Finite Group Databasebeta

← C31×#12727 #21469486

Subgroups — C31×#12727 #21469486

Order 7936.

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

normal subgroups only

This group has 810 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×#3508C211yesmaximal, characteristic
2C31×#2127C212yesmaximal, complemented
3C2×C31×#353C212yesmaximal, complemented
4C31×#1828C212yesmaximal, complemented
5C31×#559—22no—
6C31×#550—22no—
7C31×#546—24no—
8C31×#540—22no—
9C31×#535C411yescharacteristic, complemented
10C31×#535C2211yescharacteristic
11C31×#533C411yescharacteristic, complemented
12C31×#532—22no—
13C2×C31×#127C2211yescharacteristic
14C31×#518C411yescharacteristic, complemented
15C31×#518C2211yescharacteristic
16C4×#41×C31C411yescharacteristic, complemented
17C2×C31×#99C2214yes—
18C31×#354—22no—
19C31×#353—22no—
20C31×#353—22no—
21C31×#353—22no—
22C31×#342—24no—
23C2×#41×C31—24no—
24C2×#41×C31C2×C411yescharacteristic
25C22×D4×C31—22no—
26C22×D4×C31C2×C411yescharacteristic
27C22×D4×C31C2311yescharacteristic
28C31×C4⋊Q8—21no—
29C31×#127—21no—
30C31×#127—22no—
31C31×#127D412yes—
32C31×#127D412yes—
33C31×#127D412yes—
34C31×#125—21no—
35C31×#124—24no—
36C31×#124—21no—
37C31×#124D412yes—
38C31×#123—24no—
39C31×#123—24no—
40C31×C22⋊Q8—22no—
41C31×#121—24no—
42C31×#121—24no—
43C31×#121—24no—
44C31×#121—22no—
45C31×#121—22no—
46C31×#120—22no—
47C31×#120—22no—
48C4×Q8×C31—21no—
49C4×Q8×C31—22no—
50C4×D4×C31—24no—
51C4×D4×C31—21no—
52C4×D4×C31—22no—
53C4×D4×C31—22no—
54C4×D4×C31—22no—
55C4×D4×C31—22no—
56C4×D4×C31—22no—
57C31×#117—21no—
58C31×#117C2×C411yescharacteristic
59C31×#117C2×C411yescharacteristic
60C31×#117C2×C411yescharacteristic
61C2×C22⋊C4×C31—22no—
62C2×C22⋊C4×C31—24no—
63C2×C22⋊C4×C31D414yes—
64C2×C42×C31—21no—
65C2×C42×C31C2×C411yescharacteristic
66C31×#99—24no—
67C31×#99—24no—
68C24×C31—21no—
69C24×C31C2×D412yes—
70#41×C31—42no—
71#41×C31—88no—
72C2×Q8×C31—22no—
73C2×Q8×C31C22⋊C411yescharacteristic
74C2×D4×C31—21no—
75C2×D4×C31—24no—
76C2×D4×C31—24no—
77C2×D4×C31—24no—
78C2×D4×C31—24no—
79C2×D4×C31—21no—
80C2×D4×C31—42no—
81C2×D4×C31—42no—
82C2×D4×C31—84no—
83C2×D4×C31C22⋊C412yes—
84C2×D4×C31C22⋊C411yescharacteristic
85C2×D4×C31C2×D412yes—
86C2×D4×C31C2×D412yes—
87C22×C4×C31—22no—
88C22×C4×C31—21no—
89C22×C4×C31—24no—
90C22×C4×C31—24no—
91C22×C4×C31—21no—
92C22×C4×C31—44no—
93C22×C4×C31C22×C411yescharacteristic
94C4⋊C4×C31—22no—
95C4⋊C4×C31—22no—
96C4⋊C4×C31—22no—
97C4⋊C4×C31—22no—
98C4⋊C4×C31—21no—
99C4⋊C4×C31—21no—
100C4⋊C4×C31—21no—
101C22⋊C4×C31—24no—
102C22⋊C4×C31—22no—
103C22⋊C4×C31—24no—
104C22⋊C4×C31—21no—
105C22⋊C4×C31—24no—
106C22⋊C4×C31—24no—
107C22⋊C4×C31—44no—
108C22⋊C4×C31—44no—
109C22⋊C4×C31C22⋊C412yes—
110C22⋊C4×C31C22⋊C412yes—
111C42×C31—22no—
112C42×C31—21no—
113C42×C31—21no—
114C42×C31—21no—
115C42×C31C22⋊C411yescharacteristic
116C42×C31C22⋊C411yescharacteristic
117C42×C31C22⋊C412yes—
118#12727C3111yesmaximal, characteristic, direct product factor, Sylow
119C23×C31—24no—
120C23×C31—22no—
121C23×C31—21no—
122C23×C31—42no—
123C23×C31—82no—
124C23×C31—82no—
125C23×C31C2×C22⋊C411yescharacteristic
126C23×C31#12014yes—
127Q8×C31—41no—
128Q8×C31—44no—
129D4×C31—42no—
130D4×C31—42no—
131D4×C31—42no—
132D4×C31—41no—
133D4×C31—42no—
134D4×C31—48no—
135D4×C31—42no—
136D4×C31—84no—
137D4×C31—84no—
138D4×C31—84no—
139C2×C4×C31—21no—
140C2×C4×C31—22no—
141C2×C4×C31—21no—
142C2×C4×C31—22no—
143C2×C4×C31—21no—
144C2×C4×C31—22no—
145C2×C4×C31—22no—
146C2×C4×C31—42no—
147C2×C4×C31—42no—
148C2×C4×C31—44no—
149C2×C4×C31—84no—
150C2×C4×C31—84no—
151C2×C4×C31—84no—
152C2×C4×C31—164no—
153C2×C4×C31C2×C22⋊C411yescharacteristic
154C2×C4×C31C2×C22⋊C411yescharacteristic
155#3508C2×C3111yescharacteristic
156#2127C2×C3112yescomplemented
157C2×#353C2×C3112yescomplemented
158#1828C2×C3112yescomplemented
159C22×C31—24no—
160C22×C31—41no—
161C22×C31—44no—
162C22×C31—42no—
163C22×C31—44no—
164C22×C31—41no—
165C22×C31—42no—
166C22×C31—84no—
167C22×C31—162no—
168C22×C31#37911yescharacteristic
169C22×C31#41012yes—
170C4×C31—22no—
171C4×C31—21no—
172C4×C31—21no—
173C4×C31—42no—
174C4×C31—41no—
175C4×C31—42no—
176C4×C31—42no—
177C4×C31—41no—
178C4×C31—164no—
179#559—22no—
180#550—22no—
181#546—24no—
182#540—22no—
183#535C4×C3111yescharacteristic, complemented
184#535C22×C3111yescharacteristic
185#533C4×C3111yescharacteristic, complemented
186#532—22no—
187C2×#127C22×C3111yescharacteristic
188#518C4×C3111yescharacteristic, complemented
189#518C22×C3111yescharacteristic
190C4×#41C4×C3111yescharacteristic, complemented
191C2×#99C22×C3114yes—
192#354—22no—
193#353—22no—
194#353—22no—
195#353—22no—
196#342—24no—
197C2×C31—21no—
198C2×C31—22no—
199C2×C31—44no—
200C2×C31—81no—
201C2×C31—82no—
202C2×C31#178211yescharacteristic
203C2×#41—24no—
204C2×#41C2×C4×C3111yescharacteristic
205C22×D4—22no—
206C22×D4C2×C4×C3111yescharacteristic
207C22×D4C23×C3111yescharacteristic
208C4⋊Q8—21no—
209#127—21no—
210#127—22no—
211#127D4×C3112yes—
212#127D4×C3112yes—
213#127D4×C3112yes—
214#125—21no—
215#124—24no—
216#124—21no—
217#124D4×C3112yes—
218#123—24no—
219#123—24no—
220C22⋊Q8—22no—
221#121—24no—
222#121—24no—
223#121—24no—
224#121—22no—
225#121—22no—
226#120—22no—
227#120—22no—
228C4×Q8—21no—
229C4×Q8—22no—
230C4×D4—24no—
231C4×D4—21no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237#117—21no—
238#117C2×C4×C3111yescharacteristic
239#117C2×C4×C3111yescharacteristic
240#117C2×C4×C3111yescharacteristic
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4D4×C3114yes—
244C2×C42—21no—
245C2×C42C2×C4×C3111yescharacteristic
246#99—24no—
247#99—24no—
248C31#1272711yescharacteristic, direct product factor, Sylow
249C24—21no—
250C24C2×D4×C3112yes—
251#41—42no—
252#41—88no—
253C2×Q8—22no—
254C2×Q8C22⋊C4×C3111yescharacteristic
255C2×D4—21no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—21no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—84no—
264C2×D4C22⋊C4×C3112yes—
265C2×D4C22⋊C4×C3111yescharacteristic
266C2×D4C2×D4×C3112yes—
267C2×D4C2×D4×C3112yes—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—21no—
273C22×C4—44no—
274C22×C4C22×C4×C3111yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C22⋊C4—24no—
283C22⋊C4—22no—
284C22⋊C4—24no—
285C22⋊C4—21no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4C22⋊C4×C3112yes—
291C22⋊C4C22⋊C4×C3112yes—
292C42—22no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42C22⋊C4×C3111yescharacteristic
297C42C22⋊C4×C3111yescharacteristic
298C42C22⋊C4×C3112yes—
299C23—24no—
300C23—22no—
301C23—21no—
302C23—42no—
303C23—82no—
304C23—82no—
305C23C2×C22⋊C4×C3111yescharacteristic
306C23C31×#12014yes—
307Q8—41no—
308Q8—44no—
309D4—42no—
310D4—42no—
311D4—42no—
312D4—41no—
313D4—42no—
314D4—48no—
315D4—42no—
316D4—84no—
317D4—84no—
318D4—84no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4—84no—
332C2×C4—164no—
333C2×C4C2×C22⋊C4×C3111yescharacteristic
334C2×C4C2×C22⋊C4×C3111yescharacteristic
335C22—24no—
336C22—41no—
337C22—44no—
338C22—42no—
339C22—44no—
340C22—41no—
341C22—42no—
342C22—84no—
343C22—162no—
344C22C31×#37911yescharacteristic
345C22C31×#41012yes—
346C4—22no—
347C4—21no—
348C4—21no—
349C4—42no—
350C4—41no—
351C4—42no—
352C4—42no—
353C4—41no—
354C4—164no—
355C2—21no—
356C2—22no—
357C2—44no—
358C2—82no—
359C2—81no—
360C2C31×#178211yescharacteristic
subgroupG/H
center, socleC2×C31#1782
commutatorC22×C4C22×C4×C31
Frattini, cosocleC22×D4C23×C31
prime radicalC31×#3508C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C23×C31 ◁ C31×#518 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2127C2semidirect
3. C2×C31×#353C2semidirect
4. C31×#1828C2semidirect
9. C31×#535C4semidirect
11. C31×#533C4semidirect
14. C31×#518C4semidirect
16. C4×#41×C31C4semidirect
118. #12727C31direct (N × H)
156. #2127C2×C31semidirect
157. C2×#353C2×C31semidirect
158. #1828C2×C31semidirect
183. #535C4×C31semidirect
185. #533C4×C31semidirect
188. #518C4×C31semidirect
190. C4×#41C4×C31semidirect
248. C31#12727direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35082
C31×#21272
C2×C31×#3532
C31×#18282
#1272731