Finite Group Databasebeta

← C31×#13249 #21470008

Subgroups — C31×#13249 #21470008

Order 7936.

444 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#509C211yesmaximal, characteristic
2C2×C31×#365C211yesmaximal, characteristic, complemented
3C2×C31×#361C211yesmaximal, characteristic, complemented
4C2×C31×#360C211yesmaximal, characteristic, complemented
5C2×C31×#347C211yesmaximal, characteristic, complemented
6C31×#1975C211yesmaximal, characteristic, complemented
7C31×#1972C211yesmaximal, characteristic, complemented
8C2×C31×#135C2211yescharacteristic
9C22×D8×C31C2211yescharacteristic
10C31×#509—22no—
11C31×#509—22no—
12C2×C31×SD32C412yescomplemented
13C2×C31×D16C412yescomplemented
14C2×C31×M32C2211yescharacteristic
15C2×C31×M32C2211yescharacteristic
16C2×C31×#108C2211yescharacteristic
17C2×C31×#106C2211yescharacteristic
18C2×C31×#104—21no—
19C2×C31×#102—21no—
20C31×#408—21no—
21C2×C31×#97C2211yescharacteristic
22C31×#390—21no—
23C31×#365—22no—
24C31×#361—22no—
25C31×#360—22no—
26C31×#347—22no—
27C2×#41×C31—21no—
28C22×D4×C31—21no—
29C31×#135—21no—
30C31×#135—21no—
31C31×#135—21no—
32C31×#135—21no—
33C2×Q16×C31C2×C411yescharacteristic
34C2×SD16×C31—21no—
35C2×D8×C31—21no—
36C2×D8×C31—21no—
37C2×D8×C31—21no—
38C2×D8×C31—21no—
39C2×D8×C31—21no—
40C2×D8×C31C2×C411yescharacteristic
41C2×D8×C31C2×C412yes—
42C2×M16×C31—21no—
43C22×C8×C31D411yescharacteristic
44C22×C8×C31C2311yescharacteristic
45C2×C4⋊C4×C31—21no—
46C2×C22⋊C4×C31—42no—
47C2×C42×C31D411yescharacteristic
48C31×SD32—24no—
49C31×D16—24no—
50C31×M32—21no—
51C31×M32D412yes—
52C31×M32D412yes—
53C31×M32D412yes—
54C2×C16×C31—21no—
55C2×C16×C31C2×C412yes—
56C31×#108—21no—
57C31×#108—21no—
58C31×#106—21no—
59C31×#106—21no—
60C31×#104—42no—
61C31×#102—42no—
62C31×C22⋊C8—42no—
63C31×#97—21no—
64C31×#97—21no—
65C24×C31—41no—
66#41×C31—41no—
67#41×C31—41no—
68#41×C31—41no—
69#41×C31—41no—
70C2×Q8×C31—21no—
71C2×D4×C31—22no—
72C2×D4×C31—21no—
73C2×D4×C31—41no—
74C2×D4×C31—41no—
75C2×D4×C31—41no—
76C2×D4×C31—41no—
77C2×D4×C31—41no—
78C2×D4×C31—41no—
79C22×C4×C31—21no—
80C22×C4×C31—41no—
81C22×C4×C31—41no—
82C22×C4×C31C2×D411yescharacteristic
83Q16×C31—21no—
84Q16×C31—21no—
85SD16×C31—41no—
86SD16×C31—41no—
87D8×C31—21no—
88D8×C31—22no—
89D8×C31—22no—
90D8×C31—21no—
91D8×C31—41no—
92D8×C31—41no—
93M16×C31—41no—
94M16×C31—41no—
95C2×C8×C31—21no—
96C2×C8×C31—21no—
97C2×C8×C31—21no—
98C2×C8×C31—41no—
99C2×C8×C31C22×C411yescharacteristic
100C2×C8×C31C2×D411yescharacteristic
101C2×C8×C31C2×D411yescharacteristic
102C2×C8×C31C2×D411yescharacteristic
103C2×C8×C31#4111yescharacteristic
104C2×C8×C31#4111yescharacteristic
105C4⋊C4×C31—41no—
106C4⋊C4×C31—41no—
107C22⋊C4×C31—84no—
108C42×C31—21no—
109C42×C31—21no—
110C16×C31—22no—
111C16×C31C22⋊C414yescomplemented
112#13249C3111yesmaximal, characteristic, direct product factor, Sylow
113C23×C31—41no—
114C23×C31—42no—
115C23×C31—82no—
116C23×C31—82no—
117C23×C31—81no—
118C23×C31—81no—
119C23×C31AGL(1,ℤ/8)11yescharacteristic
120Q8×C31—41no—
121Q8×C31—41no—
122D4×C31—42no—
123D4×C31—41no—
124D4×C31—42no—
125D4×C31—41no—
126D4×C31—81no—
127D4×C31—81no—
128D4×C31—81no—
129D4×C31—81no—
130C2×C4×C31—42no—
131C2×C4×C31—41no—
132C2×C4×C31—41no—
133C2×C4×C31—81no—
134C2×C4×C31—81no—
135C2×C4×C31—82no—
136C2×C4×C31—81no—
137C2×C4×C31C4×D411yescharacteristic
138C2×C4×C31#12111yescharacteristic
139C2×C4×C31#12711yescharacteristic
140C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
143C8×C31—41no—
144C8×C31—81no—
145C8×C31C2×C22⋊C411yescharacteristic
146C8×C31C4×D411yescharacteristic
147C8×C31#12111yescharacteristic
148C8×C31#12411yescharacteristic
149C2×#509C2×C3111yescharacteristic
150C2×#365C2×C3111yescharacteristic, complemented
151C2×#361C2×C3111yescharacteristic, complemented
152C2×#360C2×C3111yescharacteristic, complemented
153C2×#347C2×C3111yescharacteristic, complemented
154#1975C2×C3111yescharacteristic, complemented
155#1972C2×C3111yescharacteristic, complemented
156C22×C31—22no—
157C22×C31—82no—
158C22×C31—84no—
159C22×C31—81no—
160C22×C31—82no—
161C22×C31—162no—
162C22×C31—161no—
163C22×C31—161no—
164C22×C31#44211yescharacteristic
165C22×C31#46911yescharacteristic
166C22×C31#49611yescharacteristic
167C4×C31—81no—
168C4×C31—81no—
169C4×C31—161no—
170C4×C31#39011yescharacteristic
171C4×C31#44211yescharacteristic
172C4×C31#46911yescharacteristic
173C4×C31#49611yescharacteristic
174C2×#135C22×C3111yescharacteristic
175C22×D8C22×C3111yescharacteristic
176#509—22no—
177#509—22no—
178C2×SD32C4×C3112yescomplemented
179C2×D16C4×C3112yescomplemented
180C2×M32C22×C3111yescharacteristic
181C2×M32C22×C3111yescharacteristic
182C2×#108C22×C3111yescharacteristic
183C2×#106C22×C3111yescharacteristic
184C2×#104—21no—
185C2×#102—21no—
186#408—21no—
187C2×#97C22×C3111yescharacteristic
188#390—21no—
189#365—22no—
190#361—22no—
191#360—22no—
192#347—22no—
193C2×C31—21no—
194C2×C31—21no—
195C2×C31—161no—
196C2×C31—162no—
197C2×C31#197211yescharacteristic
198C2×C31AGL(1,ℤ/16)12yes—
199C2×#41—21no—
200C22×D4—21no—
201#135—21no—
202#135—21no—
203#135—21no—
204#135—21no—
205C2×Q16C2×C4×C3111yescharacteristic
206C2×SD16—21no—
207C2×D8—21no—
208C2×D8—21no—
209C2×D8—21no—
210C2×D8—21no—
211C2×D8—21no—
212C2×D8C2×C4×C3112yes—
213C2×D8C2×C4×C3111yescharacteristic
214C2×M16—21no—
215C22×C8D4×C3111yescharacteristic
216C22×C8C23×C3111yescharacteristic
217C2×C4⋊C4—21no—
218C2×C22⋊C4—42no—
219C2×C42D4×C3111yescharacteristic
220SD32—24no—
221D16—24no—
222M32—21no—
223M32D4×C3112yes—
224M32D4×C3112yes—
225M32D4×C3112yes—
226C2×C16—21no—
227C2×C16C2×C4×C3112yes—
228#108—21no—
229#108—21no—
230#106—21no—
231#106—21no—
232#104—42no—
233#102—42no—
234C22⋊C8—42no—
235#97—21no—
236#97—21no—
237C31#1324911yescharacteristic, direct product factor, Sylow
238C24—41no—
239#41—41no—
240#41—41no—
241#41—41no—
242#41—41no—
243C2×Q8—21no—
244C2×D4—22no—
245C2×D4—21no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4C2×D4×C3111yescharacteristic
256Q16—21no—
257Q16—21no—
258SD16—41no—
259SD16—41no—
260D8—21no—
261D8—22no—
262D8—22no—
263D8—21no—
264D8—41no—
265D8—41no—
266M16—41no—
267M16—41no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—41no—
272C2×C8C22×C4×C3111yescharacteristic
273C2×C8C2×D4×C3111yescharacteristic
274C2×C8C2×D4×C3111yescharacteristic
275C2×C8C2×D4×C3111yescharacteristic
276C2×C8#41×C3111yescharacteristic
277C2×C8#41×C3111yescharacteristic
278C4⋊C4—41no—
279C4⋊C4—41no—
280C22⋊C4—84no—
281C42—21no—
282C42—21no—
283C16—22no—
284C16C22⋊C4×C3114yescomplemented
285C23—41no—
286C23—42no—
287C23—82no—
288C23—82no—
289C23—81no—
290C23—81no—
291C23C31×AGL(1,ℤ/8)11yescharacteristic
292Q8—41no—
293Q8—41no—
294D4—42no—
295D4—41no—
296D4—42no—
297D4—41no—
298D4—81no—
299D4—81no—
300D4—81no—
301D4—81no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4C4×D4×C3111yescharacteristic
310C2×C4C31×#12111yescharacteristic
311C2×C4C31×#12711yescharacteristic
312C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
313C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
314C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
315C8—41no—
316C8—81no—
317C8C2×C22⋊C4×C3111yescharacteristic
318C8C4×D4×C3111yescharacteristic
319C8C31×#12111yescharacteristic
320C8C31×#12411yescharacteristic
321C22—22no—
322C22—84no—
323C22—82no—
324C22—81no—
325C22—82no—
326C22—162no—
327C22—161no—
328C22—161no—
329C22C31×#44211yescharacteristic
330C22C31×#46911yescharacteristic
331C22C31×#49611yescharacteristic
332C4—81no—
333C4—81no—
334C4—161no—
335C4C31×#39011yescharacteristic
336C4C31×#44211yescharacteristic
337C4C31×#46911yescharacteristic
338C4C31×#49611yescharacteristic
339C2—21no—
340C2—21no—
341C2—162no—
342C2—161no—
343C2C31×#197211yescharacteristic
344C2C31×AGL(1,ℤ/16)12yes—
subgroupG/H
center, socleC22×C31#442
commutatorC2×C8C22×C4×C31
Frattini, cosocleC22×C8C23×C31
prime radicalC2×C31×#509C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C31×#97 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#365C2semidirect
3. C2×C31×#361C2semidirect
4. C2×C31×#360C2semidirect
5. C2×C31×#347C2semidirect
6. C31×#1975C2semidirect
7. C31×#1972C2semidirect
12. C2×C31×SD32C4semidirect
13. C2×C31×D16C4semidirect
111. C16×C31C22⋊C4semidirect
112. #13249C31direct (N × H)
150. C2×#365C2×C31semidirect
151. C2×#361C2×C31semidirect
152. C2×#360C2×C31semidirect
153. C2×#347C2×C31semidirect
154. #1975C2×C31semidirect
155. #1972C2×C31semidirect
178. C2×SD32C4×C31semidirect
179. C2×D16C4×C31semidirect
237. C31#13249direct (N × H)
284. C16C22⋊C4×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5092
C2×C31×#3652
C2×C31×#3612
C2×C31×#3602
C2×C31×#3472
C31×#19752
C31×#19722
#1324931