Finite Group Databasebeta

← C31×#7808 #21464567

Subgroups — C31×#7808 #21464567

Order 7936.

404 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2727C211yesmaximal, characteristic
2C31×#2099C211yesmaximal, characteristic, complemented
3C31×#2084C211yesmaximal, characteristic, complemented
4C31×#2071C211yesmaximal, characteristic
5C31×#1271C211yesmaximal, characteristic, complemented
6C31×#1271C211yesmaximal, characteristic, complemented
7C31×#1270C211yesmaximal, characteristic
8C2×C31×#126—21no—
9C2×C31×#106—21no—
10C2×C31×#102—21no—
11C31×#399—21no—
12C31×#396—21no—
13C31×#394—21no—
14C31×#390C2211yescharacteristic
15C31×#389—21no—
16C31×#389C2211yescharacteristic
17C31×#389C2211yescharacteristic
18C31×#388—21no—
19C31×#384C2211yescharacteristic
20C31×#340—21no—
21C31×#340—21no—
22C31×#340—21no—
23C31×#340—21no—
24C31×#336C2211yescharacteristic
25C31×#336C2211yescharacteristic
26C31×#336C2211yescharacteristic
27C22×D4×C31D411yescharacteristic
28C22×C8×C31—21no—
29C22×C8×C31—21no—
30C22×C8×C31—21no—
31C31×#126—42no—
32C31×#126—42no—
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×C4⋊C4×C31D411yescharacteristic
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—21no—
48C2×C22⋊C4×C31—21no—
49C2×C22⋊C4×C31—21no—
50C2×C22⋊C4×C31—21no—
51C2×C42×C31—21no—
52C2×C42×C31C2311yescharacteristic
53C31×#106—42no—
54C31×#102—42no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C24×C31—21no—
59C2×D4×C31—21no—
60C2×D4×C31—21no—
61C2×D4×C31—41no—
62C2×D4×C31—41no—
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31—21no—
66C22×C4×C31—21no—
67C22×C4×C31—21no—
68C22×C4×C31—41no—
69C22×C4×C31C2×D411yescharacteristic
70C22×C4×C31C2×D411yescharacteristic
71C22×C4×C31C2×D411yescharacteristic
72C2×C8×C31—22no—
73C2×C8×C31—41no—
74C2×C8×C31—41no—
75C2×C8×C31—41no—
76C2×C8×C31—41no—
77C2×C8×C31—41no—
78C2×C8×C31—41no—
79C2×C8×C31—41no—
80C2×C8×C31—41no—
81C4⋊C4×C31—21no—
82C4⋊C4×C31—21no—
83C4⋊C4×C31—21no—
84C4⋊C4×C31—42no—
85C4⋊C4×C31—41no—
86C4⋊C4×C31—41no—
87C4⋊C4×C31—41no—
88C4⋊C4×C31—42no—
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×C31SD1612yes—
99C22⋊C4×C31—42no—
100C22⋊C4×C31—42no—
101C22⋊C4×C31—41no—
102C22⋊C4×C31—41no—
103C22⋊C4×C31—41no—
104C22⋊C4×C31—41no—
105C22⋊C4×C31—81no—
106C22⋊C4×C31—81no—
107C42×C31—41no—
108C42×C31—42no—
109#7808C3111yesmaximal, characteristic, direct product factor, Sylow
110C23×C31—41no—
111C23×C31—41no—
112C23×C31—41no—
113C23×C31—41no—
114C23×C31—41no—
115C23×C31—41no—
116C23×C31—41no—
117C23×C31#12011yescharacteristic
118D4×C31—42no—
119D4×C31—81no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—22no—
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—41no—
138C2×C4×C31—41no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
141C2×C4×C31—81no—
142C2×C4×C31—81no—
143C2×C4×C31—81no—
144C2×C4×C31C2×SD1611yescharacteristic
145C2×C4×C31#13511yescharacteristic
146C2×C4×C31#13511yescharacteristic
147C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
150C8×C31—42no—
151C8×C31—81no—
152C8×C31—81no—
153#2727C2×C3111yescharacteristic
154#2099C2×C3111yescharacteristic, complemented
155#2084C2×C3111yescharacteristic, complemented
156#2071C2×C3111yescharacteristic
157#1271C2×C3111yescharacteristic, complemented
158#1271C2×C3111yescharacteristic, complemented
159#1270C2×C3111yescharacteristic
160C22×C31—81no—
161C22×C31—81no—
162C22×C31—81no—
163C22×C31—81no—
164C22×C31—81no—
165C22×C31—81no—
166C22×C31—81no—
167C22×C31#44911yescharacteristic
168C22×C31#44911yescharacteristic
169C22×C31#45011yescharacteristic
170C22×C31#45411yescharacteristic
171C22×C31#45511yescharacteristic
172C22×C31#45611yescharacteristic
173C22×C31#45811yescharacteristic
174C4×C31—21no—
175C4×C31—21no—
176C4×C31—21no—
177C4×C31—21no—
178C4×C31—21no—
179C4×C31—21no—
180C4×C31—42no—
181C4×C31—81no—
182C4×C31—81no—
183C4×C31—81no—
184C4×C31—81no—
185C4×C31—161no—
186C2×#126—21no—
187C2×#106—21no—
188C2×#102—21no—
189#399—21no—
190#396—21no—
191#394—21no—
192#390C22×C3111yescharacteristic
193#389—21no—
194#389C22×C3111yescharacteristic
195#389C22×C3111yescharacteristic
196#388—21no—
197#384C22×C3111yescharacteristic
198#340—21no—
199#340—21no—
200#340—21no—
201#340—21no—
202#336C22×C3111yescharacteristic
203#336C22×C3111yescharacteristic
204#336C22×C3111yescharacteristic
205C2×C31—161no—
206C2×C31#160911yescharacteristic
207C2×C31#161211yescharacteristic
208C2×C31#161611yescharacteristic
209C2×C31#162811yescharacteristic
210C2×C31#164211yescharacteristic
211C2×C31#165411yescharacteristic
212C2×C31#166411yescharacteristic
213C22×D4D4×C3111yescharacteristic
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217#126—42no—
218#126—42no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4D4×C3111yescharacteristic
227C2×C4⋊C4D4×C3111yescharacteristic
228C2×C4⋊C4D4×C3111yescharacteristic
229C2×C4⋊C4D4×C3111yescharacteristic
230C2×C4⋊C4D4×C3111yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—21no—
238C2×C42C23×C3111yescharacteristic
239#106—42no—
240#102—42no—
241#95—21no—
242#95—21no—
243#95—21no—
244C31#780811yescharacteristic, direct product factor, Sylow
245C24—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—41no—
249C2×D4—41no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4C2×D4×C3111yescharacteristic
257C22×C4C2×D4×C3111yescharacteristic
258C22×C4C2×D4×C3111yescharacteristic
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—81no—
285C4⋊C4SD16×C3112yes—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—81no—
293C22⋊C4—81no—
294C42—41no—
295C42—42no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23C31×#12011yescharacteristic
304D4—42no—
305D4—81no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
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—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4C2×SD16×C3111yescharacteristic
331C2×C4C31×#13511yescharacteristic
332C2×C4C31×#13511yescharacteristic
333C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
334C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
335C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
336C8—42no—
337C8—81no—
338C8—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22C31×#44911yescharacteristic
347C22C31×#44911yescharacteristic
348C22C31×#45011yescharacteristic
349C22C31×#45411yescharacteristic
350C22C31×#45511yescharacteristic
351C22C31×#45611yescharacteristic
352C22C31×#45811yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—42no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—161no—
365C2—161no—
366C2C31×#160911yescharacteristic
367C2C31×#161211yescharacteristic
368C2C31×#161611yescharacteristic
369C2C31×#162811yescharacteristic
370C2C31×#164211yescharacteristic
371C2C31×#165411yescharacteristic
372C2C31×#166411yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
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
2. C31×#2099C2semidirect
3. C31×#2084C2semidirect
5. C31×#1271C2semidirect
6. C31×#1271C2semidirect
109. #7808C31direct (N × H)
154. #2099C2×C31semidirect
155. #2084C2×C31semidirect
157. #1271C2×C31semidirect
158. #1271C2×C31semidirect
244. C31#7808direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27272
C31×#20992
C31×#20842
C31×#20712
C31×#12712
C31×#12702
#780831