Finite Group Databasebeta

← C31×#23899 #21480658

Subgroups — C31×#23899 #21480658

Order 7936.

1672 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1672 conjugacy classes of subgroups — too many to draw. Only its 282 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×#3486C212yesmaximal, complemented
2C31×#3020C214yesmaximal, complemented
3C31×#2010C214yesmaximal, complemented
4C31×#1912C214yesmaximal, complemented
5C31×#1784C211yesmaximal, characteristic, complemented
6C2×C31×2−1+4C2212yes—
7C23×Q8×C31C2211yescharacteristic
8C31×#575C2214yes—
9D4×Q8×C31—28no—
10C31×#548—28no—
11C31×#536—24no—
12C31×#536C2214yes—
13C31×#525—24no—
14C31×#525C2214yes—
15C31×#519C2212yes—
16C31×#456—28no—
17C31×#454—28no—
18C31×#421C2218yes—
19C2×C31×#101C2214yes—
20C31×#407C2212yes—
21C31×H(3,ℤ/4)C2214yes—
22C31×2−1+4—28no—
23C31×2−1+4—44no—
24C2×#41×C31—24no—
25C2×#41×C31—24no—
26C2×#41×C31C2314yes—
27C22×Q8×C31—28no—
28C22×Q8×C31—24no—
29C22×Q8×C31—42no—
30C22×Q8×C31D412yes—
31C22×Q8×C31C2312yes—
32C23×C4×C31D412yes—
33C23×C4×C31C2311yescharacteristic
34C31×#137—28no—
35C31×AGL(1,ℤ/8)—28no—
36C31×#135—44no—
37C2×M16×C31C2314yes—
38C31×C4⋊Q8—28no—
39C31×C4⋊Q8D418yescomplemented
40C31×#124—28no—
41C31×#124D418yescomplemented
42C31×#123—28no—
43C31×#123—24no—
44C31×C22⋊Q8—28no—
45C31×C22⋊Q8—216no—
46C31×C22⋊Q8—24no—
47C31×C22⋊Q8—28no—
48C31×#121—28no—
49C31×#121—24no—
50C31×#120—24no—
51C4×Q8×C31—28no—
52C4×D4×C31—216no—
53C4×D4×C31—28no—
54C31×#117—24no—
55C31×#117C2314yes—
56C31×#104—216no—
57C31×#101—28no—
58C31×C22⋊C8—42no—
59C24×C31C2×D411yescharacteristic
60#41×C31—28no—
61#41×C31—28no—
62#41×C31—48no—
63#41×C31—416no—
64#41×C31—48no—
65C2×Q8×C31—24no—
66C2×Q8×C31—28no—
67C2×Q8×C31—24no—
68C2×Q8×C31—24no—
69C2×Q8×C31—42no—
70C2×Q8×C31—416no—
71C2×Q8×C31—416no—
72C2×Q8×C31C2×D414yes—
73C2×Q8×C31C2×D418yes—
74C2×D4×C31—28no—
75C2×D4×C31—24no—
76C2×D4×C31—24no—
77C2×D4×C31C2×D414yes—
78C22×C4×C31—28no—
79C22×C4×C31—28no—
80C22×C4×C31—24no—
81C22×C4×C31—24no—
82C22×C4×C31—22no—
83C22×C4×C31—42no—
84C22×C4×C31C2×D411yescharacteristic
85C22×C4×C31C2×D414yes—
86C22×C4×C31C2411yescharacteristic
87Q16×C31—44no—
88SD16×C31—48no—
89D8×C31—44no—
90M16×C31—28no—
91C2×C8×C31—24no—
92C4⋊C4×C31—28no—
93C4⋊C4×C31—28no—
94C4⋊C4×C31—216no—
95C4⋊C4×C31—24no—
96C22⋊C4×C31—28no—
97C22⋊C4×C31—28no—
98C22⋊C4×C31—24no—
99C22⋊C4×C31—216no—
100C42×C31—28no—
101C42×C31C2×D418yes—
102#23899C3111yesmaximal, characteristic, direct product factor, Sylow
103C23×C31—22no—
104C23×C31—21no—
105C23×C31—24no—
106C23×C31—42no—
107C23×C31C22×D411yescharacteristic
108Q8×C31—216no—
109Q8×C31—28no—
110Q8×C31—48no—
111Q8×C31—48no—
112D4×C31—28no—
113D4×C31—48no—
114D4×C31—416no—
115D4×C31—44no—
116D4×C31—44no—
117C2×C4×C31—24no—
118C2×C4×C31—24no—
119C2×C4×C31—28no—
120C2×C4×C31—22no—
121C2×C4×C31—24no—
122C2×C4×C31—21no—
123C2×C4×C31—48no—
124C2×C4×C31—416no—
125C2×C4×C31—44no—
126C2×C4×C31—48no—
127C2×C4×C31#12018yes—
128C2×C4×C31C22×D414yes—
129C2×C4×C31H(5,𝔽2)12yes—
130C8×C31—44no—
131#3486C2×C3112yescomplemented
132#3020C2×C3114yescomplemented
133#2010C2×C3114yescomplemented
134#1912C2×C3114yescomplemented
135#1784C2×C3111yescharacteristic, complemented
136C22×C31—24no—
137C22×C31—22no—
138C22×C31—41no—
139C22×C31—42no—
140C22×C31—41no—
141C22×C31—48no—
142C22×C31—41no—
143C22×C31—44no—
144C22×C31C2×#12012yes—
145C22×C31#53411yescharacteristic
146C4×C31—28no—
147C4×C31—48no—
148C4×C31—41no—
149C4×C31—44no—
150C4×C31D4214yes—
151C2×2−1+4C22×C3112yes—
152C23×Q8C22×C3111yescharacteristic
153#575C22×C3114yes—
154D4×Q8—28no—
155#548—28no—
156#536—24no—
157#536C22×C3114yes—
158#525—24no—
159#525C22×C3114yes—
160#519C22×C3112yes—
161#456—28no—
162#454—28no—
163#421C22×C3118yes—
164C2×#101C22×C3114yes—
165#407C22×C3112yes—
166H(3,ℤ/4)C22×C3114yes—
167C2×C31—22no—
168C2×C31—21no—
169C2×C31—42no—
170C2×C31—84no—
171C2×C31#240311yescharacteristic
1722−1+4—28no—
1732−1+4—44no—
174C2×#41—24no—
175C2×#41—24no—
176C2×#41C23×C3114yes—
177C22×Q8—28no—
178C22×Q8—24no—
179C22×Q8—42no—
180C22×Q8D4×C3112yes—
181C22×Q8C23×C3112yes—
182C23×C4D4×C3112yes—
183C23×C4C23×C3111yescharacteristic
184#137—28no—
185AGL(1,ℤ/8)—28no—
186#135—44no—
187C2×M16C23×C3114yes—
188C4⋊Q8—28no—
189C4⋊Q8D4×C3118yescomplemented
190#124—28no—
191#124D4×C3118yescomplemented
192#123—28no—
193#123—24no—
194C22⋊Q8—28no—
195C22⋊Q8—216no—
196C22⋊Q8—24no—
197C22⋊Q8—28no—
198#121—28no—
199#121—24no—
200#120—24no—
201C4×Q8—28no—
202C4×D4—216no—
203C4×D4—28no—
204#117—24no—
205#117C23×C3114yes—
206#104—216no—
207#101—28no—
208C22⋊C8—42no—
209C31#2389911yescharacteristic, direct product factor, Sylow
210C24C2×D4×C3111yescharacteristic
211#41—28no—
212#41—28no—
213#41—48no—
214#41—416no—
215#41—48no—
216C2×Q8—24no—
217C2×Q8—28no—
218C2×Q8—24no—
219C2×Q8—24no—
220C2×Q8—42no—
221C2×Q8—416no—
222C2×Q8—416no—
223C2×Q8C2×D4×C3114yes—
224C2×Q8C2×D4×C3118yes—
225C2×D4—28no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4C2×D4×C3114yes—
229C22×C4—28no—
230C22×C4—28no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4C2×D4×C3111yescharacteristic
236C22×C4C2×D4×C3114yes—
237C22×C4C24×C3111yescharacteristic
238Q16—44no—
239SD16—48no—
240D8—44no—
241M16—28no—
242C2×C8—24no—
243C4⋊C4—28no—
244C4⋊C4—28no—
245C4⋊C4—216no—
246C4⋊C4—24no—
247C22⋊C4—28no—
248C22⋊C4—28no—
249C22⋊C4—24no—
250C22⋊C4—216no—
251C42—28no—
252C42C2×D4×C3118yes—
253C23—22no—
254C23—21no—
255C23—24no—
256C23—42no—
257C23C22×D4×C3111yescharacteristic
258Q8—216no—
259Q8—28no—
260Q8—48no—
261Q8—48no—
262D4—28no—
263D4—48no—
264D4—416no—
265D4—44no—
266D4—44no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—28no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—21no—
273C2×C4—48no—
274C2×C4—416no—
275C2×C4—44no—
276C2×C4—48no—
277C2×C4C31×#12018yes—
278C2×C4C22×D4×C3114yes—
279C2×C4C31×H(5,𝔽2)12yes—
280C8—44no—
281C22—24no—
282C22—22no—
283C22—41no—
284C22—42no—
285C22—41no—
286C22—48no—
287C22—41no—
288C22—44no—
289C22C2×C31×#12012yes—
290C22C31×#53411yescharacteristic
291C4—28no—
292C4—48no—
293C4—41no—
294C4—44no—
295C4D42×C3114yes—
296C2—22no—
297C2—21no—
298C2—42no—
299C2—84no—
300C2C31×#240311yescharacteristic
subgroupG/H
center, socleC2×C31#2403
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C22×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3486C2semidirect
2. C31×#3020C2semidirect
3. C31×#2010C2semidirect
4. C31×#1912C2semidirect
5. C31×#1784C2semidirect
39. C31×C4⋊Q8D4semidirect
41. C31×#124D4semidirect
102. #23899C31direct (N × H)
131. #3486C2×C31semidirect
132. #3020C2×C31semidirect
133. #2010C2×C31semidirect
134. #1912C2×C31semidirect
135. #1784C2×C31semidirect
189. C4⋊Q8D4×C31semidirect
191. #124D4×C31semidirect
209. C31#23899direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34862
C31×#30202
C31×#20102
C31×#19122
C31×#17842
#2389931