Finite Group Databasebeta

← C31×#13427 #21470186

Subgroups — C31×#13427 #21470186

Order 7936.

462 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2731C211yesmaximal, characteristic
2C2×C31×#356C211yesmaximal, characteristic, complemented
3C2×C31×#355C211yesmaximal, characteristic
4C31×#1888C211yesmaximal, characteristic
5C31×#1886C211yesmaximal, characteristic, complemented
6C31×#1857C211yesmaximal, characteristic, complemented
7C31×#1394C211yesmaximal, characteristic, complemented
8C2×C31×C4⋊Q8C2211yescharacteristic
9C2×C31×#124C2211yescharacteristic
10C31×#519C2211yescharacteristic
11C31×#428—21no—
12C31×#419—21no—
13C31×#418—21no—
14C2×C31×#101C2211yescharacteristic
15C2×C31×#101C2211yescharacteristic
16C2×C31×#101C2211yescharacteristic
17C31×#396—21no—
18C31×#394—21no—
19C31×#393—21no—
20C31×#391C2211yescharacteristic
21C31×#388—21no—
22C31×#388—21no—
23C31×#387—21no—
24C31×#356—22no—
25C31×#355—22no—
26C31×#339—21no—
27C31×#339—21no—
28C31×#339—21no—
29C22×Q8×C31D411yescharacteristic
30C22×Q8×C31C2311yescharacteristic
31C22×D4×C31D411yescharacteristic
32C2×M16×C31—21no—
33C2×M16×C31—21no—
34C2×M16×C31—21no—
35C31×C4⋊Q8—21no—
36C31×C4⋊Q8—21no—
37C31×C4⋊Q8—21no—
38C31×C4⋊Q8—21no—
39C31×#124—21no—
40C31×#124—21no—
41C31×#124—21no—
42C31×#124—21no—
43C4×Q8×C31—41no—
44C4×Q8×C31—41no—
45C31×#117—21no—
46C31×#117—21no—
47C31×#117—21no—
48C2×C4⋊C4×C31—21no—
49C2×C4⋊C4×C31—21no—
50C2×C4⋊C4×C31—21no—
51C2×C4⋊C4×C31D411yescharacteristic
52C2×C22⋊C4×C31—21no—
53C2×C22⋊C4×C31—21no—
54C2×C22⋊C4×C31—21no—
55C2×C22⋊C4×C31—21no—
56C2×C22⋊C4×C31—21no—
57C2×C22⋊C4×C31—21no—
58C2×C42×C31D411yescharacteristic
59C2×C42×C31D411yescharacteristic
60C2×C42×C31D411yescharacteristic
61C31×#107—41no—
62C31×#106—41no—
63C31×#103—41no—
64C31×#103—41no—
65C31×#102—41no—
66C31×#102—41no—
67C31×#101—21no—
68C31×#101—21no—
69C31×#101—21no—
70C31×#101—21no—
71C31×#101—21no—
72C31×#101D412yes—
73C31×#95—21no—
74C31×#95—21no—
75C31×#95—21no—
76C31×#95—21no—
77C31×#95—21no—
78C24×C31—21no—
79C2×Q8×C31—21no—
80C2×Q8×C31—21no—
81C2×Q8×C31—41no—
82C2×Q8×C31—41no—
83C2×Q8×C31—41no—
84C2×Q8×C31—41no—
85C2×Q8×C31C2×D411yescharacteristic
86C2×Q8×C31#4111yescharacteristic
87C2×Q8×C31#4111yescharacteristic
88C2×Q8×C31#4111yescharacteristic
89C2×D4×C31—21no—
90C2×D4×C31—21no—
91C2×D4×C31—41no—
92C2×D4×C31—41no—
93C22×C4×C31—21no—
94C22×C4×C31—21no—
95C22×C4×C31—21no—
96C22×C4×C31—21no—
97C22×C4×C31—21no—
98C22×C4×C31C2×D411yescharacteristic
99C22×C4×C31C2×D411yescharacteristic
100C22×C4×C31C2×D411yescharacteristic
101M16×C31—22no—
102M16×C31—41no—
103M16×C31—41no—
104M16×C31—41no—
105M16×C31—41no—
106M16×C31—41no—
107C2×C8×C31—41no—
108C2×C8×C31—41no—
109C2×C8×C31—41no—
110C4⋊C4×C31—21no—
111C4⋊C4×C31—21no—
112C4⋊C4×C31—21no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—21no—
116C4⋊C4×C31—42no—
117C4⋊C4×C31—41no—
118C4⋊C4×C31—41no—
119C4⋊C4×C31—41no—
120C4⋊C4×C31—41no—
121C4⋊C4×C31—41no—
122C22⋊C4×C31—42no—
123C22⋊C4×C31—42no—
124C22⋊C4×C31—41no—
125C22⋊C4×C31—41no—
126C22⋊C4×C31—41no—
127C22⋊C4×C31—41no—
128C22⋊C4×C31—41no—
129C22⋊C4×C31—81no—
130C22⋊C4×C31—81no—
131C42×C31—21no—
132C42×C31—21no—
133C42×C31—21no—
134C42×C31—21no—
135C42×C31—41no—
136C42×C31—41no—
137C42×C31—41no—
138C42×C31—41no—
139#13427C3111yesmaximal, characteristic, direct product factor, Sylow
140C23×C31—41no—
141C23×C31—41no—
142C23×C31—41no—
143C23×C31—41no—
144C23×C31—41no—
145C23×C31—41no—
146C23×C31—41no—
147C23×C31#12011yescharacteristic
148Q8×C31—41no—
149Q8×C31—41no—
150Q8×C31—41no—
151Q8×C31—41no—
152Q8×C31—42no—
153Q8×C31—81no—
154D4×C31—42no—
155D4×C31—81no—
156C2×C4×C31—21no—
157C2×C4×C31—21no—
158C2×C4×C31—21no—
159C2×C4×C31—21no—
160C2×C4×C31—21no—
161C2×C4×C31—21no—
162C2×C4×C31—22no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—41no—
169C2×C4×C31—41no—
170C2×C4×C31—41no—
171C2×C4×C31—41no—
172C2×C4×C31—41no—
173C2×C4×C31—41no—
174C2×C4×C31—41no—
175C2×C4×C31—41no—
176C2×C4×C31—41no—
177C2×C4×C31—41no—
178C2×C4×C31—41no—
179C2×C4×C31#12111yescharacteristic
180C2×C4×C31#12111yescharacteristic
181C2×C4×C31#12111yescharacteristic
182C2×C4×C31#12311yescharacteristic
183C2×C4×C31#12311yescharacteristic
184C2×C4×C31#12411yescharacteristic
185C8×C31—42no—
186C8×C31—81no—
187C8×C31—81no—
188#2731C2×C3111yescharacteristic
189C2×#356C2×C3111yescharacteristic, complemented
190C2×#355C2×C3111yescharacteristic
191#1888C2×C3111yescharacteristic
192#1886C2×C3111yescharacteristic, complemented
193#1857C2×C3111yescharacteristic, complemented
194#1394C2×C3111yescharacteristic, complemented
195C22×C31—21no—
196C22×C31—21no—
197C22×C31—81no—
198C22×C31—81no—
199C22×C31—81no—
200C22×C31—81no—
201C22×C31—81no—
202C22×C31—81no—
203C22×C31—81no—
204C22×C31#39411yescharacteristic
205C22×C31B(4,𝔽2)11yescharacteristic
206C22×C31#45811yescharacteristic
207C4×C31—21no—
208C4×C31—21no—
209C4×C31—21no—
210C4×C31—21no—
211C4×C31—21no—
212C4×C31—21no—
213C4×C31—42no—
214C4×C31—81no—
215C4×C31—81no—
216C4×C31—81no—
217C4×C31—81no—
218C4×C31—81no—
219C4×C31—81no—
220C2×C4⋊Q8C22×C3111yescharacteristic
221C2×#124C22×C3111yescharacteristic
222#519C22×C3111yescharacteristic
223#428—21no—
224#419—21no—
225#418—21no—
226C2×#101C22×C3111yescharacteristic
227C2×#101C22×C3111yescharacteristic
228C2×#101C22×C3111yescharacteristic
229#396—21no—
230#394—21no—
231#393—21no—
232#391C22×C3111yescharacteristic
233#388—21no—
234#388—21no—
235#387—21no—
236#356—22no—
237#355—22no—
238#339—21no—
239#339—21no—
240#339—21no—
241C2×C31—21no—
242C2×C31—21no—
243C2×C31—161no—
244C2×C31#204411yescharacteristic
245C2×C31#220311yescharacteristic
246C2×C31#220411yescharacteristic
247C22×Q8D4×C3111yescharacteristic
248C22×Q8C23×C3111yescharacteristic
249C22×D4D4×C3111yescharacteristic
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257#124—21no—
258#124—21no—
259#124—21no—
260#124—21no—
261C4×Q8—41no—
262C4×Q8—41no—
263#117—21no—
264#117—21no—
265#117—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4D4×C3111yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C42D4×C3111yescharacteristic
277C2×C42D4×C3111yescharacteristic
278C2×C42D4×C3111yescharacteristic
279#107—41no—
280#106—41no—
281#103—41no—
282#103—41no—
283#102—41no—
284#102—41no—
285#101—21no—
286#101—21no—
287#101—21no—
288#101—21no—
289#101—21no—
290#101D4×C3112yes—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296C31#1342711yescharacteristic, direct product factor, Sylow
297C24—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8C2×D4×C3111yescharacteristic
305C2×Q8#41×C3111yescharacteristic
306C2×Q8#41×C3111yescharacteristic
307C2×Q8#41×C3111yescharacteristic
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—41no—
311C2×D4—41no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4C2×D4×C3111yescharacteristic
318C22×C4C2×D4×C3111yescharacteristic
319C22×C4C2×D4×C3111yescharacteristic
320M16—22no—
321M16—41no—
322M16—41no—
323M16—41no—
324M16—41no—
325M16—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—81no—
349C22⋊C4—81no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23C31×#12011yescharacteristic
366Q8—41no—
367Q8—41no—
368Q8—41no—
369Q8—41no—
370Q8—42no—
371Q8—81no—
372D4—42no—
373D4—81no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4C31×#12111yescharacteristic
398C2×C4C31×#12111yescharacteristic
399C2×C4C31×#12111yescharacteristic
400C2×C4C31×#12311yescharacteristic
401C2×C4C31×#12311yescharacteristic
402C2×C4C31×#12411yescharacteristic
403C8—42no—
404C8—81no—
405C8—81no—
406C22—21no—
407C22—21no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22C31×#39411yescharacteristic
416C22C31×B(4,𝔽2)11yescharacteristic
417C22C31×#45811yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—42no—
425C4—81no—
426C4—81no—
427C4—81no—
428C4—81no—
429C4—81no—
430C4—81no—
431C2—21no—
432C2—21no—
433C2—161no—
434C2C31×#204411yescharacteristic
435C2C31×#220311yescharacteristic
436C2C31×#220411yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC22×Q8C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C22×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#356C2semidirect
5. C31×#1886C2semidirect
6. C31×#1857C2semidirect
7. C31×#1394C2semidirect
139. #13427C31direct (N × H)
189. C2×#356C2×C31semidirect
192. #1886C2×C31semidirect
193. #1857C2×C31semidirect
194. #1394C2×C31semidirect
296. C31#13427direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27312
C2×C31×#3562
C2×C31×#3552
C31×#18882
C31×#18862
C31×#18572
C31×#13942
#1342731