Finite Group Databasebeta

← C31×#12797 #21469556

Subgroups — C31×#12797 #21469556

Order 7936.

680 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 134 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
1C4×C31×#127C211yesmaximal, characteristic
2C2×C31×#353C212yesmaximal
3C31×#1799C212yesmaximal, complemented
4C31×#1393C212yesmaximal, complemented
5C2×C31×#127C2211yescharacteristic
6C2×C4×D4×C31—22no—
7C2×C4×D4×C31C411yescharacteristic, complemented
8C2×C4×D4×C31C2211yescharacteristic
9C2×C31×#99C2214yes—
10C31×#390—22no—
11C31×#390C411yescharacteristic, complemented
12C31×#390C2211yescharacteristic
13C31×#384C411yescharacteristic, complemented
14C31×#380—22no—
15C43×C31C411yescharacteristic, 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—41no—
27C4×D4×C31—41no—
28C4×D4×C31—48no—
29C4×D4×C31—41no—
30C4×D4×C31—41no—
31C2×C4⋊C4×C31—22no—
32C2×C4⋊C4×C31—42no—
33C2×C4⋊C4×C31C2×C411yescharacteristic
34C2×C4⋊C4×C31C2×C411yescharacteristic
35C2×C22⋊C4×C31—22no—
36C2×C22⋊C4×C31—22no—
37C2×C22⋊C4×C31—21no—
38C2×C22⋊C4×C31—24no—
39C2×C22⋊C4×C31D412yes—
40C2×C42×C31—21no—
41C2×C42×C31—21no—
42C2×C42×C31C2×C411yescharacteristic
43C2×C42×C31C2×C411yescharacteristic
44C2×C42×C31C2×C411yescharacteristic
45C31×#99—24no—
46C31×#99—24no—
47C31×#95—42no—
48C24×C31—21no—
49C24×C31C2×D412yes—
50C2×D4×C31—24no—
51C2×D4×C31—24no—
52C2×D4×C31—21no—
53C2×D4×C31—41no—
54C2×D4×C31—41no—
55C2×D4×C31—44no—
56C2×D4×C31—44no—
57C2×D4×C31—41no—
58C2×D4×C31—41no—
59C2×D4×C31C4⋊C412yescomplemented
60C2×D4×C31C2×D411yescharacteristic
61C2×D4×C31C2×Q811yescharacteristic
62C2×D4×C31#4112yes—
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31—22no—
66C22×C4×C31—22no—
67C22×C4×C31—21no—
68C22×C4×C31—44no—
69C22×C4×C31—44no—
70C22×C4×C31—41no—
71C22×C4×C31—41no—
72C22×C4×C31—44no—
73C22×C4×C31C22⋊C411yescharacteristic
74C22×C4×C31C22⋊C412yes—
75C22×C4×C31C22⋊C411yescharacteristic
76C22×C4×C31C22×C411yescharacteristic
77C4⋊C4×C31—41no—
78C4⋊C4×C31—44no—
79C4⋊C4×C31—41no—
80C4⋊C4×C31—84no—
81C22⋊C4×C31—42no—
82C22⋊C4×C31—44no—
83C22⋊C4×C31—44no—
84C22⋊C4×C31—42no—
85C22⋊C4×C31—44no—
86C22⋊C4×C31—44no—
87C42×C31—24no—
88C42×C31—21no—
89C42×C31—21no—
90C42×C31—21no—
91C42×C31—42no—
92C42×C31—41no—
93C42×C31C4⋊C412yescomplemented
94#12797C3111yesmaximal, characteristic, direct product factor, Sylow
95C23×C31—22no—
96C23×C31—22no—
97C23×C31—22no—
98C23×C31—42no—
99C23×C31—42no—
100C23×C31—42no—
101C23×C31—42no—
102C23×C31—41no—
103C23×C31—41no—
104C23×C31C2×C22⋊C411yescharacteristic
105C23×C31#12112yes—
106C23×C31C22⋊Q812yes—
107D4×C31—41no—
108D4×C31—41no—
109D4×C31—42no—
110D4×C31—42no—
111D4×C31—48no—
112D4×C31—48no—
113D4×C31—42no—
114C2×C4×C31—22no—
115C2×C4×C31—22no—
116C2×C4×C31—22no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
119C2×C4×C31—22no—
120C2×C4×C31—22no—
121C2×C4×C31—21no—
122C2×C4×C31—22no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—44no—
126C2×C4×C31—41no—
127C2×C4×C31—42no—
128C2×C4×C31—42no—
129C2×C4×C31—41no—
130C2×C4×C31—82no—
131C2×C4×C31—82no—
132C2×C4×C31—82no—
133C2×C4×C31—84no—
134C2×C4×C31—84no—
135C2×C4×C31—84no—
136C2×C4×C31#9911yescharacteristic
137C2×C4×C31#9911yescharacteristic
138C2×C4×C31C2×C4⋊C411yescharacteristic
139C2×C4×C31#11711yescharacteristic
140C4×#127C2×C3111yescharacteristic
141C2×#353C2×C3112yes—
142#1799C2×C3112yescomplemented
143#1393C2×C3112yescomplemented
144C22×C31—21no—
145C22×C31—21no—
146C22×C31—22no—
147C22×C31—22no—
148C22×C31—42no—
149C22×C31—42no—
150C22×C31—44no—
151C22×C31—42no—
152C22×C31—42no—
153C22×C31—42no—
154C22×C31—82no—
155C22×C31—82no—
156C22×C31—81no—
157C22×C31—81no—
158C22×C31#38011yescharacteristic
159C22×C31C2×#9911yescharacteristic
160C22×C31#41011yescharacteristic
161C4×C31—24no—
162C4×C31—21no—
163C4×C31—21no—
164C4×C31—21no—
165C4×C31—42no—
166C4×C31—42no—
167C4×C31—41no—
168C4×C31—81no—
169C4×C31—164no—
170C4×C31#35312yescomplemented
171C2×#127C22×C3111yescharacteristic
172C2×C4×D4—22no—
173C2×C4×D4C4×C3111yescharacteristic, complemented
174C2×C4×D4C22×C3111yescharacteristic
175C2×#99C22×C3114yes—
176#390—22no—
177#390C4×C3111yescharacteristic, complemented
178#390C22×C3111yescharacteristic
179#384C4×C3111yescharacteristic, complemented
180#380—22no—
181C43C4×C3111yescharacteristic, complemented
182#353—24no—
183#342—22no—
184C2×C31—21no—
185C2×C31—21no—
186C2×C31—42no—
187C2×C31—42no—
188C2×C31—82no—
189C2×C31#179911yescharacteristic
190C2×C31C2×#35311yescharacteristic
191C2×C31#212611yescharacteristic
192C22×D4—22no—
193C22×D4C2×C4×C3111yescharacteristic
194C22×D4C23×C3111yescharacteristic
195C23×C4—21no—
196C23×C4D4×C3112yes—
197#127—22no—
198#127D4×C3112yes—
199#127Q8×C3112yes—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—48no—
203C4×D4—41no—
204C4×D4—41no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4C2×C4×C3111yescharacteristic
208C2×C4⋊C4C2×C4×C3111yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4D4×C3112yes—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42C2×C4×C3111yescharacteristic
217C2×C42C2×C4×C3111yescharacteristic
218C2×C42C2×C4×C3111yescharacteristic
219#99—24no—
220#99—24no—
221#95—42no—
222C31#1279711yescharacteristic, direct product factor, Sylow
223C24—21no—
224C24C2×D4×C3112yes—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—21no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4C4⋊C4×C3112yescomplemented
235C2×D4C2×D4×C3111yescharacteristic
236C2×D4C2×Q8×C3111yescharacteristic
237C2×D4#41×C3112yes—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—44no—
248C22×C4C22⋊C4×C3111yescharacteristic
249C22×C4C22⋊C4×C3111yescharacteristic
250C22×C4C22⋊C4×C3112yes—
251C22×C4C22×C4×C3111yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—44no—
255C4⋊C4—84no—
256C22⋊C4—42no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—42no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C42—24no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—41no—
267C42—42no—
268C42C4⋊C4×C3112yescomplemented
269C23—22no—
270C23—22no—
271C23—22no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—41no—
277C23—41no—
278C23C2×C22⋊C4×C3111yescharacteristic
279C23C31×#12112yes—
280C23C31×C22⋊Q812yes—
281D4—41no—
282D4—41no—
283D4—42no—
284D4—42no—
285D4—48no—
286D4—48no—
287D4—42no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—84no—
308C2×C4—84no—
309C2×C4—84no—
310C2×C4C31×#9911yescharacteristic
311C2×C4C31×#9911yescharacteristic
312C2×C4C2×C4⋊C4×C3111yescharacteristic
313C2×C4C31×#11711yescharacteristic
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—82no—
325C22—82no—
326C22—81no—
327C22—81no—
328C22C31×#38011yescharacteristic
329C22C2×C31×#9911yescharacteristic
330C22C31×#41011yescharacteristic
331C4—24no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—42no—
336C4—42no—
337C4—41no—
338C4—81no—
339C4—164no—
340C4C31×#35312yescomplemented
341C2—21no—
342C2—21no—
343C2—42no—
344C2—42no—
345C2—82no—
346C2C31×#179911yescharacteristic
347C2C2×C31×#35311yescharacteristic
348C2C31×#212611yescharacteristic
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
7. C2×C4×D4×C31C4semidirect
11. C31×#390C4semidirect
13. C31×#384C4semidirect
15. C43×C31C4semidirect
59. C2×D4×C31C4⋊C4semidirect
93. C42×C31C4⋊C4semidirect
94. #12797C31direct (N × H)
142. #1799C2×C31semidirect
143. #1393C2×C31semidirect
170. C4×C31#353semidirect
173. C2×C4×D4C4×C31semidirect
177. #390C4×C31semidirect
179. #384C4×C31semidirect
181. C43C4×C31semidirect
222. C31#12797direct (N × H)
234. C2×D4C4⋊C4×C31semidirect
268. C42C4⋊C4×C31semidirect
340. C4C31×#353semidirect

Maximal subgroups

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

subgroupindex
C4×C31×#1272
C2×C31×#3532
C31×#17992
C31×#13932
#1279731