Finite Group Databasebeta

← C31×#9993 #21466752

Subgroups — C31×#9993 #21466752

Order 7936.

536 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 106 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
2C2×C31×#481C211yesmaximal, characteristic
3C2×C31×#483C211yesmaximal, characteristic, complemented
4C31×#1821C211yesmaximal, characteristic, complemented
5C31×#1472C211yesmaximal, characteristic, complemented
6C31×#1471C211yesmaximal, characteristic
7C31×#1298C211yesmaximal, characteristic, complemented
8C2×C31×C22⋊Q8C2211yescharacteristic
9C2×C31×#121C2211yescharacteristic
10C22×C4⋊C4×C31C2211yescharacteristic
11C31×#484—21no—
12C31×#483—22no—
13C31×#483—22no—
14C31×#481—22no—
15C31×#481—22no—
16C31×#480—21no—
17C31×#468—22no—
18C31×#450—21no—
19C31×#448—21no—
20C2×C31×#106—21no—
21C2×C31×#106—21no—
22C2×C31×#103—21no—
23C2×C31×#102—21no—
24C31×#407C2211yescharacteristic
25C31×#407C2211yescharacteristic
26C2×C31×C22⋊C8C2211yescharacteristic
27C2×C31×C22⋊C8C2211yescharacteristic
28C31×#343—22no—
29C31×#324—22no—
30C31×#323—22no—
31C22×Q8×C31D411yescharacteristic
32C22×D4×C31—21no—
33C22×D4×C31D411yescharacteristic
34C23×C4×C31—21no—
35C23×C4×C31C2311yescharacteristic
36C2×SD16×C31—22no—
37C2×M16×C31—21no—
38C2×M16×C31D412yes—
39C22×C8×C31—21no—
40C22×C8×C31—21no—
41C31×C22⋊Q8—21no—
42C31×C22⋊Q8—21no—
43C31×C22⋊Q8—21no—
44C31×C22⋊Q8—21no—
45C31×#121—21no—
46C31×#121—21no—
47C31×#121—21no—
48C31×#121—21no—
49C2×C4⋊C4×C31—21no—
50C2×C4⋊C4×C31—21no—
51C2×C4⋊C4×C31—21no—
52C2×C4⋊C4×C31—21no—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31D411yescharacteristic
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C4⋊C4×C31D412yes—
58C2×C22⋊C4×C31—21no—
59C2×C22⋊C4×C31—21no—
60C31×#107—22no—
61C31×#106—24no—
62C31×#106—24no—
63C31×#103—24no—
64C31×#103—22no—
65C31×#102—24no—
66C31×#102—22no—
67C31×C22⋊C8—21no—
68C31×C22⋊C8—21no—
69C31×C22⋊C8—21no—
70C31×C22⋊C8—21no—
71C31×C22⋊C8—21no—
72C31×C22⋊C8—21no—
73C31×C22⋊C8D412yes—
74C31×C22⋊C8D412yes—
75C24×C31—21no—
76C24×C31#4111yescharacteristic
77C2×Q8×C31—21no—
78C2×Q8×C31—21no—
79C2×Q8×C31—42no—
80C2×D4×C31—21no—
81C2×D4×C31—21no—
82C2×D4×C31—42no—
83C2×D4×C31—41no—
84C2×D4×C31—41no—
85C2×D4×C31—82no—
86C22×C4×C31—21no—
87C22×C4×C31—21no—
88C22×C4×C31—21no—
89C22×C4×C31—22no—
90C22×C4×C31—42no—
91C22×C4×C31—41no—
92C22×C4×C31—41no—
93C22×C4×C31—82no—
94C22×C4×C31C2×D411yescharacteristic
95C22×C4×C31C2×D411yescharacteristic
96C22×C4×C31C2×D411yescharacteristic
97C22×C4×C31C2×D411yescharacteristic
98C22×C4×C31C2×D411yescharacteristic
99C22×C4×C31C2×D411yescharacteristic
100SD16×C31—82no—
101M16×C31—42no—
102M16×C31—42no—
103C2×C8×C31—22no—
104C2×C8×C31—22no—
105C2×C8×C31—22no—
106C2×C8×C31—42no—
107C2×C8×C31—42no—
108C2×C8×C31—41no—
109C4⋊C4×C31—22no—
110C4⋊C4×C31—21no—
111C4⋊C4×C31—22no—
112C4⋊C4×C31—21no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—41no—
116C4⋊C4×C31—41no—
117C4⋊C4×C31—41no—
118C4⋊C4×C31—41no—
119C22⋊C4×C31—41no—
120C22⋊C4×C31—41no—
121C22⋊C4×C31—41no—
122C22⋊C4×C31—41no—
123#9993C3111yesmaximal, characteristic, direct product factor, Sylow
124C23×C31—22no—
125C23×C31—22no—
126C23×C31—41no—
127C23×C31—41no—
128C23×C31—42no—
129C23×C31—81no—
130C23×C31—81no—
131C23×C31#12111yescharacteristic
132C23×C31#12111yescharacteristic
133C23×C31#12111yescharacteristic
134Q8×C31—42no—
135Q8×C31—42no—
136D4×C31—42no—
137D4×C31—42no—
138D4×C31—82no—
139D4×C31—82no—
140C2×C4×C31—22no—
141C2×C4×C31—22no—
142C2×C4×C31—22no—
143C2×C4×C31—41no—
144C2×C4×C31—41no—
145C2×C4×C31—41no—
146C2×C4×C31—41no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
149C2×C4×C31—42no—
150C2×C4×C31—42no—
151C2×C4×C31—82no—
152C2×C4×C31—82no—
153C2×C4×C31—82no—
154C2×C4×C31—81no—
155C2×C4×C31—81no—
156C2×C4×C31#12011yescharacteristic
157C2×C4×C31#12011yescharacteristic
158C2×C4×C31#12011yescharacteristic
159C2×C4×C31#12711yescharacteristic
160C8×C31—42no—
161C8×C31—42no—
162C8×C31—42no—
163C8×C31—81no—
164#3193C2×C3111yescharacteristic
165C2×#481C2×C3111yescharacteristic
166C2×#483C2×C3111yescharacteristic, complemented
167#1821C2×C3111yescharacteristic, complemented
168#1472C2×C3111yescharacteristic, complemented
169#1471C2×C3111yescharacteristic
170#1298C2×C3111yescharacteristic, complemented
171C22×C31—21no—
172C22×C31—21no—
173C22×C31—21no—
174C22×C31—21no—
175C22×C31—21no—
176C22×C31—21no—
177C22×C31—21no—
178C22×C31—44no—
179C22×C31—81no—
180C22×C31—81no—
181C22×C31—81no—
182C22×C31—162no—
183C22×C31#39211yescharacteristic
184C22×C31#47112yes—
185C22×C31#47112yes—
186C4×C31—21no—
187C4×C31—21no—
188C4×C31—21no—
189C4×C31—81no—
190C4×C31—82no—
191C4×C31—81no—
192C4×C31—81no—
193C4×C31#45812yes—
194C2×C22⋊Q8C22×C3111yescharacteristic
195C2×#121C22×C3111yescharacteristic
196C22×C4⋊C4C22×C3111yescharacteristic
197#484—21no—
198#483—22no—
199#483—22no—
200#481—22no—
201#481—22no—
202#480—21no—
203#468—22no—
204#450—21no—
205#448—21no—
206C2×#106—21no—
207C2×#106—21no—
208C2×#103—21no—
209C2×#102—21no—
210#407C22×C3111yescharacteristic
211#407C22×C3111yescharacteristic
212C2×C22⋊C8C22×C3111yescharacteristic
213C2×C22⋊C8C22×C3111yescharacteristic
214#343—22no—
215#324—22no—
216#323—22no—
217C2×C31—22no—
218C2×C31—22no—
219C2×C31—41no—
220C2×C31—161no—
221C2×C31#201911yescharacteristic
222C2×C31#201911yescharacteristic
223C2×C31#202211yescharacteristic
224C22×Q8D4×C3111yescharacteristic
225C22×D4—21no—
226C22×D4D4×C3111yescharacteristic
227C23×C4—21no—
228C23×C4C23×C3111yescharacteristic
229C2×SD16—22no—
230C2×M16—21no—
231C2×M16D4×C3112yes—
232C22×C8—21no—
233C22×C8—21no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238#121—21no—
239#121—21no—
240#121—21no—
241#121—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4D4×C3112yes—
249C2×C4⋊C4D4×C3111yescharacteristic
250C2×C4⋊C4D4×C3111yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253#107—22no—
254#106—24no—
255#106—24no—
256#103—24no—
257#103—22no—
258#102—24no—
259#102—22no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265C22⋊C8—21no—
266C22⋊C8D4×C3112yes—
267C22⋊C8D4×C3112yes—
268C31#999311yescharacteristic, direct product factor, Sylow
269C24—21no—
270C24#41×C3111yescharacteristic
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—42no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—42no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—82no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—82no—
288C22×C4C2×D4×C3111yescharacteristic
289C22×C4C2×D4×C3111yescharacteristic
290C22×C4C2×D4×C3111yescharacteristic
291C22×C4C2×D4×C3111yescharacteristic
292C22×C4C2×D4×C3111yescharacteristic
293C22×C4C2×D4×C3111yescharacteristic
294SD16—82no—
295M16—42no—
296M16—42no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—41no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C23—22no—
318C23—22no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—81no—
323C23—81no—
324C23C31×#12111yescharacteristic
325C23C31×#12111yescharacteristic
326C23C31×#12111yescharacteristic
327Q8—42no—
328Q8—42no—
329D4—42no—
330D4—42no—
331D4—82no—
332D4—82no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4C31×#12011yescharacteristic
350C2×C4C31×#12011yescharacteristic
351C2×C4C31×#12011yescharacteristic
352C2×C4C31×#12711yescharacteristic
353C8—42no—
354C8—42no—
355C8—42no—
356C8—81no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—44no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—162no—
369C22C31×#39211yescharacteristic
370C22C31×#47112yes—
371C22C31×#47112yes—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—81no—
376C4—82no—
377C4—81no—
378C4—81no—
379C4C31×#45812yes—
380C2—22no—
381C2—22no—
382C2—41no—
383C2—161no—
384C2C31×#201911yescharacteristic
385C2C31×#201911yescharacteristic
386C2C31×#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. C2×C31×#483C2semidirect
4. C31×#1821C2semidirect
5. C31×#1472C2semidirect
7. C31×#1298C2semidirect
123. #9993C31direct (N × H)
166. C2×#483C2×C31semidirect
167. #1821C2×C31semidirect
168. #1472C2×C31semidirect
170. #1298C2×C31semidirect
268. C31#9993direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31932
C2×C31×#4812
C2×C31×#4832
C31×#18212
C31×#14722
C31×#14712
C31×#12982
#999331