Finite Group Databasebeta

← C31×#9992 #21466751

Subgroups — C31×#9992 #21466751

Order 7936.

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

normal subgroups only

This group has 562 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×#3192C211yesmaximal, characteristic
2C2×C31×#481C211yesmaximal, characteristic, complemented
3C2×C31×#480C211yesmaximal, characteristic, complemented
4C31×#1821C211yesmaximal, characteristic, complemented
5C31×#1471C211yesmaximal, characteristic, complemented
6C31×#1471C211yesmaximal, characteristic, complemented
7C31×#1298C211yesmaximal, characteristic, complemented
8C2×C31×#121C2211yescharacteristic
9C2×C31×#121C2211yescharacteristic
10C22×C4⋊C4×C31C2211yescharacteristic
11C31×#481—22no—
12C31×#481—21no—
13C31×#481—21no—
14C31×#481—22no—
15C31×#480—22no—
16C31×#480—22no—
17C31×#469—22no—
18C31×#447—21no—
19C31×#447—21no—
20C2×C31×#107—21no—
21C2×C31×#106—21no—
22C2×C31×#102—21no—
23C2×C31×#102—21no—
24C31×#407C2211yescharacteristic
25C31×#407C2211yescharacteristic
26C2×C31×C22⋊C8C2211yescharacteristic
27C2×C31×C22⋊C8C2211yescharacteristic
28C31×#343—22no—
29C31×#323—22no—
30C31×#323—22no—
31C22×D4×C31—21no—
32C22×D4×C31—21no—
33C22×D4×C31D411yescharacteristic
34C22×D4×C31D411yescharacteristic
35C23×C4×C31—21no—
36C23×C4×C31C2311yescharacteristic
37C2×D8×C31—22no—
38C2×M16×C31—21no—
39C2×M16×C31D412yes—
40C22×C8×C31—21no—
41C22×C8×C31—21no—
42C31×#121—21no—
43C31×#121—21no—
44C31×#121—21no—
45C31×#121—21no—
46C31×#121—21no—
47C31×#121—21no—
48C31×#121—21no—
49C31×#121—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×#102—24no—
64C31×#102—24no—
65C31×#102—22no—
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—21no—
77C24×C31#4111yescharacteristic
78C2×D4×C31—21no—
79C2×D4×C31—21no—
80C2×D4×C31—21no—
81C2×D4×C31—21no—
82C2×D4×C31—42no—
83C2×D4×C31—42no—
84C2×D4×C31—41no—
85C2×D4×C31—41no—
86C2×D4×C31—41no—
87C2×D4×C31—41no—
88C2×D4×C31—82no—
89C2×D4×C31—82no—
90C22×C4×C31—21no—
91C22×C4×C31—21no—
92C22×C4×C31—22no—
93C22×C4×C31—42no—
94C22×C4×C31—41no—
95C22×C4×C31—41no—
96C22×C4×C31—82no—
97C22×C4×C31C2×D411yescharacteristic
98C22×C4×C31C2×D411yescharacteristic
99C22×C4×C31C2×D411yescharacteristic
100C22×C4×C31C2×D411yescharacteristic
101C22×C4×C31C2×D411yescharacteristic
102C22×C4×C31C2×D411yescharacteristic
103D8×C31—82no—
104M16×C31—42no—
105M16×C31—42no—
106C2×C8×C31—22no—
107C2×C8×C31—22no—
108C2×C8×C31—22no—
109C2×C8×C31—42no—
110C2×C8×C31—42no—
111C2×C8×C31—41no—
112C4⋊C4×C31—22no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—22no—
116C4⋊C4×C31—21no—
117C4⋊C4×C31—21no—
118C4⋊C4×C31—41no—
119C4⋊C4×C31—41no—
120C22⋊C4×C31—41no—
121C22⋊C4×C31—41no—
122C22⋊C4×C31—41no—
123C22⋊C4×C31—41no—
124#9992C3111yesmaximal, characteristic, direct product factor, Sylow
125C23×C31—22no—
126C23×C31—22no—
127C23×C31—41no—
128C23×C31—41no—
129C23×C31—41no—
130C23×C31—42no—
131C23×C31—42no—
132C23×C31—81no—
133C23×C31—81no—
134C23×C31—81no—
135C23×C31—81no—
136C23×C31#12111yescharacteristic
137C23×C31#12111yescharacteristic
138C23×C31#12111yescharacteristic
139D4×C31—42no—
140D4×C31—42no—
141D4×C31—42no—
142D4×C31—42no—
143D4×C31—82no—
144D4×C31—82no—
145D4×C31—82no—
146D4×C31—82no—
147C2×C4×C31—22no—
148C2×C4×C31—22no—
149C2×C4×C31—22no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—42no—
156C2×C4×C31—82no—
157C2×C4×C31—82no—
158C2×C4×C31—82no—
159C2×C4×C31—81no—
160C2×C4×C31—81no—
161C2×C4×C31#12011yescharacteristic
162C2×C4×C31#12011yescharacteristic
163C2×C4×C31#12011yescharacteristic
164C2×C4×C31#12711yescharacteristic
165C8×C31—42no—
166C8×C31—42no—
167C8×C31—42no—
168C8×C31—81no—
169#3192C2×C3111yescharacteristic
170C2×#481C2×C3111yescharacteristic, complemented
171C2×#480C2×C3111yescharacteristic, complemented
172#1821C2×C3111yescharacteristic, complemented
173#1471C2×C3111yescharacteristic, complemented
174#1471C2×C3111yescharacteristic, complemented
175#1298C2×C3111yescharacteristic, complemented
176C22×C31—21no—
177C22×C31—21no—
178C22×C31—21no—
179C22×C31—21no—
180C22×C31—21no—
181C22×C31—21no—
182C22×C31—21no—
183C22×C31—44no—
184C22×C31—81no—
185C22×C31—81no—
186C22×C31—81no—
187C22×C31—81no—
188C22×C31—81no—
189C22×C31—81no—
190C22×C31—162no—
191C22×C31—162no—
192C22×C31#39211yescharacteristic
193C22×C31#47112yes—
194C22×C31#47212yes—
195C4×C31—21no—
196C4×C31—21no—
197C4×C31—21no—
198C4×C31—82no—
199C4×C31—81no—
200C4×C31—81no—
201C4×C31#45812yes—
202C2×#121C22×C3111yescharacteristic
203C2×#121C22×C3111yescharacteristic
204C22×C4⋊C4C22×C3111yescharacteristic
205#481—22no—
206#481—21no—
207#481—21no—
208#481—22no—
209#480—22no—
210#480—22no—
211#469—22no—
212#447—21no—
213#447—21no—
214C2×#107—21no—
215C2×#106—21no—
216C2×#102—21no—
217C2×#102—21no—
218#407C22×C3111yescharacteristic
219#407C22×C3111yescharacteristic
220C2×C22⋊C8C22×C3111yescharacteristic
221C2×C22⋊C8C22×C3111yescharacteristic
222#343—22no—
223#323—22no—
224#323—22no—
225C2×C31—22no—
226C2×C31—22no—
227C2×C31—41no—
228C2×C31—161no—
229C2×C31—161no—
230C2×C31#201811yescharacteristic
231C2×C31#201911yescharacteristic
232C2×C31#202211yescharacteristic
233C22×D4—21no—
234C22×D4—21no—
235C22×D4D4×C3111yescharacteristic
236C22×D4D4×C3111yescharacteristic
237C23×C4—21no—
238C23×C4C23×C3111yescharacteristic
239C2×D8—22no—
240C2×M16—21no—
241C2×M16D4×C3112yes—
242C22×C8—21no—
243C22×C8—21no—
244#121—21no—
245#121—21no—
246#121—21no—
247#121—21no—
248#121—21no—
249#121—21no—
250#121—21no—
251#121—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4D4×C3112yes—
258C2×C4⋊C4D4×C3111yescharacteristic
259C2×C4⋊C4D4×C3111yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262#107—24no—
263#106—24no—
264#106—22no—
265#102—24no—
266#102—24no—
267#102—22no—
268#102—22no—
269C22⋊C8—21no—
270C22⋊C8—21no—
271C22⋊C8—21no—
272C22⋊C8—21no—
273C22⋊C8—21no—
274C22⋊C8—21no—
275C22⋊C8D4×C3112yes—
276C22⋊C8D4×C3112yes—
277C31#999211yescharacteristic, direct product factor, Sylow
278C24—21no—
279C24—21no—
280C24#41×C3111yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—82no—
292C2×D4—82no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—82no—
300C22×C4C2×D4×C3111yescharacteristic
301C22×C4C2×D4×C3111yescharacteristic
302C22×C4C2×D4×C3111yescharacteristic
303C22×C4C2×D4×C3111yescharacteristic
304C22×C4C2×D4×C3111yescharacteristic
305C22×C4C2×D4×C3111yescharacteristic
306D8—82no—
307M16—42no—
308M16—42no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—41no—
315C4⋊C4—22no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—22no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C23—22no—
328C23—22no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—42no—
333C23—42no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23C31×#12111yescharacteristic
339C23C31×#12111yescharacteristic
340C23C31×#12111yescharacteristic
341D4—42no—
342D4—42no—
343D4—42no—
344D4—42no—
345D4—82no—
346D4—82no—
347D4—82no—
348D4—82no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4C31×#12011yescharacteristic
364C2×C4C31×#12011yescharacteristic
365C2×C4C31×#12011yescharacteristic
366C2×C4C31×#12711yescharacteristic
367C8—42no—
368C8—42no—
369C8—42no—
370C8—81no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—44no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—162no—
386C22—162no—
387C22C31×#39211yescharacteristic
388C22C31×#47112yes—
389C22C31×#47212yes—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—82no—
394C4—81no—
395C4—81no—
396C4C31×#45812yes—
397C2—22no—
398C2—22no—
399C2—41no—
400C2—161no—
401C2—161no—
402C2C31×#201811yescharacteristic
403C2C31×#201911yescharacteristic
404C2C31×#202211yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC31×#3192C2
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
2. C2×C31×#481C2semidirect
3. C2×C31×#480C2semidirect
4. C31×#1821C2semidirect
5. C31×#1471C2semidirect
6. C31×#1471C2semidirect
7. C31×#1298C2semidirect
124. #9992C31direct (N × H)
170. C2×#481C2×C31semidirect
171. C2×#480C2×C31semidirect
172. #1821C2×C31semidirect
173. #1471C2×C31semidirect
174. #1471C2×C31semidirect
175. #1298C2×C31semidirect
277. C31#9992direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31922
C2×C31×#4812
C2×C31×#4802
C31×#18212
C31×#14712
C31×#12982
#999231