Finite Group Databasebeta

← C83×#10321 #35238145

Subgroups — C83×#10321 #35238145

Order 21248.

726 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3186C211yesmaximal, characteristic
2C83×#2854C212yesmaximal, complemented
3C83×#1817C211yesmaximal, characteristic, complemented
4C83×#1519C212yesmaximal, complemented
5C83×#1289C211yesmaximal, characteristic, complemented
6C2×#123×C83—22no—
7C2×#121×C83C2212yes—
8C22×C42×C83C2211yescharacteristic
9#480×C83—22no—
10#466×C83—22no—
11#447×C83—22no—
12C2×C22⋊C8×C83C2211yescharacteristic
13C2×C22⋊C8×C83C2211yescharacteristic
14#390×C83—22no—
15#389×C83—22no—
16#388×C83—24no—
17#386×C83—22no—
18#380×C83C2212yes—
19#336×C83—22no—
20#327×C83—24no—
21C22×D4×C83—22no—
22C22×D4×C83D412yes—
23C23×C4×C83D411yescharacteristic
24C23×C4×C83D411yescharacteristic
25C23×C4×C83C2311yescharacteristic
26C2×D8×C83—22no—
27C22×C8×C83—21no—
28C22×C8×C83D412yes—
29#123×C83—44no—
30#123×C83—44no—
31#121×C83—22no—
32#121×C83—22no—
33#121×C83—22no—
34#121×C83—22no—
35C2×C4⋊C4×C83—24no—
36C2×C4⋊C4×C83—22no—
37C2×C4⋊C4×C83D412yes—
38C2×C22⋊C4×C83—24no—
39C2×C22⋊C4×C83—24no—
40C2×C22⋊C4×C83—22no—
41C2×C22⋊C4×C83—22no—
42C2×C42×C83—21no—
43C2×C42×C83—22no—
44C2×C42×C83—22no—
45C2×C42×C83—21no—
46#107×C83—22no—
47#102×C83—24no—
48C22⋊C8×C83—22no—
49C22⋊C8×C83—22no—
50#95×C83—24no—
51#95×C83D414yes—
52C24×C83—22no—
53C24×C83C2×D411yescharacteristic
54C2×D4×C83—22no—
55C2×D4×C83—22no—
56C2×D4×C83—44no—
57C2×D4×C83—42no—
58C2×D4×C83—42no—
59C2×D4×C83—42no—
60C2×D4×C83—42no—
61C2×D4×C83—42no—
62C2×D4×C83—42no—
63C22×C4×C83—22no—
64C22×C4×C83—22no—
65C22×C4×C83—21no—
66C22×C4×C83—21no—
67C22×C4×C83—21no—
68C22×C4×C83—22no—
69C22×C4×C83—22no—
70C22×C4×C83—24no—
71C22×C4×C83—42no—
72C22×C4×C83—42no—
73C22×C4×C83D812yescomplemented
74C22×C4×C83C2×D412yes—
75C22×C4×C83C2×D411yescharacteristic
76C22×C4×C83C2×D412yes—
77C22×C4×C83#4111yescharacteristic
78D8×C83—82no—
79C2×C8×C83—22no—
80C2×C8×C83—42no—
81C2×C8×C83—42no—
82C2×C8×C83—41no—
83C4⋊C4×C83—22no—
84C4⋊C4×C83—22no—
85C4⋊C4×C83—44no—
86C4⋊C4×C83—48no—
87C22⋊C4×C83—48no—
88C22⋊C4×C83—48no—
89C22⋊C4×C83—42no—
90C22⋊C4×C83—42no—
91C22⋊C4×C83—42no—
92C22⋊C4×C83—42no—
93C42×C83—41no—
94C42×C83—42no—
95C42×C83—41no—
96C23×C83—41no—
97C23×C83—41no—
98C23×C83—44no—
99C23×C83—44no—
100C23×C83—42no—
101C23×C83—42no—
102C23×C83—42no—
103C23×C83#12012yes—
104C23×C83#12111yescharacteristic
105C23×C83C2×D811yescharacteristic
106C23×C83#13511yescharacteristic
107C23×C83AGL(1,ℤ/8)12yes—
108D4×C83—44no—
109D4×C83—82no—
110D4×C83—82no—
111D4×C83—82no—
112D4×C83—82no—
113D4×C83—82no—
114C2×C4×C83—24no—
115C2×C4×C83—22no—
116C2×C4×C83—24no—
117C2×C4×C83—22no—
118C2×C4×C83—24no—
119C2×C4×C83—22no—
120C2×C4×C83—42no—
121C2×C4×C83—41no—
122C2×C4×C83—42no—
123C2×C4×C83—41no—
124C2×C4×C83—41no—
125C2×C4×C83—41no—
126C2×C4×C83—42no—
127C2×C4×C83—41no—
128C2×C4×C83—44no—
129C2×C4×C83—44no—
130C2×C4×C83—42no—
131C2×C4×C83—44no—
132C2×C4×C83—42no—
133C2×C4×C83—42no—
134C2×C4×C83#12011yescharacteristic
135C2×C4×C83#12112yes—
136C2×C4×C83#12711yescharacteristic
137C8×C83—42no—
138C8×C83—81no—
139C22×C83—22no—
140C22×C83—21no—
141C22×C83—22no—
142C22×C83—21no—
143C22×C83—41no—
144C22×C83—41no—
145C22×C83—42no—
146C22×C83—84no—
147C22×C83—84no—
148C22×C83—82no—
149C22×C83—82no—
150C22×C83—82no—
151C22×C83#39211yescharacteristic
152C22×C83#44712yes—
153C22×C83#44912yes—
154C22×C83#46611yescharacteristic
155C22×C83#46911yescharacteristic
156C4×C83—21no—
157C4×C83—22no—
158C4×C83—42no—
159C4×C83—42no—
160C4×C83—84no—
161C4×C83—82no—
162C4×C83#45812yes—
163#10321C8311yesmaximal, characteristic, direct product factor, Sylow
164C2×C83—22no—
165C2×C83—22no—
166C2×C83—21no—
167C2×C83—21no—
168C2×C83—162no—
169C2×C83#199911yescharacteristic
170C2×C83#201011yescharacteristic
171C2×C83#202211yescharacteristic
172#3186C2×C8311yescharacteristic
173#2854C2×C8312yescomplemented
174#1817C2×C8311yescharacteristic, complemented
175#1519C2×C8312yescomplemented
176#1289C2×C8311yescharacteristic, complemented
177C83#1032111yescharacteristic, direct product factor, Sylow
178C2×#123—22no—
179C2×#121C22×C8312yes—
180C22×C42C22×C8311yescharacteristic
181#480—22no—
182#466—22no—
183#447—22no—
184C2×C22⋊C8C22×C8311yescharacteristic
185C2×C22⋊C8C22×C8311yescharacteristic
186#390—22no—
187#389—22no—
188#388—24no—
189#386—22no—
190#380C22×C8312yes—
191#336—22no—
192#327—24no—
193C22×D4—22no—
194C22×D4D4×C8312yes—
195C23×C4D4×C8311yescharacteristic
196C23×C4D4×C8311yescharacteristic
197C23×C4C23×C8311yescharacteristic
198C2×D8—22no—
199C22×C8—21no—
200C22×C8D4×C8312yes—
201#123—44no—
202#123—44no—
203#121—22no—
204#121—22no—
205#121—22no—
206#121—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4D4×C8312yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—21no—
218#107—22no—
219#102—24no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222#95—24no—
223#95D4×C8314yes—
224C24—22no—
225C24C2×D4×C8311yescharacteristic
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—44no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4D8×C8312yescomplemented
246C22×C4C2×D4×C8312yes—
247C22×C4C2×D4×C8311yescharacteristic
248C22×C4C2×D4×C8312yes—
249C22×C4#41×C8311yescharacteristic
250D8—82no—
251C2×C8—22no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—41no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—48no—
258C4⋊C4—44no—
259C22⋊C4—48no—
260C22⋊C4—48no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C42—41no—
266C42—42no—
267C42—41no—
268C23—41no—
269C23—41no—
270C23—44no—
271C23—44no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23#120×C8312yes—
276C23#121×C8311yescharacteristic
277C23C2×D8×C8311yescharacteristic
278C23#135×C8311yescharacteristic
279C23AGL(1,ℤ/8)×C8312yes—
280D4—44no—
281D4—82no—
282D4—82no—
283D4—82no—
284D4—82no—
285D4—82no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4#120×C8311yescharacteristic
307C2×C4#121×C8312yes—
308C2×C4#127×C8311yescharacteristic
309C8—42no—
310C8—81no—
311C22—22no—
312C22—21no—
313C22—22no—
314C22—21no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—84no—
319C22—84no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22#392×C8311yescharacteristic
324C22#447×C8312yes—
325C22#449×C8312yes—
326C22#466×C8311yescharacteristic
327C22#469×C8311yescharacteristic
328C4—21no—
329C4—22no—
330C4—42no—
331C4—42no—
332C4—84no—
333C4—82no—
334C4#458×C8312yes—
335C2—22no—
336C2—22no—
337C2—21no—
338C2—21no—
339C2—162no—
340C2C83×#199911yescharacteristic
341C2C83×#201011yescharacteristic
342C2C83×#202211yescharacteristic
subgroupG/H
center, socleC22×C83#392
commutator, Frattini, cosocleC23×C4C23×C83
prime radicalC83×#3186C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C23×C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#2854C2semidirect
3. C83×#1817C2semidirect
4. C83×#1519C2semidirect
5. C83×#1289C2semidirect
73. C22×C4×C83D8semidirect
163. #10321C83direct (N × H)
173. #2854C2×C83semidirect
174. #1817C2×C83semidirect
175. #1519C2×C83semidirect
176. #1289C2×C83semidirect
177. C83#10321direct (N × H)
245. C22×C4D8×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#31862
C83×#28542
C83×#18172
C83×#15192
C83×#12892
#1032183