Finite Group Databasebeta

← C31×#13215 #21469974

Subgroups — C31×#13215 #21469974

Order 7936.

466 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3155C211yesmaximal, characteristic
2C2×C31×#361C211yesmaximal, characteristic, complemented
3C31×#2145C211yesmaximal, characteristic, complemented
4C31×#2143C211yesmaximal, characteristic, complemented
5C31×#2140C211yesmaximal, characteristic, complemented
6C31×#2110C211yesmaximal, characteristic, complemented
7C31×#1833C211yesmaximal, characteristic, complemented
8C2×C31×AGL(1,ℤ/8)—21no—
9C2×C31×#135C2211yescharacteristic
10C22×D8×C31C2211yescharacteristic
11C31×#535—21no—
12C2×C31×M32C2211yescharacteristic
13C2×C31×M32C2211yescharacteristic
14C31×#496—22no—
15C31×#495C412yescomplemented
16C31×#493C412yescomplemented
17C2×C31×#108C2211yescharacteristic
18C31×#427C2211yescharacteristic
19C31×#405C2211yescharacteristic
20C31×#361—22no—
21C31×#360—22no—
22C31×#359—22no—
23C31×#357—22no—
24C31×#347—22no—
25C31×#344—21no—
26C31×H(3,ℤ/4)—21no—
27C2×#41×C31—21no—
28C22×D4×C31—21no—
29C31×AGL(1,ℤ/8)—42no—
30C31×AGL(1,ℤ/8)—42no—
31C31×#135—21no—
32C31×#135—21no—
33C31×#135—21no—
34C31×#135—21no—
35C2×Q16×C31C2×C411yescharacteristic
36C2×SD16×C31—22no—
37C2×SD16×C31—21no—
38C2×D8×C31—22no—
39C2×D8×C31—21no—
40C2×D8×C31—21no—
41C2×D8×C31—21no—
42C2×D8×C31—21no—
43C2×D8×C31—21no—
44C2×D8×C31C2×C411yescharacteristic
45C2×D8×C31C2×C412yes—
46C2×M16×C31—21no—
47C2×M16×C31D411yescharacteristic
48C22×C8×C31C2311yescharacteristic
49C31×#127—22no—
50C31×#124—22no—
51C31×#121—42no—
52C31×#120—42no—
53C31×#117—21no—
54C31×#117D411yescharacteristic
55C31×M32—21no—
56C31×M32—21no—
57C31×M32—21no—
58C31×M32D412yescomplemented
59C2×C16×C31—21no—
60C2×C16×C31D412yescomplemented
61C31×#108—21no—
62C31×#108—21no—
63C31×#107—21no—
64C31×#106—21no—
65C31×#97D412yes—
66C4×C8×C31C2×C412yes—
67C24×C31—41no—
68#41×C31—41no—
69#41×C31—41no—
70#41×C31—41no—
71#41×C31—41no—
72C2×Q8×C31—21no—
73C2×D4×C31—22no—
74C2×D4×C31—21no—
75C2×D4×C31—44no—
76C2×D4×C31—41no—
77C2×D4×C31—41no—
78C2×D4×C31—41no—
79C2×D4×C31—41no—
80C2×D4×C31—41no—
81C2×D4×C31—41no—
82C22×C4×C31—41no—
83C22×C4×C31C2×D411yescharacteristic
84Q16×C31—21no—
85Q16×C31—21no—
86SD16×C31—44no—
87SD16×C31—41no—
88SD16×C31—41no—
89D8×C31—22no—
90D8×C31—22no—
91D8×C31—21no—
92D8×C31—21no—
93D8×C31—44no—
94D8×C31—41no—
95D8×C31—41no—
96M16×C31—21no—
97M16×C31—21no—
98M16×C31—41no—
99M16×C31—41no—
100C2×C8×C31—41no—
101C2×C8×C31C22⋊C412yes—
102C2×C8×C31C22×C411yescharacteristic
103C2×C8×C31C2×D411yescharacteristic
104C2×C8×C31C2×D411yescharacteristic
105C2×C8×C31C2×D411yescharacteristic
106C2×C8×C31#4111yescharacteristic
107C2×C8×C31#4111yescharacteristic
108C4⋊C4×C31—21no—
109C4⋊C4×C31—41no—
110C22⋊C4×C31—21no—
111C22⋊C4×C31—44no—
112C22⋊C4×C31—41no—
113C42×C31—41no—
114C42×C31C22⋊C412yes—
115C16×C31—22no—
116C16×C31—22no—
117#13215C3111yesmaximal, characteristic, direct product factor, Sylow
118C23×C31—42no—
119C23×C31—41no—
120C23×C31—81no—
121C23×C31—81no—
122C23×C31—162no—
123C23×C31AGL(1,ℤ/8)11yescharacteristic
124Q8×C31—41no—
125Q8×C31—41no—
126D4×C31—42no—
127D4×C31—41no—
128D4×C31—41no—
129D4×C31—42no—
130D4×C31—81no—
131D4×C31—81no—
132D4×C31—81no—
133D4×C31—81no—
134D4×C31—164no—
135C2×C4×C31—22no—
136C2×C4×C31—41no—
137C2×C4×C31—81no—
138C2×C4×C31—81no—
139C2×C4×C31—81no—
140C2×C4×C31C2×C22⋊C411yescharacteristic
141C2×C4×C31#12411yescharacteristic
142C2×C4×C31#12711yescharacteristic
143C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
146C8×C31—22no—
147C8×C31—81no—
148C8×C31C4×D411yescharacteristic
149C8×C31C4×D411yescharacteristic
150C8×C31#12111yescharacteristic
151C8×C31#12111yescharacteristic
152#3155C2×C3111yescharacteristic
153C2×#361C2×C3111yescharacteristic, complemented
154#2145C2×C3111yescharacteristic, complemented
155#2143C2×C3111yescharacteristic, complemented
156#2140C2×C3111yescharacteristic, complemented
157#2110C2×C3111yescharacteristic, complemented
158#1833C2×C3111yescharacteristic, complemented
159C22×C31—41no—
160C22×C31—82no—
161C22×C31—84no—
162C22×C31—81no—
163C22×C31—82no—
164C22×C31—162no—
165C22×C31—162no—
166C22×C31#41811yescharacteristic
167C22×C31#49011yescharacteristic
168C22×C31#49611yescharacteristic
169C4×C31—42no—
170C4×C31—81no—
171C4×C31—161no—
172C4×C31#39011yescharacteristic
173C4×C31#41811yescharacteristic
174C4×C31#49011yescharacteristic
175C4×C31#49611yescharacteristic
176C2×AGL(1,ℤ/8)—21no—
177C2×#135C22×C3111yescharacteristic
178C22×D8C22×C3111yescharacteristic
179#535—21no—
180C2×M32C22×C3111yescharacteristic
181C2×M32C22×C3111yescharacteristic
182#496—22no—
183#495C4×C3112yescomplemented
184#493C4×C3112yescomplemented
185C2×#108C22×C3111yescharacteristic
186#427C22×C3111yescharacteristic
187#405C22×C3111yescharacteristic
188#361—22no—
189#360—22no—
190#359—22no—
191#357—22no—
192#347—22no—
193#344—21no—
194H(3,ℤ/4)—21no—
195C2×C31—21no—
196C2×C31—21no—
197C2×C31—21no—
198C2×C31—161no—
199C2×C31—162no—
200C2×C31#196111yescharacteristic
201C2×#41—21no—
202C22×D4—21no—
203AGL(1,ℤ/8)—42no—
204AGL(1,ℤ/8)—42no—
205#135—21no—
206#135—21no—
207#135—21no—
208#135—21no—
209C2×Q16C2×C4×C3111yescharacteristic
210C2×SD16—22no—
211C2×SD16—21no—
212C2×D8—22no—
213C2×D8—21no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8—21no—
218C2×D8C2×C4×C3112yes—
219C2×D8C2×C4×C3111yescharacteristic
220C2×M16—21no—
221C2×M16D4×C3111yescharacteristic
222C22×C8C23×C3111yescharacteristic
223#127—22no—
224#124—22no—
225#121—42no—
226#120—42no—
227#117—21no—
228#117D4×C3111yescharacteristic
229M32—21no—
230M32—21no—
231M32—21no—
232M32D4×C3112yescomplemented
233C2×C16—21no—
234C2×C16D4×C3112yescomplemented
235#108—21no—
236#108—21no—
237#107—21no—
238#106—21no—
239#97D4×C3112yes—
240C4×C8C2×C4×C3112yes—
241C31#1321511yescharacteristic, direct product factor, Sylow
242C24—41no—
243#41—41no—
244#41—41no—
245#41—41no—
246#41—41no—
247C2×Q8—21no—
248C2×D4—22no—
249C2×D4—21no—
250C2×D4—44no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C22×C4—41no—
258C22×C4C2×D4×C3111yescharacteristic
259Q16—21no—
260Q16—21no—
261SD16—44no—
262SD16—41no—
263SD16—41no—
264D8—22no—
265D8—22no—
266D8—21no—
267D8—21no—
268D8—44no—
269D8—41no—
270D8—41no—
271M16—21no—
272M16—21no—
273M16—41no—
274M16—41no—
275C2×C8—41no—
276C2×C8C22⋊C4×C3112yes—
277C2×C8C22×C4×C3111yescharacteristic
278C2×C8C2×D4×C3111yescharacteristic
279C2×C8C2×D4×C3111yescharacteristic
280C2×C8C2×D4×C3111yescharacteristic
281C2×C8#41×C3111yescharacteristic
282C2×C8#41×C3111yescharacteristic
283C4⋊C4—21no—
284C4⋊C4—41no—
285C22⋊C4—21no—
286C22⋊C4—44no—
287C22⋊C4—41no—
288C42—41no—
289C42C22⋊C4×C3112yes—
290C16—22no—
291C16—22no—
292C23—42no—
293C23—41no—
294C23—81no—
295C23—81no—
296C23—162no—
297C23C31×AGL(1,ℤ/8)11yescharacteristic
298Q8—41no—
299Q8—41no—
300D4—41no—
301D4—42no—
302D4—42no—
303D4—41no—
304D4—81no—
305D4—81no—
306D4—81no—
307D4—81no—
308D4—164no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4C2×C22⋊C4×C3111yescharacteristic
315C2×C4C31×#12411yescharacteristic
316C2×C4C31×#12711yescharacteristic
317C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
318C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
319C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
320C8—22no—
321C8—81no—
322C8C4×D4×C3111yescharacteristic
323C8C4×D4×C3111yescharacteristic
324C8C31×#12111yescharacteristic
325C8C31×#12111yescharacteristic
326C22—41no—
327C22—82no—
328C22—84no—
329C22—81no—
330C22—82no—
331C22—162no—
332C22—162no—
333C22C31×#41811yescharacteristic
334C22C31×#49011yescharacteristic
335C22C31×#49611yescharacteristic
336C4—42no—
337C4—81no—
338C4—161no—
339C4C31×#39011yescharacteristic
340C4C31×#41811yescharacteristic
341C4C31×#49011yescharacteristic
342C4C31×#49611yescharacteristic
343C2—21no—
344C2—21no—
345C2—21no—
346C2—162no—
347C2—161no—
348C2C31×#196111yescharacteristic
subgroupG/H
center, socleC2×C31#1961
commutatorC2×C8C22×C4×C31
Frattini, cosocleC22×C8C23×C31
prime radicalC31×#3155C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C22×C4×C31 ◁ C31×#405 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#361C2semidirect
3. C31×#2145C2semidirect
4. C31×#2143C2semidirect
5. C31×#2140C2semidirect
6. C31×#2110C2semidirect
7. C31×#1833C2semidirect
15. C31×#495C4semidirect
16. C31×#493C4semidirect
58. C31×M32D4semidirect
60. C2×C16×C31D4semidirect
117. #13215C31direct (N × H)
153. C2×#361C2×C31semidirect
154. #2145C2×C31semidirect
155. #2143C2×C31semidirect
156. #2140C2×C31semidirect
157. #2110C2×C31semidirect
158. #1833C2×C31semidirect
183. #495C4×C31semidirect
184. #493C4×C31semidirect
232. M32D4×C31semidirect
234. C2×C16D4×C31semidirect
241. C31#13215direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31552
C2×C31×#3612
C31×#21452
C31×#21432
C31×#21402
C31×#21102
C31×#18332
#1321531