Finite Group Databasebeta

← C31×#10101 #21466860

Subgroups — C31×#10101 #21466860

Order 7936.

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

normal subgroups only

This group has 474 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×#3165C211yesmaximal, characteristic
3C31×#2660C211yesmaximal, characteristic
4C31×#1556C211yesmaximal, characteristic
5C31×#1528C211yesmaximal, characteristic
6C31×#1506C211yesmaximal, characteristic
7C31×#1300C211yesmaximal, characteristic
8C31×#533C2211yescharacteristic
9C2×C31×C4⋊Q8C2211yescharacteristic
10C2×C31×#125C2211yescharacteristic
11C31×#501—21no—
12C31×#501—22no—
13C31×#500—21no—
14C31×#498—21no—
15C31×#491—21no—
16C31×#484—21no—
17C31×#483—21no—
18C31×#479—22no—
19C31×#432C2211yescharacteristic
20C31×#432C2211yescharacteristic
21C31×#428—22no—
22C4×M16×C31C2211yescharacteristic
23C31×#398—22no—
24C31×#395—22no—
25C31×#384—21no—
26C31×#382—21no—
27C31×#376C2211yescharacteristic
28C31×#332—22no—
29C31×#330—22no—
30C31×#328—22no—
31C22×Q8×C31—21no—
32C2×M16×C31—21no—
33C31×C4⋊Q8—21no—
34C31×C4⋊Q8—21no—
35C31×C4⋊Q8—21no—
36C31×C4⋊Q8—21no—
37C31×C4⋊Q8D411yescharacteristic
38C31×C4⋊Q8D411yescharacteristic
39C31×#125—21no—
40C31×#125—21no—
41C31×#125—21no—
42C31×#125—21no—
43C31×#125—21no—
44C31×C22⋊Q8—21no—
45C31×C22⋊Q8—21no—
46C4×Q8×C31—21no—
47C4×Q8×C31—21no—
48C31×#117—21no—
49C2×C4⋊C4×C31—21no—
50C2×C4⋊C4×C31—21no—
51C2×C4⋊C4×C31—21no—
52C2×C4⋊C4×C31—24no—
53C2×C4⋊C4×C31D412yes—
54C2×C4⋊C4×C31D411yescharacteristic
55C2×C4⋊C4×C31D411yescharacteristic
56C2×C42×C31—21no—
57C2×C42×C31C2311yescharacteristic
58C31×#107—24no—
59C31×#107—22no—
60C31×#106—24no—
61C31×#106—22no—
62C31×C4⋊C8—21no—
63C31×C4⋊C8D412yes—
64C31×#103—22no—
65C31×#103—22no—
66C31×C22⋊C8—21no—
67C31×C22⋊C8—21no—
68C31×#97—21no—
69C31×#97—21no—
70C31×#97D412yes—
71C4×C8×C31—21no—
72C31×#95—24no—
73C31×#95—21no—
74C31×#95D412yes—
75C2×Q8×C31—21no—
76C2×Q8×C31—21no—
77C2×Q8×C31—41no—
78C2×Q8×C31—41no—
79C2×Q8×C31—82no—
80C22×C4×C31—22no—
81C22×C4×C31—21no—
82C22×C4×C31—22no—
83C22×C4×C31—21no—
84C22×C4×C31C2×D411yescharacteristic
85C22×C4×C31C2×D411yescharacteristic
86C22×C4×C31C2×D411yescharacteristic
87M16×C31—24no—
88C2×C8×C31—22no—
89C2×C8×C31—22no—
90C2×C8×C31—41no—
91C4⋊C4×C31—21no—
92C4⋊C4×C31—21no—
93C4⋊C4×C31—21no—
94C4⋊C4×C31—22no—
95C4⋊C4×C31—21no—
96C4⋊C4×C31—21no—
97C4⋊C4×C31—21no—
98C4⋊C4×C31—21no—
99C4⋊C4×C31—22no—
100C4⋊C4×C31—21no—
101C4⋊C4×C31—41no—
102C4⋊C4×C31—41no—
103C4⋊C4×C31—41no—
104C4⋊C4×C31—41no—
105C4⋊C4×C31—41no—
106C4⋊C4×C31—41no—
107C4⋊C4×C31—41no—
108C4⋊C4×C31—41no—
109C4⋊C4×C31—84no—
110C22⋊C4×C31—41no—
111C42×C31—24no—
112C42×C31—41no—
113C42×C31C2×D411yescharacteristic
114C42×C31C2×D411yescharacteristic
115C42×C31C2×D411yescharacteristic
116C42×C31#4111yescharacteristic
117#10101C3111yesmaximal, characteristic, direct product factor, Sylow
118C23×C31#12011yescharacteristic
119Q8×C31—42no—
120Q8×C31—81no—
121Q8×C31—82no—
122Q8×C31—82no—
123C2×C4×C31—22no—
124C2×C4×C31—22no—
125C2×C4×C31—24no—
126C2×C4×C31—41no—
127C2×C4×C31—42no—
128C2×C4×C31—42no—
129C2×C4×C31—42no—
130C2×C4×C31—44no—
131C2×C4×C31—41no—
132C2×C4×C31—41no—
133C2×C4×C31—41no—
134C2×C4×C31—41no—
135C2×C4×C31—42no—
136C2×C4×C31—42no—
137C2×C4×C31#12011yescharacteristic
138C2×C4×C31#12011yescharacteristic
139C2×C4×C31#12111yescharacteristic
140C2×C4×C31#12111yescharacteristic
141C2×C4×C31#12111yescharacteristic
142C2×C4×C31#12711yescharacteristic
143C8×C31—24no—
144C8×C31—42no—
145C8×C31—81no—
146#3236C2×C3111yescharacteristic
147#3165C2×C3111yescharacteristic
148#2660C2×C3111yescharacteristic
149#1556C2×C3111yescharacteristic
150#1528C2×C3111yescharacteristic
151#1506C2×C3111yescharacteristic
152#1300C2×C3111yescharacteristic
153C22×C31—21no—
154C22×C31—21no—
155C22×C31#39211yescharacteristic
156C22×C31#45612yes—
157C4×C31—21no—
158C4×C31—21no—
159C4×C31—21no—
160C4×C31—21no—
161C4×C31—21no—
162C4×C31—44no—
163C4×C31—81no—
164C4×C31—81no—
165C4×C31—81no—
166C4×C31—82no—
167C4×C31—81no—
168C4×C31#45612yes—
169#533C22×C3111yescharacteristic
170C2×C4⋊Q8C22×C3111yescharacteristic
171C2×#125C22×C3111yescharacteristic
172#501—21no—
173#501—22no—
174#500—21no—
175#498—21no—
176#491—21no—
177#484—21no—
178#483—21no—
179#479—22no—
180#432C22×C3111yescharacteristic
181#432C22×C3111yescharacteristic
182#428—22no—
183C4×M16C22×C3111yescharacteristic
184#398—22no—
185#395—22no—
186#384—21no—
187#382—21no—
188#376C22×C3111yescharacteristic
189#332—22no—
190#330—22no—
191#328—22no—
192C2×C31—22no—
193C2×C31#200511yescharacteristic
194C2×C31#200511yescharacteristic
195C2×C31#200811yescharacteristic
196C22×Q8—21no—
197C2×M16—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8D4×C3111yescharacteristic
203C4⋊Q8D4×C3111yescharacteristic
204#125—21no—
205#125—21no—
206#125—21no—
207#125—21no—
208#125—21no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C4×Q8—21no—
212C4×Q8—21no—
213#117—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4D4×C3111yescharacteristic
219C2×C4⋊C4D4×C3111yescharacteristic
220C2×C4⋊C4D4×C3112yes—
221C2×C42—21no—
222C2×C42C23×C3111yescharacteristic
223#107—24no—
224#107—22no—
225#106—24no—
226#106—22no—
227C4⋊C8—21no—
228C4⋊C8D4×C3112yes—
229#103—22no—
230#103—22no—
231C22⋊C8—21no—
232C22⋊C8—21no—
233#97—21no—
234#97—21no—
235#97D4×C3112yes—
236C4×C8—21no—
237#95—24no—
238#95—21no—
239#95D4×C3112yes—
240C31#1010111yescharacteristic, direct product factor, Sylow
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—82no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4C2×D4×C3111yescharacteristic
251C22×C4C2×D4×C3111yescharacteristic
252C22×C4C2×D4×C3111yescharacteristic
253M16—24no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—41no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—22no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—22no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—84no—
276C22⋊C4—41no—
277C42—24no—
278C42—41no—
279C42C2×D4×C3111yescharacteristic
280C42C2×D4×C3111yescharacteristic
281C42C2×D4×C3111yescharacteristic
282C42#41×C3111yescharacteristic
283C23C31×#12011yescharacteristic
284Q8—42no—
285Q8—81no—
286Q8—82no—
287Q8—82no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4C31×#12011yescharacteristic
303C2×C4C31×#12011yescharacteristic
304C2×C4C31×#12111yescharacteristic
305C2×C4C31×#12111yescharacteristic
306C2×C4C31×#12111yescharacteristic
307C2×C4C31×#12711yescharacteristic
308C8—24no—
309C8—42no—
310C8—81no—
311C22—21no—
312C22—21no—
313C22C31×#39211yescharacteristic
314C22C31×#45612yes—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—44no—
321C4—81no—
322C4—81no—
323C4—82no—
324C4—81no—
325C4—81no—
326C4C31×#45612yes—
327C2—22no—
328C2C31×#200511yescharacteristic
329C2C31×#200511yescharacteristic
330C2C31×#200811yescharacteristic
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
117. #10101C31direct (N × H)
240. C31#10101direct (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×#31652
C31×#26602
C31×#15562
C31×#15282
C31×#15062
C31×#13002
#1010131