Finite Group Databasebeta

← C31×#12778 #21469537

Subgroups — C31×#12778 #21469537

Order 7936.

860 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 138 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×#3549C211yesmaximal, characteristic
2C31×#2128C212yesmaximal, complemented
3C31×#2126C211yesmaximal, characteristic, complemented
4C31×#1910C212yesmaximal, complemented
5C31×#1884C211yesmaximal, characteristic, complemented
6C31×#568C2211yescharacteristic, complemented
7C31×H(3,𝔽4)—21no—
8C31×H(3,𝔽4)C412yescomplemented
9C31×#555—22no—
10C31×#555C412yescomplemented
11D42×C31—22no—
12D42×C31—21no—
13C31×#535—21no—
14C31×#535—24no—
15C31×#535C2212yes—
16C31×#532—21no—
17C31×#532C2211yescharacteristic
18C31×#421—21no—
19C31×#418—22no—
20C31×#410C2212yescomplemented
21C31×#410C2211yescharacteristic, complemented
22C31×#354—21no—
23C31×#353—21no—
24C31×#353—24no—
25C31×#339—21no—
26C31×H(3,ℤ/4)—22no—
27C2×#41×C31—22no—
28C2×#41×C31D411yescharacteristic
29C2×#41×C31C2311yescharacteristic
30C22×D4×C31—24no—
31C22×D4×C31—21no—
32C22×D4×C31D412yes—
33C31×#131—42no—
34C2×M16×C31—22no—
35C2×M16×C31—21no—
36C31×C4⋊Q8—21no—
37C31×C4⋊Q8C2×C411yescharacteristic
38C31×#127—21no—
39C31×#127—22no—
40C31×#127D411yescharacteristic
41C31×#127D414yes—
42C31×#124—24no—
43C31×#124—22no—
44C31×#124—22no—
45C31×#124—24no—
46C31×#124C2×C411yescharacteristic
47C31×#124C2×C414yes—
48C31×#124D411yescharacteristic
49C31×#121—22no—
50C31×#121—22no—
51C31×#121—24no—
52C31×#121—24no—
53C31×#121—22no—
54C31×#121—22no—
55C31×#121—22no—
56C31×#120—22no—
57C31×#120—24no—
58C31×#120—22no—
59C31×#120—24no—
60C31×#120—22no—
61C31×#120—22no—
62C4×D4×C31—22no—
63C4×D4×C31—22no—
64C4×D4×C31—22no—
65C31×#117—21no—
66C31×#117—24no—
67C31×#117—22no—
68C31×#117—21no—
69C31×#117D412yes—
70C2×C42×C31D411yescharacteristic
71C31×#104—42no—
72C31×#102—44no—
73C31×#99—22no—
74C31×#99—22no—
75C31×#99—22no—
76C24×C31—22no—
77C24×C31—22no—
78#41×C31—41no—
79#41×C31—41no—
80#41×C31—41no—
81#41×C31—41no—
82#41×C31—84no—
83C2×Q8×C31—22no—
84C2×Q8×C31C22⋊C411yescharacteristic
85C2×Q8×C31C22×C411yescharacteristic
86C2×D4×C31—22no—
87C2×D4×C31—22no—
88C2×D4×C31—22no—
89C2×D4×C31—22no—
90C2×D4×C31—24no—
91C2×D4×C31—24no—
92C2×D4×C31—21no—
93C2×D4×C31—24no—
94C2×D4×C31—22no—
95C2×D4×C31—42no—
96C2×D4×C31—42no—
97C2×D4×C31—88no—
98C2×D4×C31—81no—
99C2×D4×C31—81no—
100C2×D4×C31C22⋊C411yescharacteristic
101C2×D4×C31C22⋊C414yes—
102C2×D4×C31C2×D411yescharacteristic
103C2×D4×C31C2×D412yes—
104C22×C4×C31—24no—
105C22×C4×C31—21no—
106C22×C4×C31—21no—
107C22×C4×C31C2×D411yescharacteristic
108C22×C4×C31C2×D412yes—
109M16×C31—22no—
110M16×C31—42no—
111M16×C31—41no—
112C2×C8×C31—24no—
113C2×C8×C31—41no—
114C4⋊C4×C31—22no—
115C4⋊C4×C31—22no—
116C4⋊C4×C31—22no—
117C4⋊C4×C31—42no—
118C4⋊C4×C31—41no—
119C22⋊C4×C31—24no—
120C22⋊C4×C31—22no—
121C22⋊C4×C31—24no—
122C22⋊C4×C31—22no—
123C22⋊C4×C31—22no—
124C22⋊C4×C31—24no—
125C22⋊C4×C31—22no—
126C22⋊C4×C31—24no—
127C22⋊C4×C31—42no—
128C22⋊C4×C31—41no—
129C42×C31—22no—
130C42×C31—21no—
131C42×C31—41no—
132C42×C31—42no—
133C42×C31C22⋊C411yescharacteristic
134C42×C31C22⋊C411yescharacteristic
135C42×C31C22⋊C414yes—
136#12778C3111yesmaximal, characteristic, direct product factor, Sylow
137C23×C31—24no—
138C23×C31—21no—
139C23×C31—82no—
140C23×C31—84no—
141C23×C31—84no—
142C23×C31—82no—
143C23×C31#12012yes—
144C23×C31#12011yescharacteristic
145Q8×C31—41no—
146Q8×C31—41no—
147Q8×C31—82no—
148D4×C31—42no—
149D4×C31—41no—
150D4×C31—44no—
151D4×C31—41no—
152D4×C31—44no—
153D4×C31—44no—
154D4×C31—42no—
155D4×C31—42no—
156D4×C31—81no—
157D4×C31—84no—
158D4×C31—84no—
159D4×C31—82no—
160D4×C31—84no—
161D4×C31—82no—
162C2×C4×C31—21no—
163C2×C4×C31—21no—
164C2×C4×C31—22no—
165C2×C4×C31—22no—
166C2×C4×C31—21no—
167C2×C4×C31—21no—
168C2×C4×C31—21no—
169C2×C4×C31—24no—
170C2×C4×C31—48no—
171C2×C4×C31—42no—
172C2×C4×C31—41no—
173C2×C4×C31—41no—
174C2×C4×C31—81no—
175C2×C4×C31—82no—
176C2×C4×C31C2×C22⋊C411yescharacteristic
177C2×C4×C31C2×C22⋊C412yes—
178C2×C4×C31#12011yescharacteristic
179C8×C31—42no—
180#3549C2×C3111yescharacteristic
181#2128C2×C3112yescomplemented
182#2126C2×C3111yescharacteristic, complemented
183#1910C2×C3112yescomplemented
184#1884C2×C3111yescharacteristic, complemented
185C22×C31—21no—
186C22×C31—22no—
187C22×C31—41no—
188C22×C31—42no—
189C22×C31—44no—
190C22×C31—44no—
191C22×C31—42no—
192C22×C31—41no—
193C22×C31—44no—
194C22×C31—164no—
195C22×C31—164no—
196C22×C31#37911yescharacteristic
197C4×C31—22no—
198C4×C31—22no—
199C4×C31—21no—
200C4×C31—44no—
201C4×C31—41no—
202C4×C31—41no—
203C4×C31—161no—
204C4×C31—162no—
205C4×C31B(4,𝔽2)11yescharacteristic
206C4×C31#45811yescharacteristic
207#568C22×C3111yescharacteristic, complemented
208H(3,𝔽4)—21no—
209H(3,𝔽4)C4×C3112yescomplemented
210#555—22no—
211#555C4×C3112yescomplemented
212D42—22no—
213D42—21no—
214#535—21no—
215#535—24no—
216#535C22×C3112yes—
217#532—21no—
218#532C22×C3111yescharacteristic
219#421—21no—
220#418—22no—
221#410C22×C3112yescomplemented
222#410C22×C3111yescharacteristic, complemented
223#354—21no—
224#353—21no—
225#353—24no—
226#339—21no—
227H(3,ℤ/4)—22no—
228C2×C31—21no—
229C2×C31—41no—
230C2×C31—42no—
231C2×C31—84no—
232C2×C31—81no—
233C2×C31#179011yescharacteristic
234C2×#41—22no—
235C2×#41D4×C3111yescharacteristic
236C2×#41C23×C3111yescharacteristic
237C22×D4—24no—
238C22×D4—21no—
239C22×D4D4×C3112yes—
240#131—42no—
241C2×M16—22no—
242C2×M16—21no—
243C4⋊Q8—21no—
244C4⋊Q8C2×C4×C3111yescharacteristic
245#127—21no—
246#127—22no—
247#127D4×C3111yescharacteristic
248#127D4×C3114yes—
249#124—24no—
250#124—22no—
251#124—22no—
252#124—24no—
253#124C2×C4×C3111yescharacteristic
254#124C2×C4×C3114yes—
255#124D4×C3111yescharacteristic
256#121—22no—
257#121—22no—
258#121—24no—
259#121—24no—
260#121—22no—
261#121—22no—
262#121—22no—
263#120—22no—
264#120—24no—
265#120—22no—
266#120—24no—
267#120—22no—
268#120—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272#117—21no—
273#117—24no—
274#117—22no—
275#117—21no—
276#117D4×C3112yes—
277C2×C42D4×C3111yescharacteristic
278#104—42no—
279#102—44no—
280#99—22no—
281#99—22no—
282#99—22no—
283C31#1277811yescharacteristic, direct product factor, Sylow
284C24—22no—
285C24—22no—
286#41—41no—
287#41—41no—
288#41—41no—
289#41—41no—
290#41—84no—
291C2×Q8—22no—
292C2×Q8C22⋊C4×C3111yescharacteristic
293C2×Q8C22×C4×C3111yescharacteristic
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—21no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—88no—
306C2×D4—81no—
307C2×D4—81no—
308C2×D4C22⋊C4×C3111yescharacteristic
309C2×D4C22⋊C4×C3114yes—
310C2×D4C2×D4×C3111yescharacteristic
311C2×D4C2×D4×C3112yes—
312C22×C4—24no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4C2×D4×C3111yescharacteristic
316C22×C4C2×D4×C3112yes—
317M16—22no—
318M16—42no—
319M16—41no—
320C2×C8—24no—
321C2×C8—41no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C22⋊C4—24no—
328C22⋊C4—22no—
329C22⋊C4—24no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—24no—
333C22⋊C4—22no—
334C22⋊C4—24no—
335C22⋊C4—42no—
336C22⋊C4—41no—
337C42—22no—
338C42—21no—
339C42—41no—
340C42—42no—
341C42C22⋊C4×C3111yescharacteristic
342C42C22⋊C4×C3111yescharacteristic
343C42C22⋊C4×C3114yes—
344C23—24no—
345C23—21no—
346C23—82no—
347C23—84no—
348C23—84no—
349C23—82no—
350C23C31×#12012yes—
351C23C31×#12011yescharacteristic
352Q8—41no—
353Q8—41no—
354Q8—82no—
355D4—42no—
356D4—41no—
357D4—44no—
358D4—41no—
359D4—44no—
360D4—44no—
361D4—42no—
362D4—42no—
363D4—81no—
364D4—84no—
365D4—84no—
366D4—82no—
367D4—84no—
368D4—82no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—24no—
377C2×C4—48no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4C2×C22⋊C4×C3111yescharacteristic
384C2×C4C2×C22⋊C4×C3112yes—
385C2×C4C31×#12011yescharacteristic
386C8—42no—
387C22—21no—
388C22—22no—
389C22—41no—
390C22—42no—
391C22—44no—
392C22—44no—
393C22—42no—
394C22—41no—
395C22—44no—
396C22—164no—
397C22—164no—
398C22C31×#37911yescharacteristic
399C4—22no—
400C4—22no—
401C4—21no—
402C4—44no—
403C4—41no—
404C4—41no—
405C4—162no—
406C4—161no—
407C4C31×B(4,𝔽2)11yescharacteristic
408C4C31×#45811yescharacteristic
409C2—21no—
410C2—42no—
411C2—41no—
412C2—84no—
413C2—81no—
414C2C31×#179011yescharacteristic
subgroupG/H
center, socleC2×C31#1790
commutatorC2×Q8C22×C4×C31
Frattini, cosocleC2×#41C23×C31
prime radicalC31×#3549C2
LCSG ▷ C2×Q8 ▷ 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
2. C31×#2128C2semidirect
3. C31×#2126C2semidirect
4. C31×#1910C2semidirect
5. C31×#1884C2semidirect
6. C31×#568C22semidirect
8. C31×H(3,𝔽4)C4semidirect
10. C31×#555C4semidirect
20. C31×#410C22semidirect
21. C31×#410C22semidirect
136. #12778C31direct (N × H)
181. #2128C2×C31semidirect
182. #2126C2×C31semidirect
183. #1910C2×C31semidirect
184. #1884C2×C31semidirect
207. #568C22×C31semidirect
209. H(3,𝔽4)C4×C31semidirect
211. #555C4×C31semidirect
221. #410C22×C31semidirect
222. #410C22×C31semidirect
283. C31#12778direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35492
C31×#21282
C31×#21262
C31×#19102
C31×#18842
#1277831