Finite Group Databasebeta

← C31×#7796 #21464555

Subgroups — C31×#7796 #21464555

Order 7936.

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

normal subgroups only

This group has 478 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×#2733C211yesmaximal, characteristic
2C31×#2093C211yesmaximal, characteristic, complemented
3C31×#2072C211yesmaximal, characteristic, complemented
4C31×#2071C211yesmaximal, characteristic, complemented
5C31×#1272C211yesmaximal, characteristic, complemented
6C31×#1270C211yesmaximal, characteristic, complemented
7C31×#1270C211yesmaximal, characteristic, complemented
8C2×C31×#124—21no—
9C2×C31×#107—21no—
10C2×C31×#103—21no—
11C2×C31×#102—21no—
12C2×C31×#102—21no—
13C31×#394—21no—
14C31×#392—21no—
15C31×#392—21no—
16C31×#391C2211yescharacteristic
17C31×#390C2211yescharacteristic
18C31×#390C2211yescharacteristic
19C31×#388—21no—
20C31×#388—21no—
21C31×#384C2211yescharacteristic
22C31×#340—21no—
23C31×#340—21no—
24C31×#336C2211yescharacteristic
25C31×#336C2211yescharacteristic
26C31×#336C2211yescharacteristic
27C22×Q8×C31D411yescharacteristic
28C22×D4×C31—21no—
29C22×D4×C31—21no—
30C22×D4×C31D411yescharacteristic
31C22×D4×C31D411yescharacteristic
32C22×C8×C31—21no—
33C22×C8×C31—21no—
34C22×C8×C31—21no—
35C31×#124—42no—
36C31×#124—42no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31D411yescharacteristic
41C2×C4⋊C4×C31D411yescharacteristic
42C2×C4⋊C4×C31D411yescharacteristic
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—21no—
48C2×C22⋊C4×C31—21no—
49C2×C22⋊C4×C31—21no—
50C2×C22⋊C4×C31—21no—
51C2×C42×C31—21no—
52C2×C42×C31C2311yescharacteristic
53C31×#107—42no—
54C31×#103—42no—
55C31×#102—42no—
56C31×#102—42no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C24×C31—21no—
61C24×C31—21no—
62C2×Q8×C31—21no—
63C2×Q8×C31—41no—
64C2×Q8×C31—41no—
65C2×Q8×C31D812yes—
66C2×D4×C31—21no—
67C2×D4×C31—21no—
68C2×D4×C31—21no—
69C2×D4×C31—21no—
70C2×D4×C31—41no—
71C2×D4×C31—41no—
72C2×D4×C31—41no—
73C2×D4×C31—41no—
74C2×D4×C31—42no—
75C2×D4×C31—81no—
76C2×D4×C31—81no—
77C2×D4×C31—81no—
78C2×D4×C31—81no—
79C2×D4×C31—81no—
80C22×C4×C31—21no—
81C22×C4×C31—21no—
82C22×C4×C31—21no—
83C22×C4×C31—21no—
84C22×C4×C31—41no—
85C22×C4×C31C2×D411yescharacteristic
86C22×C4×C31C2×D411yescharacteristic
87C22×C4×C31C2×D411yescharacteristic
88C2×C8×C31—22no—
89C2×C8×C31—41no—
90C2×C8×C31—41no—
91C2×C8×C31—41no—
92C2×C8×C31—41no—
93C2×C8×C31—41no—
94C2×C8×C31—41no—
95C2×C8×C31—41no—
96C2×C8×C31—41no—
97C4⋊C4×C31—41no—
98C4⋊C4×C31—41no—
99C4⋊C4×C31—41no—
100C4⋊C4×C31—41no—
101C4⋊C4×C31—41no—
102C4⋊C4×C31—41no—
103C4⋊C4×C31—41no—
104C4⋊C4×C31—41no—
105C4⋊C4×C31—41no—
106C22⋊C4×C31—42no—
107C22⋊C4×C31—42no—
108C22⋊C4×C31—41no—
109C22⋊C4×C31—41no—
110C22⋊C4×C31—41no—
111C22⋊C4×C31—41no—
112C22⋊C4×C31—41no—
113C22⋊C4×C31—41no—
114C22⋊C4×C31—41no—
115C22⋊C4×C31—41no—
116C22⋊C4×C31—81no—
117C22⋊C4×C31—81no—
118C42×C31—41no—
119C42×C31—42no—
120#7796C3111yesmaximal, characteristic, direct product factor, Sylow
121C23×C31—41no—
122C23×C31—41no—
123C23×C31—41no—
124C23×C31—41no—
125C23×C31—41no—
126C23×C31—41no—
127C23×C31—41no—
128C23×C31—41no—
129C23×C31—41no—
130C23×C31—41no—
131C23×C31—41no—
132C23×C31—41no—
133C23×C31—41no—
134C23×C31—41no—
135C23×C31#12011yescharacteristic
136Q8×C31—42no—
137Q8×C31—81no—
138D4×C31—42no—
139D4×C31—42no—
140D4×C31—81no—
141D4×C31—81no—
142D4×C31—84no—
143D4×C31—84no—
144C2×C4×C31—21no—
145C2×C4×C31—21no—
146C2×C4×C31—21no—
147C2×C4×C31—21no—
148C2×C4×C31—21no—
149C2×C4×C31—21no—
150C2×C4×C31—22no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—81no—
163C2×C4×C31—81no—
164C2×C4×C31—81no—
165C2×C4×C31C2×D811yescharacteristic
166C2×C4×C31#13511yescharacteristic
167C2×C4×C31#13511yescharacteristic
168C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C31#13711yescharacteristic
171C8×C31—42no—
172C8×C31—81no—
173C8×C31—81no—
174#2733C2×C3111yescharacteristic
175#2093C2×C3111yescharacteristic, complemented
176#2072C2×C3111yescharacteristic, complemented
177#2071C2×C3111yescharacteristic, complemented
178#1272C2×C3111yescharacteristic, complemented
179#1270C2×C3111yescharacteristic, complemented
180#1270C2×C3111yescharacteristic, complemented
181C22×C31—81no—
182C22×C31—81no—
183C22×C31—81no—
184C22×C31—81no—
185C22×C31—81no—
186C22×C31—81no—
187C22×C31—81no—
188C22×C31—81no—
189C22×C31—81no—
190C22×C31—81no—
191C22×C31—81no—
192C22×C31—81no—
193C22×C31—81no—
194C22×C31—81no—
195C22×C31#44711yescharacteristic
196C22×C31#44911yescharacteristic
197C22×C31#45211yescharacteristic
198C22×C31#45311yescharacteristic
199C22×C31#45511yescharacteristic
200C22×C31#45511yescharacteristic
201C22×C31#45811yescharacteristic
202C4×C31—21no—
203C4×C31—21no—
204C4×C31—21no—
205C4×C31—21no—
206C4×C31—21no—
207C4×C31—21no—
208C4×C31—42no—
209C4×C31—81no—
210C4×C31—81no—
211C4×C31—81no—
212C4×C31—161no—
213C2×#124—21no—
214C2×#107—21no—
215C2×#103—21no—
216C2×#102—21no—
217C2×#102—21no—
218#394—21no—
219#392—21no—
220#392—21no—
221#391C22×C3111yescharacteristic
222#390C22×C3111yescharacteristic
223#390C22×C3111yescharacteristic
224#388—21no—
225#388—21no—
226#384C22×C3111yescharacteristic
227#340—21no—
228#340—21no—
229#336C22×C3111yescharacteristic
230#336C22×C3111yescharacteristic
231#336C22×C3111yescharacteristic
232C2×C31—161no—
233C2×C31—161no—
234C2×C31#160711yescharacteristic
235C2×C31#161211yescharacteristic
236C2×C31#161811yescharacteristic
237C2×C31#162111yescharacteristic
238C2×C31#164111yescharacteristic
239C2×C31#164611yescharacteristic
240C2×C31#165911yescharacteristic
241C22×Q8D4×C3111yescharacteristic
242C22×D4—21no—
243C22×D4—21no—
244C22×D4D4×C3111yescharacteristic
245C22×D4D4×C3111yescharacteristic
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249#124—42no—
250#124—42no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4D4×C3111yescharacteristic
255C2×C4⋊C4D4×C3111yescharacteristic
256C2×C4⋊C4D4×C3111yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C42—21no—
266C2×C42C23×C3111yescharacteristic
267#107—42no—
268#103—42no—
269#102—42no—
270#102—42no—
271#95—21no—
272#95—21no—
273#95—21no—
274C31#779611yescharacteristic, direct product factor, Sylow
275C24—21no—
276C24—21no—
277C2×Q8—21no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8D8×C3112yes—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—42no—
290C2×D4—81no—
291C2×D4—81no—
292C2×D4—81no—
293C2×D4—81no—
294C2×D4—81no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4C2×D4×C3111yescharacteristic
301C22×C4C2×D4×C3111yescharacteristic
302C22×C4C2×D4×C3111yescharacteristic
303C2×C8—22no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—81no—
332C22⋊C4—81no—
333C42—41no—
334C42—42no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23C31×#12011yescharacteristic
350Q8—42no—
351Q8—81no—
352D4—42no—
353D4—42no—
354D4—81no—
355D4—81no—
356D4—84no—
357D4—84no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4C2×D8×C3111yescharacteristic
380C2×C4C31×#13511yescharacteristic
381C2×C4C31×#13511yescharacteristic
382C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
383C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
384C2×C4C31×#13711yescharacteristic
385C8—42no—
386C8—81no—
387C8—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22C31×#44711yescharacteristic
403C22C31×#44911yescharacteristic
404C22C31×#45211yescharacteristic
405C22C31×#45311yescharacteristic
406C22C31×#45511yescharacteristic
407C22C31×#45511yescharacteristic
408C22C31×#45811yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—42no—
416C4—81no—
417C4—81no—
418C4—81no—
419C4—161no—
420C2—161no—
421C2—161no—
422C2C31×#160711yescharacteristic
423C2C31×#161211yescharacteristic
424C2C31×#161811yescharacteristic
425C2C31×#162111yescharacteristic
426C2C31×#164111yescharacteristic
427C2C31×#164611yescharacteristic
428C2C31×#165911yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#2733C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2093C2semidirect
3. C31×#2072C2semidirect
4. C31×#2071C2semidirect
5. C31×#1272C2semidirect
6. C31×#1270C2semidirect
7. C31×#1270C2semidirect
120. #7796C31direct (N × H)
175. #2093C2×C31semidirect
176. #2072C2×C31semidirect
177. #2071C2×C31semidirect
178. #1272C2×C31semidirect
179. #1270C2×C31semidirect
180. #1270C2×C31semidirect
274. C31#7796direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27332
C31×#20932
C31×#20722
C31×#20712
C31×#12722
C31×#12702
#779631