximal subgroups

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

subgroupindex
C4×#127×C832
C83×#14592
C83×#13252
C83×#13182
#1147583