Finite Group Databasebeta

← C31×#11758 #21468517

Subgroups — C31×#11758 #21468517

Order 7936.

486 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2610C211yesmaximal, characteristic
2C31×#1740C211yesmaximal, characteristic, complemented
3C31×#1464C211yesmaximal, characteristic, complemented
4C31×#1351C212yesmaximal, complemented
5C31×#1343C212yesmaximal
6C2×C31×#124C411yescharacteristic, complemented
7C2×C31×#117—21no—
8C2×C31×#117C2211yescharacteristic
9C31×#432—21no—
10C31×#395C411yescharacteristic, complemented
11C31×#394C411yescharacteristic, complemented
12C31×#393C2211yescharacteristic
13C31×#392C2211yescharacteristic
14C31×#390—21no—
15C31×#388—21no—
16C31×#388—21no—
17C31×#386—21no—
18C31×#376C411yescharacteristic, complemented
19C31×#342—21no—
20C31×#342C2212yes—
21C31×#324—21no—
22C31×#323C2212yes—
23C22×Q8×C31C2×C411yescharacteristic
24C22×D4×C31—21no—
25C22×D4×C31C2×C411yescharacteristic
26C23×C4×C31D411yescharacteristic
27C31×#124—41no—
28C31×#124—41no—
29C31×#117—24no—
30C31×#117—41no—
31C31×#117—41no—
32C31×#117—42no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31—41no—
35C2×C4⋊C4×C31C2×C411yescharacteristic
36C2×C4⋊C4×C31D411yescharacteristic
37C2×C22⋊C4×C31—22no—
38C2×C22⋊C4×C31—21no—
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31D411yescharacteristic
45C2×C22⋊C4×C31C2311yescharacteristic
46C2×C42×C31—21no—
47C2×C42×C31—41no—
48C2×C42×C31C2×C411yescharacteristic
49C2×C42×C31D411yescharacteristic
50C31×C4⋊C8—41no—
51C31×C22⋊C8—22no—
52C31×#97—41no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95C2×C411yescharacteristic
57C31×#95C2×C411yescharacteristic
58C24×C31—21no—
59C24×C31C2×D411yescharacteristic
60C2×Q8×C31—41no—
61C2×Q8×C31—81no—
62C2×D4×C31—42no—
63C2×D4×C31—41no—
64C2×D4×C31—41no—
65C2×D4×C31—41no—
66C2×D4×C31—81no—
67C2×D4×C31—81no—
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31—22no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—42no—
77C22×C4×C31C22⋊C411yescharacteristic
78C22×C4×C31C22⋊C411yescharacteristic
79C22×C4×C31C22⋊C411yescharacteristic
80C22×C4×C31C22⋊C411yescharacteristic
81C22×C4×C31C22×C411yescharacteristic
82C22×C4×C31C2×D411yescharacteristic
83C2×C8×C31—42no—
84C4⋊C4×C31—41no—
85C4⋊C4×C31—41no—
86C4⋊C4×C31—44no—
87C4⋊C4×C31—81no—
88C22⋊C4×C31—24no—
89C22⋊C4×C31—41no—
90C22⋊C4×C31—41no—
91C22⋊C4×C31—42no—
92C22⋊C4×C31—42no—
93C22⋊C4×C31—42no—
94C22⋊C4×C31—41no—
95C22⋊C4×C31—41no—
96C22⋊C4×C31—42no—
97C22⋊C4×C31—81no—
98C22⋊C4×C31—81no—
99C42×C31—21no—
100C42×C31—21no—
101C42×C31—21no—
102C42×C31—21no—
103C42×C31—42no—
104C42×C31—44no—
105#11758C3111yesmaximal, characteristic, direct product factor, Sylow
106C23×C31—21no—
107C23×C31—21no—
108C23×C31—41no—
109C23×C31—41no—
110C23×C31—41no—
111C23×C31—41no—
112C23×C31—41no—
113C23×C31—41no—
114C23×C31—41no—
115C23×C31—41no—
116C23×C31—41no—
117C23×C31C2×C22⋊C411yescharacteristic
118C23×C31AGL(1,ℤ/8)11yescharacteristic
119C23×C31#13711yescharacteristic
120Q8×C31—82no—
121D4×C31—82no—
122D4×C31—82no—
123D4×C31—82no—
124C2×C4×C31—22no—
125C2×C4×C31—22no—
126C2×C4×C31—22no—
127C2×C4×C31—22no—
128C2×C4×C31—21no—
129C2×C4×C31—21no—
130C2×C4×C31—21no—
131C2×C4×C31—21no—
132C2×C4×C31—22no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—41no—
138C2×C4×C31—41no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
141C2×C4×C31—41no—
142C2×C4×C31—41no—
143C2×C4×C31—44no—
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—82no—
151C2×C4×C31—82no—
152C2×C4×C31#9911yescharacteristic
153C2×C4×C31#9911yescharacteristic
154C2×C4×C31#10412yes—
155C8×C31—82no—
156#2610C2×C3111yescharacteristic
157#1740C2×C3111yescharacteristic, complemented
158#1464C2×C3111yescharacteristic, complemented
159#1351C2×C3112yescomplemented
160#1343C2×C3112yes—
161C22×C31—21no—
162C22×C31—21no—
163C22×C31—21no—
164C22×C31—21no—
165C22×C31—41no—
166C22×C31—41no—
167C22×C31—41no—
168C22×C31—41no—
169C22×C31—81no—
170C22×C31—81no—
171C22×C31—81no—
172C22×C31—81no—
173C22×C31—81no—
174C22×C31—81no—
175C22×C31—81no—
176C22×C31—81no—
177C22×C31C2×#9911yescharacteristic
178C22×C31#41711yescharacteristic
179C22×C31C2×#10411yescharacteristic
180C4×C31—21no—
181C4×C31—21no—
182C4×C31—22no—
183C4×C31—42no—
184C4×C31—41no—
185C4×C31—41no—
186C4×C31—41no—
187C4×C31—41no—
188C4×C31—81no—
189C4×C31—84no—
190C4×C31—81no—
191C2×#124C4×C3111yescharacteristic, complemented
192C2×#117—21no—
193C2×#117C22×C3111yescharacteristic
194#432—21no—
195#395C4×C3111yescharacteristic, complemented
196#394C4×C3111yescharacteristic, complemented
197#393C22×C3111yescharacteristic
198#392C22×C3111yescharacteristic
199#390—21no—
200#388—21no—
201#388—21no—
202#386—21no—
203#376C4×C3111yescharacteristic, complemented
204#342—21no—
205#342C22×C3112yes—
206#324—21no—
207#323C22×C3112yes—
208C2×C31—21no—
209C2×C31—21no—
210C2×C31—41no—
211C2×C31—41no—
212C2×C31—161no—
213C2×C31#150411yescharacteristic
214C2×C31#212211yescharacteristic
215C2×C31#212611yescharacteristic
216C22×Q8C2×C4×C3111yescharacteristic
217C22×D4—21no—
218C22×D4C2×C4×C3111yescharacteristic
219C23×C4D4×C3111yescharacteristic
220#124—41no—
221#124—41no—
222#117—24no—
223#117—41no—
224#117—41no—
225#117—42no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C2×C4×C3111yescharacteristic
229C2×C4⋊C4D4×C3111yescharacteristic
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4D4×C3111yescharacteristic
238C2×C22⋊C4C23×C3111yescharacteristic
239C2×C42—21no—
240C2×C42—41no—
241C2×C42C2×C4×C3111yescharacteristic
242C2×C42D4×C3111yescharacteristic
243C4⋊C8—41no—
244C22⋊C8—22no—
245#97—41no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95C2×C4×C3111yescharacteristic
250#95C2×C4×C3111yescharacteristic
251C31#1175811yescharacteristic, direct product factor, Sylow
252C24—21no—
253C24C2×D4×C3111yescharacteristic
254C2×Q8—41no—
255C2×Q8—81no—
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—81no—
261C2×D4—81no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—42no—
271C22×C4C22⋊C4×C3111yescharacteristic
272C22×C4C22⋊C4×C3111yescharacteristic
273C22×C4C22⋊C4×C3111yescharacteristic
274C22×C4C22⋊C4×C3111yescharacteristic
275C22×C4C22×C4×C3111yescharacteristic
276C22×C4C2×D4×C3111yescharacteristic
277C2×C8—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—44no—
281C4⋊C4—81no—
282C22⋊C4—24no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—42no—
298C42—44no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23C2×C22⋊C4×C3111yescharacteristic
311C23C31×AGL(1,ℤ/8)11yescharacteristic
312C23C31×#13711yescharacteristic
313Q8—82no—
314D4—82no—
315D4—82no—
316D4—82no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—44no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4C31×#9911yescharacteristic
346C2×C4C31×#9911yescharacteristic
347C2×C4C31×#10412yes—
348C8—82no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22C2×C31×#9911yescharacteristic
366C22C31×#41711yescharacteristic
367C22C2×C31×#10411yescharacteristic
368C4—21no—
369C4—21no—
370C4—22no—
371C4—42no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—81no—
377C4—84no—
378C4—81no—
379C2—21no—
380C2—21no—
381C2—41no—
382C2—41no—
383C2—161no—
384C2C31×#150411yescharacteristic
385C2C31×#212211yescharacteristic
386C2C31×#212611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC31×#392C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C31×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1740C2semidirect
3. C31×#1464C2semidirect
4. C31×#1351C2semidirect
6. C2×C31×#124C4semidirect
10. C31×#395C4semidirect
11. C31×#394C4semidirect
18. C31×#376C4semidirect
105. #11758C31direct (N × H)
157. #1740C2×C31semidirect
158. #1464C2×C31semidirect
159. #1351C2×C31semidirect
191. C2×#124C4×C31semidirect
195. #395C4×C31semidirect
196. #394C4×C31semidirect
203. #376C4×C31semidirect
251. C31#11758direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26102
C31×#17402
C31×#14642
C31×#13512
C31×#13432
#1175831