Finite Group Databasebeta

← C31×#12177 #21468936

Subgroups — C31×#12177 #21468936

Order 7936.

432 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2763C211yesmaximal, characteristic, complemented
2C31×#2044C211yesmaximal, characteristic
3C31×#1601C211yesmaximal, characteristic
4C31×#1601C211yesmaximal, characteristic
5C31×#1351C211yesmaximal, characteristic, complemented
6C31×#1351C211yesmaximal, characteristic, complemented
7C31×#1326C211yesmaximal, characteristic, complemented
8C31×#534C2211yescharacteristic
9C31×#448—21no—
10C31×#448—21no—
11C2×C31×#99—21no—
12C31×#400—21no—
13C31×#395—21no—
14C31×#393C2211yescharacteristic
15C31×#393C2211yescharacteristic
16C31×#391—21no—
17C31×#388—21no—
18C31×#386—21no—
19C31×#385—21no—
20C31×#385C2211yescharacteristic
21C31×#382—21no—
22C31×#342C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#323C2211yescharacteristic
26C31×#323C2211yescharacteristic
27C22×Q8×C31D411yescharacteristic
28C22×Q8×C31D411yescharacteristic
29C22×D4×C31—21no—
30C23×C4×C31D411yescharacteristic
31C2×SD16×C31—41no—
32C2×SD16×C31—41no—
33C31×#123—41no—
34C31×#121—21no—
35C31×#121—21no—
36C31×#120—41no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31—21no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—41no—
45C2×C22⋊C4×C31D411yescharacteristic
46C2×C22⋊C4×C31C2311yescharacteristic
47C2×C42×C31—21no—
48C31×#103—41no—
49C31×#103—41no—
50C31×#99—42no—
51C31×C22⋊C8—21no—
52C31×C22⋊C8—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95—41no—
61C31×#95D411yescharacteristic
62C31×#95D411yescharacteristic
63C24×C31—21no—
64C24×C31C2×D411yescharacteristic
65C2×Q8×C31—42no—
66C2×Q8×C31—41no—
67C2×Q8×C31—41no—
68C2×Q8×C31—81no—
69C2×D4×C31—21no—
70C2×D4×C31—21no—
71C2×D4×C31—41no—
72C2×D4×C31—41no—
73C2×D4×C31—41no—
74C2×D4×C31—41no—
75C2×D4×C31—41no—
76C2×D4×C31—41no—
77C22×C4×C31—21no—
78C22×C4×C31—21no—
79C22×C4×C31—21no—
80C22×C4×C31—21no—
81C22×C4×C31—21no—
82C22×C4×C31—21no—
83C22×C4×C31—21no—
84C22×C4×C31—41no—
85C22×C4×C31—41no—
86C22×C4×C31—81no—
87C22×C4×C31C2×D411yescharacteristic
88C22×C4×C31C2×D411yescharacteristic
89SD16×C31—82no—
90SD16×C31—82no—
91C2×C8×C31—41no—
92C2×C8×C31—41no—
93C4⋊C4×C31—21no—
94C4⋊C4×C31—21no—
95C4⋊C4×C31—81no—
96C4⋊C4×C31—81no—
97C4⋊C4×C31—81no—
98C4⋊C4×C31—81no—
99C22⋊C4×C31—21no—
100C22⋊C4×C31—41no—
101C22⋊C4×C31—42no—
102C22⋊C4×C31—41no—
103C22⋊C4×C31—41no—
104C22⋊C4×C31—41no—
105C22⋊C4×C31—41no—
106C22⋊C4×C31—81no—
107C22⋊C4×C31—82no—
108C22⋊C4×C31SD1612yescomplemented
109C42×C31—42no—
110#12177C3111yesmaximal, characteristic, direct product factor, Sylow
111C23×C31—21no—
112C23×C31—21no—
113C23×C31—41no—
114C23×C31—41no—
115C23×C31—41no—
116C23×C31—41no—
117C23×C31—41no—
118C23×C31—81no—
119C23×C31—81no—
120C23×C31#12011yescharacteristic
121C23×C31C2×SD1611yescharacteristic
122C23×C31#13711yescharacteristic
123Q8×C31—42no—
124Q8×C31—42no—
125Q8×C31—81no—
126Q8×C31—81no—
127D4×C31—81no—
128D4×C31—81no—
129D4×C31—81no—
130D4×C31—81no—
131D4×C31—81no—
132D4×C31—81no—
133C2×C4×C31—22no—
134C2×C4×C31—22no—
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—41no—
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—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—42no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—81no—
160C2×C4×C31—81no—
161C2×C4×C31—81no—
162C2×C4×C31—81no—
163C8×C31—81no—
164C8×C31—81no—
165#2763C2×C3111yescharacteristic, complemented
166#2044C2×C3111yescharacteristic
167#1601C2×C3111yescharacteristic
168#1601C2×C3111yescharacteristic
169#1351C2×C3111yescharacteristic, complemented
170#1351C2×C3111yescharacteristic, complemented
171#1326C2×C3111yescharacteristic, complemented
172C22×C31—21no—
173C22×C31—21no—
174C22×C31—21no—
175C22×C31—21no—
176C22×C31—41no—
177C22×C31—41no—
178C22×C31—41no—
179C22×C31—41no—
180C22×C31—81no—
181C22×C31—82no—
182C22×C31—81no—
183C22×C31—81no—
184C22×C31—81no—
185C22×C31—161no—
186C22×C31#44811yescharacteristic
187C22×C31#45411yescharacteristic
188C22×C31B(4,𝔽2)11yescharacteristic
189C4×C31—42no—
190C4×C31—41no—
191C4×C31—42no—
192C4×C31—41no—
193C4×C31—81no—
194C4×C31—81no—
195C4×C31—81no—
196C4×C31—81no—
197C4×C31—81no—
198C4×C31—161no—
199#534C22×C3111yescharacteristic
200#448—21no—
201#448—21no—
202C2×#99—21no—
203#400—21no—
204#395—21no—
205#393C22×C3111yescharacteristic
206#393C22×C3111yescharacteristic
207#391—21no—
208#388—21no—
209#386—21no—
210#385—21no—
211#385C22×C3111yescharacteristic
212#382—21no—
213#342C22×C3111yescharacteristic
214#327—21no—
215#327—21no—
216#323C22×C3111yescharacteristic
217#323C22×C3111yescharacteristic
218C2×C31—21no—
219C2×C31—21no—
220C2×C31—41no—
221C2×C31—41no—
222C2×C31—161no—
223C2×C31#159911yescharacteristic
224C2×C31#219711yescharacteristic
225C2×C31#219911yescharacteristic
226C22×Q8D4×C3111yescharacteristic
227C22×Q8D4×C3111yescharacteristic
228C22×D4—21no—
229C23×C4D4×C3111yescharacteristic
230C2×SD16—41no—
231C2×SD16—41no—
232#123—41no—
233#121—21no—
234#121—21no—
235#120—41no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4D4×C3111yescharacteristic
245C2×C22⋊C4C23×C3111yescharacteristic
246C2×C42—21no—
247#103—41no—
248#103—41no—
249#99—42no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—41no—
260#95D4×C3111yescharacteristic
261#95D4×C3111yescharacteristic
262C31#1217711yescharacteristic, direct product factor, Sylow
263C24—21no—
264C24C2×D4×C3111yescharacteristic
265C2×Q8—42no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—81no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—81no—
287C22×C4C2×D4×C3111yescharacteristic
288C22×C4C2×D4×C3111yescharacteristic
289SD16—82no—
290SD16—82no—
291C2×C8—41no—
292C2×C8—41no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—81no—
296C4⋊C4—81no—
297C4⋊C4—81no—
298C4⋊C4—81no—
299C22⋊C4—21no—
300C22⋊C4—41no—
301C22⋊C4—42no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—81no—
307C22⋊C4—82no—
308C22⋊C4SD16×C3112yescomplemented
309C42—42no—
310C23—21no—
311C23—21no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—81no—
318C23—81no—
319C23C31×#12011yescharacteristic
320C23C2×SD16×C3111yescharacteristic
321C23C31×#13711yescharacteristic
322Q8—42no—
323Q8—42no—
324Q8—81no—
325Q8—81no—
326D4—81no—
327D4—81no—
328D4—81no—
329D4—81no—
330D4—81no—
331D4—81no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C8—81no—
363C8—81no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—81no—
373C22—82no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—161no—
378C22C31×#44811yescharacteristic
379C22C31×#45411yescharacteristic
380C22C31×B(4,𝔽2)11yescharacteristic
381C4—42no—
382C4—41no—
383C4—42no—
384C4—41no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—161no—
391C2—21no—
392C2—21no—
393C2—41no—
394C2—41no—
395C2—161no—
396C2C31×#159911yescharacteristic
397C2C31×#219711yescharacteristic
398C2C31×#219911yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC31×#534C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C22⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2763C2semidirect
5. C31×#1351C2semidirect
6. C31×#1351C2semidirect
7. C31×#1326C2semidirect
108. C22⋊C4×C31SD16semidirect
110. #12177C31direct (N × H)
165. #2763C2×C31semidirect
169. #1351C2×C31semidirect
170. #1351C2×C31semidirect
171. #1326C2×C31semidirect
262. C31#12177direct (N × H)
308. C22⋊C4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#27632
C31×#20442
C31×#16012
C31×#13512
C31×#13262
#1217731