Finite Group Databasebeta

← C31×#11898 #21468657

Subgroups — C31×#11898 #21468657

Order 7936.

580 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2409C211yesmaximal, characteristic
2C31×#1799C211yesmaximal, characteristic
3C31×#1460C211yesmaximal, characteristic
4C31×#1351C214yesmaximal, complemented
5C23×Q8×C31C411yescharacteristic, complemented
6C2×C31×C22⋊Q8—21no—
7C2×C4×D4×C31C2211yescharacteristic
8C31×#431—21no—
9C2×C31×#99—21no—
10C31×#395C412yescomplemented
11C31×#393C2212yes—
12C31×#391—21no—
13C31×#386—21no—
14C31×#385—21no—
15C31×#380—21no—
16C31×#380C411yescharacteristic, complemented
17C31×#342C2212yes—
18C31×#323—21no—
19C31×#323C2212yes—
20C22×Q8×C31—21no—
21C22×Q8×C31—41no—
22C22×Q8×C31—41no—
23C22×Q8×C31—44no—
24C22×Q8×C31C2×C412yes—
25C22×D4×C31D411yescharacteristic
26C23×C4×C31—21no—
27C23×C4×C31C2×C411yescharacteristic
28C23×C4×C31D411yescharacteristic
29C31×C22⋊Q8—44no—
30C4×D4×C31—41no—
31C4×D4×C31—41no—
32C2×C4⋊C4×C31—22no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31—41no—
35C2×C4⋊C4×C31C2×C411yescharacteristic
36C2×C22⋊C4×C31—21no—
37C2×C22⋊C4×C31—22no—
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31D411yescharacteristic
40C2×C22⋊C4×C31C2311yescharacteristic
41C2×C42×C31D411yescharacteristic
42C31×C4⋊C8—41no—
43C31×#99—42no—
44C31×C22⋊C8—22no—
45C4×C8×C31—41no—
46C31×#95—22no—
47C31×#95—22no—
48C31×#95—41no—
49C31×#95C2×C412yes—
50C24×C31C22⋊C411yescharacteristic
51C24×C31C2×D411yescharacteristic
52C2×Q8×C31—24no—
53C2×Q8×C31—42no—
54C2×Q8×C31—44no—
55C2×Q8×C31—42no—
56C2×Q8×C31—48no—
57C2×Q8×C31—44no—
58C2×Q8×C31—44no—
59C2×Q8×C31—41no—
60C2×Q8×C31—41no—
61C2×D4×C31—41no—
62C2×D4×C31—41no—
63C2×D4×C31SD1612yes—
64C2×D4×C31Q1612yes—
65C22×C4×C31—22no—
66C22×C4×C31—21no—
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31—22no—
71C22×C4×C31—44no—
72C22×C4×C31—42no—
73C22×C4×C31—42no—
74C22×C4×C31—42no—
75C22×C4×C31—42no—
76C22×C4×C31C22⋊C411yescharacteristic
77C22×C4×C31C22⋊C411yescharacteristic
78C22×C4×C31C22⋊C411yescharacteristic
79C22×C4×C31C22×C411yescharacteristic
80C22×C4×C31C2×D411yescharacteristic
81C2×C8×C31—42no—
82C4⋊C4×C31—41no—
83C4⋊C4×C31—42no—
84C4⋊C4×C31—44no—
85C4⋊C4×C31—82no—
86C22⋊C4×C31—41no—
87C22⋊C4×C31—41no—
88C22⋊C4×C31—44no—
89C22⋊C4×C31—42no—
90C22⋊C4×C31—42no—
91C22⋊C4×C31—42no—
92C42×C31—21no—
93C42×C31—21no—
94#11898C3111yesmaximal, characteristic, direct product factor, Sylow
95C23×C31—21no—
96C23×C31—21no—
97C23×C31—21no—
98C23×C31—21no—
99C23×C31—41no—
100C23×C31—41no—
101C23×C31—41no—
102C23×C31—41no—
103C23×C31#10312yes—
104C23×C31C2×C22⋊C411yescharacteristic
105C23×C31C2×SD1611yescharacteristic
106C23×C31C2×Q1611yescharacteristic
107Q8×C31—44no—
108Q8×C31—44no—
109Q8×C31—44no—
110Q8×C31—44no—
111D4×C31—42no—
112D4×C31—42no—
113C2×C4×C31—21no—
114C2×C4×C31—21no—
115C2×C4×C31—22no—
116C2×C4×C31—24no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
120C2×C4×C31—42no—
121C2×C4×C31—42no—
122C2×C4×C31—48no—
123C2×C4×C31—42no—
124C2×C4×C31—42no—
125C2×C4×C31—42no—
126C2×C4×C31—41no—
127C2×C4×C31—41no—
128C2×C4×C31—41no—
129C2×C4×C31—42no—
130C2×C4×C31—42no—
131C2×C4×C31—42no—
132C2×C4×C31—81no—
133C2×C4×C31—82no—
134C2×C4×C31—82no—
135C2×C4×C31—82no—
136C2×C4×C31#9911yescharacteristic
137C2×C4×C31#9911yescharacteristic
138C2×C4×C31#10312yes—
139C8×C31—82no—
140#2409C2×C3111yescharacteristic
141#1799C2×C3111yescharacteristic
142#1460C2×C3111yescharacteristic
143#1351C2×C3114yescomplemented
144C22×C31—21no—
145C22×C31—21no—
146C22×C31—21no—
147C22×C31—22no—
148C22×C31—21no—
149C22×C31—42no—
150C22×C31—41no—
151C22×C31—41no—
152C22×C31—42no—
153C22×C31—41no—
154C22×C31—41no—
155C22×C31—41no—
156C22×C31—41no—
157C22×C31—81no—
158C22×C31C2×#9911yescharacteristic
159C22×C31C2×#10311yescharacteristic
160C22×C31#42111yescharacteristic
161C4×C31—21no—
162C4×C31—22no—
163C4×C31—44no—
164C4×C31—41no—
165C4×C31—41no—
166C4×C31—41no—
167C4×C31—82no—
168C4×C31—162no—
169C4×C31AΣL(1,𝔽16)12yes—
170C23×Q8C4×C3111yescharacteristic, complemented
171C2×C22⋊Q8—21no—
172C2×C4×D4C22×C3111yescharacteristic
173#431—21no—
174C2×#99—21no—
175#395C4×C3112yescomplemented
176#393C22×C3112yes—
177#391—21no—
178#386—21no—
179#385—21no—
180#380—21no—
181#380C4×C3111yescharacteristic, complemented
182#342C22×C3112yes—
183#323—21no—
184#323C22×C3112yes—
185C2×C31—21no—
186C2×C31—21no—
187C2×C31—41no—
188C2×C31—42no—
189C2×C31—41no—
190C2×C31#152011yescharacteristic
191C2×C31C2×AΣL(1,𝔽16)11yescharacteristic
192C2×C31#212911yescharacteristic
193C22×Q8—21no—
194C22×Q8—41no—
195C22×Q8—41no—
196C22×Q8—44no—
197C22×Q8C2×C4×C3112yes—
198C22×D4D4×C3111yescharacteristic
199C23×C4—21no—
200C23×C4C2×C4×C3111yescharacteristic
201C23×C4D4×C3111yescharacteristic
202C22⋊Q8—44no—
203C4×D4—41no—
204C4×D4—41no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4C2×C4×C3111yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4D4×C3111yescharacteristic
213C2×C22⋊C4C23×C3111yescharacteristic
214C2×C42D4×C3111yescharacteristic
215C4⋊C8—41no—
216#99—42no—
217C22⋊C8—22no—
218C4×C8—41no—
219#95—22no—
220#95—22no—
221#95—41no—
222#95C2×C4×C3112yes—
223C31#1189811yescharacteristic, direct product factor, Sylow
224C24C22⋊C4×C3111yescharacteristic
225C24C2×D4×C3111yescharacteristic
226C2×Q8—24no—
227C2×Q8—44no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—48no—
232C2×Q8—44no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4SD16×C3112yes—
238C2×D4Q16×C3112yes—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4C22⋊C4×C3111yescharacteristic
251C22×C4C22⋊C4×C3111yescharacteristic
252C22×C4C22⋊C4×C3111yescharacteristic
253C22×C4C22×C4×C3111yescharacteristic
254C22×C4C2×D4×C3111yescharacteristic
255C2×C8—42no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C4⋊C4—82no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—44no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C42—21no—
267C42—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23C31×#10312yes—
277C23C2×C22⋊C4×C3111yescharacteristic
278C23C2×SD16×C3111yescharacteristic
279C23C2×Q16×C3111yescharacteristic
280Q8—44no—
281Q8—44no—
282Q8—44no—
283Q8—44no—
284D4—42no—
285D4—42no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—48no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4C31×#9911yescharacteristic
310C2×C4C31×#9911yescharacteristic
311C2×C4C31×#10312yes—
312C8—82no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—42no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22C2×C31×#9911yescharacteristic
328C22C2×C31×#10311yescharacteristic
329C22C31×#42111yescharacteristic
330C4—21no—
331C4—22no—
332C4—44no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—82no—
337C4—162no—
338C4C31×AΣL(1,𝔽16)12yes—
339C2—21no—
340C2—21no—
341C2—41no—
342C2—42no—
343C2—41no—
344C2C31×#152011yescharacteristic
345C2C2×C31×AΣL(1,𝔽16)11yescharacteristic
346C2C31×#212911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#1351C2semidirect
5. C23×Q8×C31C4semidirect
10. C31×#395C4semidirect
16. C31×#380C4semidirect
94. #11898C31direct (N × H)
143. #1351C2×C31semidirect
170. C23×Q8C4×C31semidirect
175. #395C4×C31semidirect
181. #380C4×C31semidirect
223. C31#11898direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#24092
C31×#17992
C31×#14602
C31×#13512
#1189831