Finite Group Databasebeta

← C83×#7784 #35235608

Subgroups — C83×#7784 #35235608

Order 21248.

436 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2806C211yesmaximal, characteristic
2C83×#2057C211yesmaximal, characteristic
3C83×#2024C211yesmaximal, characteristic
4C83×#2016C211yesmaximal, characteristic
5C83×#1272C211yesmaximal, characteristic
6C83×#1272C211yesmaximal, characteristic
7C83×#1271C211yesmaximal, characteristic
8C22×Q16×C83—21no—
9C2×C4⋊Q8×C83C2211yescharacteristic
10C2×#107×C83—21no—
11C2×#103×C83—21no—
12C2×#103×C83—21no—
13#398×C83—21no—
14#398×C83—21no—
15#395×C83—21no—
16#391×C83—21no—
17#391×C83C2211yescharacteristic
18#391×C83C2211yescharacteristic
19#389×C83—21no—
20#389×C83C2211yescharacteristic
21#387×C83—21no—
22#340×C83—21no—
23#340×C83—21no—
24#336×C83C2211yescharacteristic
25#336×C83C2211yescharacteristic
26#336×C83C2211yescharacteristic
27C22×Q8×C83—21no—
28C22×Q8×C83D411yescharacteristic
29C22×Q8×C83D411yescharacteristic
30C2×Q16×C83—42no—
31C2×Q16×C83—42no—
32C2×Q16×C83—42no—
33C22×C8×C83—21no—
34C22×C8×C83—21no—
35C22×C8×C83—21no—
36C4⋊Q8×C83—42no—
37C2×C4⋊C4×C83—21no—
38C2×C4⋊C4×C83—21no—
39C2×C4⋊C4×C83—21no—
40C2×C4⋊C4×C83—21no—
41C2×C4⋊C4×C83—21no—
42C2×C4⋊C4×C83—21no—
43C2×C4⋊C4×C83—21no—
44C2×C4⋊C4×C83D411yescharacteristic
45C2×C4⋊C4×C83D411yescharacteristic
46C2×C4⋊C4×C83D411yescharacteristic
47C2×C4⋊C4×C83D411yescharacteristic
48C2×C42×C83—21no—
49C2×C42×C83C2311yescharacteristic
50#107×C83—42no—
51#103×C83—42no—
52#103×C83—42no—
53#95×C83—21no—
54#95×C83—21no—
55#95×C83—21no—
56#95×C83—21no—
57#95×C83—21no—
58#95×C83—21no—
59#95×C83—21no—
60#95×C83—21no—
61C2×Q8×C83—21no—
62C2×Q8×C83—22no—
63C2×Q8×C83—21no—
64C2×Q8×C83—21no—
65C2×Q8×C83—41no—
66C2×Q8×C83—41no—
67C2×Q8×C83—41no—
68C2×Q8×C83—41no—
69C2×Q8×C83—41no—
70C2×Q8×C83—41no—
71C2×Q8×C83—41no—
72C2×Q8×C83—41no—
73C2×Q8×C83—41no—
74C2×Q8×C83Q1612yescomplemented
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—21no—
81C22×C4×C83—41no—
82C22×C4×C83C2×D411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C22×C4×C83C2×D411yescharacteristic
85Q16×C83—48no—
86C2×C8×C83—22no—
87C2×C8×C83—22no—
88C2×C8×C83—41no—
89C2×C8×C83—41no—
90C2×C8×C83—41no—
91C2×C8×C83—41no—
92C2×C8×C83—41no—
93C2×C8×C83—41no—
94C2×C8×C83—41no—
95C4⋊C4×C83—21no—
96C4⋊C4×C83—22no—
97C4⋊C4×C83—22no—
98C4⋊C4×C83—41no—
99C4⋊C4×C83—41no—
100C4⋊C4×C83—41no—
101C4⋊C4×C83—41no—
102C4⋊C4×C83—41no—
103C4⋊C4×C83—41no—
104C4⋊C4×C83—41no—
105C4⋊C4×C83—81no—
106C4⋊C4×C83—81no—
107C4⋊C4×C83—81no—
108C4⋊C4×C83—81no—
109C4⋊C4×C83Q1612yescomplemented
110C42×C83—41no—
111C42×C83—81no—
112C23×C83#12011yescharacteristic
113Q8×C83—42no—
114Q8×C83—42no—
115Q8×C83—42no—
116Q8×C83—42no—
117Q8×C83—81no—
118Q8×C83—81no—
119Q8×C83—81no—
120Q8×C83—81no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—22no—
128C2×C4×C83—22no—
129C2×C4×C83—41no—
130C2×C4×C83—41no—
131C2×C4×C83—41no—
132C2×C4×C83—41no—
133C2×C4×C83—41no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83—41no—
139C2×C4×C83—41no—
140C2×C4×C83—41no—
141C2×C4×C83—41no—
142C2×C4×C83—41no—
143C2×C4×C83—42no—
144C2×C4×C83—41no—
145C2×C4×C83—41no—
146C2×C4×C83—81no—
147C2×C4×C83—81no—
148C2×C4×C83C2×Q1611yescharacteristic
149C2×C4×C83C2×Q1611yescharacteristic
150C2×C4×C83#13511yescharacteristic
151C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C83#13711yescharacteristic
153C2×C4×C83#13711yescharacteristic
154C8×C83—42no—
155C8×C83—42no—
156C8×C83—81no—
157C22×C83#44911yescharacteristic
158C22×C83#45111yescharacteristic
159C22×C83#45111yescharacteristic
160C22×C83#45311yescharacteristic
161C22×C83#45511yescharacteristic
162C22×C83#45611yescharacteristic
163C22×C83B(4,𝔽2)11yescharacteristic
164C4×C83—21no—
165C4×C83—21no—
166C4×C83—21no—
167C4×C83—21no—
168C4×C83—21no—
169C4×C83—21no—
170C4×C83—42no—
171C4×C83—42no—
172C4×C83—81no—
173C4×C83—81no—
174C4×C83—82no—
175C4×C83—81no—
176C4×C83—81no—
177#7784C8311yesmaximal, characteristic, direct product factor, Sylow
178C2×C83#160011yescharacteristic
179C2×C83#160211yescharacteristic
180C2×C83#160511yescharacteristic
181C2×C83#163311yescharacteristic
182C2×C83#164411yescharacteristic
183C2×C83#164811yescharacteristic
184C2×C83#166111yescharacteristic
185#2806C2×C8311yescharacteristic
186#2057C2×C8311yescharacteristic
187#2024C2×C8311yescharacteristic
188#2016C2×C8311yescharacteristic
189#1272C2×C8311yescharacteristic
190#1272C2×C8311yescharacteristic
191#1271C2×C8311yescharacteristic
192C83#778411yescharacteristic, direct product factor, Sylow
193C22×Q16—21no—
194C2×C4⋊Q8C22×C8311yescharacteristic
195C2×#107—21no—
196C2×#103—21no—
197C2×#103—21no—
198#398—21no—
199#398—21no—
200#395—21no—
201#391—21no—
202#391C22×C8311yescharacteristic
203#391C22×C8311yescharacteristic
204#389—21no—
205#389C22×C8311yescharacteristic
206#387—21no—
207#340—21no—
208#340—21no—
209#336C22×C8311yescharacteristic
210#336C22×C8311yescharacteristic
211#336C22×C8311yescharacteristic
212C22×Q8—21no—
213C22×Q8D4×C8311yescharacteristic
214C22×Q8D4×C8311yescharacteristic
215C2×Q16—42no—
216C2×Q16—42no—
217C2×Q16—42no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C4⋊Q8—42no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4D4×C8311yescharacteristic
230C2×C4⋊C4D4×C8311yescharacteristic
231C2×C4⋊C4D4×C8311yescharacteristic
232C2×C4⋊C4D4×C8311yescharacteristic
233C2×C42—21no—
234C2×C42C23×C8311yescharacteristic
235#107—42no—
236#103—42no—
237#103—42no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8Q16×C8312yescomplemented
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—41no—
267C22×C4C2×D4×C8311yescharacteristic
268C22×C4C2×D4×C8311yescharacteristic
269C22×C4C2×D4×C8311yescharacteristic
270Q16—48no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C4⋊C4—21no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—81no—
291C4⋊C4—81no—
292C4⋊C4—81no—
293C4⋊C4—81no—
294C4⋊C4Q16×C8312yescomplemented
295C42—41no—
296C42—81no—
297C23#120×C8311yescharacteristic
298Q8—42no—
299Q8—42no—
300Q8—42no—
301Q8—42no—
302Q8—81no—
303Q8—81no—
304Q8—81no—
305Q8—81no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4C2×Q16×C8311yescharacteristic
334C2×C4C2×Q16×C8311yescharacteristic
335C2×C4#135×C8311yescharacteristic
336C2×C4AGL(1,ℤ/8)×C8311yescharacteristic
337C2×C4#137×C8311yescharacteristic
338C2×C4#137×C8311yescharacteristic
339C8—42no—
340C8—42no—
341C8—81no—
342C22#449×C8311yescharacteristic
343C22#451×C8311yescharacteristic
344C22#451×C8311yescharacteristic
345C22#453×C8311yescharacteristic
346C22#455×C8311yescharacteristic
347C22#456×C8311yescharacteristic
348C22B(4,𝔽2)×C8311yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—42no—
356C4—42no—
357C4—81no—
358C4—81no—
359C4—82no—
360C4—81no—
361C4—81no—
362C2C83×#160011yescharacteristic
363C2C83×#160211yescharacteristic
364C2C83×#160511yescharacteristic
365C2C83×#163311yescharacteristic
366C2C83×#164411yescharacteristic
367C2C83×#164811yescharacteristic
368C2C83×#166111yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C83#120
commutator, Frattini, cosocleC2×C42C23×C83
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
74. C2×Q8×C83Q16semidirect
109. C4⋊C4×C83Q16semidirect
177. #7784C83direct (N × H)
192. C83#7784direct (N × H)
259. C2×Q8Q16×C83semidirect
294. C4⋊C4Q16×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#28062
C83×#20572
C83×#20242
C83×#20162
C83×#12722
C83×#12712
#778483