Finite Group Databasebeta

← C31×#7809 #21464568

Subgroups — C31×#7809 #21464568

Order 7936.

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

normal subgroups only

This group has 378 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×#2806C211yesmaximal, characteristic
2C31×#2096C211yesmaximal, characteristic
3C31×#2083C211yesmaximal, characteristic
4C31×#2072C211yesmaximal, characteristic
5C31×#1272C211yesmaximal, characteristic
6C31×#1272C211yesmaximal, characteristic
7C31×#1271C211yesmaximal, characteristic
8C2×C31×#107—21no—
9C2×C31×#106—21no—
10C2×C31×#103—21no—
11C2×C31×#103—21no—
12C31×#398—21no—
13C31×#398—21no—
14C31×#395—21no—
15C31×#391—21no—
16C31×#391C2211yescharacteristic
17C31×#391C2211yescharacteristic
18C31×#389—21no—
19C31×#389C2211yescharacteristic
20C31×#387—21no—
21C31×#384C2211yescharacteristic
22C31×#340—21no—
23C31×#340—21no—
24C31×#336C2211yescharacteristic
25C31×#336C2211yescharacteristic
26C31×#336C2211yescharacteristic
27C22×Q8×C31D411yescharacteristic
28C22×Q8×C31D411yescharacteristic
29C22×C8×C31—21no—
30C22×C8×C31—21no—
31C22×C8×C31—21no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31—21no—
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×C31D411yescharacteristic
41C2×C4⋊C4×C31D411yescharacteristic
42C2×C4⋊C4×C31D411yescharacteristic
43C2×C4⋊C4×C31D411yescharacteristic
44C2×C42×C31—21no—
45C2×C42×C31C2311yescharacteristic
46C31×#107—42no—
47C31×#106—42no—
48C31×#103—42no—
49C31×#103—42no—
50C31×#95—21no—
51C31×#95—21no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C2×Q8×C31—21no—
59C2×Q8×C31—21no—
60C2×Q8×C31—21no—
61C2×Q8×C31—41no—
62C2×Q8×C31—41no—
63C2×Q8×C31—41no—
64C2×Q8×C31—41no—
65C2×Q8×C31Q1612yes—
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—41no—
73C22×C4×C31C2×D411yescharacteristic
74C22×C4×C31C2×D411yescharacteristic
75C22×C4×C31C2×D411yescharacteristic
76C2×C8×C31—22no—
77C2×C8×C31—22no—
78C2×C8×C31—41no—
79C2×C8×C31—41no—
80C2×C8×C31—41no—
81C2×C8×C31—41no—
82C2×C8×C31—41no—
83C2×C8×C31—41no—
84C2×C8×C31—41no—
85C4⋊C4×C31—21no—
86C4⋊C4×C31—41no—
87C4⋊C4×C31—41no—
88C4⋊C4×C31—41no—
89C4⋊C4×C31—41no—
90C4⋊C4×C31—41no—
91C4⋊C4×C31—41no—
92C4⋊C4×C31—41no—
93C4⋊C4×C31—41no—
94C4⋊C4×C31—41no—
95C4⋊C4×C31—41no—
96C4⋊C4×C31—41no—
97C4⋊C4×C31—81no—
98C4⋊C4×C31—81no—
99C4⋊C4×C31—81no—
100C4⋊C4×C31—81no—
101C4⋊C4×C31SD1612yes—
102C42×C31—41no—
103C42×C31—81no—
104#7809C3111yesmaximal, characteristic, direct product factor, Sylow
105C23×C31#12011yescharacteristic
106Q8×C31—42no—
107Q8×C31—42no—
108Q8×C31—81no—
109Q8×C31—81no—
110C2×C4×C31—21no—
111C2×C4×C31—21no—
112C2×C4×C31—21no—
113C2×C4×C31—21no—
114C2×C4×C31—21no—
115C2×C4×C31—21no—
116C2×C4×C31—22no—
117C2×C4×C31—22no—
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—41no—
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—81no—
135C2×C4×C31—81no—
136C2×C4×C31—81no—
137C2×C4×C31C2×SD1611yescharacteristic
138C2×C4×C31C2×Q1611yescharacteristic
139C2×C4×C31#13511yescharacteristic
140C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C31#13711yescharacteristic
142C2×C4×C31#13711yescharacteristic
143C8×C31—42no—
144C8×C31—42no—
145C8×C31—81no—
146#2806C2×C3111yescharacteristic
147#2096C2×C3111yescharacteristic
148#2083C2×C3111yescharacteristic
149#2072C2×C3111yescharacteristic
150#1272C2×C3111yescharacteristic
151#1272C2×C3111yescharacteristic
152#1271C2×C3111yescharacteristic
153C22×C31#44811yescharacteristic
154C22×C31#44911yescharacteristic
155C22×C31#45111yescharacteristic
156C22×C31#45311yescharacteristic
157C22×C31#45511yescharacteristic
158C22×C31#45611yescharacteristic
159C22×C31#45811yescharacteristic
160C4×C31—21no—
161C4×C31—21no—
162C4×C31—21no—
163C4×C31—21no—
164C4×C31—21no—
165C4×C31—21no—
166C4×C31—42no—
167C4×C31—42no—
168C4×C31—81no—
169C4×C31—81no—
170C4×C31—81no—
171C4×C31—81no—
172C4×C31—161no—
173C2×#107—21no—
174C2×#106—21no—
175C2×#103—21no—
176C2×#103—21no—
177#398—21no—
178#398—21no—
179#395—21no—
180#391—21no—
181#391C22×C3111yescharacteristic
182#391C22×C3111yescharacteristic
183#389—21no—
184#389C22×C3111yescharacteristic
185#387—21no—
186#384C22×C3111yescharacteristic
187#340—21no—
188#340—21no—
189#336C22×C3111yescharacteristic
190#336C22×C3111yescharacteristic
191#336C22×C3111yescharacteristic
192C2×C31#161111yescharacteristic
193C2×C31#161211yescharacteristic
194C2×C31#161411yescharacteristic
195C2×C31#162211yescharacteristic
196C2×C31#164511yescharacteristic
197C2×C31#164811yescharacteristic
198C2×C31#166111yescharacteristic
199C22×Q8D4×C3111yescharacteristic
200C22×Q8D4×C3111yescharacteristic
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4D4×C3111yescharacteristic
213C2×C4⋊C4D4×C3111yescharacteristic
214C2×C4⋊C4D4×C3111yescharacteristic
215C2×C4⋊C4D4×C3111yescharacteristic
216C2×C42—21no—
217C2×C42C23×C3111yescharacteristic
218#107—42no—
219#106—42no—
220#103—42no—
221#103—42no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230C31#780911yescharacteristic, direct product factor, Sylow
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8Q16×C3112yes—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—41no—
246C22×C4C2×D4×C3111yescharacteristic
247C22×C4C2×D4×C3111yescharacteristic
248C22×C4C2×D4×C3111yescharacteristic
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C4⋊C4—21no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—81no—
271C4⋊C4—81no—
272C4⋊C4—81no—
273C4⋊C4—81no—
274C4⋊C4SD16×C3112yes—
275C42—41no—
276C42—81no—
277C23C31×#12011yescharacteristic
278Q8—42no—
279Q8—42no—
280Q8—81no—
281Q8—81no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
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—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4C2×SD16×C3111yescharacteristic
310C2×C4C2×Q16×C3111yescharacteristic
311C2×C4C31×#13511yescharacteristic
312C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
313C2×C4C31×#13711yescharacteristic
314C2×C4C31×#13711yescharacteristic
315C8—42no—
316C8—42no—
317C8—81no—
318C22C31×#44811yescharacteristic
319C22C31×#44911yescharacteristic
320C22C31×#45111yescharacteristic
321C22C31×#45311yescharacteristic
322C22C31×#45511yescharacteristic
323C22C31×#45611yescharacteristic
324C22C31×#45811yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—42no—
332C4—42no—
333C4—81no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—161no—
338C2C31×#161111yescharacteristic
339C2C31×#161211yescharacteristic
340C2C31×#161411yescharacteristic
341C2C31×#162211yescharacteristic
342C2C31×#164511yescharacteristic
343C2C31×#164811yescharacteristic
344C2C31×#166111yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
104. #7809C31direct (N × H)
230. C31#7809direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28062
C31×#20962
C31×#20832
C31×#20722
C31×#12722
C31×#12712
#780931