Finite Group Databasebeta

← C31×#13054 #21469813

Subgroups — C31×#13054 #21469813

Order 7936.

632 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C31×#3018C211yesmaximal, characteristic
2C31×#2979C211yesmaximal, characteristic, complemented
3C31×#2126C211yesmaximal, characteristic, complemented
4C31×#1794C212yesmaximal, complemented
5C31×#1395C212yesmaximal, complemented
6C31×#585C2211yescharacteristic
7C31×#575—21no—
8C31×#532C2211yescharacteristic
9C31×#456C412yescomplemented
10C31×#455—21no—
11C31×#454C412yescomplemented
12C31×#453—21no—
13C31×#436—22no—
14C8×D4×C31—22no—
15C31×#431—21no—
16C31×#421—22no—
17C2×C31×#104—21no—
18C31×#416—22no—
19C31×#413C2211yescharacteristic
20C31×#410C2212yes—
21C31×#408—21no—
22C31×#408C2212yes—
23C31×#354—21no—
24C31×#353—21no—
25C31×2−1+4—41no—
26C31×2−1+4C2×C412yes—
27C31×H(5,𝔽2)—21no—
28C31×H(5,𝔽2)—22no—
29C31×H(5,𝔽2)—41no—
30C2×#41×C31—22no—
31C2×#41×C31—21no—
32C2×#41×C31—42no—
33C2×#41×C31D411yescharacteristic
34C2×#41×C31C2311yescharacteristic
35C31×#137—22no—
36C31×#137—41no—
37C31×AGL(1,ℤ/8)—22no—
38C31×AGL(1,ℤ/8)—41no—
39C31×#135—42no—
40C2×M16×C31—21no—
41C2×M16×C31—21no—
42C2×M16×C31D412yes—
43C22×C8×C31—21no—
44C22×C8×C31D412yes—
45C31×C4⋊Q8C2×C411yescharacteristic
46C31×#127D411yescharacteristic
47C31×#124C2×C411yescharacteristic
48C31×#124D411yescharacteristic
49C31×#121—22no—
50C4×D4×C31—22no—
51C31×#117—22no—
52C2×C42×C31D411yescharacteristic
53C31×C4⋊C8—21no—
54C31×C4⋊C8—21no—
55C31×#104—22no—
56C31×#104—22no—
57C31×#104—44no—
58C31×#103—42no—
59C31×#102—42no—
60C31×#101C2×C412yes—
61C31×#100—21no—
62C31×#99—22no—
63C31×#99—22no—
64C31×C22⋊C8—22no—
65C31×C22⋊C8—22no—
66C31×C22⋊C8—22no—
67C4×C8×C31—22no—
68#41×C31—24no—
69#41×C31—22no—
70#41×C31—22no—
71#41×C31—24no—
72#41×C31—41no—
73#41×C31—41no—
74#41×C31—42no—
75#41×C31—41no—
76#41×C31—41no—
77#41×C31—41no—
78#41×C31—42no—
79#41×C31—42no—
80#41×C31—44no—
81#41×C31—41no—
82C2×Q8×C31—22no—
83C2×Q8×C31—21no—
84C2×Q8×C31—42no—
85C2×Q8×C31C22⋊C411yescharacteristic
86C2×Q8×C31C22×C411yescharacteristic
87C2×D4×C31—22no—
88C2×D4×C31—21no—
89C2×D4×C31—22no—
90C2×D4×C31—21no—
91C2×D4×C31—42no—
92C2×D4×C31—44no—
93C2×D4×C31—42no—
94C2×D4×C31—41no—
95C2×D4×C31C22⋊C411yescharacteristic
96C2×D4×C31C2×D411yescharacteristic
97C2×D4×C31#4112yes—
98C22×C4×C31—21no—
99C22×C4×C31—21no—
100C22×C4×C31—42no—
101C22×C4×C31—41no—
102C22×C4×C31C2×D411yescharacteristic
103C22×C4×C31C2×D412yes—
104Q16×C31—42no—
105SD16×C31—42no—
106SD16×C31—42no—
107D8×C31—42no—
108M16×C31—22no—
109M16×C31—41no—
110M16×C31—42no—
111M16×C31—81no—
112C2×C8×C31—22no—
113C2×C8×C31—22no—
114C2×C8×C31—41no—
115C2×C8×C31—42no—
116C2×C8×C31—81no—
117C4⋊C4×C31—22no—
118C4⋊C4×C31—42no—
119C22⋊C4×C31—22no—
120C22⋊C4×C31—22no—
121C22⋊C4×C31—42no—
122C42×C31—21no—
123C42×C31—42no—
124C42×C31C22⋊C411yescharacteristic
125C42×C31C22⋊C411yescharacteristic
126#13054C3111yesmaximal, characteristic, direct product factor, Sylow
127C23×C31—21no—
128C23×C31—42no—
129C23×C31—41no—
130C23×C31#12112yes—
131C23×C31#12311yescharacteristic
132Q8×C31—22no—
133Q8×C31—24no—
134Q8×C31—41no—
135Q8×C31—41no—
136Q8×C31—41no—
137D4×C31—22no—
138D4×C31—24no—
139D4×C31—41no—
140D4×C31—42no—
141D4×C31—41no—
142D4×C31—44no—
143D4×C31—44no—
144D4×C31—42no—
145D4×C31—42no—
146D4×C31—41no—
147D4×C31—81no—
148C2×C4×C31—21no—
149C2×C4×C31—22no—
150C2×C4×C31—21no—
151C2×C4×C31—22no—
152C2×C4×C31—21no—
153C2×C4×C31—21no—
154C2×C4×C31—21no—
155C2×C4×C31—42no—
156C2×C4×C31—42no—
157C2×C4×C31—44no—
158C2×C4×C31—42no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—42no—
163C2×C4×C31—82no—
164C2×C4×C31C2×C22⋊C411yescharacteristic
165C2×C4×C31C4×D412yes—
166C2×C4×C31#12011yescharacteristic
167C8×C31—42no—
168C8×C31—42no—
169C8×C31—42no—
170#3018C2×C3111yescharacteristic
171#2979C2×C3111yescharacteristic, complemented
172#2126C2×C3111yescharacteristic, complemented
173#1794C2×C3112yescomplemented
174#1395C2×C3112yescomplemented
175C22×C31—22no—
176C22×C31—21no—
177C22×C31—41no—
178C22×C31—42no—
179C22×C31—42no—
180C22×C31—42no—
181C22×C31—41no—
182C22×C31—162no—
183C22×C31—161no—
184C22×C31#38611yescharacteristic
185C4×C31—22no—
186C4×C31—21no—
187C4×C31—22no—
188C4×C31—41no—
189C4×C31—41no—
190C4×C31—42no—
191C4×C31—162no—
192C4×C31B(4,𝔽2)11yescharacteristic
193C4×C31B(4,𝔽2)11yescharacteristic
194#585C22×C3111yescharacteristic
195#575—21no—
196#532C22×C3111yescharacteristic
197#456C4×C3112yescomplemented
198#455—21no—
199#454C4×C3112yescomplemented
200#453—21no—
201#436—22no—
202C8×D4—22no—
203#431—21no—
204#421—22no—
205C2×#104—21no—
206#416—22no—
207#413C22×C3111yescharacteristic
208#410C22×C3112yes—
209#408—21no—
210#408C22×C3112yes—
211#354—21no—
212#353—21no—
213C2×C31—21no—
214C2×C31—42no—
215C2×C31—41no—
216C2×C31—81no—
217C2×C31—82no—
218C2×C31#188911yescharacteristic
2192−1+4—41no—
2202−1+4C2×C4×C3112yes—
221H(5,𝔽2)—21no—
222H(5,𝔽2)—22no—
223H(5,𝔽2)—41no—
224C2×#41—21no—
225C2×#41—22no—
226C2×#41—42no—
227C2×#41D4×C3111yescharacteristic
228C2×#41C23×C3111yescharacteristic
229#137—22no—
230#137—41no—
231AGL(1,ℤ/8)—22no—
232AGL(1,ℤ/8)—41no—
233#135—42no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16D4×C3112yes—
237C22×C8—21no—
238C22×C8D4×C3112yes—
239C4⋊Q8C2×C4×C3111yescharacteristic
240#127D4×C3111yescharacteristic
241#124C2×C4×C3111yescharacteristic
242#124D4×C3111yescharacteristic
243#121—22no—
244C4×D4—22no—
245#117—22no—
246C2×C42D4×C3111yescharacteristic
247C4⋊C8—21no—
248C4⋊C8—21no—
249#104—22no—
250#104—22no—
251#104—44no—
252#103—42no—
253#102—42no—
254#101C2×C4×C3112yes—
255#100—21no—
256#99—22no—
257#99—22no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C4×C8—22no—
262C31#1305411yescharacteristic, direct product factor, Sylow
263#41—24no—
264#41—22no—
265#41—22no—
266#41—24no—
267#41—41no—
268#41—41no—
269#41—42no—
270#41—41no—
271#41—41no—
272#41—41no—
273#41—42no—
274#41—42no—
275#41—44no—
276#41—41no—
277C2×Q8—22no—
278C2×Q8—21no—
279C2×Q8—42no—
280C2×Q8C22⋊C4×C3111yescharacteristic
281C2×Q8C22×C4×C3111yescharacteristic
282C2×D4—22no—
283C2×D4—21no—
284C2×D4—22no—
285C2×D4—21no—
286C2×D4—42no—
287C2×D4—44no—
288C2×D4—42no—
289C2×D4—41no—
290C2×D4C22⋊C4×C3111yescharacteristic
291C2×D4C2×D4×C3111yescharacteristic
292C2×D4#41×C3112yes—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4C2×D4×C3111yescharacteristic
298C22×C4C2×D4×C3112yes—
299Q16—42no—
300SD16—42no—
301SD16—42no—
302D8—42no—
303M16—22no—
304M16—41no—
305M16—42no—
306M16—81no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—81no—
312C4⋊C4—22no—
313C4⋊C4—42no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—42no—
317C42—21no—
318C42—42no—
319C42C22⋊C4×C3111yescharacteristic
320C42C22⋊C4×C3111yescharacteristic
321C23—21no—
322C23—42no—
323C23—41no—
324C23C31×#12112yes—
325C23C31×#12311yescharacteristic
326Q8—22no—
327Q8—24no—
328Q8—41no—
329Q8—41no—
330Q8—41no—
331D4—22no—
332D4—24no—
333D4—41no—
334D4—42no—
335D4—41no—
336D4—44no—
337D4—44no—
338D4—42no—
339D4—42no—
340D4—41no—
341D4—81no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4C2×C22⋊C4×C3111yescharacteristic
359C2×C4C4×D4×C3112yes—
360C2×C4C31×#12011yescharacteristic
361C8—42no—
362C8—42no—
363C8—42no—
364C22—22no—
365C22—21no—
366C22—41no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—41no—
371C22—162no—
372C22—161no—
373C22C31×#38611yescharacteristic
374C4—22no—
375C4—21no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—42no—
380C4—162no—
381C4C31×B(4,𝔽2)11yescharacteristic
382C4C31×B(4,𝔽2)11yescharacteristic
383C2—21no—
384C2—42no—
385C2—41no—
386C2—81no—
387C2—82no—
388C2C31×#188911yescharacteristic
subgroupG/H
centerC4×C31B(4,𝔽2)
commutatorC2×Q8C22×C4×C31
Frattini, cosocleC2×#41C23×C31
socleC2×C31#1889
prime radicalC31×#3018C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C31 ◁ C2×C4×C31 ◁ C2×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2979C2semidirect
3. C31×#2126C2semidirect
4. C31×#1794C2semidirect
5. C31×#1395C2semidirect
9. C31×#456C4semidirect
11. C31×#454C4semidirect
126. #13054C31direct (N × H)
171. #2979C2×C31semidirect
172. #2126C2×C31semidirect
173. #1794C2×C31semidirect
174. #1395C2×C31semidirect
197. #456C4×C31semidirect
199. #454C4×C31semidirect
262. C31#13054direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#30182
C31×#29792
C31×#21262
C31×#17942
C31×#13952
#1305431