Finite Group Databasebeta

← C31×#32927 #21489686

Subgroups — C31×#32927 #21489686

Order 7936.

1482 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 1482 conjugacy classes of subgroups — too many to draw. Only its 298 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
1C4×C31×H(5,𝔽2)C211yesmaximal, characteristic
2C2×C31×#458C212yesmaximal
3C31×#2486C213yesmaximal, complemented
4C31×#1890C216yesmaximal, complemented
5C31×#1799C213yesmaximal, complemented
6C2×C31×H(5,𝔽2)C2211yescharacteristic
7C2×C31×#123C2216yes—
8C31×#520—23no—
9C31×#518—23no—
10C31×#518C2213yes—
11C4×#41×C31—23no—
12C2×C4×D4×C31—23no—
13C2×C4×D4×C31C2213yes—
14C22×C4⋊C4×C31C2211yescharacteristic, complemented
15C31×#458—28no—
16C2×C31×#99C2216yes—
17C31×#398—23no—
18C31×#397—23no—
19C31×#389—23no—
20C31×#381C2216yescomplemented
21C31×#380—23no—
22C31×#380C2213yescomplemented
23C31×#342C2216yescomplemented
24C31×H(5,𝔽2)—26no—
25C31×H(5,𝔽2)D414yes—
26C2×#41×C31—23no—
27C22×D4×C31—23no—
28C22×D4×C31C2313yes—
29C23×C4×C31—21no—
30C23×C4×C31D413yes—
31C23×C4×C31C2312yes—
32C23×C4×C31C2311yescharacteristic
33C31×#123—212no—
34C31×#123—212no—
35C4×Q8×C31—21no—
36C4×Q8×C31—23no—
37C4×D4×C31—23no—
38C4×D4×C31—26no—
39C4×D4×C31—23no—
40C4×D4×C31—26no—
41C4×D4×C31—26no—
42C4×D4×C31—26no—
43C4×D4×C31—26no—
44C31×#117—26no—
45C31×#117—23no—
46C2×C4⋊C4×C31—23no—
47C2×C4⋊C4×C31—23no—
48C2×C4⋊C4×C31—26no—
49C2×C4⋊C4×C31—23no—
50C2×C4⋊C4×C31—23no—
51C2×C4⋊C4×C31D413yes—
52C2×C22⋊C4×C31—23no—
53C2×C22⋊C4×C31—23no—
54C2×C22⋊C4×C31—26no—
55C2×C22⋊C4×C31D413yes—
56C2×C22⋊C4×C31C2316yes—
57C2×C22⋊C4×C31C2313yes—
58C2×C42×C31—23no—
59C2×C42×C31D413yes—
60C31×#99—212no—
61C31×#95—23no—
62C31×#95—26no—
63C31×#95—26no—
64C24×C31—21no—
65C24×C31C2×D413yes—
66C24×C31C2411yescharacteristic
67#41×C31—212no—
68#41×C31—212no—
69C2×Q8×C31—21no—
70C2×D4×C31—23no—
71C2×D4×C31—212no—
72C2×D4×C31—212no—
73C2×D4×C31—26no—
74C2×D4×C31—26no—
75C2×D4×C31C2×D416yes—
76C2×D4×C31#4116yes—
77C22×C4×C31—23no—
78C22×C4×C31—23no—
79C22×C4×C31—23no—
80C22×C4×C31—23no—
81C22×C4×C31—23no—
82C22×C4×C31—26no—
83C22×C4×C31—26no—
84C22×C4×C31—22no—
85C22×C4×C31—26no—
86C22×C4×C31—42no—
87C22×C4×C31—46no—
88C22×C4×C31—82no—
89C22×C4×C31C2×D413yes—
90C22×C4×C31C2×D413yes—
91C22×C4×C31C2×D413yes—
92C22×C4×C31C2×D413yes—
93C22×C4×C31C2×D413yes—
94C4⋊C4×C31—23no—
95C4⋊C4×C31—23no—
96C4⋊C4×C31—26no—
97C4⋊C4×C31—46no—
98C4⋊C4×C31—43no—
99C4⋊C4×C31—41no—
100C4⋊C4×C31—412no—
101C22⋊C4×C31—26no—
102C22⋊C4×C31—26no—
103C22⋊C4×C31—26no—
104C22⋊C4×C31—26no—
105C22⋊C4×C31—26no—
106C22⋊C4×C31—412no—
107C42×C31—26no—
108C42×C31—23no—
109C42×C31—23no—
110#32927C3111yesmaximal, characteristic, direct product factor, Sylow
111C23×C31—24no—
112C23×C31—23no—
113C23×C31—23no—
114C23×C31—41no—
115C23×C31—41no—
116C23×C31—43no—
117C23×C31—43no—
118C23×C31—81no—
119C23×C31—81no—
120C23×C31#12116yes—
121C23×C31C22×D413yes—
122C23×C31C22×D411yescharacteristic
123C23×C31C2×#4113yes—
124Q8×C31—24no—
125D4×C31—212no—
126D4×C31—212no—
127D4×C31—212no—
128C2×C4×C31—212no—
129C2×C4×C31—23no—
130C2×C4×C31—23no—
131C2×C4×C31—26no—
132C2×C4×C31—23no—
133C2×C4×C31—23no—
134C2×C4×C31—43no—
135C2×C4×C31—46no—
136C2×C4×C31—43no—
137C2×C4×C31—43no—
138C2×C4×C31—46no—
139C2×C4×C31—46no—
140C2×C4×C31—42no—
141C2×C4×C31—46no—
142C2×C4×C31—46no—
143C2×C4×C31—46no—
144C2×C4×C31—83no—
145C2×C4×C31—83no—
146C2×C4×C31—83no—
147C2×C4×C31#12013yes—
148C2×C4×C31#12011yescharacteristic
149C2×C4×C31#12116yes—
150C4×H(5,𝔽2)C2×C3111yescharacteristic
151C2×#458C2×C3112yes—
152#2486C2×C3113yescomplemented
153#1890C2×C3116yescomplemented
154#1799C2×C3113yescomplemented
155C22×C31—23no—
156C22×C31—23no—
157C22×C31—26no—
158C22×C31—41no—
159C22×C31—43no—
160C22×C31—46no—
161C22×C31—43no—
162C22×C31—43no—
163C22×C31—83no—
164C22×C31—84no—
165C22×C31C2×#12011yescharacteristic
166C22×C31C2×#12113yes—
167C22×C31#52513yes—
168C4×C31—26no—
169C4×C31—23no—
170C4×C31—43no—
171C4×C31—43no—
172C4×C31—86no—
173C4×C31—82no—
174C4×C31#45812yescomplemented
175C2×H(5,𝔽2)C22×C3111yescharacteristic
176C2×#123C22×C3116yes—
177#520—23no—
178#518—23no—
179#518C22×C3113yes—
180C4×#41—23no—
181C2×C4×D4—23no—
182C2×C4×D4C22×C3113yes—
183C22×C4⋊C4C22×C3111yescharacteristic, complemented
184#458—28no—
185C2×#99C22×C3116yes—
186#398—23no—
187#397—23no—
188#389—23no—
189#381C22×C3116yescomplemented
190#380—23no—
191#380C22×C3113yescomplemented
192#342C22×C3116yescomplemented
193C2×C31—23no—
194C2×C31—23no—
195C2×C31—46no—
196C2×C31#284711yescharacteristic
197C2×C31C2×#45811yescharacteristic
198C2×C31#302811yescharacteristic
199H(5,𝔽2)—26no—
200H(5,𝔽2)D4×C3114yes—
201C2×#41—23no—
202C22×D4—23no—
203C22×D4C23×C3113yes—
204C23×C4—21no—
205C23×C4D4×C3113yes—
206C23×C4C23×C3112yes—
207C23×C4C23×C3111yescharacteristic
208#123—212no—
209#123—212no—
210C4×Q8—21no—
211C4×Q8—23no—
212C4×D4—23no—
213C4×D4—26no—
214C4×D4—23no—
215C4×D4—26no—
216C4×D4—26no—
217C4×D4—26no—
218C4×D4—26no—
219#117—26no—
220#117—23no—
221C2×C4⋊C4—23no—
222C2×C4⋊C4—23no—
223C2×C4⋊C4—23no—
224C2×C4⋊C4—26no—
225C2×C4⋊C4—23no—
226C2×C4⋊C4D4×C3113yes—
227C2×C22⋊C4—23no—
228C2×C22⋊C4—23no—
229C2×C22⋊C4—26no—
230C2×C22⋊C4D4×C3113yes—
231C2×C22⋊C4C23×C3116yes—
232C2×C22⋊C4C23×C3113yes—
233C2×C42—23no—
234C2×C42D4×C3113yes—
235#99—212no—
236#95—26no—
237#95—23no—
238#95—26no—
239C31#3292711yescharacteristic, direct product factor, Sylow
240C24—21no—
241C24C2×D4×C3113yes—
242C24C24×C3111yescharacteristic
243#41—212no—
244#41—212no—
245C2×Q8—21no—
246C2×D4—23no—
247C2×D4—212no—
248C2×D4—212no—
249C2×D4—26no—
250C2×D4—26no—
251C2×D4C2×D4×C3116yes—
252C2×D4#41×C3116yes—
253C22×C4—23no—
254C22×C4—23no—
255C22×C4—23no—
256C22×C4—23no—
257C22×C4—23no—
258C22×C4—26no—
259C22×C4—26no—
260C22×C4—22no—
261C22×C4—26no—
262C22×C4—42no—
263C22×C4—46no—
264C22×C4—82no—
265C22×C4C2×D4×C3113yes—
266C22×C4C2×D4×C3113yes—
267C22×C4C2×D4×C3113yes—
268C22×C4C2×D4×C3113yes—
269C22×C4C2×D4×C3113yes—
270C4⋊C4—23no—
271C4⋊C4—23no—
272C4⋊C4—26no—
273C4⋊C4—46no—
274C4⋊C4—43no—
275C4⋊C4—41no—
276C4⋊C4—412no—
277C22⋊C4—26no—
278C22⋊C4—26no—
279C22⋊C4—26no—
280C22⋊C4—26no—
281C22⋊C4—26no—
282C22⋊C4—412no—
283C42—26no—
284C42—23no—
285C42—23no—
286C23—24no—
287C23—23no—
288C23—23no—
289C23—41no—
290C23—41no—
291C23—43no—
292C23—43no—
293C23—81no—
294C23—81no—
295C23C31×#12116yes—
296C23C22×D4×C3113yes—
297C23C22×D4×C3111yescharacteristic
298C23C2×#41×C3113yes—
299Q8—24no—
300D4—212no—
301D4—212no—
302D4—212no—
303C2×C4—212no—
304C2×C4—23no—
305C2×C4—23no—
306C2×C4—26no—
307C2×C4—23no—
308C2×C4—23no—
309C2×C4—43no—
310C2×C4—46no—
311C2×C4—43no—
312C2×C4—43no—
313C2×C4—46no—
314C2×C4—46no—
315C2×C4—42no—
316C2×C4—46no—
317C2×C4—46no—
318C2×C4—46no—
319C2×C4—83no—
320C2×C4—83no—
321C2×C4—83no—
322C2×C4C31×#12013yes—
323C2×C4C31×#12011yescharacteristic
324C2×C4C31×#12116yes—
325C22—23no—
326C22—23no—
327C22—26no—
328C22—41no—
329C22—43no—
330C22—46no—
331C22—43no—
332C22—43no—
333C22—83no—
334C22—84no—
335C22C2×C31×#12011yescharacteristic
336C22C2×C31×#12113yes—
337C22C31×#52513yes—
338C4—26no—
339C4—23no—
340C4—43no—
341C4—43no—
342C4—86no—
343C4—82no—
344C4C31×#45812yescomplemented
345C2—23no—
346C2—23no—
347C2—46no—
348C2C31×#284711yescharacteristic
349C2C2×C31×#45811yescharacteristic
350C2C31×#302811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC24C24×C31
prime radicalC2×C31×H(5,𝔽2)C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#2486C2semidirect
4. C31×#1890C2semidirect
5. C31×#1799C2semidirect
14. C22×C4⋊C4×C31C22semidirect
20. C31×#381C22semidirect
22. C31×#380C22semidirect
23. C31×#342C22semidirect
110. #32927C31direct (N × H)
152. #2486C2×C31semidirect
153. #1890C2×C31semidirect
154. #1799C2×C31semidirect
174. C4×C31#458semidirect
183. C22×C4⋊C4C22×C31semidirect
189. #381C22×C31semidirect
191. #380C22×C31semidirect
192. #342C22×C31semidirect
239. C31#32927direct (N × H)
344. C4C31×#458semidirect

Maximal subgroups

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

subgroupindex
C4×C31×H(5,𝔽2)2
C2×C31×#4582
C31×#24862
C31×#18902
C31×#17992
#3292731