Finite Group Databasebeta

← C31×#12155 #21468914

Subgroups — C31×#12155 #21468914

Order 7936.

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

normal subgroups only

This group has 416 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×#2823C211yesmaximal, characteristic, complemented
2C31×#2039C211yesmaximal, characteristic, complemented
3C31×#1610C211yesmaximal, characteristic, complemented
4C31×#1598C211yesmaximal, characteristic
5C31×#1346C211yesmaximal, characteristic, complemented
6C31×#1345C211yesmaximal, characteristic, complemented
7C31×#1325C211yesmaximal, characteristic, complemented
8C31×#537C2211yescharacteristic
9C2×C31×#126—21no—
10C31×#480—21no—
11C31×#447—21no—
12C2×C31×#104—21no—
13C31×#417—21no—
14C31×#398—21no—
15C31×#398C2211yescharacteristic
16C31×#396—21no—
17C31×#396—21no—
18C31×#396—21no—
19C31×#394—21no—
20C31×#394C2211yescharacteristic
21C31×#384C2211yescharacteristic
22C31×#343C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C2×#41×C31—21no—
28C22×D4×C31D411yescharacteristic
29C2×D8×C31—41no—
30C2×M16×C31—21no—
31C31×#126—42no—
32C31×#126—42no—
33C31×C22⋊Q8—41no—
34C31×#121—21no—
35C31×#121—21no—
36C31×#121—41no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31—21no—
41C2×C4⋊C4×C31—21no—
42C2×C4⋊C4×C31—21no—
43C2×C4⋊C4×C31D411yescharacteristic
44C2×C4⋊C4×C31D411yescharacteristic
45C2×C4⋊C4×C31C2311yescharacteristic
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—21no—
48C2×C22⋊C4×C31—21no—
49C2×C22⋊C4×C31—21no—
50C2×C22⋊C4×C31—21no—
51C2×C22⋊C4×C31—21no—
52C2×C42×C31D411yescharacteristic
53C31×#107—41no—
54C31×#104—42no—
55C31×#103—41no—
56C31×#102—41no—
57C31×#102—41no—
58C31×#102—41no—
59C31×C22⋊C8—21no—
60C31×C22⋊C8—21no—
61C31×#95—21no—
62C31×#95—21no—
63C31×#95D411yescharacteristic
64C31×#95D411yescharacteristic
65C24×C31—21no—
66#41×C31—41no—
67#41×C31—41no—
68#41×C31—41no—
69#41×C31—41no—
70C2×Q8×C31—21no—
71C2×D4×C31—21no—
72C2×D4×C31—21no—
73C2×D4×C31—21no—
74C2×D4×C31—41no—
75C2×D4×C31—41no—
76C2×D4×C31—81no—
77C22×C4×C31—21no—
78C22×C4×C31—21no—
79C22×C4×C31—21no—
80C22×C4×C31—21no—
81C22×C4×C31—21no—
82C22×C4×C31—41no—
83C22×C4×C31C2×D411yescharacteristic
84C22×C4×C31C2×D411yescharacteristic
85C22×C4×C31C2×D411yescharacteristic
86D8×C31—82no—
87M16×C31—41no—
88M16×C31—41no—
89C2×C8×C31—41no—
90C2×C8×C31—41no—
91C2×C8×C31—41no—
92C4⋊C4×C31—21no—
93C4⋊C4×C31—21no—
94C4⋊C4×C31—41no—
95C4⋊C4×C31—42no—
96C4⋊C4×C31—42no—
97C4⋊C4×C31—41no—
98C4⋊C4×C31—41no—
99C4⋊C4×C31—41no—
100C4⋊C4×C31—41no—
101C4⋊C4×C31—81no—
102C4⋊C4×C31—81no—
103C4⋊C4×C31—81no—
104C22⋊C4×C31—42no—
105C22⋊C4×C31—42no—
106C22⋊C4×C31—41no—
107C22⋊C4×C31—41no—
108C22⋊C4×C31—41no—
109C22⋊C4×C31—41no—
110C22⋊C4×C31—81no—
111C22⋊C4×C31—81no—
112C22⋊C4×C31—81no—
113C42×C31—21no—
114C42×C31D812yescomplemented
115#12155C3111yesmaximal, characteristic, direct product factor, Sylow
116C23×C31—41no—
117C23×C31—41no—
118C23×C31—41no—
119C23×C31—41no—
120C23×C31—41no—
121C23×C31—41no—
122C23×C31—41no—
123C23×C31—41no—
124C23×C31#12011yescharacteristic
125Q8×C31—41no—
126Q8×C31—41no—
127D4×C31—41no—
128D4×C31—41no—
129D4×C31—42no—
130D4×C31—81no—
131D4×C31—81no—
132D4×C31—81no—
133D4×C31—81no—
134D4×C31—81no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—22no—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
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—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31—81no—
158C2×C4×C31—81no—
159C2×C4×C31C2×D811yescharacteristic
160C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
161C8×C31—81no—
162C8×C31—81no—
163C8×C31—81no—
164#2823C2×C3111yescharacteristic, complemented
165#2039C2×C3111yescharacteristic, complemented
166#1610C2×C3111yescharacteristic, complemented
167#1598C2×C3111yescharacteristic
168#1346C2×C3111yescharacteristic, complemented
169#1345C2×C3111yescharacteristic, complemented
170#1325C2×C3111yescharacteristic, complemented
171C22×C31—21no—
172C22×C31—21no—
173C22×C31—81no—
174C22×C31—81no—
175C22×C31—81no—
176C22×C31—81no—
177C22×C31—81no—
178C22×C31—81no—
179C22×C31—81no—
180C22×C31—81no—
181C22×C31—81no—
182C22×C31—81no—
183C22×C31#44711yescharacteristic
184C22×C31#45411yescharacteristic
185C22×C31B(4,𝔽2)11yescharacteristic
186C4×C31—21no—
187C4×C31—21no—
188C4×C31—42no—
189C4×C31—41no—
190C4×C31—41no—
191C4×C31—81no—
192C4×C31—81no—
193C4×C31—81no—
194C4×C31—81no—
195C4×C31—81no—
196#537C22×C3111yescharacteristic
197C2×#126—21no—
198#480—21no—
199#447—21no—
200C2×#104—21no—
201#417—21no—
202#398—21no—
203#398C22×C3111yescharacteristic
204#396—21no—
205#396—21no—
206#396—21no—
207#394—21no—
208#394C22×C3111yescharacteristic
209#384C22×C3111yescharacteristic
210#343C22×C3111yescharacteristic
211#327—21no—
212#327—21no—
213#324C22×C3111yescharacteristic
214#324C22×C3111yescharacteristic
215C2×C31—21no—
216C2×C31—21no—
217C2×C31—161no—
218C2×C31—161no—
219C2×C31#159811yescharacteristic
220C2×C31#220011yescharacteristic
221C2×C31#220311yescharacteristic
222C2×#41—21no—
223C22×D4D4×C3111yescharacteristic
224C2×D8—41no—
225C2×M16—21no—
226#126—42no—
227#126—42no—
228C22⋊Q8—41no—
229#121—21no—
230#121—21no—
231#121—41no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4D4×C3111yescharacteristic
239C2×C4⋊C4D4×C3111yescharacteristic
240C2×C4⋊C4C23×C3111yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C42D4×C3111yescharacteristic
248#107—41no—
249#104—42no—
250#103—41no—
251#102—41no—
252#102—41no—
253#102—41no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256#95—21no—
257#95—21no—
258#95D4×C3111yescharacteristic
259#95D4×C3111yescharacteristic
260C31#1215511yescharacteristic, direct product factor, Sylow
261C24—21no—
262#41—41no—
263#41—41no—
264#41—41no—
265#41—41no—
266C2×Q8—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—81no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4C2×D4×C3111yescharacteristic
280C22×C4C2×D4×C3111yescharacteristic
281C22×C4C2×D4×C3111yescharacteristic
282D8—82no—
283M16—41no—
284M16—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—81no—
298C4⋊C4—81no—
299C4⋊C4—81no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—81no—
307C22⋊C4—81no—
308C22⋊C4—81no—
309C42—21no—
310C42D8×C3112yescomplemented
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23C31×#12011yescharacteristic
320Q8—41no—
321Q8—41no—
322D4—41no—
323D4—41no—
324D4—42no—
325D4—81no—
326D4—81no—
327D4—81no—
328D4—81no—
329D4—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—41no—
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—81no—
353C2×C4—81no—
354C2×C4C2×D8×C3111yescharacteristic
355C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
356C8—81no—
357C8—81no—
358C8—81no—
359C22—21no—
360C22—21no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22C31×#44711yescharacteristic
372C22C31×#45411yescharacteristic
373C22C31×B(4,𝔽2)11yescharacteristic
374C4—21no—
375C4—21no—
376C4—42no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—81no—
384C2—21no—
385C2—21no—
386C2—161no—
387C2—161no—
388C2C31×#159811yescharacteristic
389C2C31×#220011yescharacteristic
390C2C31×#220311yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC31×#1598C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C4⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2823C2semidirect
2. C31×#2039C2semidirect
3. C31×#1610C2semidirect
5. C31×#1346C2semidirect
6. C31×#1345C2semidirect
7. C31×#1325C2semidirect
114. C42×C31D8semidirect
115. #12155C31direct (N × H)
164. #2823C2×C31semidirect
165. #2039C2×C31semidirect
166. #1610C2×C31semidirect
168. #1346C2×C31semidirect
169. #1345C2×C31semidirect
170. #1325C2×C31semidirect
260. C31#12155direct (N × H)
310. C42D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#28232
C31×#20392
C31×#16102
C31×#15982
C31×#13462
C31×#13452
C31×#13252
#1215531