Finite Group Databasebeta

← C31×#12265 #21469024

Subgroups — C31×#12265 #21469024

Order 7936.

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

normal subgroups only

This group has 362 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×#2771C211yesmaximal, characteristic
2C31×#2030C211yesmaximal, characteristic
3C31×#1614C211yesmaximal, characteristic
4C31×#1608C211yesmaximal, characteristic
5C31×#1352C211yesmaximal, characteristic
6C31×#1344C211yesmaximal, characteristic
7C31×#1317C211yesmaximal, characteristic
8C31×#543C2211yescharacteristic
9C31×#484—21no—
10C31×#483—21no—
11C31×#400—21no—
12C31×#400—21no—
13C31×#399C2211yescharacteristic
14C31×#397C2211yescharacteristic
15C31×#389—21no—
16C31×#388—21no—
17C31×#387—21no—
18C31×#385—21no—
19C31×#381—21no—
20C31×#381C2211yescharacteristic
21C31×#342—21no—
22C31×#342C2211yescharacteristic
23C31×#328—21no—
24C31×#328—21no—
25C31×#323C2211yescharacteristic
26C31×#323C2211yescharacteristic
27C23×C4×C31D411yescharacteristic
28C31×C22⋊Q8—21no—
29C31×C22⋊Q8—21no—
30C31×C22⋊Q8—41no—
31C31×C22⋊Q8—41no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31D411yescharacteristic
37C2×C4⋊C4×C31D411yescharacteristic
38C2×C22⋊C4×C31—21no—
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31D411yescharacteristic
43C2×C22⋊C4×C31C2311yescharacteristic
44C2×C42×C31—21no—
45C31×#107—41no—
46C31×#106—41no—
47C31×#103—41no—
48C31×#103—41no—
49C31×C22⋊C8—21no—
50C31×C22⋊C8—21no—
51C31×#95—21no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—41no—
59C31×#95—41no—
60C31×#95D411yescharacteristic
61C31×#95D411yescharacteristic
62C24×C31C2×D411yescharacteristic
63C2×Q8×C31—21no—
64C2×Q8×C31—21no—
65C22×C4×C31—21no—
66C22×C4×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—41no—
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—41no—
87C4⋊C4×C31—41no—
88C4⋊C4×C31—41no—
89C4⋊C4×C31—41no—
90C4⋊C4×C31—81no—
91C4⋊C4×C31—81no—
92C22⋊C4×C31—41no—
93C22⋊C4×C31—42no—
94C22⋊C4×C31—41no—
95C22⋊C4×C31—41no—
96C22⋊C4×C31—41no—
97C22⋊C4×C31—41no—
98C22⋊C4×C31—41no—
99C22⋊C4×C31—41no—
100C22⋊C4×C31—41no—
101C42×C31—81no—
102#12265C3111yesmaximal, characteristic, direct product factor, Sylow
103C23×C31—21no—
104C23×C31—21no—
105C23×C31—41no—
106C23×C31—41no—
107C23×C31#12011yescharacteristic
108C23×C31#13511yescharacteristic
109C23×C31#13711yescharacteristic
110Q8×C31—81no—
111Q8×C31—81no—
112C2×C4×C31—22no—
113C2×C4×C31—21no—
114C2×C4×C31—21no—
115C2×C4×C31—41no—
116C2×C4×C31—41no—
117C2×C4×C31—41no—
118C2×C4×C31—41no—
119C2×C4×C31—41no—
120C2×C4×C31—41no—
121C2×C4×C31—41no—
122C2×C4×C31—41no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—42no—
126C2×C4×C31—41no—
127C2×C4×C31—41no—
128C2×C4×C31—41no—
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—81no—
138C2×C4×C31—81no—
139C2×C4×C31—81no—
140C2×C4×C31—81no—
141C2×C4×C31—81no—
142C2×C4×C31—81no—
143C8×C31—81no—
144C8×C31—81no—
145#2771C2×C3111yescharacteristic
146#2030C2×C3111yescharacteristic
147#1614C2×C3111yescharacteristic
148#1608C2×C3111yescharacteristic
149#1352C2×C3111yescharacteristic
150#1344C2×C3111yescharacteristic
151#1317C2×C3111yescharacteristic
152C22×C31—21no—
153C22×C31—21no—
154C22×C31—21no—
155C22×C31—21no—
156C22×C31—41no—
157C22×C31—41no—
158C22×C31—41no—
159C22×C31—41no—
160C22×C31—81no—
161C22×C31#45211yescharacteristic
162C22×C31#45511yescharacteristic
163C22×C31B(4,𝔽2)11yescharacteristic
164C4×C31—42no—
165C4×C31—41no—
166C4×C31—41no—
167C4×C31—81no—
168C4×C31—81no—
169C4×C31—81no—
170C4×C31—81no—
171C4×C31—81no—
172C4×C31—81no—
173C4×C31—81no—
174C4×C31—161no—
175#543C22×C3111yescharacteristic
176#484—21no—
177#483—21no—
178#400—21no—
179#400—21no—
180#399C22×C3111yescharacteristic
181#397C22×C3111yescharacteristic
182#389—21no—
183#388—21no—
184#387—21no—
185#385—21no—
186#381—21no—
187#381C22×C3111yescharacteristic
188#342—21no—
189#342C22×C3111yescharacteristic
190#328—21no—
191#328—21no—
192#323C22×C3111yescharacteristic
193#323C22×C3111yescharacteristic
194C2×C31—21no—
195C2×C31—21no—
196C2×C31—41no—
197C2×C31—41no—
198C2×C31#160611yescharacteristic
199C2×C31#219811yescharacteristic
200C2×C31#220111yescharacteristic
201C23×C4D4×C3111yescharacteristic
202C22⋊Q8—21no—
203C22⋊Q8—21no—
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4D4×C3111yescharacteristic
211C2×C4⋊C4D4×C3111yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4D4×C3111yescharacteristic
217C2×C22⋊C4C23×C3111yescharacteristic
218C2×C42—21no—
219#107—41no—
220#106—41no—
221#103—41no—
222#103—41no—
223C22⋊C8—21no—
224C22⋊C8—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—41no—
233#95—41no—
234#95D4×C3111yescharacteristic
235#95D4×C3111yescharacteristic
236C31#1226511yescharacteristic, direct product factor, Sylow
237C24C2×D4×C3111yescharacteristic
238C2×Q8—21no—
239C2×Q8—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4C2×D4×C3111yescharacteristic
254C22×C4C2×D4×C3111yescharacteristic
255C2×C8—41no—
256C2×C8—41no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—81no—
266C4⋊C4—81no—
267C22⋊C4—41no—
268C22⋊C4—42no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C42—81no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23C31×#12011yescharacteristic
282C23C31×#13511yescharacteristic
283C23C31×#13711yescharacteristic
284Q8—81no—
285Q8—81no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C8—81no—
318C8—81no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22C31×#45211yescharacteristic
329C22C31×#45511yescharacteristic
330C22C31×B(4,𝔽2)11yescharacteristic
331C4—42no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—161no—
342C2—21no—
343C2—21no—
344C2—41no—
345C2—41no—
346C2C31×#160611yescharacteristic
347C2C31×#219811yescharacteristic
348C2C31×#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
102. #12265C31direct (N × H)
236. C31#12265direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27712
C31×#20302
C31×#16142
C31×#16082
C31×#13522
C31×#13442
C31×#13172
#1226531