Finite Group Databasebeta

← C31×#13360 #21470119

Subgroups — C31×#13360 #21470119

Order 7936.

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

normal subgroups only

This group has 702 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×#3021C212yesmaximal, complemented
3C31×#2122C212yesmaximal, complemented
4C31×#1794C211yesmaximal, characteristic, complemented
5C31×#1392C211yesmaximal, characteristic, complemented
6C31×#585C2211yescharacteristic
7C2×C31×AGL(1,ℤ/8)—22no—
8C31×#535C2212yes—
9C31×#482—22no—
10C31×#469—21no—
11C31×#466—21no—
12C31×#454—24no—
13C31×#453—24no—
14C31×#447—22no—
15C31×#421—22no—
16C31×#421—21no—
17C31×#416—21no—
18C31×#413C2212yes—
19C31×#410C2211yescharacteristic
20C31×#408C2211yescharacteristic
21C31×AΣL(1,𝔽16)—24no—
22C31×2−1+4—41no—
23C31×2−1+4D412yes—
24C31×H(5,𝔽2)—22no—
25C31×H(5,𝔽2)—21no—
26C31×H(5,𝔽2)—41no—
27C2×#41×C31—21no—
28C2×#41×C31—22no—
29C2×#41×C31—42no—
30C2×#41×C31C2311yescharacteristic
31C22×D4×C31D412yes—
32C31×AGL(1,ℤ/8)—28no—
33C31×AGL(1,ℤ/8)—44no—
34C2×SD16×C31—42no—
35C2×D8×C31—21no—
36C2×D8×C31—21no—
37C2×D8×C31—42no—
38C2×M16×C31—22no—
39C2×M16×C31D411yescharacteristic
40C22×C8×C31D411yescharacteristic
41C31×#127—22no—
42C31×#124—22no—
43C31×#121—22no—
44C31×#121—22no—
45C31×#120—22no—
46C31×#120—22no—
47C31×#117—21no—
48C31×#117D412yes—
49C31×#107—21no—
50C31×#106—21no—
51C31×#104—28no—
52C31×#104—42no—
53C31×#103—41no—
54C31×#102—22no—
55C31×#102—22no—
56C31×#102—41no—
57C31×#101—22no—
58C31×#100D414yes—
59C31×#99—22no—
60C31×C22⋊C8—22no—
61C24×C31—22no—
62#41×C31—22no—
63#41×C31—24no—
64#41×C31—24no—
65#41×C31—22no—
66#41×C31—41no—
67#41×C31—41no—
68#41×C31—41no—
69#41×C31—42no—
70#41×C31—42no—
71#41×C31—44no—
72#41×C31—42no—
73#41×C31—41no—
74C2×Q8×C31—21no—
75C2×Q8×C31—22no—
76C2×Q8×C31—42no—
77C2×Q8×C31C2×D411yescharacteristic
78C2×D4×C31—21no—
79C2×D4×C31—22no—
80C2×D4×C31—22no—
81C2×D4×C31—22no—
82C2×D4×C31—42no—
83C2×D4×C31—42no—
84C2×D4×C31—44no—
85C2×D4×C31—42no—
86C2×D4×C31—42no—
87C2×D4×C31—41no—
88C2×D4×C31—42no—
89C2×D4×C31—42no—
90C2×D4×C31C2×D412yes—
91C2×D4×C31#4111yescharacteristic
92C22×C4×C31—41no—
93C22×C4×C31—42no—
94C22×C4×C31C2×D412yes—
95C22×C4×C31C2×D411yescharacteristic
96SD16×C31—48no—
97D8×C31—48no—
98D8×C31—81no—
99D8×C31—81no—
100M16×C31—24no—
101M16×C31—42no—
102M16×C31—41no—
103C2×C8×C31—21no—
104C2×C8×C31—21no—
105C2×C8×C31—42no—
106C2×C8×C31—41no—
107C4⋊C4×C31—22no—
108C4⋊C4×C31—41no—
109C22⋊C4×C31—22no—
110C22⋊C4×C31—42no—
111C22⋊C4×C31—42no—
112C22⋊C4×C31—41no—
113C42×C31—22no—
114C42×C31—41no—
115#13360C3111yesmaximal, characteristic, direct product factor, Sylow
116C23×C31—41no—
117C23×C31—42no—
118C23×C31—42no—
119C23×C31—82no—
120C23×C31—162no—
121C23×C31#12112yes—
122C23×C31#12711yescharacteristic
123Q8×C31—24no—
124Q8×C31—22no—
125Q8×C31—41no—
126Q8×C31—41no—
127D4×C31—24no—
128D4×C31—22no—
129D4×C31—42no—
130D4×C31—41no—
131D4×C31—44no—
132D4×C31—42no—
133D4×C31—44no—
134D4×C31—42no—
135D4×C31—41no—
136D4×C31—42no—
137D4×C31—42no—
138D4×C31—162no—
139D4×C31—162no—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31—22no—
143C2×C4×C31—22no—
144C2×C4×C31—44no—
145C2×C4×C31—42no—
146C2×C4×C31—41no—
147C2×C4×C31—42no—
148C2×C4×C31—42no—
149C2×C4×C31—42no—
150C2×C4×C31—81no—
151C2×C4×C31#12012yes—
152C2×C4×C31#12011yescharacteristic
153C2×C4×C31#12111yescharacteristic
154C8×C31—44no—
155C8×C31—41no—
156C8×C31—41no—
157#3199C2×C3111yescharacteristic, complemented
158#3021C2×C3112yescomplemented
159#2122C2×C3112yescomplemented
160#1794C2×C3111yescharacteristic, complemented
161#1392C2×C3111yescharacteristic, complemented
162C22×C31—21no—
163C22×C31—22no—
164C22×C31—41no—
165C22×C31—42no—
166C22×C31—42no—
167C22×C31—82no—
168C22×C31—82no—
169C22×C31—82no—
170C22×C31—161no—
171C22×C31—162no—
172C22×C31—164no—
173C22×C31#39211yescharacteristic
174C4×C31—22no—
175C4×C31—21no—
176C4×C31—42no—
177C4×C31—82no—
178C4×C31—161no—
179C4×C31B(4,𝔽2)11yescharacteristic
180C4×C31B(4,𝔽2)11yescharacteristic
181#585C22×C3111yescharacteristic
182C2×AGL(1,ℤ/8)—22no—
183#535C22×C3112yes—
184#482—22no—
185#469—21no—
186#466—21no—
187#454—24no—
188#453—24no—
189#447—22no—
190#421—22no—
191#421—21no—
192#416—21no—
193#413C22×C3112yes—
194#410C22×C3111yescharacteristic
195#408C22×C3111yescharacteristic
196AΣL(1,𝔽16)—24no—
197C2×C31—21no—
198C2×C31—41no—
199C2×C31—42no—
200C2×C31—82no—
201C2×C31—162no—
202C2×C31#202111yescharacteristic
2032−1+4—41no—
2042−1+4D4×C3112yes—
205H(5,𝔽2)—22no—
206H(5,𝔽2)—21no—
207H(5,𝔽2)—41no—
208C2×#41—22no—
209C2×#41—21no—
210C2×#41—42no—
211C2×#41C23×C3111yescharacteristic
212C22×D4D4×C3112yes—
213AGL(1,ℤ/8)—28no—
214AGL(1,ℤ/8)—44no—
215C2×SD16—42no—
216C2×D8—21no—
217C2×D8—21no—
218C2×D8—42no—
219C2×M16—22no—
220C2×M16D4×C3111yescharacteristic
221C22×C8D4×C3111yescharacteristic
222#127—22no—
223#124—22no—
224#121—22no—
225#121—22no—
226#120—22no—
227#120—22no—
228#117—21no—
229#117D4×C3112yes—
230#107—21no—
231#106—21no—
232#104—28no—
233#104—42no—
234#103—41no—
235#102—22no—
236#102—22no—
237#102—41no—
238#101—22no—
239#100D4×C3114yes—
240#99—22no—
241C22⋊C8—22no—
242C31#1336011yescharacteristic, direct product factor, Sylow
243C24—22no—
244#41—22no—
245#41—24no—
246#41—24no—
247#41—22no—
248#41—41no—
249#41—41no—
250#41—41no—
251#41—42no—
252#41—42no—
253#41—44no—
254#41—42no—
255#41—41no—
256C2×Q8—21no—
257C2×Q8—22no—
258C2×Q8—42no—
259C2×Q8C2×D4×C3111yescharacteristic
260C2×D4—21no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—44no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4C2×D4×C3112yes—
273C2×D4#41×C3111yescharacteristic
274C22×C4—41no—
275C22×C4—42no—
276C22×C4C2×D4×C3112yes—
277C22×C4C2×D4×C3111yescharacteristic
278SD16—48no—
279D8—48no—
280D8—81no—
281D8—81no—
282M16—24no—
283M16—42no—
284M16—41no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—42no—
288C2×C8—41no—
289C4⋊C4—22no—
290C4⋊C4—41no—
291C22⋊C4—22no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—41no—
295C42—22no—
296C42—41no—
297C23—41no—
298C23—42no—
299C23—42no—
300C23—82no—
301C23—162no—
302C23C31×#12112yes—
303C23C31×#12711yescharacteristic
304Q8—24no—
305Q8—22no—
306Q8—41no—
307Q8—41no—
308D4—24no—
309D4—22no—
310D4—42no—
311D4—41no—
312D4—44no—
313D4—42no—
314D4—44no—
315D4—42no—
316D4—41no—
317D4—42no—
318D4—42no—
319D4—162no—
320D4—162no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—81no—
332C2×C4C31×#12012yes—
333C2×C4C31×#12011yescharacteristic
334C2×C4C31×#12111yescharacteristic
335C8—44no—
336C8—41no—
337C8—41no—
338C22—21no—
339C22—22no—
340C22—41no—
341C22—42no—
342C22—42no—
343C22—82no—
344C22—82no—
345C22—82no—
346C22—161no—
347C22—162no—
348C22—164no—
349C22C31×#39211yescharacteristic
350C4—22no—
351C4—21no—
352C4—42no—
353C4—82no—
354C4—161no—
355C4C31×B(4,𝔽2)11yescharacteristic
356C4C31×B(4,𝔽2)11yescharacteristic
357C2—21no—
358C2—41no—
359C2—42no—
360C2—82no—
361C2—162no—
362C2C31×#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×#3021C2semidirect
3. C31×#2122C2semidirect
4. C31×#1794C2semidirect
5. C31×#1392C2semidirect
115. #13360C31direct (N × H)
157. #3199C2×C31semidirect
158. #3021C2×C31semidirect
159. #2122C2×C31semidirect
160. #1794C2×C31semidirect
161. #1392C2×C31semidirect
242. C31#13360direct (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×#30212
C31×#21222
C31×#17942
C31×#13922
#1336031