Finite Group Databasebeta

← C31×#61375 #21518134

Subgroups — C31×#61375 #21518134

Order 7936.

1706 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3504C212yesmaximal, complemented
2C31×#3390C214yesmaximal, complemented
3C31×#3389C214yesmaximal, complemented
4C31×#3381C218yesmaximal, complemented
5C31×#3163C212yesmaximal, complemented
6C2×C31×#499C211yesmaximal, characteristic, complemented
7C31×#3083C214yesmaximal, complemented
8C31×#3080C212yesmaximal, complemented
9C31×#2960C211yesmaximal, characteristic, complemented
10C31×#2952C212yesmaximal
11C4×C31×#135C211yesmaximal, characteristic
12C2×C31×#135C2211yescharacteristic
13C31×#557C2218yes—
14C31×#554—28no—
15C31×#554C2218yes—
16C31×#544—24no—
17C31×#544C2214yes—
18C31×#533—22no—
19C31×#533C2211yescharacteristic
20C2×C31×#125—22no—
21C2×C31×#125C2212yes—
22C31×#520C2214yes—
23C4×#41×C31C2212yes—
24C31×#501C2218yes—
25C31×#499C2214yes—
26C31×#499C2214yes—
27C31×#499C2214yes—
28C31×#498C2214yes—
29C31×#479C2218yes—
30C31×#479C2218yes—
31C31×#478C2218yes—
32C31×#478C2218yes—
33C31×#477C2218yes—
34C31×#474C2218yes—
35C31×#446C2214yes—
36C8×Q8×C31C2214yes—
37C31×#442C2214yes—
38C31×#441C2214yes—
39C4×Q16×C31C2212yes—
40C4×SD16×C31C2214yes—
41C4×SD16×C31C2218yes—
42C4×D8×C31C2212yes—
43C31×#428C2214yes—
44C31×#427C2211yescharacteristic
45C2×C31×#107C2212yes—
46C2×C31×#106C2212yes—
47C31×#424C2212yes—
48C31×#423C2211yescharacteristic
49C31×#417C2214yes—
50C31×#416C2212yes—
51C31×#405C2212yes—
52C2×C4×C8×C31C2211yescharacteristic
53C2×#41×C31C2312yes—
54C31×#135Q818yescomplemented
55C2×Q16×C31C2311yescharacteristic
56C2×SD16×C31C2312yes—
57C2×D8×C31C2311yescharacteristic
58C2×M16×C31C2312yes—
59C22×C8×C31C2311yescharacteristic
60C22×C8×C31C2311yescharacteristic
61C31×C4⋊Q8—28no—
62C31×C4⋊Q8—24no—
63C31×C4⋊Q8D412yes—
64C31×C4⋊Q8C2318yes—
65C31×#125—24no—
66C31×#125—28no—
67C31×#125—28no—
68C31×#125—22no—
69C31×#125—22no—
70C31×#125D412yes—
71C31×#125C2318yes—
72C31×#125C2318yes—
73C31×C22⋊Q8—28no—
74C31×C22⋊Q8—28no—
75C31×C22⋊Q8—28no—
76C31×C22⋊Q8—28no—
77C31×C22⋊Q8D414yes—
78C4×Q8×C31—28no—
79C4×Q8×C31C2314yes—
80C4×Q8×C31C2318yes—
81C4×Q8×C31C2314yes—
82C4×D4×C31—24no—
83C4×D4×C31—28no—
84C4×D4×C31C2314yes—
85C4×D4×C31C2318yes—
86C31×#117—24no—
87C31×#117—22no—
88C31×#117C2312yes—
89C31×#117C2312yes—
90C2×C4⋊C4×C31—24no—
91C2×C4⋊C4×C31—24no—
92C2×C4⋊C4×C31—28no—
93C2×C4⋊C4×C31C2314yes—
94C2×C42×C31—22no—
95C2×C42×C31C2311yescharacteristic
96C31×#107C2314yes—
97C31×#107C2314yes—
98C31×#107C2312yes—
99C31×#107C2318yes—
100C31×#106C2314yes—
101C31×#106C2314yes—
102C31×#106C2312yes—
103C31×#106C2318yes—
104C31×C4⋊C8C2314yes—
105C31×C4⋊C8C2318yes—
106C31×#103C2314yes—
107C31×#103C2318yes—
108C31×#102C2314yes—
109C31×#102C2318yes—
110C31×#97C2314yes—
111C4×C8×C31C2312yes—
112C4×C8×C31C2312yes—
113C4×C8×C31C2314yes—
114#41×C31—24no—
115#41×C31C2×Q818yes—
116C2×Q8×C31—28no—
117C2×Q8×C31C2×D414yes—
118C2×Q8×C31C2412yes—
119C2×D4×C31—22no—
120C2×D4×C31C2412yes—
121C22×C4×C31—22no—
122C22×C4×C31—24no—
123C22×C4×C31—24no—
124C22×C4×C31—28no—
125C22×C4×C31C2×D412yes—
126C22×C4×C31C2411yescharacteristic
127Q16×C31C2×Q814yes—
128SD16×C31C2×Q818yes—
129D8×C31C2×Q814yes—
130M16×C31—24no—
131C2×C8×C31—22no—
132C2×C8×C31C2×Q814yes—
133C2×C8×C31C2412yes—
134C2×C8×C31C2414yes—
135C2×C8×C31C2411yescharacteristic
136C2×C8×C31C2411yescharacteristic
137C4⋊C4×C31—28no—
138C4⋊C4×C31—24no—
139C4⋊C4×C31—28no—
140C4⋊C4×C31—28no—
141C4⋊C4×C31—28no—
142C4⋊C4×C31—28no—
143C4⋊C4×C31—24no—
144C4⋊C4×C31—28no—
145C4⋊C4×C31C2×D414yes—
146C4⋊C4×C31C2×D418yes—
147C4⋊C4×C31C2×D414yes—
148C4⋊C4×C31C2418yes—
149C4⋊C4×C31C2414yes—
150C22⋊C4×C31—22no—
151C22⋊C4×C31—24no—
152C22⋊C4×C31—28no—
153C22⋊C4×C31C2×D414yes—
154C42×C31—22no—
155C42×C31—24no—
156C42×C31—28no—
157C42×C31C2×D411yescharacteristic
158C42×C31C2×D411yescharacteristic
159C42×C31C2412yes—
160C42×C31C2414yes—
161#61375C3111yesmaximal, characteristic, direct product factor, Sylow
162C23×C31—22no—
163C23×C31C22×D411yescharacteristic
164Q8×C31—22no—
165Q8×C31—44no—
166Q8×C31—88no—
167Q8×C31C22×Q814yes—
168D4×C31—28no—
169D4×C31—22no—
170D4×C31C22×Q814yes—
171C2×C4×C31—22no—
172C2×C4×C31—28no—
173C2×C4×C31—24no—
174C2×C4×C31—28no—
175C2×C4×C31—42no—
176C2×C4×C31—84no—
177C2×C4×C31—88no—
178C2×C4×C31—84no—
179C2×C4×C31C22×D412yes—
180C2×C4×C31C22×D418yes—
181C2×C4×C31C22×D412yes—
182C2×C4×C31C22×D411yescharacteristic
183C2×C4×C31C22×Q812yes—
184C2×C4×C312−1+412yes—
185C2×C4×C31C2511yescharacteristic
186C8×C31—22no—
187C8×C31—24no—
188C8×C31C22×Q812yes—
189C8×C31C22×Q812yes—
190#3504C2×C3112yescomplemented
191#3390C2×C3114yescomplemented
192#3389C2×C3114yescomplemented
193#3381C2×C3118yescomplemented
194#3163C2×C3112yescomplemented
195C2×#499C2×C3111yescharacteristic, complemented
196#3083C2×C3114yescomplemented
197#3080C2×C3112yescomplemented
198#2960C2×C3111yescharacteristic, complemented
199#2952C2×C3112yes—
200C4×#135C2×C3111yescharacteristic
201C22×C31—22no—
202C22×C31—24no—
203C22×C31—44no—
204C22×C31D4×Q812yes—
205C22×C31C23×D411yescharacteristic
206C4×C31—22no—
207C4×C31—22no—
208C4×C31—28no—
209C4×C31—24no—
210C4×C31—48no—
211C4×C31—44no—
212C4×C31D4×Q812yes—
213C4×C31C23×Q811yescharacteristic
214C4×C31C2×2−1+411yescharacteristic
215C2×#135C22×C3111yescharacteristic
216#557C22×C3118yes—
217#554—28no—
218#554C22×C3118yes—
219#544—24no—
220#544C22×C3114yes—
221#533—22no—
222#533C22×C3111yescharacteristic
223C2×#125—22no—
224C2×#125C22×C3112yes—
225#520C22×C3114yes—
226C4×#41C22×C3112yes—
227#501C22×C3118yes—
228#499C22×C3114yes—
229#499C22×C3114yes—
230#499C22×C3114yes—
231#498C22×C3114yes—
232#479C22×C3118yes—
233#479C22×C3118yes—
234#478C22×C3118yes—
235#478C22×C3118yes—
236#477C22×C3118yes—
237#474C22×C3118yes—
238#446C22×C3114yes—
239C8×Q8C22×C3114yes—
240#442C22×C3114yes—
241#441C22×C3114yes—
242C4×Q16C22×C3112yes—
243C4×SD16C22×C3114yes—
244C4×SD16C22×C3118yes—
245C4×D8C22×C3112yes—
246#428C22×C3114yes—
247#427C22×C3111yescharacteristic
248C2×#107C22×C3112yes—
249C2×#106C22×C3112yes—
250#424C22×C3112yes—
251#423C22×C3111yescharacteristic
252#417C22×C3114yes—
253#416C22×C3112yes—
254#405C22×C3112yes—
255C2×C4×C8C22×C3111yescharacteristic
256C2×C31—22no—
257C2×C31—44no—
258C2×C31C2×D4×Q811yescharacteristic
259C2×C31#358412yes—
260C2×#41C23×C3112yes—
261#135Q8×C3118yescomplemented
262C2×Q16C23×C3111yescharacteristic
263C2×SD16C23×C3112yes—
264C2×D8C23×C3111yescharacteristic
265C2×M16C23×C3112yes—
266C22×C8C23×C3111yescharacteristic
267C22×C8C23×C3111yescharacteristic
268C4⋊Q8—28no—
269C4⋊Q8—24no—
270C4⋊Q8D4×C3112yes—
271C4⋊Q8C23×C3118yes—
272#125—24no—
273#125—28no—
274#125—28no—
275#125—22no—
276#125—22no—
277#125D4×C3112yes—
278#125C23×C3118yes—
279#125C23×C3118yes—
280C22⋊Q8—28no—
281C22⋊Q8—28no—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8D4×C3114yes—
285C4×Q8—28no—
286C4×Q8C23×C3114yes—
287C4×Q8C23×C3118yes—
288C4×Q8C23×C3114yes—
289C4×D4—24no—
290C4×D4—28no—
291C4×D4C23×C3114yes—
292C4×D4C23×C3118yes—
293#117—24no—
294#117—22no—
295#117C23×C3112yes—
296#117C23×C3112yes—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4C23×C3114yes—
301C2×C42—22no—
302C2×C42C23×C3111yescharacteristic
303#107C23×C3114yes—
304#107C23×C3114yes—
305#107C23×C3112yes—
306#107C23×C3118yes—
307#106C23×C3114yes—
308#106C23×C3114yes—
309#106C23×C3112yes—
310#106C23×C3118yes—
311C4⋊C8C23×C3114yes—
312C4⋊C8C23×C3118yes—
313#103C23×C3114yes—
314#103C23×C3118yes—
315#102C23×C3114yes—
316#102C23×C3118yes—
317#97C23×C3114yes—
318C4×C8C23×C3112yes—
319C4×C8C23×C3112yes—
320C4×C8C23×C3114yes—
321C31#6137511yescharacteristic, direct product factor, Sylow
322#41—24no—
323#41C2×Q8×C3118yes—
324C2×Q8—28no—
325C2×Q8C2×D4×C3114yes—
326C2×Q8C24×C3112yes—
327C2×D4—22no—
328C2×D4C24×C3112yes—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—28no—
333C22×C4C2×D4×C3112yes—
334C22×C4C24×C3111yescharacteristic
335Q16C2×Q8×C3114yes—
336SD16C2×Q8×C3118yes—
337D8C2×Q8×C3114yes—
338M16—24no—
339C2×C8—22no—
340C2×C8C2×Q8×C3114yes—
341C2×C8C24×C3112yes—
342C2×C8C24×C3114yes—
343C2×C8C24×C3111yescharacteristic
344C2×C8C24×C3111yescharacteristic
345C4⋊C4—28no—
346C4⋊C4—24no—
347C4⋊C4—28no—
348C4⋊C4—28no—
349C4⋊C4—28no—
350C4⋊C4—28no—
351C4⋊C4—24no—
352C4⋊C4—28no—
353C4⋊C4C2×D4×C3114yes—
354C4⋊C4C2×D4×C3118yes—
355C4⋊C4C2×D4×C3114yes—
356C4⋊C4C24×C3118yes—
357C4⋊C4C24×C3114yes—
358C22⋊C4—22no—
359C22⋊C4—24no—
360C22⋊C4—28no—
361C22⋊C4C2×D4×C3114yes—
362C42—22no—
363C42—24no—
364C42—28no—
365C42C2×D4×C3111yescharacteristic
366C42C2×D4×C3111yescharacteristic
367C42C24×C3112yes—
368C42C24×C3114yes—
369C23—22no—
370C23C22×D4×C3111yescharacteristic
371Q8—22no—
372Q8—44no—
373Q8—88no—
374Q8C22×Q8×C3114yes—
375D4—28no—
376D4—22no—
377D4C22×Q8×C3114yes—
378C2×C4—22no—
379C2×C4—28no—
380C2×C4—24no—
381C2×C4—28no—
382C2×C4—42no—
383C2×C4—84no—
384C2×C4—88no—
385C2×C4—84no—
386C2×C4C22×D4×C3112yes—
387C2×C4C22×D4×C3118yes—
388C2×C4C22×D4×C3112yes—
389C2×C4C22×D4×C3111yescharacteristic
390C2×C4C22×Q8×C3112yes—
391C2×C4C31×2−1+412yes—
392C2×C4C25×C3111yescharacteristic
393C8—22no—
394C8—24no—
395C8C22×Q8×C3112yes—
396C8C22×Q8×C3112yes—
397C22—22no—
398C22—24no—
399C22—44no—
400C22D4×Q8×C3112yes—
401C22C23×D4×C3111yescharacteristic
402C4—24no—
403C4—22no—
404C4—22no—
405C4—28no—
406C4—48no—
407C4—44no—
408C4D4×Q8×C3112yes—
409C4C23×Q8×C3111yescharacteristic
410C4C2×C31×2−1+411yescharacteristic
411C2—22no—
412C2—44no—
413C2C2×D4×Q8×C3111yescharacteristic
414C2C31×#358412yes—
subgroupG/H
center, socleC22×C31C23×D4
commutator, Frattini, cosocleC2×C4C25×C31
prime radicalC2×C31×#135C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C31×#533 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3504C2semidirect
2. C31×#3390C2semidirect
3. C31×#3389C2semidirect
4. C31×#3381C2semidirect
5. C31×#3163C2semidirect
6. C2×C31×#499C2semidirect
7. C31×#3083C2semidirect
8. C31×#3080C2semidirect
9. C31×#2960C2semidirect
54. C31×#135Q8semidirect
161. #61375C31direct (N × H)
190. #3504C2×C31semidirect
191. #3390C2×C31semidirect
192. #3389C2×C31semidirect
193. #3381C2×C31semidirect
194. #3163C2×C31semidirect
195. C2×#499C2×C31semidirect
196. #3083C2×C31semidirect
197. #3080C2×C31semidirect
198. #2960C2×C31semidirect
261. #135Q8×C31semidirect
321. C31#61375direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35042
C31×#33902
C31×#33892
C31×#33812
C31×#31632
C2×C31×#4992
C31×#30832
C31×#30802
C31×#29602
C31×#29522
C4×C31×#1352
#6137531