Finite Group Databasebeta

← C31×#9995 #21466754

Subgroups — C31×#9995 #21466754

Order 7936.

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

normal subgroups only

This group has 510 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×#3195C211yesmaximal, characteristic
2C2×C31×#484C211yesmaximal, characteristic
3C2×C31×#483C211yesmaximal, characteristic
4C31×#1821C211yesmaximal, characteristic
5C31×#1472C211yesmaximal, characteristic
6C31×#1472C211yesmaximal, characteristic
7C31×#1298C211yesmaximal, characteristic
8C2×C31×C22⋊Q8C2211yescharacteristic
9C2×C31×C22⋊Q8C2211yescharacteristic
10C22×C4⋊C4×C31C2211yescharacteristic
11C31×#484—22no—
12C31×#484—22no—
13C31×#483—22no—
14C31×#483—21no—
15C31×#483—21no—
16C31×#483—22no—
17C31×#470—22no—
18C31×#451—21no—
19C31×#451—21no—
20C2×C31×#107—21no—
21C2×C31×#106—21no—
22C2×C31×#103—21no—
23C2×C31×#103—21no—
24C31×#407C2211yescharacteristic
25C31×#407C2211yescharacteristic
26C2×C31×C22⋊C8C2211yescharacteristic
27C2×C31×C22⋊C8C2211yescharacteristic
28C31×#343—22no—
29C31×#324—22no—
30C31×#324—22no—
31C22×Q8×C31D411yescharacteristic
32C22×Q8×C31D411yescharacteristic
33C23×C4×C31—21no—
34C23×C4×C31C2311yescharacteristic
35C2×Q16×C31—22no—
36C2×M16×C31—21no—
37C2×M16×C31D412yes—
38C22×C8×C31—21no—
39C22×C8×C31—21no—
40C31×C22⋊Q8—21no—
41C31×C22⋊Q8—21no—
42C31×C22⋊Q8—21no—
43C31×C22⋊Q8—21no—
44C31×C22⋊Q8—21no—
45C31×C22⋊Q8—21no—
46C31×C22⋊Q8—21no—
47C31×C22⋊Q8—21no—
48C2×C4⋊C4×C31—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—24no—
61C31×#106—24no—
62C31×#106—22no—
63C31×#103—24no—
64C31×#103—24no—
65C31×#103—22no—
66C31×#103—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#4111yescharacteristic
76C2×Q8×C31—21no—
77C2×Q8×C31—21no—
78C2×Q8×C31—21no—
79C2×Q8×C31—21no—
80C2×Q8×C31—42no—
81C2×Q8×C31—42no—
82C22×C4×C31—21no—
83C22×C4×C31—21no—
84C22×C4×C31—21no—
85C22×C4×C31—21no—
86C22×C4×C31—22no—
87C22×C4×C31—42no—
88C22×C4×C31—41no—
89C22×C4×C31—41no—
90C22×C4×C31—82no—
91C22×C4×C31C2×D411yescharacteristic
92C22×C4×C31C2×D411yescharacteristic
93C22×C4×C31C2×D411yescharacteristic
94C22×C4×C31C2×D411yescharacteristic
95C22×C4×C31C2×D411yescharacteristic
96C22×C4×C31C2×D411yescharacteristic
97Q16×C31—82no—
98M16×C31—42no—
99M16×C31—42no—
100C2×C8×C31—22no—
101C2×C8×C31—22no—
102C2×C8×C31—22no—
103C2×C8×C31—42no—
104C2×C8×C31—42no—
105C2×C8×C31—41no—
106C4⋊C4×C31—22no—
107C4⋊C4×C31—21no—
108C4⋊C4×C31—21no—
109C4⋊C4×C31—22no—
110C4⋊C4×C31—21no—
111C4⋊C4×C31—21no—
112C4⋊C4×C31—41no—
113C4⋊C4×C31—41no—
114C4⋊C4×C31—41no—
115C4⋊C4×C31—41no—
116C4⋊C4×C31—41no—
117C4⋊C4×C31—41no—
118C22⋊C4×C31—41no—
119C22⋊C4×C31—41no—
120C22⋊C4×C31—41no—
121C22⋊C4×C31—41no—
122#9995C3111yesmaximal, characteristic, direct product factor, Sylow
123C23×C31—22no—
124C23×C31—22no—
125C23×C31—41no—
126C23×C31#12111yescharacteristic
127C23×C31#12111yescharacteristic
128C23×C31#12111yescharacteristic
129Q8×C31—42no—
130Q8×C31—42no—
131Q8×C31—42no—
132Q8×C31—42no—
133C2×C4×C31—22no—
134C2×C4×C31—22no—
135C2×C4×C31—22no—
136C2×C4×C31—41no—
137C2×C4×C31—41no—
138C2×C4×C31—41no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
141C2×C4×C31—41no—
142C2×C4×C31—41no—
143C2×C4×C31—42no—
144C2×C4×C31—42no—
145C2×C4×C31—42no—
146C2×C4×C31—82no—
147C2×C4×C31—82no—
148C2×C4×C31—82no—
149C2×C4×C31—81no—
150C2×C4×C31—81no—
151C2×C4×C31#12011yescharacteristic
152C2×C4×C31#12011yescharacteristic
153C2×C4×C31#12011yescharacteristic
154C2×C4×C31#12711yescharacteristic
155C8×C31—42no—
156C8×C31—42no—
157C8×C31—42no—
158C8×C31—81no—
159#3195C2×C3111yescharacteristic
160C2×#484C2×C3111yescharacteristic
161C2×#483C2×C3111yescharacteristic
162#1821C2×C3111yescharacteristic
163#1472C2×C3111yescharacteristic
164#1472C2×C3111yescharacteristic
165#1298C2×C3111yescharacteristic
166C22×C31—21no—
167C22×C31—21no—
168C22×C31—21no—
169C22×C31—21no—
170C22×C31—21no—
171C22×C31—21no—
172C22×C31—21no—
173C22×C31—44no—
174C22×C31#39211yescharacteristic
175C22×C31#47112yes—
176C22×C31#47312yes—
177C4×C31—21no—
178C4×C31—21no—
179C4×C31—21no—
180C4×C31—81no—
181C4×C31—82no—
182C4×C31—81no—
183C4×C31—81no—
184C4×C31—81no—
185C4×C31#45812yes—
186C2×C22⋊Q8C22×C3111yescharacteristic
187C2×C22⋊Q8C22×C3111yescharacteristic
188C22×C4⋊C4C22×C3111yescharacteristic
189#484—22no—
190#484—22no—
191#483—22no—
192#483—21no—
193#483—21no—
194#483—22no—
195#470—22no—
196#451—21no—
197#451—21no—
198C2×#107—21no—
199C2×#106—21no—
200C2×#103—21no—
201C2×#103—21no—
202#407C22×C3111yescharacteristic
203#407C22×C3111yescharacteristic
204C2×C22⋊C8C22×C3111yescharacteristic
205C2×C22⋊C8C22×C3111yescharacteristic
206#343—22no—
207#324—22no—
208#324—22no—
209C2×C31—22no—
210C2×C31—22no—
211C2×C31—41no—
212C2×C31#201911yescharacteristic
213C2×C31#202011yescharacteristic
214C2×C31#202211yescharacteristic
215C22×Q8D4×C3111yescharacteristic
216C22×Q8D4×C3111yescharacteristic
217C23×C4—21no—
218C23×C4C23×C3111yescharacteristic
219C2×Q16—22no—
220C2×M16—21no—
221C2×M16D4×C3112yes—
222C22×C8—21no—
223C22×C8—21no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4D4×C3112yes—
240C2×C4⋊C4D4×C3111yescharacteristic
241C2×C4⋊C4D4×C3111yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244#107—24no—
245#106—24no—
246#106—22no—
247#103—24no—
248#103—24no—
249#103—22no—
250#103—22no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8D4×C3112yes—
258C22⋊C8D4×C3112yes—
259C31#999511yescharacteristic, direct product factor, Sylow
260C24#41×C3111yescharacteristic
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—42no—
266C2×Q8—42no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—82no—
276C22×C4C2×D4×C3111yescharacteristic
277C22×C4C2×D4×C3111yescharacteristic
278C22×C4C2×D4×C3111yescharacteristic
279C22×C4C2×D4×C3111yescharacteristic
280C22×C4C2×D4×C3111yescharacteristic
281C22×C4C2×D4×C3111yescharacteristic
282Q16—82no—
283M16—42no—
284M16—42no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—41no—
291C4⋊C4—22no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C23—22no—
308C23—22no—
309C23—41no—
310C23C31×#12111yescharacteristic
311C23C31×#12111yescharacteristic
312C23C31×#12111yescharacteristic
313Q8—42no—
314Q8—42no—
315Q8—42no—
316Q8—42no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4C31×#12011yescharacteristic
336C2×C4C31×#12011yescharacteristic
337C2×C4C31×#12011yescharacteristic
338C2×C4C31×#12711yescharacteristic
339C8—42no—
340C8—42no—
341C8—42no—
342C8—81no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—44no—
351C22C31×#39211yescharacteristic
352C22C31×#47112yes—
353C22C31×#47312yes—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—81no—
358C4—82no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4C31×#45812yes—
363C2—22no—
364C2—22no—
365C2—41no—
366C2C31×#201911yescharacteristic
367C2C31×#202011yescharacteristic
368C2C31×#202211yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC24×C31#41
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
122. #9995C31direct (N × H)
259. C31#9995direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31952
C2×C31×#4842
C2×C31×#4832
C31×#18212
C31×#14722
C31×#12982
#999531