Finite Group Databasebeta

← C31×#12244 #21469003

Subgroups — C31×#12244 #21469003

Order 7936.

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

normal subgroups only

This group has 398 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×#2722C211yesmaximal, characteristic
2C31×#2062C211yesmaximal, characteristic
3C31×#1608C211yesmaximal, characteristic
4C31×#1608C211yesmaximal, characteristic
5C31×#1344C211yesmaximal, characteristic
6C31×#1344C211yesmaximal, characteristic
7C31×#1326C211yesmaximal, characteristic
8C31×#543C2211yescharacteristic
9C2×C31×#126—21no—
10C31×#483—21no—
11C31×#483—21no—
12C31×#398—21no—
13C31×#398—21no—
14C31×#397C2211yescharacteristic
15C31×#397C2211yescharacteristic
16C31×#389—21no—
17C31×#387—21no—
18C31×#385—21no—
19C31×#385C2211yescharacteristic
20C31×#380—21no—
21C31×#342—21no—
22C31×#342C2211yescharacteristic
23C31×#328—21no—
24C31×#328—21no—
25C31×#323C2211yescharacteristic
26C31×#323C2211yescharacteristic
27C23×C4×C31D411yescharacteristic
28C31×#126—42no—
29C31×#126—42no—
30C31×C22⋊Q8—21no—
31C31×C22⋊Q8—21no—
32C31×C22⋊Q8—41no—
33C31×C22⋊Q8—41no—
34C2×C4⋊C4×C31—21no—
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×C31—41no—
41C2×C4⋊C4×C31D411yescharacteristic
42C2×C4⋊C4×C31D411yescharacteristic
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31D411yescharacteristic
48C2×C22⋊C4×C31C2311yescharacteristic
49C2×C42×C31—21no—
50C31×#106—41no—
51C31×#106—41no—
52C31×#103—41no—
53C31×#103—41no—
54C31×C22⋊C8—21no—
55C31×C22⋊C8—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95—21no—
61C31×#95—41no—
62C31×#95D411yescharacteristic
63C31×#95D411yescharacteristic
64C24×C31C2×D411yescharacteristic
65C2×Q8×C31—21no—
66C2×Q8×C31—21no—
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—42no—
76C22×C4×C31—41no—
77C22×C4×C31—41no—
78C22×C4×C31C2×D411yescharacteristic
79C22×C4×C31C2×D411yescharacteristic
80C2×C8×C31—41no—
81C2×C8×C31—41no—
82C4⋊C4×C31—21no—
83C4⋊C4×C31—21no—
84C4⋊C4×C31—42no—
85C4⋊C4×C31—42no—
86C4⋊C4×C31—42no—
87C4⋊C4×C31—42no—
88C4⋊C4×C31—42no—
89C4⋊C4×C31—41no—
90C4⋊C4×C31—41no—
91C4⋊C4×C31—41no—
92C4⋊C4×C31—41no—
93C4⋊C4×C31—81no—
94C4⋊C4×C31—82no—
95C22⋊C4×C31—21no—
96C22⋊C4×C31—22no—
97C22⋊C4×C31—41no—
98C22⋊C4×C31—41no—
99C22⋊C4×C31—41no—
100C22⋊C4×C31—41no—
101C22⋊C4×C31—41no—
102C22⋊C4×C31—41no—
103C22⋊C4×C31—41no—
104C22⋊C4×C31—41no—
105C22⋊C4×C31SD1612yes—
106C42×C31—42no—
107#12244C3111yesmaximal, characteristic, direct product factor, Sylow
108C23×C31—21no—
109C23×C31—21no—
110C23×C31—41no—
111C23×C31—41no—
112C23×C31#12011yescharacteristic
113C23×C31C2×SD1611yescharacteristic
114C23×C31AGL(1,ℤ/8)11yescharacteristic
115Q8×C31—81no—
116Q8×C31—81no—
117C2×C4×C31—22no—
118C2×C4×C31—22no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—41no—
122C2×C4×C31—41no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31—41no—
127C2×C4×C31—42no—
128C2×C4×C31—42no—
129C2×C4×C31—41no—
130C2×C4×C31—41no—
131C2×C4×C31—41no—
132C2×C4×C31—41no—
133C2×C4×C31—41no—
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—41no—
141C2×C4×C31—41no—
142C2×C4×C31—41no—
143C2×C4×C31—81no—
144C2×C4×C31—81no—
145C2×C4×C31—81no—
146C2×C4×C31—81no—
147C2×C4×C31—81no—
148C8×C31—81no—
149C8×C31—81no—
150#2722C2×C3111yescharacteristic
151#2062C2×C3111yescharacteristic
152#1608C2×C3111yescharacteristic
153#1608C2×C3111yescharacteristic
154#1344C2×C3111yescharacteristic
155#1344C2×C3111yescharacteristic
156#1326C2×C3111yescharacteristic
157C22×C31—21no—
158C22×C31—21no—
159C22×C31—21no—
160C22×C31—21no—
161C22×C31—41no—
162C22×C31—41no—
163C22×C31—41no—
164C22×C31—41no—
165C22×C31—81no—
166C22×C31#45011yescharacteristic
167C22×C31#45611yescharacteristic
168C22×C31B(4,𝔽2)11yescharacteristic
169C4×C31—42no—
170C4×C31—41no—
171C4×C31—42no—
172C4×C31—41no—
173C4×C31—81no—
174C4×C31—81no—
175C4×C31—82no—
176C4×C31—81no—
177C4×C31—81no—
178C4×C31—81no—
179C4×C31—81no—
180#543C22×C3111yescharacteristic
181C2×#126—21no—
182#483—21no—
183#483—21no—
184#398—21no—
185#398—21no—
186#397C22×C3111yescharacteristic
187#397C22×C3111yescharacteristic
188#389—21no—
189#387—21no—
190#385—21no—
191#385C22×C3111yescharacteristic
192#380—21no—
193#342—21no—
194#342C22×C3111yescharacteristic
195#328—21no—
196#328—21no—
197#323C22×C3111yescharacteristic
198#323C22×C3111yescharacteristic
199C2×C31—21no—
200C2×C31—21no—
201C2×C31—41no—
202C2×C31—41no—
203C2×C31#160411yescharacteristic
204C2×C31#219711yescharacteristic
205C2×C31#220111yescharacteristic
206C23×C4D4×C3111yescharacteristic
207#126—42no—
208#126—42no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8—41no—
212C22⋊Q8—41no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4D4×C3111yescharacteristic
221C2×C4⋊C4D4×C3111yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4D4×C3111yescharacteristic
227C2×C22⋊C4C23×C3111yescharacteristic
228C2×C42—21no—
229#106—41no—
230#106—41no—
231#103—41no—
232#103—41no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—41no—
241#95D4×C3111yescharacteristic
242#95D4×C3111yescharacteristic
243C31#1224411yescharacteristic, direct product factor, Sylow
244C24C2×D4×C3111yescharacteristic
245C2×Q8—21no—
246C2×Q8—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C2×D4×C3111yescharacteristic
259C22×C4C2×D4×C3111yescharacteristic
260C2×C8—41no—
261C2×C8—41no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—81no—
274C4⋊C4—82no—
275C22⋊C4—21no—
276C22⋊C4—22no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4SD16×C3112yes—
286C42—42no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23C31×#12011yescharacteristic
292C23C2×SD16×C3111yescharacteristic
293C23C31×AGL(1,ℤ/8)11yescharacteristic
294Q8—81no—
295Q8—81no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C8—81no—
328C8—81no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22C31×#45011yescharacteristic
339C22C31×#45611yescharacteristic
340C22C31×B(4,𝔽2)11yescharacteristic
341C4—42no—
342C4—41no—
343C4—42no—
344C4—41no—
345C4—81no—
346C4—81no—
347C4—82no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—81no—
352C2—21no—
353C2—21no—
354C2—41no—
355C2—41no—
356C2C31×#160411yescharacteristic
357C2C31×#219711yescharacteristic
358C2C31×#220111yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC24×C31C2×D4
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
107. #12244C31direct (N × H)
243. C31#12244direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27222
C31×#20622
C31×#16082
C31×#13442
C31×#13262
#1224431