Finite Group Databasebeta

← C31×#12799 #21469558

Subgroups — C31×#12799 #21469558

Order 7936.

588 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2328C211yesmaximal, characteristic
2C2×C31×#353C212yesmaximal
3C31×#1799C212yesmaximal, complemented
4C31×#1393C212yesmaximal, complemented
5C2×C31×#127C2211yescharacteristic
6C2×C4×D4×C31C411yescharacteristic, complemented
7C2×C4×D4×C31C2211yescharacteristic
8C2×C31×#99C2214yes—
9C31×#390C411yescharacteristic, complemented
10C31×#390C2211yescharacteristic
11C31×#386—22no—
12C31×#386—22no—
13C31×#382C411yescharacteristic, complemented
14C31×#380—22no—
15C31×#376C411yescharacteristic, complemented
16C31×#353—24no—
17C31×#342—22no—
18C22×D4×C31—22no—
19C22×D4×C31C2×C411yescharacteristic
20C22×D4×C31C2311yescharacteristic
21C23×C4×C31—21no—
22C23×C4×C31D412yes—
23C31×#127—22no—
24C31×#127D412yes—
25C31×#127Q812yes—
26C4×D4×C31—42no—
27C4×D4×C31—42no—
28C2×C4⋊C4×C31—42no—
29C2×C4⋊C4×C31C2×C411yescharacteristic
30C2×C4⋊C4×C31C2×C411yescharacteristic
31C2×C22⋊C4×C31—21no—
32C2×C22⋊C4×C31—24no—
33C2×C22⋊C4×C31—22no—
34C2×C22⋊C4×C31—22no—
35C2×C22⋊C4×C31D412yes—
36C2×C42×C31C2×C411yescharacteristic
37C2×C42×C31C2×C411yescharacteristic
38C2×C42×C31C2×C411yescharacteristic
39C31×#99—24no—
40C31×#99—24no—
41C31×#95—21no—
42C31×#95—22no—
43C31×#95—21no—
44C31×#95—42no—
45C24×C31—21no—
46C24×C31C2×D412yes—
47C2×D4×C31—24no—
48C2×D4×C31—24no—
49C2×D4×C31—21no—
50C2×D4×C31—41no—
51C2×D4×C31—41no—
52C2×D4×C31—41no—
53C2×D4×C31—41no—
54C2×D4×C31—82no—
55C2×D4×C31—82no—
56C2×D4×C31C4⋊C412yescomplemented
57C2×D4×C31C2×D411yescharacteristic
58C2×D4×C31C2×Q811yescharacteristic
59C2×D4×C31#4112yes—
60C22×C4×C31—21no—
61C22×C4×C31—22no—
62C22×C4×C31—22no—
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31—42no—
66C22×C4×C31—44no—
67C22×C4×C31—82no—
68C22×C4×C31—82no—
69C22×C4×C31C22⋊C411yescharacteristic
70C22×C4×C31C22⋊C412yes—
71C22×C4×C31C22⋊C411yescharacteristic
72C22×C4×C31C22×C411yescharacteristic
73C4⋊C4×C31—41no—
74C4⋊C4×C31—41no—
75C4⋊C4×C31—84no—
76C22⋊C4×C31—42no—
77C22⋊C4×C31—42no—
78C22⋊C4×C31—44no—
79C22⋊C4×C31—44no—
80C22⋊C4×C31—82no—
81C22⋊C4×C31—82no—
82C42×C31—22no—
83C42×C31—21no—
84C42×C31—41no—
85C42×C31C4⋊C412yescomplemented
86#12799C3111yesmaximal, characteristic, direct product factor, Sylow
87C23×C31—22no—
88C23×C31—22no—
89C23×C31—22no—
90C23×C31—42no—
91C23×C31—42no—
92C23×C31—42no—
93C23×C31—42no—
94C23×C31—41no—
95C23×C31—41no—
96C23×C31C2×C22⋊C411yescharacteristic
97C23×C31#12112yes—
98C23×C31C22⋊Q812yes—
99D4×C31—41no—
100D4×C31—41no—
101D4×C31—42no—
102D4×C31—42no—
103D4×C31—42no—
104D4×C31—84no—
105D4×C31—84no—
106C2×C4×C31—21no—
107C2×C4×C31—21no—
108C2×C4×C31—22no—
109C2×C4×C31—21no—
110C2×C4×C31—22no—
111C2×C4×C31—41no—
112C2×C4×C31—44no—
113C2×C4×C31—41no—
114C2×C4×C31—41no—
115C2×C4×C31—42no—
116C2×C4×C31—41no—
117C2×C4×C31—41no—
118C2×C4×C31—41no—
119C2×C4×C31—42no—
120C2×C4×C31—41no—
121C2×C4×C31—41no—
122C2×C4×C31—82no—
123C2×C4×C31—84no—
124C2×C4×C31—84no—
125C2×C4×C31—84no—
126C2×C4×C31—84no—
127C2×C4×C31#9912yes—
128C2×C4×C31C2×C4⋊C411yescharacteristic
129C2×C4×C31#11711yescharacteristic
130#2328C2×C3111yescharacteristic
131C2×#353C2×C3112yes—
132#1799C2×C3112yescomplemented
133#1393C2×C3112yescomplemented
134C22×C31—21no—
135C22×C31—21no—
136C22×C31—22no—
137C22×C31—22no—
138C22×C31—42no—
139C22×C31—42no—
140C22×C31—44no—
141C22×C31—42no—
142C22×C31—42no—
143C22×C31—42no—
144C22×C31—82no—
145C22×C31—82no—
146C22×C31—81no—
147C22×C31—81no—
148C22×C31#38011yescharacteristic
149C22×C31C2×#9911yescharacteristic
150C22×C31#41011yescharacteristic
151C4×C31—21no—
152C4×C31—21no—
153C4×C31—22no—
154C4×C31—42no—
155C4×C31—42no—
156C4×C31—41no—
157C4×C31—81no—
158C4×C31—81no—
159C4×C31—164no—
160C2×#127C22×C3111yescharacteristic
161C2×C4×D4C4×C3111yescharacteristic, complemented
162C2×C4×D4C22×C3111yescharacteristic
163C2×#99C22×C3114yes—
164#390C4×C3111yescharacteristic, complemented
165#390C22×C3111yescharacteristic
166#386—22no—
167#386—22no—
168#382C4×C3111yescharacteristic, complemented
169#380—22no—
170#376C4×C3111yescharacteristic, complemented
171#353—24no—
172#342—22no—
173C2×C31—21no—
174C2×C31—21no—
175C2×C31—42no—
176C2×C31—42no—
177C2×C31—82no—
178C2×C31#179911yescharacteristic
179C2×C31#212711yescharacteristic
180C2×C31#212811yescharacteristic
181C22×D4—22no—
182C22×D4C2×C4×C3111yescharacteristic
183C22×D4C23×C3111yescharacteristic
184C23×C4—21no—
185C23×C4D4×C3112yes—
186#127—22no—
187#127D4×C3112yes—
188#127Q8×C3112yes—
189C4×D4—42no—
190C4×D4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4C2×C4×C3111yescharacteristic
193C2×C4⋊C4C2×C4×C3111yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4D4×C3112yes—
199C2×C42C2×C4×C3111yescharacteristic
200C2×C42C2×C4×C3111yescharacteristic
201C2×C42C2×C4×C3111yescharacteristic
202#99—24no—
203#99—24no—
204#95—21no—
205#95—21no—
206#95—22no—
207#95—42no—
208C31#1279911yescharacteristic, direct product factor, Sylow
209C24—21no—
210C24C2×D4×C3112yes—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—21no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—82no—
219C2×D4—82no—
220C2×D4C4⋊C4×C3112yescomplemented
221C2×D4C2×D4×C3111yescharacteristic
222C2×D4C2×Q8×C3111yescharacteristic
223C2×D4#41×C3112yes—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4C22⋊C4×C3111yescharacteristic
234C22×C4C22⋊C4×C3112yes—
235C22×C4C22⋊C4×C3111yescharacteristic
236C22×C4C22×C4×C3111yescharacteristic
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—84no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C22⋊C4—82no—
245C22⋊C4—82no—
246C42—22no—
247C42—21no—
248C42—41no—
249C42C4⋊C4×C3112yescomplemented
250C23—22no—
251C23—22no—
252C23—22no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—41no—
258C23—41no—
259C23C2×C22⋊C4×C3111yescharacteristic
260C23C31×#12112yes—
261C23C31×C22⋊Q812yes—
262D4—41no—
263D4—41no—
264D4—42no—
265D4—42no—
266D4—42no—
267D4—84no—
268D4—84no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—82no—
286C2×C4—84no—
287C2×C4—84no—
288C2×C4—84no—
289C2×C4—84no—
290C2×C4C31×#9912yes—
291C2×C4C2×C4⋊C4×C3111yescharacteristic
292C2×C4C31×#11711yescharacteristic
293C22—21no—
294C22—21no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—44no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—82no—
304C22—82no—
305C22—81no—
306C22—81no—
307C22C31×#38011yescharacteristic
308C22C2×C31×#9911yescharacteristic
309C22C31×#41011yescharacteristic
310C4—21no—
311C4—21no—
312C4—22no—
313C4—42no—
314C4—42no—
315C4—41no—
316C4—81no—
317C4—81no—
318C4—164no—
319C2—21no—
320C2—21no—
321C2—42no—
322C2—42no—
323C2—82no—
324C2C31×#179911yescharacteristic
325C2C31×#212711yescharacteristic
326C2C31×#212811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC22×D4C23×C31
prime radicalC2×C31×#127C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#1799C2semidirect
4. C31×#1393C2semidirect
6. C2×C4×D4×C31C4semidirect
9. C31×#390C4semidirect
13. C31×#382C4semidirect
15. C31×#376C4semidirect
56. C2×D4×C31C4⋊C4semidirect
85. C42×C31C4⋊C4semidirect
86. #12799C31direct (N × H)
132. #1799C2×C31semidirect
133. #1393C2×C31semidirect
161. C2×C4×D4C4×C31semidirect
164. #390C4×C31semidirect
168. #382C4×C31semidirect
170. #376C4×C31semidirect
208. C31#12799direct (N × H)
220. C2×D4C4⋊C4×C31semidirect
249. C42C4⋊C4×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#23282
C2×C31×#3532
C31×#17992
C31×#13932
#1279931