Finite Group Databasebeta

← C31×#13350 #21470109

Subgroups — C31×#13350 #21470109

Order 7936.

738 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3199C211yesmaximal, characteristic, complemented
2C31×#3027C212yesmaximal, complemented
3C31×#2128C212yesmaximal, complemented
4C31×#1794C211yesmaximal, characteristic, complemented
5C31×#1391C211yesmaximal, characteristic, complemented
6C31×#585C2211yescharacteristic
7C31×#535—22no—
8C31×#535C2212yes—
9C31×#482—22no—
10C31×#469—21no—
11C31×#466—21no—
12C31×B(4,𝔽2)—24no—
13C31×B(4,𝔽2)—24no—
14C31×#447—22no—
15C31×#421—21no—
16C31×#416—21no—
17C31×#410—22no—
18C31×#410C2211yescharacteristic
19C31×#410C2212yes—
20C31×#408C2211yescharacteristic
21C31×#353—24no—
22C31×2−1+4—21no—
23C31×2−1+4—41no—
24C31×H(5,𝔽2)—22no—
25C31×H(5,𝔽2)—41no—
26C31×H(5,𝔽2)D412yes—
27C2×#41×C31—22no—
28C2×#41×C31—21no—
29C2×#41×C31—42no—
30C2×#41×C31C2311yescharacteristic
31C22×D4×C31D412yescomplemented
32C2×D8×C31—21no—
33C2×D8×C31—21no—
34C2×M16×C31D411yescharacteristic, complemented
35C22×C8×C31D411yescharacteristic, complemented
36C31×#127—22no—
37C31×#127—42no—
38C31×#124—22no—
39C31×#124—42no—
40C31×#121—22no—
41C31×#121—22no—
42C31×#121—44no—
43C31×#120—28no—
44C31×#120—22no—
45C31×#120—22no—
46C31×#117—22no—
47C31×#117—21no—
48C31×#117D412yescomplemented
49C31×#107—21no—
50C31×#106—21no—
51C31×#104—42no—
52C31×#103—41no—
53C31×#102—22no—
54C31×#102—22no—
55C31×#102—41no—
56C31×#99—28no—
57C31×#99—22no—
58C31×#99—22no—
59C31×#99D414yescomplemented
60C31×C22⋊C8—22no—
61C24×C31—22no—
62#41×C31—22no—
63#41×C31—22no—
64#41×C31—41no—
65#41×C31—41no—
66#41×C31—42no—
67#41×C31—42no—
68#41×C31—42no—
69#41×C31—41no—
70#41×C31—44no—
71#41×C31—42no—
72#41×C31—41no—
73#41×C31—42no—
74C2×Q8×C31—22no—
75C2×Q8×C31—21no—
76C2×Q8×C31—42no—
77C2×Q8×C31#4111yescharacteristic
78C2×D4×C31—21no—
79C2×D4×C31—22no—
80C2×D4×C31—24no—
81C2×D4×C31—22no—
82C2×D4×C31—22no—
83C2×D4×C31—42no—
84C2×D4×C31—42no—
85C2×D4×C31—42no—
86C2×D4×C31—44no—
87C2×D4×C31—41no—
88C2×D4×C31—48no—
89C2×D4×C31—42no—
90C2×D4×C31—42no—
91C2×D4×C31C2×D411yescharacteristic
92C2×D4×C31C2×D412yes—
93C22×C4×C31—24no—
94C22×C4×C31—41no—
95C22×C4×C31C2×D412yes—
96C22×C4×C31C2×D411yescharacteristic
97D8×C31—81no—
98D8×C31—81no—
99M16×C31—41no—
100C2×C8×C31—21no—
101C2×C8×C31—21no—
102C2×C8×C31—41no—
103C4⋊C4×C31—22no—
104C4⋊C4×C31—42no—
105C4⋊C4×C31—41no—
106C22⋊C4×C31—22no—
107C22⋊C4×C31—24no—
108C22⋊C4×C31—48no—
109C22⋊C4×C31—42no—
110C22⋊C4×C31—42no—
111C22⋊C4×C31—41no—
112C42×C31—22no—
113C42×C31—42no—
114C42×C31—41no—
115#13350C3111yesmaximal, characteristic, direct product factor, Sylow
116C23×C31—24no—
117C23×C31—41no—
118C23×C31—44no—
119C23×C31—42no—
120C23×C31—162no—
121C23×C31#12012yes—
122C23×C31#12711yescharacteristic
123Q8×C31—22no—
124Q8×C31—41no—
125Q8×C31—42no—
126Q8×C31—41no—
127D4×C31—22no—
128D4×C31—41no—
129D4×C31—42no—
130D4×C31—42no—
131D4×C31—44no—
132D4×C31—42no—
133D4×C31—42no—
134D4×C31—41no—
135D4×C31—42no—
136D4×C31—42no—
137D4×C31—82no—
138D4×C31—162no—
139D4×C31—162no—
140D4×C31—168no—
141C2×C4×C31—21no—
142C2×C4×C31—22no—
143C2×C4×C31—21no—
144C2×C4×C31—22no—
145C2×C4×C31—24no—
146C2×C4×C31—44no—
147C2×C4×C31—42no—
148C2×C4×C31—41no—
149C2×C4×C31—42no—
150C2×C4×C31—44no—
151C2×C4×C31—42no—
152C2×C4×C31—81no—
153C2×C4×C31#12011yescharacteristic
154C2×C4×C31#12112yes—
155C2×C4×C31#12111yescharacteristic
156C8×C31—41no—
157C8×C31—41no—
158#3199C2×C3111yescharacteristic, complemented
159#3027C2×C3112yescomplemented
160#2128C2×C3112yescomplemented
161#1794C2×C3111yescharacteristic, complemented
162#1391C2×C3111yescharacteristic, complemented
163C22×C31—22no—
164C22×C31—21no—
165C22×C31—41no—
166C22×C31—42no—
167C22×C31—42no—
168C22×C31—82no—
169C22×C31—82no—
170C22×C31—84no—
171C22×C31—88no—
172C22×C31—82no—
173C22×C31—161no—
174C22×C31#39211yescharacteristic
175C4×C31—21no—
176C4×C31—22no—
177C4×C31—42no—
178C4×C31—82no—
179C4×C31—84no—
180C4×C31—161no—
181C4×C31B(4,𝔽2)11yescharacteristic
182C4×C31B(4,𝔽2)11yescharacteristic
183#585C22×C3111yescharacteristic
184#535—22no—
185#535C22×C3112yes—
186#482—22no—
187#469—21no—
188#466—21no—
189B(4,𝔽2)—24no—
190B(4,𝔽2)—24no—
191#447—22no—
192#421—21no—
193#416—21no—
194#410—22no—
195#410C22×C3111yescharacteristic
196#410C22×C3112yes—
197#408C22×C3111yescharacteristic
198#353—24no—
199C2×C31—21no—
200C2×C31—42no—
201C2×C31—41no—
202C2×C31—82no—
203C2×C31—162no—
204C2×C31#202111yescharacteristic
2052−1+4—21no—
2062−1+4—41no—
207H(5,𝔽2)—22no—
208H(5,𝔽2)—41no—
209H(5,𝔽2)D4×C3112yes—
210C2×#41—22no—
211C2×#41—21no—
212C2×#41—42no—
213C2×#41C23×C3111yescharacteristic
214C22×D4D4×C3112yescomplemented
215C2×D8—21no—
216C2×D8—21no—
217C2×M16D4×C3111yescharacteristic, complemented
218C22×C8D4×C3111yescharacteristic, complemented
219#127—22no—
220#127—42no—
221#124—22no—
222#124—42no—
223#121—22no—
224#121—22no—
225#121—44no—
226#120—28no—
227#120—22no—
228#120—22no—
229#117—22no—
230#117—21no—
231#117D4×C3112yescomplemented
232#107—21no—
233#106—21no—
234#104—42no—
235#103—41no—
236#102—22no—
237#102—22no—
238#102—41no—
239#99—28no—
240#99—22no—
241#99—22no—
242#99D4×C3114yescomplemented
243C22⋊C8—22no—
244C31#1335011yescharacteristic, direct product factor, Sylow
245C24—22no—
246#41—22no—
247#41—22no—
248#41—41no—
249#41—41no—
250#41—42no—
251#41—42no—
252#41—42no—
253#41—41no—
254#41—44no—
255#41—42no—
256#41—41no—
257#41—42no—
258C2×Q8—22no—
259C2×Q8—21no—
260C2×Q8—42no—
261C2×Q8#41×C3111yescharacteristic
262C2×D4—21no—
263C2×D4—22no—
264C2×D4—24no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—44no—
271C2×D4—41no—
272C2×D4—48no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4C2×D4×C3111yescharacteristic
276C2×D4C2×D4×C3112yes—
277C22×C4—24no—
278C22×C4—41no—
279C22×C4C2×D4×C3112yes—
280C22×C4C2×D4×C3111yescharacteristic
281D8—81no—
282D8—81no—
283M16—41no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—41no—
287C4⋊C4—22no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C22⋊C4—22no—
291C22⋊C4—24no—
292C22⋊C4—48no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—41no—
296C42—22no—
297C42—42no—
298C42—41no—
299C23—24no—
300C23—41no—
301C23—44no—
302C23—42no—
303C23—162no—
304C23C31×#12012yes—
305C23C31×#12711yescharacteristic
306Q8—22no—
307Q8—41no—
308Q8—42no—
309Q8—41no—
310D4—22no—
311D4—41no—
312D4—42no—
313D4—42no—
314D4—44no—
315D4—42no—
316D4—42no—
317D4—41no—
318D4—42no—
319D4—42no—
320D4—82no—
321D4—162no—
322D4—162no—
323D4—168no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4C31×#12011yescharacteristic
337C2×C4C31×#12112yes—
338C2×C4C31×#12111yescharacteristic
339C8—41no—
340C8—41no—
341C22—22no—
342C22—21no—
343C22—41no—
344C22—42no—
345C22—42no—
346C22—82no—
347C22—82no—
348C22—84no—
349C22—88no—
350C22—82no—
351C22—161no—
352C22C31×#39211yescharacteristic
353C4—21no—
354C4—22no—
355C4—42no—
356C4—82no—
357C4—84no—
358C4—161no—
359C4C31×B(4,𝔽2)11yescharacteristic
360C4C31×B(4,𝔽2)11yescharacteristic
361C2—21no—
362C2—42no—
363C2—41no—
364C2—82no—
365C2—162no—
366C2C31×#202111yescharacteristic
subgroupG/H
center, socleC2×C31#2021
commutator, Frattini, cosocleC2×#41C23×C31
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3199C2semidirect
2. C31×#3027C2semidirect
3. C31×#2128C2semidirect
4. C31×#1794C2semidirect
5. C31×#1391C2semidirect
31. C22×D4×C31D4semidirect
34. C2×M16×C31D4semidirect
35. C22×C8×C31D4semidirect
48. C31×#117D4semidirect
59. C31×#99D4semidirect
115. #13350C31direct (N × H)
158. #3199C2×C31semidirect
159. #3027C2×C31semidirect
160. #2128C2×C31semidirect
161. #1794C2×C31semidirect
162. #1391C2×C31semidirect
214. C22×D4D4×C31semidirect
217. C2×M16D4×C31semidirect
218. C22×C8D4×C31semidirect
231. #117D4×C31semidirect
242. #99D4×C31semidirect
244. C31#13350direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31992
C31×#30272
C31×#21282
C31×#17942
C31×#13912
#1335031