Finite Group Databasebeta

← C31×#12178 #21468937

Subgroups — C31×#12178 #21468937

Order 7936.

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

normal subgroups only

This group has 422 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×#2742C211yesmaximal, characteristic, complemented
2C31×#2044C211yesmaximal, characteristic
3C31×#1613C211yesmaximal, characteristic
4C31×#1601C211yesmaximal, characteristic
5C31×#1352C211yesmaximal, characteristic, complemented
6C31×#1351C211yesmaximal, characteristic, complemented
7C31×#1326C211yesmaximal, characteristic, complemented
8C31×#534C2211yescharacteristic
9C31×#481—21no—
10C31×#448—21no—
11C2×C31×#99—21no—
12C31×#400—21no—
13C31×#399C2211yescharacteristic
14C31×#395—21no—
15C31×#393C2211yescharacteristic
16C31×#389—21no—
17C31×#388—21no—
18C31×#387—21no—
19C31×#386—21no—
20C31×#385C2211yescharacteristic
21C31×#380—21no—
22C31×#342C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#323C2211yescharacteristic
26C31×#323C2211yescharacteristic
27C22×Q8×C31D411yescharacteristic
28C22×D4×C31—21no—
29C23×C4×C31D411yescharacteristic
30C2×SD16×C31—41no—
31C31×#123—41no—
32C31×#121—21no—
33C31×#121—21no—
34C31×#120—41no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31D411yescharacteristic
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×#106—41no—
49C31×#103—41no—
50C31×#102—41no—
51C31×#99—42no—
52C31×C22⋊C8—21no—
53C31×C22⋊C8—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—41no—
66C2×Q8×C31—81no—
67C2×D4×C31—21no—
68C2×D4×C31—21no—
69C2×D4×C31—41no—
70C2×D4×C31—41no—
71C2×D4×C31—41no—
72C2×D4×C31—41no—
73C2×D4×C31—41no—
74C2×D4×C31—41no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—21no—
79C22×C4×C31—21no—
80C22×C4×C31—21no—
81C22×C4×C31—21no—
82C22×C4×C31—42no—
83C22×C4×C31—41no—
84C22×C4×C31—41no—
85C22×C4×C31C2×D411yescharacteristic
86C22×C4×C31C2×D411yescharacteristic
87SD16×C31—82no—
88C2×C8×C31—41no—
89C2×C8×C31—41no—
90C4⋊C4×C31—21no—
91C4⋊C4×C31—21no—
92C4⋊C4×C31—42no—
93C4⋊C4×C31—42no—
94C4⋊C4×C31—41no—
95C4⋊C4×C31—81no—
96C4⋊C4×C31—81no—
97C4⋊C4×C31—81no—
98C22⋊C4×C31—21no—
99C22⋊C4×C31—41no—
100C22⋊C4×C31—41no—
101C22⋊C4×C31—41no—
102C22⋊C4×C31—41no—
103C22⋊C4×C31—41no—
104C22⋊C4×C31—81no—
105C22⋊C4×C31—81no—
106C22⋊C4×C31—82no—
107C22⋊C4×C31SD1612yescomplemented
108C42×C31—81no—
109#12178C3111yesmaximal, characteristic, direct product factor, Sylow
110C23×C31—21no—
111C23×C31—21no—
112C23×C31—41no—
113C23×C31—41no—
114C23×C31—41no—
115C23×C31—41no—
116C23×C31—41no—
117C23×C31—81no—
118C23×C31—81no—
119C23×C31#12011yescharacteristic
120C23×C31C2×SD1611yescharacteristic
121C23×C31#13711yescharacteristic
122Q8×C31—42no—
123Q8×C31—81no—
124D4×C31—81no—
125D4×C31—81no—
126D4×C31—81no—
127D4×C31—81no—
128D4×C31—81no—
129D4×C31—81no—
130C2×C4×C31—22no—
131C2×C4×C31—22no—
132C2×C4×C31—21no—
133C2×C4×C31—21no—
134C2×C4×C31—41no—
135C2×C4×C31—41no—
136C2×C4×C31—41no—
137C2×C4×C31—41no—
138C2×C4×C31—41no—
139C2×C4×C31—41no—
140C2×C4×C31—42no—
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—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—81no—
157C2×C4×C31—81no—
158C2×C4×C31—81no—
159C2×C4×C31—81no—
160C8×C31—81no—
161C8×C31—81no—
162#2742C2×C3111yescharacteristic, complemented
163#2044C2×C3111yescharacteristic
164#1613C2×C3111yescharacteristic
165#1601C2×C3111yescharacteristic
166#1352C2×C3111yescharacteristic, complemented
167#1351C2×C3111yescharacteristic, complemented
168#1326C2×C3111yescharacteristic, complemented
169C22×C31—21no—
170C22×C31—21no—
171C22×C31—21no—
172C22×C31—21no—
173C22×C31—41no—
174C22×C31—41no—
175C22×C31—41no—
176C22×C31—41no—
177C22×C31—81no—
178C22×C31—82no—
179C22×C31—81no—
180C22×C31—81no—
181C22×C31—81no—
182C22×C31—161no—
183C22×C31#44811yescharacteristic
184C22×C31#45411yescharacteristic
185C22×C31B(4,𝔽2)11yescharacteristic
186C4×C31—42no—
187C4×C31—41no—
188C4×C31—42no—
189C4×C31—41no—
190C4×C31—81no—
191C4×C31—81no—
192C4×C31—81no—
193C4×C31—81no—
194C4×C31—81no—
195C4×C31—161no—
196#534C22×C3111yescharacteristic
197#481—21no—
198#448—21no—
199C2×#99—21no—
200#400—21no—
201#399C22×C3111yescharacteristic
202#395—21no—
203#393C22×C3111yescharacteristic
204#389—21no—
205#388—21no—
206#387—21no—
207#386—21no—
208#385C22×C3111yescharacteristic
209#380—21no—
210#342C22×C3111yescharacteristic
211#327—21no—
212#327—21no—
213#323C22×C3111yescharacteristic
214#323C22×C3111yescharacteristic
215C2×C31—21no—
216C2×C31—21no—
217C2×C31—41no—
218C2×C31—41no—
219C2×C31—161no—
220C2×C31#159911yescharacteristic
221C2×C31#219711yescharacteristic
222C2×C31#220011yescharacteristic
223C22×Q8D4×C3111yescharacteristic
224C22×D4—21no—
225C23×C4D4×C3111yescharacteristic
226C2×SD16—41no—
227#123—41no—
228#121—21no—
229#121—21no—
230#120—41no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4D4×C3111yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4D4×C3111yescharacteristic
242C2×C22⋊C4C23×C3111yescharacteristic
243C2×C42—21no—
244#106—41no—
245#103—41no—
246#102—41no—
247#99—42no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—41no—
257#95D4×C3111yescharacteristic
258#95D4×C3111yescharacteristic
259C31#1217811yescharacteristic, direct product factor, Sylow
260C24—21no—
261C24C2×D4×C3111yescharacteristic
262C2×Q8—41no—
263C2×Q8—81no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4C2×D4×C3111yescharacteristic
283C22×C4C2×D4×C3111yescharacteristic
284SD16—82no—
285C2×C8—41no—
286C2×C8—41no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—81no—
293C4⋊C4—81no—
294C4⋊C4—81no—
295C22⋊C4—21no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—81no—
302C22⋊C4—81no—
303C22⋊C4—82no—
304C22⋊C4SD16×C3112yescomplemented
305C42—81no—
306C23—21no—
307C23—21no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—81no—
314C23—81no—
315C23C31×#12011yescharacteristic
316C23C2×SD16×C3111yescharacteristic
317C23C31×#13711yescharacteristic
318Q8—42no—
319Q8—81no—
320D4—81no—
321D4—81no—
322D4—81no—
323D4—81no—
324D4—81no—
325D4—81no—
326C2×C4—22no—
327C2×C4—22no—
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—42no—
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—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C8—81no—
357C8—81no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—81no—
367C22—82no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—161no—
372C22C31×#44811yescharacteristic
373C22C31×#45411yescharacteristic
374C22C31×B(4,𝔽2)11yescharacteristic
375C4—42no—
376C4—41no—
377C4—42no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4—161no—
385C2—21no—
386C2—21no—
387C2—41no—
388C2—41no—
389C2—161no—
390C2C31×#159911yescharacteristic
391C2C31×#219711yescharacteristic
392C2C31×#220011yescharacteristic
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×#2742C2semidirect
5. C31×#1352C2semidirect
6. C31×#1351C2semidirect
7. C31×#1326C2semidirect
107. C22⋊C4×C31SD16semidirect
109. #12178C31direct (N × H)
162. #2742C2×C31semidirect
166. #1352C2×C31semidirect
167. #1351C2×C31semidirect
168. #1326C2×C31semidirect
259. C31#12178direct (N × H)
304. C22⋊C4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#27422
C31×#20442
C31×#16132
C31×#16012
C31×#13522
C31×#13512
C31×#13262
#1217831