Finite Group Databasebeta

← C31×#10011 #21466770

Subgroups — C31×#10011 #21466770

Order 7936.

496 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3236C211yesmaximal, characteristic
2C31×#3113C211yesmaximal, characteristic
3C31×#2584C211yesmaximal, characteristic
4C31×#1986C211yesmaximal, characteristic
5C31×#1528C211yesmaximal, characteristic
6C31×#1491C211yesmaximal, characteristic
7C31×#1285C211yesmaximal, characteristic
8C31×#533C2211yescharacteristic
9C2×C31×#125C2211yescharacteristic
10C2×C4×Q8×C31C2211yescharacteristic
11C31×#501—21no—
12C31×#491—21no—
13C31×#484—21no—
14C31×#483—21no—
15C31×#483—21no—
16C31×#479—22no—
17C31×#475—21no—
18C31×#462—21no—
19C31×#451—21no—
20C31×#441—22no—
21C31×#432C2211yescharacteristic
22C31×#432C2211yescharacteristic
23C4×M16×C31C2211yescharacteristic
24C31×#398—21no—
25C31×#395—21no—
26C31×#389—21no—
27C31×#389—21no—
28C31×#389—22no—
29C4×C4⋊C4×C31C2211yescharacteristic
30C31×#339—22no—
31C31×#330—22no—
32C31×Q8⋊C8—22no—
33C22×Q8×C31D411yescharacteristic
34C2×Q16×C31—22no—
35C2×M16×C31—21no—
36C31×C4⋊Q8D411yescharacteristic
37C31×C4⋊Q8D411yescharacteristic
38C31×#125—21no—
39C31×#125—21no—
40C31×#125—21no—
41C31×#125—21no—
42C31×#125—21no—
43C31×C22⋊Q8—21no—
44C31×C22⋊Q8—21no—
45C4×Q8×C31—21no—
46C4×Q8×C31—21no—
47C4×Q8×C31—21no—
48C4×Q8×C31—21no—
49C4×Q8×C31—21no—
50C4×Q8×C31—21no—
51C31×#117—21no—
52C2×C4⋊C4×C31—21no—
53C2×C4⋊C4×C31—22no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31—22no—
58C2×C4⋊C4×C31D411yescharacteristic
59C2×C4⋊C4×C31D411yescharacteristic
60C2×C4⋊C4×C31D411yescharacteristic
61C2×C42×C31—21no—
62C2×C42×C31D412yes—
63C2×C42×C31C2311yescharacteristic
64C31×#107—22no—
65C31×#106—22no—
66C31×#106—22no—
67C31×C4⋊C8—21no—
68C31×C4⋊C8D412yes—
69C31×#103—22no—
70C31×#103—22no—
71C31×#103—22no—
72C31×#103—22no—
73C31×C22⋊C8—21no—
74C31×C22⋊C8—21no—
75C31×#97—21no—
76C31×#97—21no—
77C31×#97D412yes—
78C4×C8×C31—21no—
79C31×#95—22no—
80C31×#95—22no—
81C31×#95—21no—
82C2×Q8×C31—21no—
83C2×Q8×C31—21no—
84C2×Q8×C31—21no—
85C2×Q8×C31—21no—
86C2×Q8×C31—42no—
87C22×C4×C31—22no—
88C22×C4×C31—21no—
89C22×C4×C31—21no—
90C22×C4×C31—21no—
91C22×C4×C31—21no—
92C22×C4×C31C2×D411yescharacteristic
93C22×C4×C31C2×D411yescharacteristic
94C22×C4×C31C2×D411yescharacteristic
95Q16×C31—44no—
96M16×C31—22no—
97M16×C31—41no—
98C2×C8×C31—22no—
99C2×C8×C31—22no—
100C2×C8×C31—41no—
101C4⋊C4×C31—24no—
102C4⋊C4×C31—21no—
103C4⋊C4×C31—21no—
104C4⋊C4×C31—21no—
105C4⋊C4×C31—21no—
106C4⋊C4×C31—21no—
107C4⋊C4×C31—21no—
108C4⋊C4×C31—21no—
109C4⋊C4×C31—21no—
110C4⋊C4×C31—21no—
111C4⋊C4×C31—21no—
112C4⋊C4×C31—44no—
113C4⋊C4×C31—41no—
114C4⋊C4×C31—41no—
115C4⋊C4×C31—41no—
116C4⋊C4×C31—41no—
117C4⋊C4×C31—41no—
118C4⋊C4×C31—41no—
119C4⋊C4×C31—41no—
120C4⋊C4×C31—41no—
121C4⋊C4×C31—82no—
122C22⋊C4×C31—41no—
123C42×C31—42no—
124C42×C31—41no—
125C42×C31—41no—
126C42×C31—41no—
127C42×C31C2×D411yescharacteristic
128C42×C31C2×D411yescharacteristic
129C42×C31C2×D411yescharacteristic
130C42×C31#4111yescharacteristic
131#10011C3111yesmaximal, characteristic, direct product factor, Sylow
132C23×C31#12011yescharacteristic
133Q8×C31—22no—
134Q8×C31—22no—
135Q8×C31—41no—
136Q8×C31—41no—
137Q8×C31—42no—
138Q8×C31—42no—
139C2×C4×C31—21no—
140C2×C4×C31—21no—
141C2×C4×C31—22no—
142C2×C4×C31—24no—
143C2×C4×C31—22no—
144C2×C4×C31—41no—
145C2×C4×C31—42no—
146C2×C4×C31—41no—
147C2×C4×C31—41no—
148C2×C4×C31—42no—
149C2×C4×C31—42no—
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—41no—
158C2×C4×C31—41no—
159C2×C4×C31#12011yescharacteristic
160C2×C4×C31#12011yescharacteristic
161C2×C4×C31#12111yescharacteristic
162C2×C4×C31#12111yescharacteristic
163C2×C4×C31#12111yescharacteristic
164C2×C4×C31#12711yescharacteristic
165C8×C31—42no—
166C8×C31—42no—
167C8×C31—42no—
168#3236C2×C3111yescharacteristic
169#3113C2×C3111yescharacteristic
170#2584C2×C3111yescharacteristic
171#1986C2×C3111yescharacteristic
172#1528C2×C3111yescharacteristic
173#1491C2×C3111yescharacteristic
174#1285C2×C3111yescharacteristic
175C22×C31—21no—
176C22×C31—21no—
177C22×C31#39211yescharacteristic
178C22×C31#45511yescharacteristic
179C22×C31#45611yescharacteristic
180C4×C31—21no—
181C4×C31—21no—
182C4×C31—21no—
183C4×C31—21no—
184C4×C31—21no—
185C4×C31—44no—
186C4×C31—42no—
187C4×C31—81no—
188C4×C31—81no—
189C4×C31—81no—
190C4×C31—81no—
191C4×C31—81no—
192C4×C31#45612yes—
193#533C22×C3111yescharacteristic
194C2×#125C22×C3111yescharacteristic
195C2×C4×Q8C22×C3111yescharacteristic
196#501—21no—
197#491—21no—
198#484—21no—
199#483—21no—
200#483—21no—
201#479—22no—
202#475—21no—
203#462—21no—
204#451—21no—
205#441—22no—
206#432C22×C3111yescharacteristic
207#432C22×C3111yescharacteristic
208C4×M16C22×C3111yescharacteristic
209#398—21no—
210#395—21no—
211#389—21no—
212#389—21no—
213#389—22no—
214C4×C4⋊C4C22×C3111yescharacteristic
215#339—22no—
216#330—22no—
217Q8⋊C8—22no—
218C2×C31—21no—
219C2×C31—21no—
220C2×C31#200511yescharacteristic
221C2×C31#200711yescharacteristic
222C2×C31#201011yescharacteristic
223C22×Q8D4×C3111yescharacteristic
224C2×Q16—22no—
225C2×M16—21no—
226C4⋊Q8D4×C3111yescharacteristic
227C4⋊Q8D4×C3111yescharacteristic
228#125—21no—
229#125—21no—
230#125—21no—
231#125—21no—
232#125—21no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241#117—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4D4×C3111yescharacteristic
249C2×C4⋊C4D4×C3111yescharacteristic
250C2×C4⋊C4D4×C3111yescharacteristic
251C2×C42—21no—
252C2×C42D4×C3112yes—
253C2×C42C23×C3111yescharacteristic
254#107—22no—
255#106—22no—
256#106—22no—
257C4⋊C8—21no—
258C4⋊C8D4×C3112yes—
259#103—22no—
260#103—22no—
261#103—22no—
262#103—22no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265#97—21no—
266#97—21no—
267#97D4×C3112yes—
268C4×C8—21no—
269#95—22no—
270#95—22no—
271#95—21no—
272C31#1001111yescharacteristic, direct product factor, Sylow
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—42no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4C2×D4×C3111yescharacteristic
284C22×C4C2×D4×C3111yescharacteristic
285C22×C4C2×D4×C3111yescharacteristic
286Q16—44no—
287M16—22no—
288M16—41no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—41no—
292C4⋊C4—24no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—44no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—82no—
313C22⋊C4—41no—
314C42—42no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42C2×D4×C3111yescharacteristic
319C42C2×D4×C3111yescharacteristic
320C42C2×D4×C3111yescharacteristic
321C42#41×C3111yescharacteristic
322C23C31×#12011yescharacteristic
323Q8—22no—
324Q8—22no—
325Q8—41no—
326Q8—41no—
327Q8—42no—
328Q8—42no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—42no—
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×C4C31×#12011yescharacteristic
350C2×C4C31×#12011yescharacteristic
351C2×C4C31×#12111yescharacteristic
352C2×C4C31×#12111yescharacteristic
353C2×C4C31×#12111yescharacteristic
354C2×C4C31×#12711yescharacteristic
355C8—42no—
356C8—42no—
357C8—42no—
358C22—21no—
359C22—21no—
360C22C31×#39211yescharacteristic
361C22C31×#45511yescharacteristic
362C22C31×#45611yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—44no—
369C4—42no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4C31×#45612yes—
376C2—21no—
377C2—21no—
378C2C31×#200511yescharacteristic
379C2C31×#200711yescharacteristic
380C2C31×#201011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC23×C31#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
131. #10011C31direct (N × H)
272. C31#10011direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32362
C31×#31132
C31×#25842
C31×#19862
C31×#15282
C31×#14912
C31×#12852
#1001131