Finite Group Databasebeta

← C31×#12230 #21468989

Subgroups — C31×#12230 #21468989

Order 7936.

422 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2824C211yesmaximal, characteristic
2C31×#2062C211yesmaximal, characteristic, complemented
3C31×#1602C211yesmaximal, characteristic, complemented
4C31×#1596C211yesmaximal, characteristic
5C31×#1351C211yesmaximal, characteristic, complemented
6C31×#1343C211yesmaximal, characteristic
7C31×#1326C211yesmaximal, characteristic, complemented
8C31×#543C2211yescharacteristic
9C31×#451—21no—
10C31×#450—21no—
11C31×#400—21no—
12C31×#397—21no—
13C31×#394—21no—
14C31×#394—21no—
15C31×#393—21no—
16C31×#393C2211yescharacteristic
17C31×#392C2211yescharacteristic
18C31×#386—21no—
19C31×#385—21no—
20C31×#385C2211yescharacteristic
21C31×#342—21no—
22C31×#342C2211yescharacteristic
23C31×#328—21no—
24C31×#328—21no—
25C31×#323C2211yescharacteristic
26C31×#323C2211yescharacteristic
27C22×Q8×C31D411yescharacteristic
28C22×D4×C31—21no—
29C22×D4×C31D411yescharacteristic
30C23×C4×C31D411yescharacteristic
31C2×Q16×C31—41no—
32C2×SD16×C31—41no—
33C31×C22⋊Q8—21no—
34C31×C22⋊Q8—21no—
35C31×C22⋊Q8—41no—
36C31×C22⋊Q8—41no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31—41no—
40C2×C22⋊C4×C31—21no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—21no—
48C2×C22⋊C4×C31D411yescharacteristic
49C2×C22⋊C4×C31C2311yescharacteristic
50C31×#103—41no—
51C31×#102—41no—
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—41no—
60C31×#95D411yescharacteristic
61C31×#95D411yescharacteristic
62C24×C31—21no—
63C24×C31C2×D411yescharacteristic
64C2×Q8×C31—21no—
65C2×Q8×C31—21no—
66C2×Q8×C31—41no—
67C2×Q8×C31—81no—
68C2×D4×C31—41no—
69C2×D4×C31—41no—
70C2×D4×C31—41no—
71C2×D4×C31—81no—
72C2×D4×C31—81no—
73C2×D4×C31—81no—
74C22×C4×C31—21no—
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—41no—
82C22×C4×C31—41no—
83C22×C4×C31—81no—
84C22×C4×C31C2×D411yescharacteristic
85C22×C4×C31C2×D411yescharacteristic
86Q16×C31—82no—
87SD16×C31—82no—
88C2×C8×C31—41no—
89C2×C8×C31—41no—
90C4⋊C4×C31—21no—
91C4⋊C4×C31—21no—
92C4⋊C4×C31—41no—
93C4⋊C4×C31—41no—
94C4⋊C4×C31—81no—
95C4⋊C4×C31—81no—
96C4⋊C4×C31—82no—
97C22⋊C4×C31—22no—
98C22⋊C4×C31—21no—
99C22⋊C4×C31—21no—
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—81no—
107C22⋊C4×C31—81no—
108C22⋊C4×C31—81no—
109C22⋊C4×C31—81no—
110#12230C3111yesmaximal, 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—41no—
119C23×C31—41no—
120C23×C31—41no—
121C23×C31—41no—
122C23×C31#12011yescharacteristic
123C23×C31#13511yescharacteristic
124C23×C31#13711yescharacteristic
125Q8×C31—42no—
126Q8×C31—81no—
127Q8×C31—81no—
128Q8×C31—81no—
129D4×C31—42no—
130D4×C31—81no—
131D4×C31—82no—
132D4×C31—82no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—41no—
136C2×C4×C31—41no—
137C2×C4×C31—42no—
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—42no—
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—
160C2×C4×C31—81no—
161C8×C31—81no—
162C8×C31—81no—
163#2824C2×C3111yescharacteristic
164#2062C2×C3111yescharacteristic, complemented
165#1602C2×C3111yescharacteristic, complemented
166#1596C2×C3111yescharacteristic
167#1351C2×C3111yescharacteristic, complemented
168#1343C2×C3111yescharacteristic
169#1326C2×C3111yescharacteristic, complemented
170C22×C31—21no—
171C22×C31—21no—
172C22×C31—21no—
173C22×C31—21no—
174C22×C31—41no—
175C22×C31—41no—
176C22×C31—41no—
177C22×C31—41no—
178C22×C31—81no—
179C22×C31—81no—
180C22×C31—81no—
181C22×C31—81no—
182C22×C31—81no—
183C22×C31—81no—
184C22×C31—81no—
185C22×C31—81no—
186C22×C31#45211yescharacteristic
187C22×C31#45611yescharacteristic
188C22×C31B(4,𝔽2)11yescharacteristic
189C4×C31—41no—
190C4×C31—41no—
191C4×C31—81no—
192C4×C31—81no—
193C4×C31—81no—
194C4×C31—82no—
195C4×C31—81no—
196C4×C31—81no—
197C4×C31—81no—
198C4×C31—81no—
199#543C22×C3111yescharacteristic
200#451—21no—
201#450—21no—
202#400—21no—
203#397—21no—
204#394—21no—
205#394—21no—
206#393—21no—
207#393C22×C3111yescharacteristic
208#392C22×C3111yescharacteristic
209#386—21no—
210#385—21no—
211#385C22×C3111yescharacteristic
212#342—21no—
213#342C22×C3111yescharacteristic
214#328—21no—
215#328—21no—
216#323C22×C3111yescharacteristic
217#323C22×C3111yescharacteristic
218C2×C31—21no—
219C2×C31—21no—
220C2×C31—41no—
221C2×C31—41no—
222C2×C31—161no—
223C2×C31#160311yescharacteristic
224C2×C31#219911yescharacteristic
225C2×C31#220011yescharacteristic
226C22×Q8D4×C3111yescharacteristic
227C22×D4—21no—
228C22×D4D4×C3111yescharacteristic
229C23×C4D4×C3111yescharacteristic
230C2×Q16—41no—
231C2×SD16—41no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C22⋊Q8—41no—
235C22⋊Q8—41no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—41no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
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×C22⋊C4D4×C3111yescharacteristic
248C2×C22⋊C4C23×C3111yescharacteristic
249#103—41no—
250#102—41no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—41no—
259#95D4×C3111yescharacteristic
260#95D4×C3111yescharacteristic
261C31#1223011yescharacteristic, direct product factor, Sylow
262C24—21no—
263C24C2×D4×C3111yescharacteristic
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—41no—
267C2×Q8—81no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—81no—
272C2×D4—81no—
273C2×D4—81no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—81no—
284C22×C4C2×D4×C3111yescharacteristic
285C22×C4C2×D4×C3111yescharacteristic
286Q16—82no—
287SD16—82no—
288C2×C8—41no—
289C2×C8—41no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C4⋊C4—82no—
297C22⋊C4—22no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—81no—
305C22⋊C4—81no—
306C22⋊C4—81no—
307C22⋊C4—81no—
308C22⋊C4—81no—
309C22⋊C4—81no—
310C23—21no—
311C23—21no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23C31×#12011yescharacteristic
322C23C31×#13511yescharacteristic
323C23C31×#13711yescharacteristic
324Q8—42no—
325Q8—81no—
326Q8—81no—
327Q8—81no—
328D4—42no—
329D4—81no—
330D4—82no—
331D4—82no—
332C2×C4—21no—
333C2×C4—21no—
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—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C8—81no—
361C8—81no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22C31×#45211yescharacteristic
379C22C31×#45611yescharacteristic
380C22C31×B(4,𝔽2)11yescharacteristic
381C4—41no—
382C4—41no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—82no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C2—21no—
392C2—21no—
393C2—41no—
394C2—41no—
395C2—161no—
396C2C31×#160311yescharacteristic
397C2C31×#219911yescharacteristic
398C2C31×#220011yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC31×#392C22
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
2. C31×#2062C2semidirect
3. C31×#1602C2semidirect
5. C31×#1351C2semidirect
7. C31×#1326C2semidirect
110. #12230C31direct (N × H)
164. #2062C2×C31semidirect
165. #1602C2×C31semidirect
167. #1351C2×C31semidirect
169. #1326C2×C31semidirect
261. C31#12230direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28242
C31×#20622
C31×#16022
C31×#15962
C31×#13512
C31×#13432
C31×#13262
#1223031