Finite Group Databasebeta

← C31×#9984 #21466743

Subgroups — C31×#9984 #21466743

Order 7936.

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

normal subgroups only

This group has 544 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
1C31×#3095C211yesmaximal, characteristic, complemented
2C31×#3094C211yesmaximal, characteristic
3C31×#2816C211yesmaximal, characteristic
4C31×#1820C211yesmaximal, characteristic, complemented
5C31×#1511C211yesmaximal, characteristic, complemented
6C31×#1510C211yesmaximal, characteristic
7C31×#1306C211yesmaximal, characteristic, complemented
8C2×C31×C22⋊Q8C2211yescharacteristic
9C2×C31×#121C2211yescharacteristic
10C2×C31×#117C2211yescharacteristic
11C31×#485—22no—
12C31×#484—21no—
13C31×#483—21no—
14C31×#482—22no—
15C31×#481—21no—
16C31×#480—21no—
17C31×#428—21no—
18C31×#428—21no—
19C31×#419—21no—
20C31×#418—21no—
21C31×#407C2211yescharacteristic
22C31×#407C2211yescharacteristic
23C31×#399—21no—
24C31×#398—21no—
25C31×#394—21no—
26C31×#394—22no—
27C31×#393—21no—
28C31×#381C2211yescharacteristic
29C31×#381C2211yescharacteristic
30C31×#342—22no—
31C31×#328—22no—
32C31×#327—22no—
33C22×Q8×C31D411yescharacteristic
34C22×D4×C31—21no—
35C22×D4×C31D411yescharacteristic
36C23×C4×C31C2311yescharacteristic
37C2×M16×C31—21no—
38C2×M16×C31—21no—
39C31×C22⋊Q8—21no—
40C31×C22⋊Q8—21no—
41C31×C22⋊Q8—21no—
42C31×C22⋊Q8—21no—
43C31×#121—21no—
44C31×#121—21no—
45C31×#121—21no—
46C31×#121—21no—
47C31×#117—21no—
48C31×#117—21no—
49C31×#117—21no—
50C31×#117—21no—
51C2×C4⋊C4×C31—21no—
52C2×C4⋊C4×C31—22no—
53C2×C4⋊C4×C31D411yescharacteristic
54C2×C4⋊C4×C31D411yescharacteristic
55C2×C4⋊C4×C31D411yescharacteristic
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C22⋊C4×C31—21no—
58C2×C22⋊C4×C31—21no—
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31—21no—
61C2×C22⋊C4×C31—22no—
62C2×C22⋊C4×C31—22no—
63C2×C22⋊C4×C31D412yes—
64C2×C42×C31—21no—
65C31×#107—22no—
66C31×#107—22no—
67C31×#106—22no—
68C31×#106—22no—
69C31×#103—22no—
70C31×#103—22no—
71C31×#102—22no—
72C31×#102—22no—
73C31×C22⋊C8—21no—
74C31×C22⋊C8—21no—
75C31×C22⋊C8D412yes—
76C31×C22⋊C8D412yes—
77C31×#95—22no—
78C31×#95—21no—
79C31×#95—21no—
80C31×#95—21no—
81C31×#95—21no—
82C24×C31—21no—
83C24×C31C2×D411yescharacteristic
84C2×Q8×C31—21no—
85C2×Q8×C31—21no—
86C2×Q8×C31—42no—
87C2×D4×C31—21no—
88C2×D4×C31—21no—
89C2×D4×C31—42no—
90C2×D4×C31—41no—
91C2×D4×C31—41no—
92C2×D4×C31—82no—
93C22×C4×C31—22no—
94C22×C4×C31—21no—
95C22×C4×C31—21no—
96C22×C4×C31—21no—
97C22×C4×C31—21no—
98C22×C4×C31—21no—
99C22×C4×C31—42no—
100C22×C4×C31—41no—
101C22×C4×C31C2×D411yescharacteristic
102C22×C4×C31C2×D411yescharacteristic
103C22×C4×C31C2×D411yescharacteristic
104C22×C4×C31C2×D411yescharacteristic
105C22×C4×C31C2×D411yescharacteristic
106C22×C4×C31#4111yescharacteristic
107M16×C31—42no—
108M16×C31—42no—
109C2×C8×C31—22no—
110C2×C8×C31—22no—
111C4⋊C4×C31—21no—
112C4⋊C4×C31—21no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—21no—
116C4⋊C4×C31—21no—
117C4⋊C4×C31—21no—
118C4⋊C4×C31—21no—
119C4⋊C4×C31—41no—
120C4⋊C4×C31—41no—
121C4⋊C4×C31—82no—
122C22⋊C4×C31—41no—
123C22⋊C4×C31—41no—
124C22⋊C4×C31—41no—
125C22⋊C4×C31—41no—
126C22⋊C4×C31—41no—
127C22⋊C4×C31—41no—
128C22⋊C4×C31—42no—
129C22⋊C4×C31—42no—
130C22⋊C4×C31—82no—
131C22⋊C4×C31—82no—
132C42×C31—41no—
133C42×C31—41no—
134#9984C3111yesmaximal, characteristic, direct product factor, Sylow
135C23×C31—22no—
136C23×C31—41no—
137C23×C31—41no—
138C23×C31—42no—
139C23×C31—42no—
140C23×C31—41no—
141C23×C31—42no—
142C23×C31#12011yescharacteristic
143C23×C31#12111yescharacteristic
144C23×C31#12711yescharacteristic
145Q8×C31—42no—
146Q8×C31—81no—
147D4×C31—42no—
148D4×C31—81no—
149D4×C31—82no—
150D4×C31—82no—
151C2×C4×C31—22no—
152C2×C4×C31—22no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—42no—
159C2×C4×C31—41no—
160C2×C4×C31—42no—
161C2×C4×C31—42no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—42no—
168C2×C4×C31—42no—
169C2×C4×C31—81no—
170C2×C4×C31—81no—
171C2×C4×C31—81no—
172C2×C4×C31—81no—
173C2×C4×C31—81no—
174C2×C4×C31#12011yescharacteristic
175C2×C4×C31#12011yescharacteristic
176C2×C4×C31#12111yescharacteristic
177C2×C4×C31#12111yescharacteristic
178C8×C31—42no—
179C8×C31—42no—
180#3095C2×C3111yescharacteristic, complemented
181#3094C2×C3111yescharacteristic
182#2816C2×C3111yescharacteristic
183#1820C2×C3111yescharacteristic, complemented
184#1511C2×C3111yescharacteristic, complemented
185#1510C2×C3111yescharacteristic
186#1306C2×C3111yescharacteristic, complemented
187C22×C31—21no—
188C22×C31—21no—
189C22×C31—21no—
190C22×C31—21no—
191C22×C31—21no—
192C22×C31—21no—
193C22×C31—21no—
194C22×C31—21no—
195C22×C31—82no—
196C22×C31—84no—
197C22×C31—81no—
198C22×C31—81no—
199C22×C31—81no—
200C22×C31#39211yescharacteristic
201C22×C31#45812yes—
202C4×C31—21no—
203C4×C31—21no—
204C4×C31—21no—
205C4×C31—21no—
206C4×C31—82no—
207C4×C31—81no—
208C4×C31—81no—
209C4×C31—81no—
210C4×C31—81no—
211C4×C31—81no—
212C4×C31—161no—
213C2×C22⋊Q8C22×C3111yescharacteristic
214C2×#121C22×C3111yescharacteristic
215C2×#117C22×C3111yescharacteristic
216#485—22no—
217#484—21no—
218#483—21no—
219#482—22no—
220#481—21no—
221#480—21no—
222#428—21no—
223#428—21no—
224#419—21no—
225#418—21no—
226#407C22×C3111yescharacteristic
227#407C22×C3111yescharacteristic
228#399—21no—
229#398—21no—
230#394—21no—
231#394—22no—
232#393—21no—
233#381C22×C3111yescharacteristic
234#381C22×C3111yescharacteristic
235#342—22no—
236#328—22no—
237#327—22no—
238C2×C31—22no—
239C2×C31—41no—
240C2×C31—41no—
241C2×C31—161no—
242C2×C31#200811yescharacteristic
243C2×C31#200911yescharacteristic
244C2×C31#202211yescharacteristic
245C22×Q8D4×C3111yescharacteristic
246C22×D4—21no—
247C22×D4D4×C3111yescharacteristic
248C23×C4C23×C3111yescharacteristic
249C2×M16—21no—
250C2×M16—21no—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254C22⋊Q8—21no—
255#121—21no—
256#121—21no—
257#121—21no—
258#121—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4D4×C3111yescharacteristic
266C2×C4⋊C4D4×C3111yescharacteristic
267C2×C4⋊C4D4×C3111yescharacteristic
268C2×C4⋊C4D4×C3111yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4D4×C3112yes—
276C2×C42—21no—
277#107—22no—
278#107—22no—
279#106—22no—
280#106—22no—
281#103—22no—
282#103—22no—
283#102—22no—
284#102—22no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8D4×C3112yes—
288C22⋊C8D4×C3112yes—
289#95—22no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294C31#998411yescharacteristic, direct product factor, Sylow
295C24—21no—
296C24C2×D4×C3111yescharacteristic
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—42no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—82no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4C2×D4×C3111yescharacteristic
315C22×C4C2×D4×C3111yescharacteristic
316C22×C4C2×D4×C3111yescharacteristic
317C22×C4C2×D4×C3111yescharacteristic
318C22×C4C2×D4×C3111yescharacteristic
319C22×C4#41×C3111yescharacteristic
320M16—42no—
321M16—42no—
322C2×C8—22no—
323C2×C8—22no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—82no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C42—41no—
346C42—41no—
347C23—22no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—41no—
354C23C31×#12011yescharacteristic
355C23C31×#12111yescharacteristic
356C23C31×#12711yescharacteristic
357Q8—42no—
358Q8—81no—
359D4—42no—
360D4—81no—
361D4—82no—
362D4—82no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4C31×#12011yescharacteristic
387C2×C4C31×#12011yescharacteristic
388C2×C4C31×#12111yescharacteristic
389C2×C4C31×#12111yescharacteristic
390C8—42no—
391C8—42no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—82no—
401C22—84no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22C31×#39211yescharacteristic
406C22C31×#45812yes—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—82no—
412C4—81no—
413C4—81no—
414C4—81no—
415C4—81no—
416C4—81no—
417C4—161no—
418C2—22no—
419C2—41no—
420C2—41no—
421C2—161no—
422C2C31×#200811yescharacteristic
423C2C31×#200911yescharacteristic
424C2C31×#202211yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC2×C31×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3095C2semidirect
4. C31×#1820C2semidirect
5. C31×#1511C2semidirect
7. C31×#1306C2semidirect
134. #9984C31direct (N × H)
180. #3095C2×C31semidirect
183. #1820C2×C31semidirect
184. #1511C2×C31semidirect
186. #1306C2×C31semidirect
294. C31#9984direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#30952
C31×#30942
C31×#28162
C31×#18202
C31×#15112
C31×#15102
C31×#13062
#998431