Finite Group Databasebeta

← C31×#11700 #21468459

Subgroups — C31×#11700 #21468459

Order 7936.

452 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2621C211yesmaximal, characteristic
2C31×#1742C211yesmaximal, characteristic
3C31×#1458C211yesmaximal, characteristic
4C31×#1346C214yesmaximal, complemented
5C22×M16×C31—21no—
6C2×C31×C4⋊Q8C411yescharacteristic
7C2×C31×C22⋊Q8—21no—
8C2×C31×#117C2211yescharacteristic
9C31×#431—21no—
10C31×#398—21no—
11C31×#398C2212yes—
12C31×#397—21no—
13C31×#397C412yes—
14C31×#391—21no—
15C31×#382C411yescharacteristic
16C31×#343—21no—
17C31×#343C2212yes—
18C31×#324C2212yes—
19C31×#323—21no—
20C22×Q8×C31—21no—
21C23×C4×C31D411yescharacteristic
22C2×M16×C31—24no—
23C2×M16×C31—22no—
24C2×M16×C31—41no—
25C2×M16×C31—42no—
26C22×C8×C31—41no—
27C31×C4⋊Q8—41no—
28C31×C4⋊Q8—41no—
29C31×C22⋊Q8—42no—
30C31×C22⋊Q8—42no—
31C31×#117—41no—
32C31×#117—41no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31—22no—
36C2×C4⋊C4×C31C2×C412yes—
37C2×C4⋊C4×C31D411yescharacteristic
38C2×C4⋊C4×C31C2311yescharacteristic
39C2×C22⋊C4×C31—22no—
40C2×C22⋊C4×C31C2×C411yescharacteristic
41C2×C22⋊C4×C31D411yescharacteristic
42C2×C42×C31C2×C411yescharacteristic
43C2×C42×C31D411yescharacteristic
44C31×C4⋊C8—41no—
45C31×C22⋊C8—22no—
46C4×C8×C31—41no—
47C31×#95—21no—
48C31×#95—22no—
49C31×#95—22no—
50C31×#95C2×C412yes—
51C24×C31C22⋊C411yescharacteristic
52C2×Q8×C31—41no—
53C2×Q8×C31—41no—
54C2×Q8×C31—42no—
55C2×Q8×C31—42no—
56C22×C4×C31—21no—
57C22×C4×C31—21no—
58C22×C4×C31—22no—
59C22×C4×C31—22no—
60C22×C4×C31—22no—
61C22×C4×C31C22⋊C411yescharacteristic
62C22×C4×C31C22⋊C411yescharacteristic
63C22×C4×C31C22⋊C411yescharacteristic
64C22×C4×C31D812yes—
65C22×C4×C31SD1612yes—
66C22×C4×C31C22×C411yescharacteristic
67C22×C4×C31C2×D411yescharacteristic
68C22×C4×C31C2×D411yescharacteristic
69M16×C31—24no—
70M16×C31—44no—
71M16×C31—42no—
72C2×C8×C31—42no—
73C2×C8×C31—44no—
74C2×C8×C31—42no—
75C4⋊C4×C31—41no—
76C4⋊C4×C31—41no—
77C4⋊C4×C31—42no—
78C4⋊C4×C31—42no—
79C4⋊C4×C31—42no—
80C4⋊C4×C31—42no—
81C4⋊C4×C31—42no—
82C22⋊C4×C31—41no—
83C22⋊C4×C31—41no—
84C22⋊C4×C31—44no—
85C42×C31—21no—
86C42×C31—21no—
87C42×C31—21no—
88C42×C31—21no—
89#11700C3111yesmaximal, characteristic, direct product factor, Sylow
90C23×C31—21no—
91C23×C31—21no—
92C23×C31—41no—
93C23×C31—41no—
94C23×C31#10212yes—
95C23×C31C2×C22⋊C411yescharacteristic
96Q8×C31—82no—
97Q8×C31—82no—
98C2×C4×C31—21no—
99C2×C4×C31—21no—
100C2×C4×C31—22no—
101C2×C4×C31—22no—
102C2×C4×C31—22no—
103C2×C4×C31—22no—
104C2×C4×C31—21no—
105C2×C4×C31—21no—
106C2×C4×C31—21no—
107C2×C4×C31—21no—
108C2×C4×C31—41no—
109C2×C4×C31—41no—
110C2×C4×C31—41no—
111C2×C4×C31—41no—
112C2×C4×C31—42no—
113C2×C4×C31—42no—
114C2×C4×C31—42no—
115C2×C4×C31—42no—
116C2×C4×C31—42no—
117C2×C4×C31—42no—
118C2×C4×C31#9911yescharacteristic
119C2×C4×C31#9911yescharacteristic
120C2×C4×C31#10212yes—
121C2×C4×C31#10412yes—
122C2×C4×C31C2×D811yescharacteristic
123C2×C4×C31C2×SD1611yescharacteristic
124C8×C31—44no—
125C8×C31—82no—
126#2621C2×C3111yescharacteristic
127#1742C2×C3111yescharacteristic
128#1458C2×C3111yescharacteristic
129#1346C2×C3114yescomplemented
130C22×C31—21no—
131C22×C31—21no—
132C22×C31—22no—
133C22×C31—42no—
134C22×C31—41no—
135C22×C31—41no—
136C22×C31—81no—
137C22×C31C2×#9911yescharacteristic
138C22×C31C2×#10211yescharacteristic
139C22×C31C2×#10411yescharacteristic
140C4×C31—21no—
141C4×C31—21no—
142C4×C31—41no—
143C4×C31—41no—
144C4×C31—41no—
145C4×C31—41no—
146C4×C31—82no—
147C4×C31—82no—
148C4×C31#32712yes—
149C4×C31#32712yes—
150C22×M16—21no—
151C2×C4⋊Q8C4×C3111yescharacteristic
152C2×C22⋊Q8—21no—
153C2×#117C22×C3111yescharacteristic
154#431—21no—
155#398—21no—
156#398C22×C3112yes—
157#397—21no—
158#397C4×C3112yes—
159#391—21no—
160#382C4×C3111yescharacteristic
161#343—21no—
162#343C22×C3112yes—
163#324C22×C3112yes—
164#323—21no—
165C2×C31—21no—
166C2×C31—21no—
167C2×C31—42no—
168C2×C31C2×#32711yescharacteristic
169C2×C31#212311yescharacteristic
170C2×C31#213511yescharacteristic
171C22×Q8—21no—
172C23×C4D4×C3111yescharacteristic
173C2×M16—24no—
174C2×M16—22no—
175C2×M16—41no—
176C2×M16—42no—
177C22×C8—41no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182#117—41no—
183#117—41no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4C2×C4×C3112yes—
188C2×C4⋊C4D4×C3111yescharacteristic
189C2×C4⋊C4C23×C3111yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4C2×C4×C3111yescharacteristic
192C2×C22⋊C4D4×C3111yescharacteristic
193C2×C42C2×C4×C3111yescharacteristic
194C2×C42D4×C3111yescharacteristic
195C4⋊C8—41no—
196C22⋊C8—22no—
197C4×C8—41no—
198#95—21no—
199#95—22no—
200#95—22no—
201#95C2×C4×C3112yes—
202C31#1170011yescharacteristic, direct product factor, Sylow
203C24C22⋊C4×C3111yescharacteristic
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—42no—
207C2×Q8—42no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4C22⋊C4×C3111yescharacteristic
214C22×C4C22⋊C4×C3111yescharacteristic
215C22×C4C22⋊C4×C3111yescharacteristic
216C22×C4D8×C3112yes—
217C22×C4SD16×C3112yes—
218C22×C4C22×C4×C3111yescharacteristic
219C22×C4C2×D4×C3111yescharacteristic
220C22×C4C2×D4×C3111yescharacteristic
221M16—24no—
222M16—44no—
223M16—42no—
224C2×C8—42no—
225C2×C8—44no—
226C2×C8—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C22⋊C4—44no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23C31×#10212yes—
246C23C2×C22⋊C4×C3111yescharacteristic
247Q8—82no—
248Q8—82no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4C31×#9911yescharacteristic
270C2×C4C31×#9911yescharacteristic
271C2×C4C31×#10212yes—
272C2×C4C31×#10412yes—
273C2×C4C2×D8×C3111yescharacteristic
274C2×C4C2×SD16×C3111yescharacteristic
275C8—44no—
276C8—82no—
277C22—21no—
278C22—21no—
279C22—22no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—81no—
284C22C2×C31×#9911yescharacteristic
285C22C2×C31×#10211yescharacteristic
286C22C2×C31×#10411yescharacteristic
287C4—21no—
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—82no—
294C4—82no—
295C4C31×#32712yes—
296C4C31×#32712yes—
297C2—21no—
298C2—21no—
299C2—42no—
300C2C2×C31×#32711yescharacteristic
301C2C31×#212311yescharacteristic
302C2C31×#213511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC24×C31C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C31×#117 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#1346C2semidirect
89. #11700C31direct (N × H)
129. #1346C2×C31semidirect
202. C31#11700direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26212
C31×#17422
C31×#14582
C31×#13462
#1170031