Finite Group Databasebeta

← C31×#10170 #21466929

Subgroups — C31×#10170 #21466929

Order 7936.

574 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3206C211yesmaximal, characteristic
2C31×#3152C211yesmaximal, characteristic
3C2×C31×#487C211yesmaximal, characteristic, complemented
4C31×#1997C211yesmaximal, characteristic, complemented
5C31×#1546C211yesmaximal, characteristic, complemented
6C31×#1492C211yesmaximal, characteristic
7C31×#1278C211yesmaximal, characteristic, complemented
8C2×C31×#137—22no—
9C31×#532C2211yescharacteristic
10C2×C31×C4⋊Q8C2211yescharacteristic
11C2×C31×C4⋊Q8C2211yescharacteristic
12C31×#497—22no—
13C31×#494—21no—
14C31×#492—21no—
15C31×#492—21no—
16C31×#487—22no—
17C31×#487—21no—
18C31×#487—22no—
19C31×#461—22no—
20C31×#448—22no—
21C31×#431C2211yescharacteristic
22C31×#423C2211yescharacteristic
23C2×C31×#103—22no—
24C4×M16×C31C2211yescharacteristic
25C2×C4×C8×C31C2211yescharacteristic
26C31×#343—22no—
27C31×#332—22no—
28C31×D4⋊C8—22no—
29C2×#41×C31—21no—
30C22×Q8×C31—21no—
31C22×Q8×C31D412yes—
32C31×#137—44no—
33C31×#137—44no—
34C2×Q16×C31—24no—
35C2×SD16×C31—24no—
36C2×SD16×C31—24no—
37C2×M16×C31—21no—
38C2×M16×C31D412yes—
39C22×C8×C31—21no—
40C31×C4⋊Q8—21no—
41C31×C4⋊Q8—21no—
42C31×C4⋊Q8—21no—
43C31×C4⋊Q8—21no—
44C31×C4⋊Q8—21no—
45C31×C4⋊Q8—21no—
46C31×C4⋊Q8—21no—
47C31×C4⋊Q8D411yescharacteristic
48C31×C4⋊Q8D412yes—
49C31×#127D411yescharacteristic
50C31×#124—21no—
51C31×#121—21no—
52C31×#121—21no—
53C4×D4×C31—21no—
54C4×D4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—21no—
59C2×C42×C31C2311yescharacteristic
60C31×C4⋊C8—21no—
61C31×C4⋊C8—21no—
62C31×C4⋊C8D412yes—
63C31×#103—28no—
64C31×#103—24no—
65C31×C22⋊C8—21no—
66C31×#97—21no—
67C4×C8×C31—21no—
68C4×C8×C31—21no—
69C4×C8×C31—21no—
70C4×C8×C31D412yes—
71#41×C31—42no—
72#41×C31—42no—
73C2×Q8×C31—22no—
74C2×Q8×C31—22no—
75C2×Q8×C31—21no—
76C2×Q8×C31—22no—
77C2×Q8×C31—44no—
78C2×Q8×C31—41no—
79C2×Q8×C31—42no—
80C2×Q8×C31—42no—
81C2×D4×C31—21no—
82C2×D4×C31—21no—
83C2×D4×C31—21no—
84C2×D4×C31—41no—
85C22×C4×C31—22no—
86C22×C4×C31—41no—
87C22×C4×C31—41no—
88C22×C4×C31C2×D411yescharacteristic
89C22×C4×C31C2×D411yescharacteristic
90C22×C4×C31#4111yescharacteristic
91Q16×C31—48no—
92SD16×C31—48no—
93SD16×C31—84no—
94M16×C31—22no—
95M16×C31—42no—
96M16×C31SD1614yescomplemented
97C2×C8×C31—22no—
98C2×C8×C31—22no—
99C2×C8×C31—22no—
100C2×C8×C31—42no—
101C2×C8×C31—41no—
102C4⋊C4×C31—22no—
103C4⋊C4×C31—21no—
104C4⋊C4×C31—22no—
105C4⋊C4×C31—21no—
106C4⋊C4×C31—41no—
107C4⋊C4×C31—41no—
108C4⋊C4×C31—41no—
109C4⋊C4×C31—41no—
110C4⋊C4×C31—41no—
111C4⋊C4×C31—41no—
112C22⋊C4×C31—41no—
113C22⋊C4×C31—41no—
114C42×C31C2×D411yescharacteristic
115C42×C31C2×D411yescharacteristic
116C42×C31C2×D411yescharacteristic
117C42×C31C2×D411yescharacteristic
118#10170C3111yesmaximal, characteristic, direct product factor, Sylow
119C23×C31—41no—
120C23×C31#12111yescharacteristic
121Q8×C31—44no—
122Q8×C31—44no—
123Q8×C31—42no—
124Q8×C31—44no—
125Q8×C31—81no—
126Q8×C31—81no—
127D4×C31—42no—
128D4×C31—42no—
129D4×C31—42no—
130D4×C31—82no—
131C2×C4×C31—22no—
132C2×C4×C31—22no—
133C2×C4×C31—41no—
134C2×C4×C31—42no—
135C2×C4×C31—42no—
136C2×C4×C31—82no—
137C2×C4×C31—81no—
138C2×C4×C31—81no—
139C2×C4×C31—82no—
140C2×C4×C31#12011yescharacteristic
141C2×C4×C31#12011yescharacteristic
142C2×C4×C31#12011yescharacteristic
143C2×C4×C31#12111yescharacteristic
144C2×C4×C31#12111yescharacteristic
145C2×C4×C31#12711yescharacteristic
146C2×C4×C31C2×SD1612yes—
147C2×C4×C31AGL(1,ℤ/8)12yes—
148C8×C31—24no—
149C8×C31—42no—
150C8×C31—42no—
151C8×C31—81no—
152#3206C2×C3111yescharacteristic
153#3152C2×C3111yescharacteristic
154C2×#487C2×C3111yescharacteristic, complemented
155#1997C2×C3111yescharacteristic, complemented
156#1546C2×C3111yescharacteristic, complemented
157#1492C2×C3111yescharacteristic
158#1278C2×C3111yescharacteristic, complemented
159C22×C31—21no—
160C22×C31—21no—
161C22×C31—81no—
162C22×C31—81no—
163C22×C31—81no—
164C22×C31#39211yescharacteristic
165C22×C31#46012yes—
166C4×C31—21no—
167C4×C31—21no—
168C4×C31—82no—
169C4×C31—81no—
170C4×C31—82no—
171C4×C31#45012yes—
172C4×C31#45512yes—
173C4×C31#47312yes—
174C4×C31#49211yescharacteristic
175C4×C31#49611yescharacteristic
176C2×#137—22no—
177#532C22×C3111yescharacteristic
178C2×C4⋊Q8C22×C3111yescharacteristic
179C2×C4⋊Q8C22×C3111yescharacteristic
180#497—22no—
181#494—21no—
182#492—21no—
183#492—21no—
184#487—22no—
185#487—21no—
186#487—22no—
187#461—22no—
188#448—22no—
189#431C22×C3111yescharacteristic
190#423C22×C3111yescharacteristic
191C2×#103—22no—
192C4×M16C22×C3111yescharacteristic
193C2×C4×C8C22×C3111yescharacteristic
194#343—22no—
195#332—22no—
196D4⋊C8—22no—
197C2×C31—22no—
198C2×C31—161no—
199C2×C31#200411yescharacteristic
200C2×C31#201411yescharacteristic
201C2×C31#202011yescharacteristic
202C2×#41—21no—
203C22×Q8—21no—
204C22×Q8D4×C3112yes—
205#137—44no—
206#137—44no—
207C2×Q16—24no—
208C2×SD16—24no—
209C2×SD16—24no—
210C2×M16—21no—
211C2×M16D4×C3112yes—
212C22×C8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8D4×C3112yes—
221C4⋊Q8D4×C3111yescharacteristic
222#127D4×C3111yescharacteristic
223#124—21no—
224#121—21no—
225#121—21no—
226C4×D4—21no—
227C4×D4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C42C23×C3111yescharacteristic
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8D4×C3112yes—
236#103—28no—
237#103—24no—
238C22⋊C8—21no—
239#97—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8D4×C3112yes—
244C31#1017011yescharacteristic, direct product factor, Sylow
245#41—42no—
246#41—42no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—21no—
250C2×Q8—22no—
251C2×Q8—44no—
252C2×Q8—41no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4C2×D4×C3111yescharacteristic
263C22×C4C2×D4×C3111yescharacteristic
264C22×C4#41×C3111yescharacteristic
265Q16—48no—
266SD16—48no—
267SD16—84no—
268M16—22no—
269M16—42no—
270M16SD16×C3114yescomplemented
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—42no—
275C2×C8—41no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C42C2×D4×C3111yescharacteristic
289C42C2×D4×C3111yescharacteristic
290C42C2×D4×C3111yescharacteristic
291C42C2×D4×C3111yescharacteristic
292C23—41no—
293C23C31×#12111yescharacteristic
294Q8—44no—
295Q8—44no—
296Q8—42no—
297Q8—44no—
298Q8—81no—
299Q8—81no—
300D4—42no—
301D4—42no—
302D4—42no—
303D4—82no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4C31×#12011yescharacteristic
314C2×C4C31×#12011yescharacteristic
315C2×C4C31×#12011yescharacteristic
316C2×C4C31×#12111yescharacteristic
317C2×C4C31×#12111yescharacteristic
318C2×C4C31×#12711yescharacteristic
319C2×C4C2×SD16×C3112yes—
320C2×C4C31×AGL(1,ℤ/8)12yes—
321C8—24no—
322C8—42no—
323C8—42no—
324C8—81no—
325C22—21no—
326C22—21no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22C31×#39211yescharacteristic
331C22C31×#46012yes—
332C4—21no—
333C4—21no—
334C4—82no—
335C4—81no—
336C4—82no—
337C4C31×#45012yes—
338C4C31×#45512yes—
339C4C31×#47312yes—
340C4C31×#49211yescharacteristic
341C4C31×#49611yescharacteristic
342C2—22no—
343C2—161no—
344C2C31×#200411yescharacteristic
345C2C31×#201411yescharacteristic
346C2C31×#202011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C31×#487C2semidirect
4. C31×#1997C2semidirect
5. C31×#1546C2semidirect
7. C31×#1278C2semidirect
96. M16×C31SD16semidirect
118. #10170C31direct (N × H)
154. C2×#487C2×C31semidirect
155. #1997C2×C31semidirect
156. #1546C2×C31semidirect
158. #1278C2×C31semidirect
244. C31#10170direct (N × H)
270. M16SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#32062
C31×#31522
C2×C31×#4872
C31×#19972
C31×#15462
C31×#14922
C31×#12782
#1017031