Finite Group Databasebeta

← C31×#13608 #21470367

Subgroups — C31×#13608 #21470367

Order 7936.

308 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 86 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×#3223C211yesmaximal, characteristic
2C31×#2133C211yesmaximal, characteristic
3C31×#2129C211yesmaximal, characteristic
4C31×#1414C212yesmaximal, complemented
5C31×#1407C212yesmaximal, complemented
6C31×#541C411yescharacteristic, complemented
7C31×#536C2211yescharacteristic
8C31×#497C2211yescharacteristic
9C31×#494C411yescharacteristic, complemented
10C31×#491C411yescharacteristic, complemented
11C31×#488C2211yescharacteristic
12C31×#464—21no—
13C31×#462—21no—
14C31×#452—21no—
15C31×#451—21no—
16C31×#433C411yescharacteristic, complemented
17C2×C31×#101—21no—
18C31×#410—21no—
19C31×#356—21no—
20C31×#355C2212yes—
21C31×#354—21no—
22C31×#354C2212yes—
23C2×#41×C31D411yescharacteristic
24C22×Q8×C31D411yescharacteristic
25C2×Q16×C31—21no—
26C2×Q16×C31—21no—
27C2×Q16×C31—21no—
28C2×SD16×C31—21no—
29C2×M16×C31—41no—
30C31×C4⋊Q8C2×C411yescharacteristic
31C31×C4⋊Q8C2×C411yescharacteristic
32C31×C4⋊Q8D411yescharacteristic
33C31×#126—41no—
34C31×#125—41no—
35C31×#125C2×C411yescharacteristic
36C31×#124D411yescharacteristic
37C31×#124C2311yescharacteristic
38C31×#123—41no—
39C31×C22⋊Q8—21no—
40C31×C22⋊Q8—41no—
41C4×Q8×C31—21no—
42C31×#117—41no—
43C31×#117C2×C411yescharacteristic
44C31×C4⋊C8—21no—
45C31×#103—21no—
46C31×#103—21no—
47C31×#103—21no—
48C31×#102—21no—
49C31×#101—22no—
50C31×#101—41no—
51C31×#99—22no—
52C31×#99—41no—
53C31×C22⋊C8—21no—
54C31×#97C2×C411yescharacteristic
55C4×C8×C31C2×C411yescharacteristic
56#41×C31—81no—
57C2×Q8×C31—21no—
58C2×Q8×C31—21no—
59C2×Q8×C31—81no—
60C2×Q8×C31C22⋊C411yescharacteristic
61C2×Q8×C31C22⋊C411yescharacteristic
62C2×Q8×C31C2×D411yescharacteristic
63C2×D4×C31—21no—
64C2×D4×C31C2×D411yescharacteristic
65C22×C4×C31—21no—
66C22×C4×C31—21no—
67C22×C4×C31C22⋊C411yescharacteristic
68Q16×C31—81no—
69Q16×C31—81no—
70Q16×C31—81no—
71SD16×C31—81no—
72M16×C31—42no—
73M16×C31—81no—
74C2×C8×C31—21no—
75C2×C8×C31—21no—
76C2×C8×C31—81no—
77C4⋊C4×C31—21no—
78C4⋊C4×C31—21no—
79C4⋊C4×C31—41no—
80C4⋊C4×C31—41no—
81C4⋊C4×C31—41no—
82C4⋊C4×C31—41no—
83C4⋊C4×C31—41no—
84C4⋊C4×C31—41no—
85C4⋊C4×C31—81no—
86C22⋊C4×C31—21no—
87C22⋊C4×C31—41no—
88C22⋊C4×C31—41no—
89C22⋊C4×C31—41no—
90C22⋊C4×C31—42no—
91C42×C31—41no—
92C42×C31—81no—
93C42×C31C22⋊C411yescharacteristic
94C42×C31C22×C411yescharacteristic
95#13608C3111yesmaximal, characteristic, direct product factor, Sylow
96C23×C31—21no—
97C23×C31#9911yescharacteristic
98Q8×C31—41no—
99Q8×C31—41no—
100Q8×C31—41no—
101Q8×C31—41no—
102Q8×C31—41no—
103Q8×C31—81no—
104D4×C31—41no—
105D4×C31—81no—
106C2×C4×C31—21no—
107C2×C4×C31—21no—
108C2×C4×C31—21no—
109C2×C4×C31—21no—
110C2×C4×C31—21no—
111C2×C4×C31—41no—
112C2×C4×C31—41no—
113C2×C4×C31—41no—
114C2×C4×C31—41no—
115C2×C4×C31—41no—
116C2×C4×C31—41no—
117C2×C4×C31—41no—
118C2×C4×C31—82no—
119C2×C4×C31#9911yescharacteristic
120C2×C4×C31C2×C22⋊C411yescharacteristic
121C8×C31—41no—
122C8×C31—41no—
123C8×C31—82no—
124#3223C2×C3111yescharacteristic
125#2133C2×C3111yescharacteristic
126#2129C2×C3111yescharacteristic
127#1414C2×C3112yescomplemented
128#1407C2×C3112yescomplemented
129C22×C31—21no—
130C22×C31—41no—
131C22×C31—41no—
132C22×C31—81no—
133C22×C31C2×#9911yescharacteristic
134C4×C31—21no—
135C4×C31—21no—
136C4×C31—41no—
137C4×C31—41no—
138C4×C31—41no—
139C4×C31—41no—
140C4×C31—41no—
141C4×C31—81no—
142C4×C31—81no—
143C4×C31—162no—
144#541C4×C3111yescharacteristic, complemented
145#536C22×C3111yescharacteristic
146#497C22×C3111yescharacteristic
147#494C4×C3111yescharacteristic, complemented
148#491C4×C3111yescharacteristic, complemented
149#488C22×C3111yescharacteristic
150#464—21no—
151#462—21no—
152#452—21no—
153#451—21no—
154#433C4×C3111yescharacteristic, complemented
155C2×#101—21no—
156#410—21no—
157#356—21no—
158#355C22×C3112yes—
159#354—21no—
160#354C22×C3112yes—
161C2×C31—21no—
162C2×C31—41no—
163C2×C31—81no—
164C2×C31#212711yescharacteristic
165C2×#41D4×C3111yescharacteristic
166C22×Q8D4×C3111yescharacteristic
167C2×Q16—21no—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×SD16—21no—
171C2×M16—41no—
172C4⋊Q8C2×C4×C3111yescharacteristic
173C4⋊Q8C2×C4×C3111yescharacteristic
174C4⋊Q8D4×C3111yescharacteristic
175#126—41no—
176#125—41no—
177#125C2×C4×C3111yescharacteristic
178#124D4×C3111yescharacteristic
179#124C23×C3111yescharacteristic
180#123—41no—
181C22⋊Q8—21no—
182C22⋊Q8—41no—
183C4×Q8—21no—
184#117—41no—
185#117C2×C4×C3111yescharacteristic
186C4⋊C8—21no—
187#103—21no—
188#103—21no—
189#103—21no—
190#102—21no—
191#101—22no—
192#101—41no—
193#99—22no—
194#99—41no—
195C22⋊C8—21no—
196#97C2×C4×C3111yescharacteristic
197C4×C8C2×C4×C3111yescharacteristic
198C31#1360811yescharacteristic, direct product factor, Sylow
199#41—81no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—81no—
203C2×Q8C22⋊C4×C3111yescharacteristic
204C2×Q8C22⋊C4×C3111yescharacteristic
205C2×Q8C2×D4×C3111yescharacteristic
206C2×D4—21no—
207C2×D4C2×D4×C3111yescharacteristic
208C22×C4—21no—
209C22×C4—21no—
210C22×C4C22⋊C4×C3111yescharacteristic
211Q16—81no—
212Q16—81no—
213Q16—81no—
214SD16—81no—
215M16—42no—
216M16—81no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—81no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—81no—
229C22⋊C4—21no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—42no—
234C42—41no—
235C42—81no—
236C42C22⋊C4×C3111yescharacteristic
237C42C22×C4×C3111yescharacteristic
238C23—21no—
239C23C31×#9911yescharacteristic
240Q8—41no—
241Q8—41no—
242Q8—41no—
243Q8—41no—
244Q8—41no—
245Q8—81no—
246D4—41no—
247D4—81no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—82no—
261C2×C4C31×#9911yescharacteristic
262C2×C4C2×C22⋊C4×C3111yescharacteristic
263C8—41no—
264C8—41no—
265C8—82no—
266C22—21no—
267C22—41no—
268C22—41no—
269C22—81no—
270C22C2×C31×#9911yescharacteristic
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—81no—
279C4—81no—
280C4—162no—
281C2—21no—
282C2—41no—
283C2—81no—
284C2C31×#212711yescharacteristic
subgroupG/H
center, socleC2×C31#2127
commutatorC42C22×C4×C31
Frattini, cosocle#124C23×C31
prime radicalC2×#41×C31D4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C22×C31 ◁ C22×C4×C31 ◁ C31×#536 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#1414C2semidirect
5. C31×#1407C2semidirect
6. C31×#541C4semidirect
9. C31×#494C4semidirect
10. C31×#491C4semidirect
16. C31×#433C4semidirect
95. #13608C31direct (N × H)
127. #1414C2×C31semidirect
128. #1407C2×C31semidirect
144. #541C4×C31semidirect
147. #494C4×C31semidirect
148. #491C4×C31semidirect
154. #433C4×C31semidirect
198. C31#13608direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32232
C31×#21332
C31×#21292
C31×#14142
C31×#14072
#1360831