Finite Group Databasebeta

← C31×#9933 #21466692

Subgroups — C31×#9933 #21466692

Order 7936.

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

normal subgroups only

This group has 568 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
1C31×#3193C211yesmaximal, characteristic
2C31×#3094C211yesmaximal, characteristic
3C31×#2625C211yesmaximal, characteristic, complemented
4C31×#1964C211yesmaximal, characteristic
5C31×#1826C211yesmaximal, characteristic, complemented
6C31×#1508C211yesmaximal, characteristic, complemented
7C31×#1305C211yesmaximal, characteristic, complemented
8C2×C31×C22⋊Q8C2211yescharacteristic
9C2×C31×#121C2211yescharacteristic
10C2×C31×#117C2211yescharacteristic
11C31×#484—21no—
12C31×#482—22no—
13C31×#481—21no—
14C31×#480—21no—
15C31×#480—21no—
16C31×#468—22no—
17C31×#450—21no—
18C31×#448—21no—
19C31×#428—21no—
20C31×#418—21no—
21C2×C31×#99—22no—
22C31×#407C2211yescharacteristic
23C31×#407C2211yescharacteristic
24C31×#400—21no—
25C31×#398—21no—
26C31×#397—22no—
27C31×#391—21no—
28C31×#389—21no—
29C31×#380C2211yescharacteristic
30C4×C22⋊C4×C31C2211yescharacteristic
31C31×#339—22no—
32C31×#328—22no—
33C22×Q8×C31D411yescharacteristic
34C22×D4×C31—21no—
35C22×D4×C31D411yescharacteristic
36C23×C4×C31C2311yescharacteristic
37C2×SD16×C31—22no—
38C2×M16×C31—21no—
39C2×M16×C31D412yes—
40C31×C22⋊Q8—21no—
41C31×C22⋊Q8—21no—
42C31×C22⋊Q8—21no—
43C31×C22⋊Q8—21no—
44C31×#121—21no—
45C31×#121—21no—
46C31×#121—21no—
47C31×#121—21no—
48C31×#117—21no—
49C31×#117—21no—
50C31×#117—21no—
51C31×#117—21no—
52C2×C4⋊C4×C31—21no—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31—22no—
55C2×C4⋊C4×C31D411yescharacteristic
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C4⋊C4×C31D411yescharacteristic
58C2×C4⋊C4×C31D411yescharacteristic
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31—21no—
61C2×C22⋊C4×C31—21no—
62C2×C22⋊C4×C31—21no—
63C2×C22⋊C4×C31D412yes—
64C2×C42×C31—21no—
65C2×C42×C31—21no—
66C31×#107—22no—
67C31×#107—22no—
68C31×#106—22no—
69C31×#103—22no—
70C31×#102—22no—
71C31×#102—22no—
72C31×#102—22no—
73C31×#99—44no—
74C31×C22⋊C8—21no—
75C31×C22⋊C8—21no—
76C31×C22⋊C8—21no—
77C31×C22⋊C8D412yes—
78C31×#95—22no—
79C31×#95—22no—
80C31×#95—22no—
81C31×#95—21no—
82C31×#95—21no—
83C24×C31—21no—
84C24×C31C2×D411yescharacteristic
85C2×Q8×C31—21no—
86C2×Q8×C31—21no—
87C2×Q8×C31—42no—
88C2×D4×C31—21no—
89C2×D4×C31—21no—
90C2×D4×C31—42no—
91C2×D4×C31—42no—
92C2×D4×C31—42no—
93C2×D4×C31—41no—
94C2×D4×C31—41no—
95C22×C4×C31—21no—
96C22×C4×C31—21no—
97C22×C4×C31—22no—
98C22×C4×C31—21no—
99C22×C4×C31—21no—
100C22×C4×C31—21no—
101C22×C4×C31—42no—
102C22×C4×C31—41no—
103C22×C4×C31C2×D411yescharacteristic
104C22×C4×C31C2×D411yescharacteristic
105C22×C4×C31C2×D411yescharacteristic
106C22×C4×C31C2×D411yescharacteristic
107C22×C4×C31C2×D411yescharacteristic
108C22×C4×C31#4111yescharacteristic
109SD16×C31—82no—
110M16×C31—42no—
111M16×C31—42no—
112C2×C8×C31—22no—
113C2×C8×C31—22no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—21no—
116C4⋊C4×C31—21no—
117C4⋊C4×C31—21no—
118C4⋊C4×C31—21no—
119C4⋊C4×C31—21no—
120C4⋊C4×C31—21no—
121C4⋊C4×C31—21no—
122C4⋊C4×C31—42no—
123C4⋊C4×C31—41no—
124C4⋊C4×C31—41no—
125C4⋊C4×C31—82no—
126C22⋊C4×C31—22no—
127C22⋊C4×C31—41no—
128C22⋊C4×C31—41no—
129C22⋊C4×C31—41no—
130C22⋊C4×C31—41no—
131C22⋊C4×C31—41no—
132C22⋊C4×C31—41no—
133C22⋊C4×C31—41no—
134C22⋊C4×C31—42no—
135C42×C31—42no—
136C42×C31—41no—
137C42×C31—41no—
138#9933C3111yesmaximal, characteristic, direct product factor, Sylow
139C23×C31—21no—
140C23×C31—21no—
141C23×C31—41no—
142C23×C31—41no—
143C23×C31—42no—
144C23×C31—41no—
145C23×C31—81no—
146C23×C31—81no—
147C23×C31#12011yescharacteristic
148C23×C31#12111yescharacteristic
149C23×C31#12711yescharacteristic
150Q8×C31—42no—
151Q8×C31—81no—
152D4×C31—42no—
153D4×C31—42no—
154D4×C31—82no—
155D4×C31—81no—
156D4×C31—81no—
157C2×C4×C31—22no—
158C2×C4×C31—22no—
159C2×C4×C31—22no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—42no—
164C2×C4×C31—42no—
165C2×C4×C31—42no—
166C2×C4×C31—42no—
167C2×C4×C31—42no—
168C2×C4×C31—42no—
169C2×C4×C31—41no—
170C2×C4×C31—41no—
171C2×C4×C31—41no—
172C2×C4×C31—41no—
173C2×C4×C31—41no—
174C2×C4×C31—42no—
175C2×C4×C31—41no—
176C2×C4×C31—41no—
177C2×C4×C31—41no—
178C2×C4×C31—41no—
179C2×C4×C31—81no—
180C2×C4×C31—81no—
181C2×C4×C31#12011yescharacteristic
182C2×C4×C31#12011yescharacteristic
183C2×C4×C31#12111yescharacteristic
184C2×C4×C31#12111yescharacteristic
185C8×C31—42no—
186C8×C31—42no—
187#3193C2×C3111yescharacteristic
188#3094C2×C3111yescharacteristic
189#2625C2×C3111yescharacteristic, complemented
190#1964C2×C3111yescharacteristic
191#1826C2×C3111yescharacteristic, complemented
192#1508C2×C3111yescharacteristic, complemented
193#1305C2×C3111yescharacteristic, complemented
194C22×C31—21no—
195C22×C31—21no—
196C22×C31—21no—
197C22×C31—21no—
198C22×C31—21no—
199C22×C31—21no—
200C22×C31—21no—
201C22×C31—21no—
202C22×C31—81no—
203C22×C31—81no—
204C22×C31—81no—
205C22×C31—81no—
206C22×C31—81no—
207C22×C31—162no—
208C22×C31#39211yescharacteristic
209C22×C31#45311yescharacteristic
210C22×C31#45411yescharacteristic
211C4×C31—21no—
212C4×C31—21no—
213C4×C31—21no—
214C4×C31—82no—
215C4×C31—82no—
216C4×C31—81no—
217C4×C31—81no—
218C4×C31—81no—
219C4×C31—81no—
220C4×C31—81no—
221C4×C31#45812yes—
222C2×C22⋊Q8C22×C3111yescharacteristic
223C2×#121C22×C3111yescharacteristic
224C2×#117C22×C3111yescharacteristic
225#484—21no—
226#482—22no—
227#481—21no—
228#480—21no—
229#480—21no—
230#468—22no—
231#450—21no—
232#448—21no—
233#428—21no—
234#418—21no—
235C2×#99—22no—
236#407C22×C3111yescharacteristic
237#407C22×C3111yescharacteristic
238#400—21no—
239#398—21no—
240#397—22no—
241#391—21no—
242#389—21no—
243#380C22×C3111yescharacteristic
244C4×C22⋊C4C22×C3111yescharacteristic
245#339—22no—
246#328—22no—
247C2×C31—21no—
248C2×C31—21no—
249C2×C31—41no—
250C2×C31—41no—
251C2×C31—161no—
252C2×C31#200611yescharacteristic
253C2×C31#200911yescharacteristic
254C2×C31#202211yescharacteristic
255C22×Q8D4×C3111yescharacteristic
256C22×D4—21no—
257C22×D4D4×C3111yescharacteristic
258C23×C4C23×C3111yescharacteristic
259C2×SD16—22no—
260C2×M16—21no—
261C2×M16D4×C3112yes—
262C22⋊Q8—21no—
263C22⋊Q8—21no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266#121—21no—
267#121—21no—
268#121—21no—
269#121—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4D4×C3111yescharacteristic
278C2×C4⋊C4D4×C3111yescharacteristic
279C2×C4⋊C4D4×C3111yescharacteristic
280C2×C4⋊C4D4×C3111yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4D4×C3112yes—
286C2×C42—21no—
287C2×C42—21no—
288#107—22no—
289#107—22no—
290#106—22no—
291#103—22no—
292#102—22no—
293#102—22no—
294#102—22no—
295#99—44no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8D4×C3112yes—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—21no—
304#95—21no—
305C31#993311yescharacteristic, direct product factor, Sylow
306C24—21no—
307C24C2×D4×C3111yescharacteristic
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—42no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—41no—
317C2×D4—41no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4C2×D4×C3111yescharacteristic
327C22×C4C2×D4×C3111yescharacteristic
328C22×C4C2×D4×C3111yescharacteristic
329C22×C4C2×D4×C3111yescharacteristic
330C22×C4C2×D4×C3111yescharacteristic
331C22×C4#41×C3111yescharacteristic
332SD16—82no—
333M16—42no—
334M16—42no—
335C2×C8—22no—
336C2×C8—22no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—82no—
349C22⋊C4—22no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—42no—
358C42—42no—
359C42—41no—
360C42—41no—
361C23—21no—
362C23—21no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—41no—
367C23—81no—
368C23—81no—
369C23C31×#12011yescharacteristic
370C23C31×#12111yescharacteristic
371C23C31×#12711yescharacteristic
372Q8—42no—
373Q8—81no—
374D4—42no—
375D4—42no—
376D4—82no—
377D4—81no—
378D4—81no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4C31×#12011yescharacteristic
404C2×C4C31×#12011yescharacteristic
405C2×C4C31×#12111yescharacteristic
406C2×C4C31×#12111yescharacteristic
407C8—42no—
408C8—42no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—162no—
423C22C31×#39211yescharacteristic
424C22C31×#45311yescharacteristic
425C22C31×#45411yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—82no—
430C4—82no—
431C4—81no—
432C4—81no—
433C4—81no—
434C4—81no—
435C4—81no—
436C4C31×#45812yes—
437C2—21no—
438C2—21no—
439C2—41no—
440C2—41no—
441C2—161no—
442C2C31×#200611yescharacteristic
443C2C31×#200911yescharacteristic
444C2C31×#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
3. C31×#2625C2semidirect
5. C31×#1826C2semidirect
6. C31×#1508C2semidirect
7. C31×#1305C2semidirect
138. #9933C31direct (N × H)
189. #2625C2×C31semidirect
191. #1826C2×C31semidirect
192. #1508C2×C31semidirect
193. #1305C2×C31semidirect
305. C31#9933direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31932
C31×#30942
C31×#26252
C31×#19642
C31×#18262
C31×#15082
C31×#13052
#993331