Finite Group Databasebeta

← C31×#12198 #21468957

Subgroups — C31×#12198 #21468957

Order 7936.

474 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2828C211yesmaximal, characteristic, complemented
2C31×#2006C211yesmaximal, characteristic, complemented
3C31×#1616C211yesmaximal, characteristic, complemented
4C31×#1598C211yesmaximal, characteristic, complemented
5C31×#1353C211yesmaximal, characteristic, complemented
6C31×#1345C211yesmaximal, characteristic, complemented
7C31×#1318C211yesmaximal, characteristic, complemented
8C2×C31×AGL(1,ℤ/8)—21no—
9C31×#537C2211yescharacteristic
10C31×#481—21no—
11C31×#447—21no—
12C2×C31×#104—21no—
13C31×#400C2211yescharacteristic
14C31×#397—21no—
15C31×#394—21no—
16C31×#394—21no—
17C31×#394C2211yescharacteristic
18C31×#392—21no—
19C31×#392—21no—
20C31×#390C2211yescharacteristic
21C31×#388—21no—
22C31×#343C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C2×#41×C31—21no—
28C22×D4×C31—21no—
29C22×D4×C31—21no—
30C22×D4×C31D411yescharacteristic
31C22×D4×C31D411yescharacteristic
32C31×AGL(1,ℤ/8)—42no—
33C31×AGL(1,ℤ/8)—42no—
34C2×SD16×C31—41no—
35C2×D8×C31—41no—
36C2×D8×C31—41no—
37C2×M16×C31—21no—
38C31×C22⋊Q8—41no—
39C31×#121—21no—
40C31×#121—21no—
41C31×#121—41no—
42C2×C4⋊C4×C31—21no—
43C2×C4⋊C4×C31D411yescharacteristic
44C2×C4⋊C4×C31C2311yescharacteristic
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—21no—
48C2×C22⋊C4×C31—21no—
49C2×C22⋊C4×C31—21no—
50C2×C22⋊C4×C31—21no—
51C2×C22⋊C4×C31—21no—
52C2×C22⋊C4×C31—21no—
53C2×C42×C31D411yescharacteristic
54C31×#106—41no—
55C31×#104—42no—
56C31×#102—41no—
57C31×#102—41no—
58C31×C22⋊C8—21no—
59C31×C22⋊C8—21no—
60C31×#95—21no—
61C31×#95—21no—
62C31×#95—21no—
63C31×#95D411yescharacteristic
64C31×#95D411yescharacteristic
65C24×C31—21no—
66C24×C31—21no—
67#41×C31—41no—
68#41×C31—41no—
69#41×C31—41no—
70#41×C31—41no—
71C2×Q8×C31—21no—
72C2×D4×C31—21no—
73C2×D4×C31—21no—
74C2×D4×C31—21no—
75C2×D4×C31—21no—
76C2×D4×C31—21no—
77C2×D4×C31—41no—
78C2×D4×C31—41no—
79C2×D4×C31—41no—
80C2×D4×C31—41no—
81C2×D4×C31—81no—
82C2×D4×C31—81no—
83C2×D4×C31—81no—
84C2×D4×C31—81no—
85C2×D4×C31—81no—
86C2×D4×C31—81no—
87C2×D4×C31—81no—
88C22×C4×C31—21no—
89C22×C4×C31—21no—
90C22×C4×C31—21no—
91C22×C4×C31—21no—
92C22×C4×C31—41no—
93C22×C4×C31C2×D411yescharacteristic
94C22×C4×C31C2×D411yescharacteristic
95C22×C4×C31C2×D411yescharacteristic
96SD16×C31—44no—
97D8×C31—44no—
98D8×C31—82no—
99M16×C31—22no—
100M16×C31—41no—
101C2×C8×C31—41no—
102C2×C8×C31—41no—
103C2×C8×C31—41no—
104C4⋊C4×C31—21no—
105C4⋊C4×C31—21no—
106C4⋊C4×C31—41no—
107C4⋊C4×C31—41no—
108C4⋊C4×C31—81no—
109C22⋊C4×C31—41no—
110C22⋊C4×C31—41no—
111C22⋊C4×C31—41no—
112C22⋊C4×C31—41no—
113C22⋊C4×C31—81no—
114C22⋊C4×C31—81no—
115C22⋊C4×C31—81no—
116C22⋊C4×C31—81no—
117C22⋊C4×C31—81no—
118C22⋊C4×C31—81no—
119C22⋊C4×C31—81no—
120C42×C31—41no—
121#12198C3111yesmaximal, characteristic, direct product factor, Sylow
122C23×C31—41no—
123C23×C31—41no—
124C23×C31—41no—
125C23×C31—41no—
126C23×C31—41no—
127C23×C31—41no—
128C23×C31—41no—
129C23×C31—41no—
130C23×C31—41no—
131C23×C31—41no—
132C23×C31—41no—
133C23×C31—41no—
134C23×C31—41no—
135C23×C31—41no—
136C23×C31—41no—
137C23×C31#12011yescharacteristic
138Q8×C31—41no—
139Q8×C31—41no—
140D4×C31—41no—
141D4×C31—42no—
142D4×C31—41no—
143D4×C31—42no—
144D4×C31—81no—
145D4×C31—81no—
146D4×C31—81no—
147D4×C31—81no—
148D4×C31—84no—
149D4×C31—84no—
150D4×C31—81no—
151D4×C31—81no—
152C2×C4×C31—21no—
153C2×C4×C31—21no—
154C2×C4×C31—21no—
155C2×C4×C31—21no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—41no—
169C2×C4×C31—41no—
170C2×C4×C31—41no—
171C2×C4×C31—81no—
172C2×C4×C31—81no—
173C2×C4×C31#13511yescharacteristic
174C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
175C8×C31—42no—
176C8×C31—81no—
177C8×C31—81no—
178#2828C2×C3111yescharacteristic, complemented
179#2006C2×C3111yescharacteristic, complemented
180#1616C2×C3111yescharacteristic, complemented
181#1598C2×C3111yescharacteristic, complemented
182#1353C2×C3111yescharacteristic, complemented
183#1345C2×C3111yescharacteristic, complemented
184#1318C2×C3111yescharacteristic, complemented
185C22×C31—21no—
186C22×C31—21no—
187C22×C31—81no—
188C22×C31—81no—
189C22×C31—81no—
190C22×C31—81no—
191C22×C31—81no—
192C22×C31—81no—
193C22×C31—81no—
194C22×C31—81no—
195C22×C31—81no—
196C22×C31—81no—
197C22×C31—81no—
198C22×C31—81no—
199C22×C31—81no—
200C22×C31—81no—
201C22×C31—81no—
202C22×C31—81no—
203C22×C31—81no—
204C22×C31#44911yescharacteristic
205C22×C31#45311yescharacteristic
206C22×C31B(4,𝔽2)11yescharacteristic
207C4×C31—21no—
208C4×C31—21no—
209C4×C31—41no—
210C4×C31—41no—
211C4×C31—81no—
212C4×C31—81no—
213C4×C31—81no—
214C4×C31—81no—
215C4×C31—81no—
216C2×AGL(1,ℤ/8)—21no—
217#537C22×C3111yescharacteristic
218#481—21no—
219#447—21no—
220C2×#104—21no—
221#400C22×C3111yescharacteristic
222#397—21no—
223#394—21no—
224#394—21no—
225#394C22×C3111yescharacteristic
226#392—21no—
227#392—21no—
228#390C22×C3111yescharacteristic
229#388—21no—
230#343C22×C3111yescharacteristic
231#327—21no—
232#327—21no—
233#324C22×C3111yescharacteristic
234#324C22×C3111yescharacteristic
235C2×C31—21no—
236C2×C31—21no—
237C2×C31—161no—
238C2×C31—161no—
239C2×C31—161no—
240C2×C31#160011yescharacteristic
241C2×C31#220011yescharacteristic
242C2×C31#220211yescharacteristic
243C2×#41—21no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4D4×C3111yescharacteristic
247C22×D4D4×C3111yescharacteristic
248AGL(1,ℤ/8)—42no—
249AGL(1,ℤ/8)—42no—
250C2×SD16—41no—
251C2×D8—41no—
252C2×D8—41no—
253C2×M16—21no—
254C22⋊Q8—41no—
255#121—21no—
256#121—21no—
257#121—41no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4D4×C3111yescharacteristic
260C2×C4⋊C4C23×C3111yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C42D4×C3111yescharacteristic
270#106—41no—
271#104—42no—
272#102—41no—
273#102—41no—
274C22⋊C8—21no—
275C22⋊C8—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95D4×C3111yescharacteristic
280#95D4×C3111yescharacteristic
281C31#1219811yescharacteristic, direct product factor, Sylow
282C24—21no—
283C24—21no—
284#41—41no—
285#41—41no—
286#41—41no—
287#41—41no—
288C2×Q8—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—81no—
299C2×D4—81no—
300C2×D4—81no—
301C2×D4—81no—
302C2×D4—81no—
303C2×D4—81no—
304C2×D4—81no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4C2×D4×C3111yescharacteristic
311C22×C4C2×D4×C3111yescharacteristic
312C22×C4C2×D4×C3111yescharacteristic
313SD16—44no—
314D8—44no—
315D8—82no—
316M16—22no—
317M16—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—81no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—81no—
331C22⋊C4—81no—
332C22⋊C4—81no—
333C22⋊C4—81no—
334C22⋊C4—81no—
335C22⋊C4—81no—
336C22⋊C4—81no—
337C42—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23C31×#12011yescharacteristic
354Q8—41no—
355Q8—41no—
356D4—41no—
357D4—42no—
358D4—41no—
359D4—42no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4—81no—
364D4—84no—
365D4—84no—
366D4—81no—
367D4—81no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4C31×#13511yescharacteristic
390C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
391C8—42no—
392C8—81no—
393C8—81no—
394C22—21no—
395C22—21no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22C31×#44911yescharacteristic
414C22C31×#45311yescharacteristic
415C22C31×B(4,𝔽2)11yescharacteristic
416C4—21no—
417C4—21no—
418C4—41no—
419C4—41no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—81no—
425C2—21no—
426C2—21no—
427C2—161no—
428C2—161no—
429C2—161no—
430C2C31×#160011yescharacteristic
431C2C31×#220011yescharacteristic
432C2C31×#220211yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C4⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2828C2semidirect
2. C31×#2006C2semidirect
3. C31×#1616C2semidirect
4. C31×#1598C2semidirect
5. C31×#1353C2semidirect
6. C31×#1345C2semidirect
7. C31×#1318C2semidirect
121. #12198C31direct (N × H)
178. #2828C2×C31semidirect
179. #2006C2×C31semidirect
180. #1616C2×C31semidirect
181. #1598C2×C31semidirect
182. #1353C2×C31semidirect
183. #1345C2×C31semidirect
184. #1318C2×C31semidirect
281. C31#12198direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28282
C31×#20062
C31×#16162
C31×#15982
C31×#13532
C31×#13452
C31×#13182
#1219831