Finite Group Databasebeta

← C31×#13412 #21470171

Subgroups — C31×#13412 #21470171

Order 7936.

430 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3156C211yesmaximal, characteristic
2C2×C31×#362C211yesmaximal, characteristic, complemented
3C2×C31×#359C211yesmaximal, characteristic
4C31×#2139C211yesmaximal, characteristic
5C2×C31×#358C211yesmaximal, characteristic, complemented
6C31×#1952C211yesmaximal, characteristic, complemented
7C31×#1379C211yesmaximal, characteristic, complemented
8C2×C31×#137—21no—
9C2×C31×#135C2211yescharacteristic
10C22×Q16×C31C2211yescharacteristic
11C31×#536—21no—
12C2×C31×M32C2211yescharacteristic
13C22×C16×C31C2211yescharacteristic
14C31×#497—22no—
15C31×#495—21no—
16C31×#494—21no—
17C2×C31×#108C2211yescharacteristic
18C31×#428—21no—
19C2×C31×#107C2211yescharacteristic
20C31×#405C2211yescharacteristic
21C31×#362—22no—
22C31×#359—22no—
23C31×#358—22no—
24C31×#358—21no—
25C31×#357—21no—
26C31×#343—21no—
27C2×#41×C31—21no—
28C22×Q8×C31—21no—
29C31×#137—42no—
30C31×#137—42no—
31C31×#135—21no—
32C31×#135—21no—
33C31×#135—21no—
34C31×#135—21no—
35C2×Q16×C31—22no—
36C2×Q16×C31—21no—
37C2×Q16×C31—21no—
38C2×Q16×C31—21no—
39C2×Q16×C31—21no—
40C2×Q16×C31—21no—
41C2×Q16×C31D411yescharacteristic
42C2×Q16×C31D412yes—
43C2×SD16×C31—22no—
44C2×SD16×C31—21no—
45C2×D8×C31D411yescharacteristic
46C2×M16×C31—21no—
47C2×M16×C31D411yescharacteristic
48C22×C8×C31C2311yescharacteristic
49C31×C4⋊Q8—22no—
50C31×#124—22no—
51C31×#123—42no—
52C31×C22⋊Q8—41no—
53C31×C22⋊Q8—41no—
54C31×#117D411yescharacteristic
55C2×C4⋊C4×C31—21no—
56C31×M32—21no—
57C31×M32—21no—
58C2×C16×C31—21no—
59C2×C16×C31—21no—
60C2×C16×C31—21no—
61C2×C16×C31—21no—
62C31×#108—21no—
63C31×#108D412yes—
64C31×#107—21no—
65C31×#107—21no—
66C31×#107—41no—
67C31×#106—41no—
68C31×#97—21no—
69C4×C8×C31—21no—
70#41×C31—41no—
71#41×C31—41no—
72#41×C31—41no—
73#41×C31—41no—
74C2×Q8×C31—22no—
75C2×Q8×C31—21no—
76C2×Q8×C31—41no—
77C2×Q8×C31—41no—
78C2×Q8×C31—41no—
79C2×Q8×C31—41no—
80C2×Q8×C31—41no—
81C2×D4×C31—21no—
82C2×D4×C31—41no—
83C22×C4×C31—41no—
84C22×C4×C31—41no—
85C22×C4×C31—41no—
86C22×C4×C31C2×D411yescharacteristic
87Q16×C31—22no—
88Q16×C31—21no—
89Q16×C31—22no—
90Q16×C31—21no—
91Q16×C31—44no—
92Q16×C31—41no—
93Q16×C31—41no—
94SD16×C31—44no—
95SD16×C31—41no—
96SD16×C31—41no—
97D8×C31—21no—
98D8×C31—21no—
99M16×C31—21no—
100M16×C31—22no—
101M16×C31—41no—
102M16×C31SD1612yes—
103C2×C8×C31—21no—
104C2×C8×C31—41no—
105C2×C8×C31C2×D411yescharacteristic
106C2×C8×C31C2×D411yescharacteristic
107C2×C8×C31C2×D411yescharacteristic
108C2×C8×C31#4111yescharacteristic
109C2×C8×C31#4111yescharacteristic
110C2×C8×C31#4111yescharacteristic
111C4⋊C4×C31—42no—
112C4⋊C4×C31—42no—
113C4⋊C4×C31—41no—
114C4⋊C4×C31—41no—
115C4⋊C4×C31D811yescharacteristic
116C4⋊C4×C31D811yescharacteristic
117C22⋊C4×C31—21no—
118C22⋊C4×C31—42no—
119C22⋊C4×C31—42no—
120C42×C31—21no—
121C16×C31—41no—
122C16×C31—41no—
123#13412C3111yesmaximal, characteristic, direct product factor, Sylow
124C23×C31—41no—
125C23×C31AGL(1,ℤ/8)11yescharacteristic
126Q8×C31—41no—
127Q8×C31—42no—
128Q8×C31—42no—
129Q8×C31—41no—
130Q8×C31—81no—
131Q8×C31—81no—
132D4×C31—41no—
133D4×C31—41no—
134D4×C31—81no—
135D4×C31—81no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—42no—
139C2×C4×C31—41no—
140C2×C4×C31—81no—
141C2×C4×C31—81no—
142C2×C4×C31—81no—
143C2×C4×C31—81no—
144C2×C4×C31—81no—
145C2×C4×C31—161no—
146C2×C4×C31#12011yescharacteristic
147C2×C4×C31#12111yescharacteristic
148C2×C4×C31#12411yescharacteristic
149C2×C4×C31C2×D811yescharacteristic
150C2×C4×C31C2×SD1611yescharacteristic
151C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
152C8×C31—22no—
153C8×C31—42no—
154C8×C31#12111yescharacteristic
155C8×C31#12111yescharacteristic
156C8×C31#12311yescharacteristic
157C8×C31#12311yescharacteristic
158#3156C2×C3111yescharacteristic
159C2×#362C2×C3111yescharacteristic, complemented
160C2×#359C2×C3111yescharacteristic
161#2139C2×C3111yescharacteristic
162C2×#358C2×C3111yescharacteristic, complemented
163#1952C2×C3111yescharacteristic, complemented
164#1379C2×C3111yescharacteristic, complemented
165C22×C31—21no—
166C22×C31—21no—
167C22×C31—81no—
168C22×C31—81no—
169C22×C31—81no—
170C22×C31#44711yescharacteristic
171C22×C31#46011yescharacteristic
172C22×C31#49011yescharacteristic
173C4×C31—41no—
174C4×C31—41no—
175C4×C31—81no—
176C4×C31—82no—
177C4×C31—161no—
178C4×C31#39411yescharacteristic
179C4×C31#45011yescharacteristic
180C4×C31#45911yescharacteristic
181C4×C31#48611yescharacteristic
182C2×#137—21no—
183C2×#135C22×C3111yescharacteristic
184C22×Q16C22×C3111yescharacteristic
185#536—21no—
186C2×M32C22×C3111yescharacteristic
187C22×C16C22×C3111yescharacteristic
188#497—22no—
189#495—21no—
190#494—21no—
191C2×#108C22×C3111yescharacteristic
192#428—21no—
193C2×#107C22×C3111yescharacteristic
194#405C22×C3111yescharacteristic
195#362—22no—
196#359—22no—
197#358—22no—
198#358—21no—
199#357—21no—
200#343—21no—
201C2×C31—21no—
202C2×C31—21no—
203C2×C31—161no—
204C2×C31#204111yescharacteristic
205C2×C31#219311yescharacteristic
206C2×C31#219511yescharacteristic
207C2×#41—21no—
208C22×Q8—21no—
209#137—42no—
210#137—42no—
211#135—21no—
212#135—21no—
213#135—21no—
214#135—21no—
215C2×Q16—22no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16—21no—
220C2×Q16—21no—
221C2×Q16D4×C3112yes—
222C2×Q16D4×C3111yescharacteristic
223C2×SD16—22no—
224C2×SD16—21no—
225C2×D8D4×C3111yescharacteristic
226C2×M16—21no—
227C2×M16D4×C3111yescharacteristic
228C22×C8C23×C3111yescharacteristic
229C4⋊Q8—22no—
230#124—22no—
231#123—42no—
232C22⋊Q8—41no—
233C22⋊Q8—41no—
234#117D4×C3111yescharacteristic
235C2×C4⋊C4—21no—
236M32—21no—
237M32—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242#108—21no—
243#108D4×C3112yes—
244#107—21no—
245#107—21no—
246#107—41no—
247#106—41no—
248#97—21no—
249C4×C8—21no—
250C31#1341211yescharacteristic, direct product factor, Sylow
251#41—41no—
252#41—41no—
253#41—41no—
254#41—41no—
255C2×Q8—22no—
256C2×Q8—21no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×D4—21no—
263C2×D4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4C2×D4×C3111yescharacteristic
268Q16—22no—
269Q16—21no—
270Q16—22no—
271Q16—21no—
272Q16—44no—
273Q16—41no—
274Q16—41no—
275SD16—44no—
276SD16—41no—
277SD16—41no—
278D8—21no—
279D8—21no—
280M16—21no—
281M16—22no—
282M16—41no—
283M16SD16×C3112yes—
284C2×C8—21no—
285C2×C8—41no—
286C2×C8C2×D4×C3111yescharacteristic
287C2×C8C2×D4×C3111yescharacteristic
288C2×C8C2×D4×C3111yescharacteristic
289C2×C8#41×C3111yescharacteristic
290C2×C8#41×C3111yescharacteristic
291C2×C8#41×C3111yescharacteristic
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4D8×C3111yescharacteristic
297C4⋊C4D8×C3111yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C42—21no—
302C16—41no—
303C16—41no—
304C23—41no—
305C23C31×AGL(1,ℤ/8)11yescharacteristic
306Q8—42no—
307Q8—41no—
308Q8—41no—
309Q8—42no—
310Q8—81no—
311Q8—81no—
312D4—41no—
313D4—41no—
314D4—81no—
315D4—81no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—161no—
326C2×C4C31×#12011yescharacteristic
327C2×C4C31×#12111yescharacteristic
328C2×C4C31×#12411yescharacteristic
329C2×C4C2×D8×C3111yescharacteristic
330C2×C4C2×SD16×C3111yescharacteristic
331C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
332C8—22no—
333C8—42no—
334C8C31×#12111yescharacteristic
335C8C31×#12111yescharacteristic
336C8C31×#12311yescharacteristic
337C8C31×#12311yescharacteristic
338C22—21no—
339C22—21no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22C31×#44711yescharacteristic
344C22C31×#46011yescharacteristic
345C22C31×#49011yescharacteristic
346C4—41no—
347C4—41no—
348C4—81no—
349C4—82no—
350C4—161no—
351C4C31×#39411yescharacteristic
352C4C31×#45011yescharacteristic
353C4C31×#45911yescharacteristic
354C4C31×#48611yescharacteristic
355C2—21no—
356C2—21no—
357C2—161no—
358C2C31×#204111yescharacteristic
359C2C31×#219311yescharacteristic
360C2C31×#219511yescharacteristic
subgroupG/H
center, socleC22×C31#447
commutator, Frattini, cosocleC22×C8C23×C31
prime radicalC2×C31×#135C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C31×#405 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#362C2semidirect
5. C2×C31×#358C2semidirect
6. C31×#1952C2semidirect
7. C31×#1379C2semidirect
123. #13412C31direct (N × H)
159. C2×#362C2×C31semidirect
162. C2×#358C2×C31semidirect
163. #1952C2×C31semidirect
164. #1379C2×C31semidirect
250. C31#13412direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31562
C2×C31×#3622
C2×C31×#3592
C31×#21392
C2×C31×#3582
C31×#19522
C31×#13792
#1341231