Finite Group Databasebeta

← C31×#10161 #21466920

Subgroups — C31×#10161 #21466920

Order 7936.

556 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
1C31×#3208C211yesmaximal, characteristic
2C31×#3165C211yesmaximal, characteristic, complemented
3C2×C31×#486C211yesmaximal, characteristic
4C31×#1965C211yesmaximal, characteristic
5C31×#1548C211yesmaximal, characteristic, complemented
6C31×#1493C211yesmaximal, characteristic, complemented
7C31×#1278C211yesmaximal, characteristic, complemented
8C31×#533C2211yescharacteristic
9C2×C31×C4⋊Q8C2211yescharacteristic
10C2×C31×#127C2211yescharacteristic
11C31×#501—22no—
12C31×#500—21no—
13C31×#498—21no—
14C31×#492—21no—
15C31×#486—22no—
16C31×#486—21no—
17C31×#486—22no—
18C31×#460—22no—
19C31×#450—22no—
20C31×#431C2211yescharacteristic
21C31×#428—22no—
22C31×#423C2211yescharacteristic
23C2×C31×#102—22no—
24C2×C31×#100—22no—
25C4×M16×C31C2211yescharacteristic
26C2×C4×C8×C31C2211yescharacteristic
27C31×#333—22no—
28C31×Q8⋊C8—22no—
29C22×Q8×C31—21no—
30C22×D4×C31—21no—
31C22×D4×C31—21no—
32C22×D4×C31D412yes—
33C2×SD16×C31—24no—
34C2×M16×C31—21no—
35C2×M16×C31D412yes—
36C22×C8×C31—21no—
37C31×C4⋊Q8—21no—
38C31×C4⋊Q8—21no—
39C31×C4⋊Q8—21no—
40C31×C4⋊Q8D411yescharacteristic
41C31×C4⋊Q8D412yes—
42C31×C4⋊Q8D411yescharacteristic
43C31×#127—21no—
44C31×#127—21no—
45C31×#127—21no—
46C31×#127—21no—
47C31×#125—21no—
48C31×C22⋊Q8—21no—
49C31×C22⋊Q8—21no—
50C4×Q8×C31—21no—
51C4×Q8×C31—21no—
52C31×#117—21no—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31—21no—
55C2×C42×C31C2311yescharacteristic
56C31×#107—24no—
57C31×#106—24no—
58C31×C4⋊C8—21no—
59C31×C4⋊C8—21no—
60C31×C4⋊C8D412yes—
61C31×#102—28no—
62C31×#102—24no—
63C31×#100—44no—
64C31×C22⋊C8—21no—
65C31×#97—21no—
66C4×C8×C31—21no—
67C4×C8×C31—21no—
68C4×C8×C31—21no—
69C4×C8×C31D412yes—
70C24×C31—22no—
71C2×Q8×C31—21no—
72C2×Q8×C31—21no—
73C2×Q8×C31—22no—
74C2×Q8×C31—41no—
75C2×Q8×C31—82no—
76C2×D4×C31—22no—
77C2×D4×C31—22no—
78C2×D4×C31—44no—
79C2×D4×C31—41no—
80C2×D4×C31—41no—
81C2×D4×C31—41no—
82C2×D4×C31—41no—
83C2×D4×C31—42no—
84C2×D4×C31—42no—
85C2×D4×C31—82no—
86C22×C4×C31—41no—
87C22×C4×C31C2×D411yescharacteristic
88C22×C4×C31C2×D411yescharacteristic
89C22×C4×C31#4111yescharacteristic
90SD16×C31—84no—
91M16×C31—22no—
92M16×C31—42no—
93M16×C31SD1614yescomplemented
94C2×C8×C31—22no—
95C2×C8×C31—22no—
96C2×C8×C31—22no—
97C2×C8×C31—42no—
98C2×C8×C31—41no—
99C4⋊C4×C31—22no—
100C4⋊C4×C31—21no—
101C4⋊C4×C31—21no—
102C4⋊C4×C31—22no—
103C4⋊C4×C31—21no—
104C4⋊C4×C31—21no—
105C4⋊C4×C31—41no—
106C4⋊C4×C31—41no—
107C4⋊C4×C31—41no—
108C4⋊C4×C31—41no—
109C22⋊C4×C31—41no—
110C42×C31—41no—
111C42×C31C2×D411yescharacteristic
112C42×C31C2×D411yescharacteristic
113C42×C31C2×D411yescharacteristic
114C42×C31C2×D411yescharacteristic
115#10161C3111yesmaximal, characteristic, direct product factor, Sylow
116C23×C31—42no—
117C23×C31—44no—
118C23×C31—82no—
119C23×C31—82no—
120C23×C31#12111yescharacteristic
121Q8×C31—42no—
122Q8×C31—42no—
123Q8×C31—82no—
124Q8×C31—82no—
125D4×C31—44no—
126D4×C31—44no—
127D4×C31—82no—
128D4×C31—82no—
129D4×C31—82no—
130D4×C31—81no—
131D4×C31—81no—
132C2×C4×C31—22no—
133C2×C4×C31—22no—
134C2×C4×C31—41no—
135C2×C4×C31—41no—
136C2×C4×C31—42no—
137C2×C4×C31—82no—
138C2×C4×C31#12011yescharacteristic
139C2×C4×C31#12011yescharacteristic
140C2×C4×C31#12011yescharacteristic
141C2×C4×C31#12111yescharacteristic
142C2×C4×C31#12111yescharacteristic
143C2×C4×C31#12711yescharacteristic
144C2×C4×C31C2×SD1612yes—
145C2×C4×C31#13712yes—
146C8×C31—24no—
147C8×C31—42no—
148C8×C31—42no—
149C8×C31—81no—
150#3208C2×C3111yescharacteristic
151#3165C2×C3111yescharacteristic, complemented
152C2×#486C2×C3111yescharacteristic
153#1965C2×C3111yescharacteristic
154#1548C2×C3111yescharacteristic, complemented
155#1493C2×C3111yescharacteristic, complemented
156#1278C2×C3111yescharacteristic, complemented
157C22×C31—21no—
158C22×C31—21no—
159C22×C31—82no—
160C22×C31—82no—
161C22×C31—82no—
162C22×C31—164no—
163C22×C31#39211yescharacteristic
164C22×C31#46112yes—
165C4×C31—21no—
166C4×C31—21no—
167C4×C31—81no—
168C4×C31—82no—
169C4×C31—81no—
170C4×C31#44812yes—
171C4×C31#45512yes—
172C4×C31#47212yes—
173C4×C31#49211yescharacteristic
174C4×C31#49711yescharacteristic
175#533C22×C3111yescharacteristic
176C2×C4⋊Q8C22×C3111yescharacteristic
177C2×#127C22×C3111yescharacteristic
178#501—22no—
179#500—21no—
180#498—21no—
181#492—21no—
182#486—22no—
183#486—21no—
184#486—22no—
185#460—22no—
186#450—22no—
187#431C22×C3111yescharacteristic
188#428—22no—
189#423C22×C3111yescharacteristic
190C2×#102—22no—
191C2×#100—22no—
192C4×M16C22×C3111yescharacteristic
193C2×C4×C8C22×C3111yescharacteristic
194#333—22no—
195Q8⋊C8—22no—
196C2×C31—22no—
197C2×C31—162no—
198C2×C31#200411yescharacteristic
199C2×C31#201311yescharacteristic
200C2×C31#201811yescharacteristic
201C22×Q8—21no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4D4×C3112yes—
205C2×SD16—24no—
206C2×M16—21no—
207C2×M16D4×C3112yes—
208C22×C8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8D4×C3112yes—
213C4⋊Q8D4×C3111yescharacteristic
214C4⋊Q8D4×C3111yescharacteristic
215#127—21no—
216#127—21no—
217#127—21no—
218#127—21no—
219#125—21no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224#117—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C42C23×C3111yescharacteristic
228#107—24no—
229#106—24no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8D4×C3112yes—
233#102—28no—
234#102—24no—
235#100—44no—
236C22⋊C8—21no—
237#97—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8D4×C3112yes—
242C31#1016111yescharacteristic, direct product factor, Sylow
243C24—22no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—82no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—44no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—82no—
259C22×C4—41no—
260C22×C4C2×D4×C3111yescharacteristic
261C22×C4C2×D4×C3111yescharacteristic
262C22×C4#41×C3111yescharacteristic
263SD16—84no—
264M16—22no—
265M16—42no—
266M16SD16×C3114yescomplemented
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—42no—
271C2×C8—41no—
272C4⋊C4—22no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C22⋊C4—41no—
283C42—41no—
284C42C2×D4×C3111yescharacteristic
285C42C2×D4×C3111yescharacteristic
286C42C2×D4×C3111yescharacteristic
287C42C2×D4×C3111yescharacteristic
288C23—42no—
289C23—44no—
290C23—82no—
291C23—82no—
292C23C31×#12111yescharacteristic
293Q8—42no—
294Q8—42no—
295Q8—82no—
296Q8—82no—
297D4—44no—
298D4—44no—
299D4—82no—
300D4—82no—
301D4—82no—
302D4—81no—
303D4—81no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—82no—
310C2×C4C31×#12011yescharacteristic
311C2×C4C31×#12011yescharacteristic
312C2×C4C31×#12011yescharacteristic
313C2×C4C31×#12111yescharacteristic
314C2×C4C31×#12111yescharacteristic
315C2×C4C31×#12711yescharacteristic
316C2×C4C2×SD16×C3112yes—
317C2×C4C31×#13712yes—
318C8—24no—
319C8—42no—
320C8—42no—
321C8—81no—
322C22—21no—
323C22—21no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—164no—
328C22C31×#39211yescharacteristic
329C22C31×#46112yes—
330C4—21no—
331C4—21no—
332C4—81no—
333C4—82no—
334C4—81no—
335C4C31×#44812yes—
336C4C31×#45512yes—
337C4C31×#47212yes—
338C4C31×#49211yescharacteristic
339C4C31×#49711yescharacteristic
340C2—22no—
341C2—162no—
342C2C31×#200411yescharacteristic
343C2C31×#201311yescharacteristic
344C2C31×#201811yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#127C22
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
2. C31×#3165C2semidirect
5. C31×#1548C2semidirect
6. C31×#1493C2semidirect
7. C31×#1278C2semidirect
93. M16×C31SD16semidirect
115. #10161C31direct (N × H)
151. #3165C2×C31semidirect
154. #1548C2×C31semidirect
155. #1493C2×C31semidirect
156. #1278C2×C31semidirect
242. C31#10161direct (N × H)
266. M16SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#32082
C31×#31652
C2×C31×#4862
C31×#19652
C31×#15482
C31×#14932
C31×#12782
#1016131