Finite Group Databasebeta

← C31×#13680 #21470439

Subgroups — C31×#13680 #21470439

Order 7936.

378 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 58 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×#3289C211yesmaximal, characteristic
2C31×#2203C211yesmaximal, characteristic, complemented
3C31×#2201C211yesmaximal, characteristic, complemented
4C31×#1660C211yesmaximal, characteristic, complemented
5C31×#1407C211yesmaximal, characteristic, complemented
6C31×#1404C211yesmaximal, characteristic, complemented
7C31×#1403C211yesmaximal, characteristic, complemented
8C31×#578—21no—
9C31×#577—21no—
10C31×#563C2211yescharacteristic
11C31×#501—21no—
12C31×#497C2211yescharacteristic
13C31×#473—21no—
14C31×#471—21no—
15C31×#458—21no—
16C31×#456—21no—
17C31×#456—21no—
18C31×#455C2211yescharacteristic
19C31×#454C2211yescharacteristic
20C31×#444—21no—
21C31×#356—21no—
22C31×#355C2211yescharacteristic
23C31×#354C2211yescharacteristic
24C31×#352—21no—
25C31×#330—21no—
26C31×#329C2211yescharacteristic
27C31×2−1+4—21no—
28C31×2−1+4D411yescharacteristic
29C31×H(5,𝔽2)D411yescharacteristic
30C31×#137—21no—
31C31×#137—21no—
32C31×#137—21no—
33C31×#137—41no—
34C31×#137—41no—
35C31×#137—41no—
36C31×#137—41no—
37C31×AGL(1,ℤ/8)—21no—
38C31×AGL(1,ℤ/8)—41no—
39C31×#135—21no—
40C31×#135—41no—
41C31×#135—41no—
42C2×Q16×C31—21no—
43C2×Q16×C31—41no—
44C2×SD16×C31—21no—
45C2×SD16×C31—41no—
46C31×#131—21no—
47C31×#131—21no—
48C31×C4⋊Q8—21no—
49C31×C4⋊Q8D411yescharacteristic
50C31×#126—41no—
51C31×#125—21no—
52C31×#124C2311yescharacteristic
53C31×#123—21no—
54C31×#123—41no—
55C31×C22⋊Q8—41no—
56C31×#108—21no—
57C31×#107—41no—
58C31×#106—41no—
59C31×#104—21no—
60C31×#104—21no—
61C31×#104—41no—
62C31×#101—21no—
63C31×#101—21no—
64C31×#101D411yescharacteristic
65C31×#100D411yescharacteristic
66C31×#99—21no—
67C31×#99—41no—
68C31×#97—21no—
69C31×#97D411yescharacteristic
70#41×C31—21no—
71#41×C31—21no—
72#41×C31—41no—
73#41×C31—41no—
74#41×C31—41no—
75#41×C31—41no—
76#41×C31—41no—
77#41×C31—81no—
78C2×Q8×C31—21no—
79C2×Q8×C31—21no—
80C2×Q8×C31—41no—
81C2×Q8×C31—41no—
82C2×Q8×C31—41no—
83C2×Q8×C31C2×D411yescharacteristic
84C2×D4×C31—41no—
85C2×D4×C31—41no—
86C2×D4×C31C2×D411yescharacteristic
87C22×C4×C31—21no—
88Q16×C31—21no—
89Q16×C31—41no—
90Q16×C31—41no—
91Q16×C31—41no—
92Q16×C31—41no—
93Q16×C31—41no—
94Q16×C31—81no—
95SD16×C31—21no—
96SD16×C31—21no—
97SD16×C31—41no—
98SD16×C31—41no—
99SD16×C31—41no—
100SD16×C31—41no—
101SD16×C31—41no—
102SD16×C31—41no—
103SD16×C31—81no—
104D8×C31—21no—
105D8×C31—41no—
106M16×C31—21no—
107M16×C31—21no—
108M16×C31—41no—
109M16×C31—41no—
110M16×C31—41no—
111C2×C8×C31—21no—
112C2×C8×C31—41no—
113C2×C8×C31—41no—
114C2×C8×C31—41no—
115C4⋊C4×C31—21no—
116C4⋊C4×C31—21no—
117C4⋊C4×C31—41no—
118C4⋊C4×C31—41no—
119C4⋊C4×C31—41no—
120C4⋊C4×C31—41no—
121C4⋊C4×C31—81no—
122C22⋊C4×C31—21no—
123C22⋊C4×C31—41no—
124C22⋊C4×C31—41no—
125C22⋊C4×C31—41no—
126C22⋊C4×C31—81no—
127C42×C31—21no—
128C42×C31C2×D411yescharacteristic
129#13680C3111yesmaximal, characteristic, direct product factor, Sylow
130C23×C31—21no—
131C23×C31—41no—
132Q8×C31—21no—
133Q8×C31—21no—
134Q8×C31—41no—
135Q8×C31—41no—
136Q8×C31—41no—
137Q8×C31—41no—
138Q8×C31—81no—
139Q8×C31—81no—
140D4×C31—21no—
141D4×C31—21no—
142D4×C31—41no—
143D4×C31—41no—
144D4×C31—41no—
145D4×C31—41no—
146D4×C31—81no—
147D4×C31—81no—
148C2×C4×C31—21no—
149C2×C4×C31—21no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—81no—
157C2×C4×C31—81no—
158C2×C4×C31—81no—
159C2×C4×C31—81no—
160C2×C4×C31—81no—
161C2×C4×C31#12011yescharacteristic
162C8×C31—21no—
163C8×C31—21no—
164C8×C31—41no—
165C8×C31—41no—
166C8×C31—42no—
167C8×C31—81no—
168#3289C2×C3111yescharacteristic
169#2203C2×C3111yescharacteristic, complemented
170#2201C2×C3111yescharacteristic, complemented
171#1660C2×C3111yescharacteristic, complemented
172#1407C2×C3111yescharacteristic, complemented
173#1404C2×C3111yescharacteristic, complemented
174#1403C2×C3111yescharacteristic, complemented
175C22×C31—41no—
176C22×C31—41no—
177C22×C31—41no—
178C22×C31—81no—
179C22×C31—161no—
180C22×C31B(4,𝔽2)11yescharacteristic
181C4×C31—21no—
182C4×C31—41no—
183C4×C31—41no—
184C4×C31—41no—
185C4×C31—41no—
186C4×C31—81no—
187C4×C31—81no—
188C4×C31—81no—
189C4×C31—161no—
190#578—21no—
191#577—21no—
192#563C22×C3111yescharacteristic
193#501—21no—
194#497C22×C3111yescharacteristic
195#473—21no—
196#471—21no—
197#458—21no—
198#456—21no—
199#456—21no—
200#455C22×C3111yescharacteristic
201#454C22×C3111yescharacteristic
202#444—21no—
203#356—21no—
204#355C22×C3111yescharacteristic
205#354C22×C3111yescharacteristic
206#352—21no—
207#330—21no—
208#329C22×C3111yescharacteristic
209C2×C31—21no—
210C2×C31—81no—
211C2×C31—81no—
212C2×C31—81no—
213C2×C31#219611yescharacteristic
2142−1+4—21no—
2152−1+4D4×C3111yescharacteristic
216H(5,𝔽2)D4×C3111yescharacteristic
217#137—21no—
218#137—21no—
219#137—21no—
220#137—41no—
221#137—41no—
222#137—41no—
223#137—41no—
224AGL(1,ℤ/8)—21no—
225AGL(1,ℤ/8)—41no—
226#135—21no—
227#135—41no—
228#135—41no—
229C2×Q16—21no—
230C2×Q16—41no—
231C2×SD16—21no—
232C2×SD16—41no—
233#131—21no—
234#131—21no—
235C4⋊Q8—21no—
236C4⋊Q8D4×C3111yescharacteristic
237#126—41no—
238#125—21no—
239#124C23×C3111yescharacteristic
240#123—21no—
241#123—41no—
242C22⋊Q8—41no—
243#108—21no—
244#107—41no—
245#106—41no—
246#104—21no—
247#104—21no—
248#104—41no—
249#101—21no—
250#101—21no—
251#101D4×C3111yescharacteristic
252#100D4×C3111yescharacteristic
253#99—21no—
254#99—41no—
255#97—21no—
256#97D4×C3111yescharacteristic
257C31#1368011yescharacteristic, direct product factor, Sylow
258#41—21no—
259#41—21no—
260#41—41no—
261#41—41no—
262#41—41no—
263#41—41no—
264#41—41no—
265#41—81no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8C2×D4×C3111yescharacteristic
272C2×D4—41no—
273C2×D4—41no—
274C2×D4C2×D4×C3111yescharacteristic
275C22×C4—21no—
276Q16—21no—
277Q16—41no—
278Q16—41no—
279Q16—41no—
280Q16—41no—
281Q16—41no—
282Q16—81no—
283SD16—21no—
284SD16—21no—
285SD16—41no—
286SD16—41no—
287SD16—41no—
288SD16—41no—
289SD16—41no—
290SD16—41no—
291SD16—81no—
292D8—21no—
293D8—41no—
294M16—21no—
295M16—21no—
296M16—41no—
297M16—41no—
298M16—41no—
299C2×C8—21no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—81no—
310C22⋊C4—21no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—81no—
315C42—21no—
316C42C2×D4×C3111yescharacteristic
317C23—21no—
318C23—41no—
319Q8—21no—
320Q8—21no—
321Q8—41no—
322Q8—41no—
323Q8—41no—
324Q8—41no—
325Q8—81no—
326Q8—81no—
327D4—21no—
328D4—21no—
329D4—41no—
330D4—41no—
331D4—41no—
332D4—41no—
333D4—81no—
334D4—81no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4C31×#12011yescharacteristic
349C8—21no—
350C8—21no—
351C8—41no—
352C8—41no—
353C8—42no—
354C8—81no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—81no—
359C22—161no—
360C22C31×B(4,𝔽2)11yescharacteristic
361C4—21no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—161no—
370C2—21no—
371C2—81no—
372C2—81no—
373C2—81no—
374C2C31×#219611yescharacteristic
subgroupG/H
center, socleC2×C31#2196
commutator, Frattini, cosocle#124C23×C31
prime radicalC31×#3289C2
LCSG ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C22×C31 ◁ C2×C4×C31 ◁ C31×#124 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2203C2semidirect
3. C31×#2201C2semidirect
4. C31×#1660C2semidirect
5. C31×#1407C2semidirect
6. C31×#1404C2semidirect
7. C31×#1403C2semidirect
129. #13680C31direct (N × H)
169. #2203C2×C31semidirect
170. #2201C2×C31semidirect
171. #1660C2×C31semidirect
172. #1407C2×C31semidirect
173. #1404C2×C31semidirect
174. #1403C2×C31semidirect
257. C31#13680direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32892
C31×#22032
C31×#22012
C31×#16602
C31×#14072
C31×#14042
C31×#14032
#1368031