Finite Group Databasebeta

← C31×#13357 #21470116

Subgroups — C31×#13357 #21470116

Order 7936.

772 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C2×C31×#454C212yesmaximal, complemented
2C31×#2733C211yesmaximal, characteristic, complemented
3C2×C31×#355C212yesmaximal, complemented
4C31×#1793C211yesmaximal, characteristic, complemented
5C31×#1394C211yesmaximal, characteristic, complemented
6C2×C31×2−1+4C2211yescharacteristic
7C2×C31×AGL(1,ℤ/8)—22no—
8C2×C31×#124C2212yes—
9C31×#454—24no—
10C31×#454—24no—
11C2×C31×#104—22no—
12C2×C31×#104—21no—
13C31×#417—21no—
14C2×C31×#101C2211yescharacteristic
15C2×C31×#101C2212yes—
16C31×#394—21no—
17C31×#392—22no—
18C31×#391C2211yescharacteristic
19C31×#390—21no—
20C31×#388—22no—
21C31×#355—24no—
22C31×2−1+4—22no—
23C31×2−1+4—21no—
24C31×2−1+4—42no—
25C31×2−1+4D412yescomplemented
26C2×#41×C31—21no—
27C2×#41×C31—22no—
28C2×#41×C31—41no—
29C22×Q8×C31—41no—
30C22×Q8×C31C2311yescharacteristic
31C22×D4×C31—21no—
32C22×D4×C31—21no—
33C22×D4×C31D412yes—
34C31×AGL(1,ℤ/8)—28no—
35C31×AGL(1,ℤ/8)—44no—
36C2×SD16×C31—42no—
37C2×D8×C31—42no—
38C2×M16×C31—22no—
39C2×M16×C31—21no—
40C31×#124—22no—
41C31×#124—22no—
42C31×#124—22no—
43C31×#124—22no—
44C2×C4⋊C4×C31—21no—
45C2×C4⋊C4×C31D411yescharacteristic
46C2×C22⋊C4×C31—22no—
47C2×C22⋊C4×C31—22no—
48C2×C22⋊C4×C31—22no—
49C2×C22⋊C4×C31—22no—
50C2×C42×C31D412yes—
51C2×C42×C31D411yescharacteristic
52C31×#104—28no—
53C31×#104—42no—
54C31×#103—41no—
55C31×#102—41no—
56C31×#101—22no—
57C31×#101—22no—
58C31×#101D414yescomplemented
59C31×#95—21no—
60C31×#95—22no—
61C24×C31—22no—
62#41×C31—22no—
63#41×C31—24no—
64#41×C31—24no—
65#41×C31—22no—
66#41×C31—41no—
67#41×C31—42no—
68#41×C31—42no—
69#41×C31—44no—
70#41×C31—42no—
71#41×C31—42no—
72#41×C31—41no—
73C2×Q8×C31—21no—
74C2×Q8×C31—22no—
75C2×Q8×C31—41no—
76C2×Q8×C31—41no—
77C2×Q8×C31—42no—
78C2×Q8×C31—44no—
79C2×Q8×C31—42no—
80C2×Q8×C31C2×D411yescharacteristic
81C2×Q8×C31C2×D412yes—
82C2×Q8×C31#4111yescharacteristic
83C2×D4×C31—21no—
84C2×D4×C31—22no—
85C2×D4×C31—22no—
86C2×D4×C31—22no—
87C2×D4×C31—42no—
88C2×D4×C31—42no—
89C2×D4×C31—42no—
90C2×D4×C31—41no—
91C2×D4×C31—81no—
92C2×D4×C31—81no—
93C2×D4×C31—82no—
94C2×D4×C31—81no—
95C22×C4×C31—21no—
96C22×C4×C31—21no—
97C22×C4×C31—22no—
98C22×C4×C31—41no—
99C22×C4×C31—42no—
100C22×C4×C31C2×D412yes—
101C22×C4×C31C2×D411yescharacteristic
102SD16×C31—48no—
103D8×C31—48no—
104M16×C31—24no—
105M16×C31—42no—
106M16×C31—41no—
107M16×C31—41no—
108C2×C8×C31—42no—
109C2×C8×C31—41no—
110C4⋊C4×C31—41no—
111C4⋊C4×C31—81no—
112C22⋊C4×C31—44no—
113C22⋊C4×C31—42no—
114C22⋊C4×C31—42no—
115C22⋊C4×C31—42no—
116C22⋊C4×C31—42no—
117C22⋊C4×C31—82no—
118C42×C31—22no—
119C42×C31—22no—
120C42×C31—41no—
121#13357C3111yesmaximal, characteristic, direct product factor, Sylow
122C23×C31—42no—
123C23×C31—42no—
124C23×C31—42no—
125C23×C31—42no—
126C23×C31—41no—
127C23×C31—42no—
128C23×C31—42no—
129C23×C31—42no—
130C23×C31#12011yescharacteristic
131Q8×C31—24no—
132Q8×C31—22no—
133Q8×C31—41no—
134Q8×C31—42no—
135Q8×C31—41no—
136Q8×C31—41no—
137Q8×C31—42no—
138D4×C31—24no—
139D4×C31—22no—
140D4×C31—41no—
141D4×C31—42no—
142D4×C31—44no—
143D4×C31—42no—
144D4×C31—42no—
145D4×C31—82no—
146D4×C31—84no—
147D4×C31—84no—
148C2×C4×C31—21no—
149C2×C4×C31—22no—
150C2×C4×C31—21no—
151C2×C4×C31—22no—
152C2×C4×C31—22no—
153C2×C4×C31—44no—
154C2×C4×C31—42no—
155C2×C4×C31—42no—
156C2×C4×C31—44no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—42no—
161C2×C4×C31—42no—
162C2×C4×C31—42no—
163C2×C4×C31—42no—
164C2×C4×C31—41no—
165C2×C4×C31#12012yes—
166C2×C4×C31#12112yes—
167C2×C4×C31#12111yescharacteristic
168C2×C4×C31#12711yescharacteristic
169C8×C31—44no—
170C8×C31—81no—
171C2×#454C2×C3112yescomplemented
172#2733C2×C3111yescharacteristic, complemented
173C2×#355C2×C3112yescomplemented
174#1793C2×C3111yescharacteristic, complemented
175#1394C2×C3111yescharacteristic, complemented
176C22×C31—21no—
177C22×C31—21no—
178C22×C31—42no—
179C22×C31—82no—
180C22×C31—82no—
181C22×C31—82no—
182C22×C31—82no—
183C22×C31—81no—
184C22×C31—81no—
185C22×C31—82no—
186C22×C31—82no—
187C22×C31—82no—
188C22×C31#39211yescharacteristic
189C22×C31B(4,𝔽2)11yescharacteristic
190C22×C31B(4,𝔽2)11yescharacteristic
191C4×C31—22no—
192C4×C31—21no—
193C4×C31—21no—
194C4×C31—22no—
195C4×C31—42no—
196C4×C31—42no—
197C4×C31—82no—
198C4×C31—81no—
199C2×2−1+4C22×C3111yescharacteristic
200C2×AGL(1,ℤ/8)—22no—
201C2×#124C22×C3112yes—
202#454—24no—
203#454—24no—
204C2×#104—22no—
205C2×#104—21no—
206#417—21no—
207C2×#101C22×C3111yescharacteristic
208C2×#101C22×C3112yes—
209#394—21no—
210#392—22no—
211#391C22×C3111yescharacteristic
212#390—21no—
213#388—22no—
214#355—24no—
215C2×C31—21no—
216C2×C31—21no—
217C2×C31—82no—
218C2×C31—162no—
219C2×C31#202111yescharacteristic
220C2×C31#220211yescharacteristic
221C2×C31#220411yescharacteristic
2222−1+4—22no—
2232−1+4—21no—
2242−1+4—42no—
2252−1+4D4×C3112yescomplemented
226C2×#41—22no—
227C2×#41—21no—
228C2×#41—41no—
229C22×Q8—41no—
230C22×Q8C23×C3111yescharacteristic
231C22×D4—21no—
232C22×D4—21no—
233C22×D4D4×C3112yes—
234AGL(1,ℤ/8)—28no—
235AGL(1,ℤ/8)—44no—
236C2×SD16—42no—
237C2×D8—42no—
238C2×M16—22no—
239C2×M16—21no—
240#124—22no—
241#124—22no—
242#124—22no—
243#124—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4D4×C3111yescharacteristic
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C42D4×C3112yes—
251C2×C42D4×C3111yescharacteristic
252#104—28no—
253#104—42no—
254#103—41no—
255#102—41no—
256#101—22no—
257#101—22no—
258#101D4×C3114yescomplemented
259#95—21no—
260#95—22no—
261C31#1335711yescharacteristic, direct product factor, Sylow
262C24—22no—
263#41—22no—
264#41—24no—
265#41—24no—
266#41—22no—
267#41—41no—
268#41—42no—
269#41—42no—
270#41—44no—
271#41—42no—
272#41—42no—
273#41—41no—
274C2×Q8—21no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—42no—
279C2×Q8—44no—
280C2×Q8—42no—
281C2×Q8C2×D4×C3111yescharacteristic
282C2×Q8C2×D4×C3112yes—
283C2×Q8#41×C3111yescharacteristic
284C2×D4—21no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—41no—
292C2×D4—81no—
293C2×D4—81no—
294C2×D4—82no—
295C2×D4—81no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4C2×D4×C3112yes—
302C22×C4C2×D4×C3111yescharacteristic
303SD16—48no—
304D8—48no—
305M16—24no—
306M16—42no—
307M16—41no—
308M16—41no—
309C2×C8—42no—
310C2×C8—41no—
311C4⋊C4—41no—
312C4⋊C4—81no—
313C22⋊C4—44no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—82no—
319C42—22no—
320C42—22no—
321C42—41no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—41no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23C31×#12011yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—41no—
334Q8—42no—
335Q8—41no—
336Q8—41no—
337Q8—42no—
338D4—24no—
339D4—22no—
340D4—41no—
341D4—42no—
342D4—44no—
343D4—42no—
344D4—42no—
345D4—82no—
346D4—84no—
347D4—84no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4C31×#12012yes—
366C2×C4C31×#12112yes—
367C2×C4C31×#12111yescharacteristic
368C2×C4C31×#12711yescharacteristic
369C8—44no—
370C8—81no—
371C22—21no—
372C22—21no—
373C22—42no—
374C22—82no—
375C22—82no—
376C22—82no—
377C22—82no—
378C22—81no—
379C22—81no—
380C22—82no—
381C22—82no—
382C22—82no—
383C22C31×#39211yescharacteristic
384C22C31×B(4,𝔽2)11yescharacteristic
385C22C31×B(4,𝔽2)11yescharacteristic
386C4—22no—
387C4—21no—
388C4—21no—
389C4—22no—
390C4—42no—
391C4—42no—
392C4—82no—
393C4—81no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—162no—
398C2C31×#202111yescharacteristic
399C2C31×#220211yescharacteristic
400C2C31×#220411yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC22×Q8C23×C31
LCSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C22×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#454C2semidirect
2. C31×#2733C2semidirect
3. C2×C31×#355C2semidirect
4. C31×#1793C2semidirect
5. C31×#1394C2semidirect
25. C31×2−1+4D4semidirect
58. C31×#101D4semidirect
121. #13357C31direct (N × H)
171. C2×#454C2×C31semidirect
172. #2733C2×C31semidirect
173. C2×#355C2×C31semidirect
174. #1793C2×C31semidirect
175. #1394C2×C31semidirect
225. 2−1+4D4×C31semidirect
258. #101D4×C31semidirect
261. C31#13357direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4542
C31×#27332
C2×C31×#3552
C31×#17932
C31×#13942
#1335731