Finite Group Databasebeta

← C31×#13056 #21469815

Subgroups — C31×#13056 #21469815

Order 7936.

632 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 118 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
1C2×C31×#456C211yesmaximal, characteristic
2C31×#2381C211yesmaximal, characteristic, complemented
3C2×C31×#356C211yesmaximal, characteristic, complemented
4C31×#1793C212yesmaximal
5C31×#1394C212yesmaximal, complemented
6C2×C31×2−1+4C2211yescharacteristic
7C2×C31×#137—21no—
8C2×C31×C4⋊Q8C2211yescharacteristic
9C31×#456—22no—
10C31×#456C414yescomplemented
11C2×C31×#104—21no—
12C2×C31×#104—22no—
13C31×#417—22no—
14C2×C31×#101C2212yes—
15C2×C31×#101C2211yescharacteristic
16C31×#391—21no—
17C31×#391C2212yes—
18C31×#387—22no—
19C31×#387—22no—
20C31×#383—21no—
21C31×#356—22no—
22C31×2−1+4—21no—
23C31×2−1+4—22no—
24C31×2−1+4—42no—
25C31×2−1+4C2×C412yes—
26C2×#41×C31—22no—
27C2×#41×C31—21no—
28C2×#41×C31—41no—
29C22×Q8×C31—41no—
30C22×Q8×C31D411yescharacteristic
31C22×Q8×C31C2311yescharacteristic
32C31×#137—24no—
33C31×#137—42no—
34C2×Q16×C31—41no—
35C2×SD16×C31—41no—
36C2×M16×C31—21no—
37C2×M16×C31—22no—
38C31×C4⋊Q8—22no—
39C31×C4⋊Q8C2×C412yes—
40C31×C4⋊Q8D412yes—
41C2×C4⋊C4×C31—22no—
42C2×C4⋊C4×C31—21no—
43C2×C4⋊C4×C31D412yes—
44C2×C42×C31—21no—
45C2×C42×C31D411yescharacteristic
46C2×C42×C31D412yes—
47C31×#104—24no—
48C31×#104—44no—
49C31×#103—42no—
50C31×#102—42no—
51C31×#101—21no—
52C31×#101—22no—
53C31×#101—22no—
54C31×#101C2×C412yes—
55C31×#95—21no—
56C31×#95—22no—
57C31×#95—21no—
58C31×#95—22no—
59C31×#95—22no—
60C31×#95—22no—
61#41×C31—24no—
62#41×C31—22no—
63#41×C31—22no—
64#41×C31—24no—
65#41×C31—42no—
66#41×C31—41no—
67#41×C31—42no—
68#41×C31—42no—
69#41×C31—44no—
70#41×C31—42no—
71#41×C31—41no—
72C2×Q8×C31—21no—
73C2×Q8×C31—22no—
74C2×Q8×C31—21no—
75C2×Q8×C31—41no—
76C2×Q8×C31—41no—
77C2×Q8×C31—41no—
78C2×Q8×C31—41no—
79C2×Q8×C31—42no—
80C2×Q8×C31—42no—
81C2×Q8×C31—44no—
82C2×Q8×C31C22⋊C412yes—
83C2×Q8×C31C22×C411yescharacteristic
84C2×Q8×C31C2×D411yescharacteristic
85C2×Q8×C31#4112yes—
86C2×D4×C31—22no—
87C2×D4×C31—21no—
88C2×D4×C31—41no—
89C22×C4×C31—21no—
90C22×C4×C31—21no—
91C22×C4×C31—22no—
92C22×C4×C31—22no—
93C22×C4×C31—42no—
94C22×C4×C31—41no—
95C22×C4×C31C2×D411yescharacteristic
96C22×C4×C31C2×D412yes—
97Q16×C31—44no—
98SD16×C31—44no—
99M16×C31—22no—
100M16×C31—41no—
101M16×C31—42no—
102M16×C31—42no—
103C2×C8×C31—41no—
104C2×C8×C31—42no—
105C4⋊C4×C31—24no—
106C4⋊C4×C31—24no—
107C4⋊C4×C31—42no—
108C4⋊C4×C31—81no—
109C42×C31—21no—
110C42×C31—42no—
111C42×C31—81no—
112C42×C31C22⋊C412yes—
113#13056C3111yesmaximal, characteristic, direct product factor, Sylow
114C23×C31—41no—
115C23×C31#12011yescharacteristic
116Q8×C31—22no—
117Q8×C31—24no—
118Q8×C31—41no—
119Q8×C31—41no—
120Q8×C31—42no—
121Q8×C31—42no—
122Q8×C31—42no—
123Q8×C31—41no—
124Q8×C31—81no—
125D4×C31—22no—
126D4×C31—24no—
127D4×C31—42no—
128D4×C31—41no—
129D4×C31—42no—
130D4×C31—42no—
131C2×C4×C31—22no—
132C2×C4×C31—21no—
133C2×C4×C31—22no—
134C2×C4×C31—21no—
135C2×C4×C31—22no—
136C2×C4×C31—22no—
137C2×C4×C31—42no—
138C2×C4×C31—44no—
139C2×C4×C31—44no—
140C2×C4×C31—42no—
141C2×C4×C31—42no—
142C2×C4×C31—41no—
143C2×C4×C31—41no—
144C2×C4×C31—41no—
145C2×C4×C31—44no—
146C2×C4×C31—44no—
147C2×C4×C31—42no—
148C2×C4×C31—42no—
149C2×C4×C31C2×C22⋊C411yescharacteristic
150C2×C4×C31C4×D412yes—
151C2×C4×C31#12112yes—
152C2×C4×C31#12311yescharacteristic
153C8×C31—42no—
154C8×C31—82no—
155C2×#456C2×C3111yescharacteristic
156#2381C2×C3111yescharacteristic, complemented
157C2×#356C2×C3111yescharacteristic, complemented
158#1793C2×C3112yes—
159#1394C2×C3112yescomplemented
160C22×C31—22no—
161C22×C31—42no—
162C22×C31—82no—
163C22×C31#38611yescharacteristic
164C22×C31B(4,𝔽2)11yescharacteristic
165C22×C31B(4,𝔽2)11yescharacteristic
166C4×C31—21no—
167C4×C31—22no—
168C4×C31—22no—
169C4×C31—21no—
170C4×C31—42no—
171C4×C31—42no—
172C4×C31—42no—
173C4×C31—82no—
174C4×C31—82no—
175C2×2−1+4C22×C3111yescharacteristic
176C2×#137—21no—
177C2×C4⋊Q8C22×C3111yescharacteristic
178#456—22no—
179#456C4×C3114yescomplemented
180C2×#104—21no—
181C2×#104—22no—
182#417—22no—
183C2×#101C22×C3112yes—
184C2×#101C22×C3111yescharacteristic
185#391—21no—
186#391C22×C3112yes—
187#387—22no—
188#387—22no—
189#383—21no—
190#356—22no—
191C2×C31—21no—
192C2×C31—21no—
193C2×C31—82no—
194C2×C31#188911yescharacteristic
195C2×C31#220312yes—
1962−1+4—21no—
1972−1+4—22no—
1982−1+4—42no—
1992−1+4C2×C4×C3112yes—
200C2×#41—21no—
201C2×#41—22no—
202C2×#41—41no—
203C22×Q8—41no—
204C22×Q8D4×C3111yescharacteristic
205C22×Q8C23×C3111yescharacteristic
206#137—24no—
207#137—42no—
208C2×Q16—41no—
209C2×SD16—41no—
210C2×M16—21no—
211C2×M16—22no—
212C4⋊Q8—22no—
213C4⋊Q8C2×C4×C3112yes—
214C4⋊Q8D4×C3112yes—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4D4×C3112yes—
218C2×C42—21no—
219C2×C42D4×C3111yescharacteristic
220C2×C42D4×C3112yes—
221#104—24no—
222#104—44no—
223#103—42no—
224#102—42no—
225#101—21no—
226#101—22no—
227#101—22no—
228#101C2×C4×C3112yes—
229#95—21no—
230#95—22no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95—22no—
235C31#1305611yescharacteristic, direct product factor, Sylow
236#41—24no—
237#41—22no—
238#41—22no—
239#41—24no—
240#41—42no—
241#41—41no—
242#41—42no—
243#41—42no—
244#41—44no—
245#41—42no—
246#41—41no—
247C2×Q8—21no—
248C2×Q8—22no—
249C2×Q8—21no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—44no—
257C2×Q8C22⋊C4×C3112yes—
258C2×Q8C22×C4×C3111yescharacteristic
259C2×Q8C2×D4×C3111yescharacteristic
260C2×Q8#41×C3112yes—
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—41no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4C2×D4×C3111yescharacteristic
271C22×C4C2×D4×C3112yes—
272Q16—44no—
273SD16—44no—
274M16—22no—
275M16—41no—
276M16—42no—
277M16—42no—
278C2×C8—41no—
279C2×C8—42no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—42no—
283C4⋊C4—81no—
284C42—21no—
285C42—42no—
286C42—81no—
287C42C22⋊C4×C3112yes—
288C23—41no—
289C23C31×#12011yescharacteristic
290Q8—22no—
291Q8—24no—
292Q8—41no—
293Q8—41no—
294Q8—42no—
295Q8—42no—
296Q8—42no—
297Q8—41no—
298Q8—81no—
299D4—22no—
300D4—24no—
301D4—42no—
302D4—41no—
303D4—42no—
304D4—42no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4C2×C22⋊C4×C3111yescharacteristic
324C2×C4C4×D4×C3112yes—
325C2×C4C31×#12112yes—
326C2×C4C31×#12311yescharacteristic
327C8—42no—
328C8—82no—
329C22—22no—
330C22—42no—
331C22—82no—
332C22C31×#38611yescharacteristic
333C22C31×B(4,𝔽2)11yescharacteristic
334C22C31×B(4,𝔽2)11yescharacteristic
335C4—21no—
336C4—22no—
337C4—22no—
338C4—21no—
339C4—42no—
340C4—42no—
341C4—42no—
342C4—82no—
343C4—82no—
344C2—21no—
345C2—21no—
346C2—82no—
347C2C31×#188911yescharacteristic
348C2C31×#220312yes—
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutatorC2×Q8C22×C4×C31
Frattini, cosocleC22×Q8C23×C31
prime radicalC2×C31×2−1+4C22
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C22×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2381C2semidirect
3. C2×C31×#356C2semidirect
5. C31×#1394C2semidirect
10. C31×#456C4semidirect
113. #13056C31direct (N × H)
156. #2381C2×C31semidirect
157. C2×#356C2×C31semidirect
159. #1394C2×C31semidirect
179. #456C4×C31semidirect
235. C31#13056direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4562
C31×#23812
C2×C31×#3562
C31×#17932
C31×#13942
#1305631