Finite Group Databasebeta

← C31×#12195 #21468954

Subgroups — C31×#12195 #21468954

Order 7936.

458 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2572C211yesmaximal, characteristic, complemented
2C31×#2007C211yesmaximal, characteristic, complemented
3C31×#1616C211yesmaximal, characteristic, complemented
4C31×#1598C211yesmaximal, characteristic
5C31×#1353C211yesmaximal, characteristic, complemented
6C31×#1345C211yesmaximal, characteristic, complemented
7C31×#1319C211yesmaximal, characteristic, complemented
8C2×C31×#137—21no—
9C31×#537C2211yescharacteristic
10C31×#481—21no—
11C31×#447—21no—
12C2×C31×#104—21no—
13C31×#400C2211yescharacteristic
14C31×#394C2211yescharacteristic
15C31×#393—21no—
16C31×#391C2211yescharacteristic
17C31×#388—21no—
18C31×#386—21no—
19C31×#383—21no—
20C31×#381—21no—
21C4×C22⋊C4×C31—21no—
22C31×#343C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C2×#41×C31—21no—
28C22×Q8×C31D411yescharacteristic
29C22×D4×C31D411yescharacteristic
30C23×C4×C31—21no—
31C31×#137—42no—
32C31×#137—42no—
33C2×Q16×C31—41no—
34C2×SD16×C31—41no—
35C2×D8×C31—41no—
36C2×M16×C31—21no—
37C31×C22⋊Q8—41no—
38C31×#121—21no—
39C31×#121—21no—
40C31×#121—41no—
41C2×C4⋊C4×C31D411yescharacteristic
42C2×C4⋊C4×C31C2311yescharacteristic
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—21no—
48C2×C42×C31—21no—
49C2×C42×C31D411yescharacteristic
50C31×#106—41no—
51C31×#104—42no—
52C31×#102—41no—
53C31×#102—41no—
54C31×C22⋊C8—21no—
55C31×C22⋊C8—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95—21no—
61C31×#95—21no—
62C31×#95—21no—
63C31×#95D411yescharacteristic
64C31×#95D411yescharacteristic
65C24×C31—21no—
66#41×C31—41no—
67#41×C31—41no—
68#41×C31—41no—
69#41×C31—41no—
70C2×Q8×C31—21no—
71C2×Q8×C31—21no—
72C2×Q8×C31—21no—
73C2×Q8×C31—41no—
74C2×Q8×C31—41no—
75C2×D4×C31—21no—
76C2×D4×C31—21no—
77C2×D4×C31—21no—
78C2×D4×C31—41no—
79C2×D4×C31—42no—
80C2×D4×C31—41no—
81C22×C4×C31—21no—
82C22×C4×C31—21no—
83C22×C4×C31—21no—
84C22×C4×C31—21no—
85C22×C4×C31—21no—
86C22×C4×C31—41no—
87C22×C4×C31—42no—
88C22×C4×C31—42no—
89C22×C4×C31—81no—
90C22×C4×C31C2×D411yescharacteristic
91C22×C4×C31C2×D411yescharacteristic
92C22×C4×C31C2×D411yescharacteristic
93Q16×C31—44no—
94SD16×C31—44no—
95D8×C31—82no—
96M16×C31—22no—
97M16×C31—41no—
98C2×C8×C31—41no—
99C2×C8×C31—41no—
100C2×C8×C31—41no—
101C4⋊C4×C31—21no—
102C4⋊C4×C31—21no—
103C4⋊C4×C31—41no—
104C4⋊C4×C31—41no—
105C22⋊C4×C31—42no—
106C22⋊C4×C31—41no—
107C22⋊C4×C31—41no—
108C22⋊C4×C31—42no—
109C22⋊C4×C31—41no—
110C22⋊C4×C31—41no—
111C22⋊C4×C31—81no—
112C22⋊C4×C31—81no—
113C42×C31—42no—
114C42×C31—41no—
115#12195C3111yesmaximal, characteristic, direct product factor, Sylow
116C23×C31—22no—
117C23×C31—41no—
118C23×C31—41no—
119C23×C31—41no—
120C23×C31—41no—
121C23×C31—41no—
122C23×C31—41no—
123C23×C31—41no—
124C23×C31#12011yescharacteristic
125Q8×C31—42no—
126Q8×C31—41no—
127Q8×C31—41no—
128Q8×C31—81no—
129D4×C31—41no—
130D4×C31—41no—
131D4×C31—42no—
132D4×C31—81no—
133D4×C31—81no—
134D4×C31—81no—
135D4×C31—81no—
136D4×C31—81no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—22no—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31—41no—
143C2×C4×C31—41no—
144C2×C4×C31—41no—
145C2×C4×C31—41no—
146C2×C4×C31—41no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
149C2×C4×C31—41no—
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—41no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—81no—
160C2×C4×C31—81no—
161C2×C4×C31—84no—
162C2×C4×C31#13511yescharacteristic
163C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
164C8×C31—42no—
165C8×C31—81no—
166C8×C31—81no—
167#2572C2×C3111yescharacteristic, complemented
168#2007C2×C3111yescharacteristic, complemented
169#1616C2×C3111yescharacteristic, complemented
170#1598C2×C3111yescharacteristic
171#1353C2×C3111yescharacteristic, complemented
172#1345C2×C3111yescharacteristic, complemented
173#1319C2×C3111yescharacteristic, complemented
174C22×C31—21no—
175C22×C31—21no—
176C22×C31—42no—
177C22×C31—42no—
178C22×C31—42no—
179C22×C31—81no—
180C22×C31—81no—
181C22×C31—81no—
182C22×C31—81no—
183C22×C31—81no—
184C22×C31—81no—
185C22×C31—81no—
186C22×C31#44911yescharacteristic
187C22×C31#45311yescharacteristic
188C22×C31B(4,𝔽2)11yescharacteristic
189C4×C31—21no—
190C4×C31—21no—
191C4×C31—42no—
192C4×C31—41no—
193C4×C31—41no—
194C4×C31—81no—
195C4×C31—81no—
196C4×C31—81no—
197C4×C31—81no—
198C4×C31—81no—
199C2×#137—21no—
200#537C22×C3111yescharacteristic
201#481—21no—
202#447—21no—
203C2×#104—21no—
204#400C22×C3111yescharacteristic
205#394C22×C3111yescharacteristic
206#393—21no—
207#391C22×C3111yescharacteristic
208#388—21no—
209#386—21no—
210#383—21no—
211#381—21no—
212C4×C22⋊C4—21no—
213#343C22×C3111yescharacteristic
214#327—21no—
215#327—21no—
216#324C22×C3111yescharacteristic
217#324C22×C3111yescharacteristic
218C2×C31—21no—
219C2×C31—21no—
220C2×C31—82no—
221C2×C31—161no—
222C2×C31#160011yescharacteristic
223C2×C31#219711yescharacteristic
224C2×C31#220411yescharacteristic
225C2×#41—21no—
226C22×Q8D4×C3111yescharacteristic
227C22×D4D4×C3111yescharacteristic
228C23×C4—21no—
229#137—42no—
230#137—42no—
231C2×Q16—41no—
232C2×SD16—41no—
233C2×D8—41no—
234C2×M16—21no—
235C22⋊Q8—41no—
236#121—21no—
237#121—21no—
238#121—41no—
239C2×C4⋊C4D4×C3111yescharacteristic
240C2×C4⋊C4C23×C3111yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C42—21no—
247C2×C42D4×C3111yescharacteristic
248#106—41no—
249#104—42no—
250#102—41no—
251#102—41no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95D4×C3111yescharacteristic
262#95D4×C3111yescharacteristic
263C31#1219511yescharacteristic, direct product factor, Sylow
264C24—21no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—41no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—81no—
289C22×C4C2×D4×C3111yescharacteristic
290C22×C4C2×D4×C3111yescharacteristic
291C22×C4C2×D4×C3111yescharacteristic
292Q16—44no—
293SD16—44no—
294D8—82no—
295M16—22no—
296M16—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C22⋊C4—42no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—81no—
311C22⋊C4—81no—
312C42—41no—
313C42—42no—
314C23—22no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23C31×#12011yescharacteristic
323Q8—42no—
324Q8—41no—
325Q8—41no—
326Q8—81no—
327D4—41no—
328D4—41no—
329D4—42no—
330D4—81no—
331D4—81no—
332D4—81no—
333D4—81no—
334D4—81no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—84no—
360C2×C4C31×#13511yescharacteristic
361C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
362C8—42no—
363C8—81no—
364C8—81no—
365C22—21no—
366C22—21no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22C31×#44911yescharacteristic
378C22C31×#45311yescharacteristic
379C22C31×B(4,𝔽2)11yescharacteristic
380C4—21no—
381C4—21no—
382C4—42no—
383C4—41no—
384C4—41no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C2—21no—
391C2—21no—
392C2—82no—
393C2—161no—
394C2C31×#160011yescharacteristic
395C2C31×#219711yescharacteristic
396C2C31×#220411yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC31×#1598C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C4⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2572C2semidirect
2. C31×#2007C2semidirect
3. C31×#1616C2semidirect
5. C31×#1353C2semidirect
6. C31×#1345C2semidirect
7. C31×#1319C2semidirect
115. #12195C31direct (N × H)
167. #2572C2×C31semidirect
168. #2007C2×C31semidirect
169. #1616C2×C31semidirect
171. #1353C2×C31semidirect
172. #1345C2×C31semidirect
173. #1319C2×C31semidirect
263. C31#12195direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25722
C31×#20072
C31×#16162
C31×#15982
C31×#13532
C31×#13452
C31×#13192
#1219531