Finite Group Databasebeta

← C31×#7742 #21464501

Subgroups — C31×#7742 #21464501

Order 7936.

668 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2733C211yesmaximal, characteristic, complemented
2C31×#2014C211yesmaximal, characteristic, complemented
3C31×#2011C212yesmaximal, complemented
4C31×#1272C211yesmaximal, characteristic, complemented
5C31×#1270C212yesmaximal, complemented
6C22×SD16×C31—21no—
7C22×D8×C31—22no—
8C2×C31×#127C2211yescharacteristic
9C2×C31×#124—21no—
10C2×C31×#102—22no—
11C2×C31×#102—21no—
12C31×#394—21no—
13C31×#392—22no—
14C31×#391C2211yescharacteristic
15C31×#390C2212yes—
16C31×#388—22no—
17C31×#336C2212yes—
18C31×#336C2211yescharacteristic
19C22×Q8×C31D411yescharacteristic
20C22×D4×C31—21no—
21C22×D4×C31—21no—
22C22×D4×C31—22no—
23C22×D4×C31—21no—
24C22×D4×C31D412yes—
25C2×SD16×C31—42no—
26C2×SD16×C31—42no—
27C2×SD16×C31—42no—
28C2×D8×C31—44no—
29C2×D8×C31—44no—
30C2×D8×C31—44no—
31C22×C8×C31—22no—
32C22×C8×C31—21no—
33C31×#127—42no—
34C31×#124—42no—
35C31×#124—42no—
36C2×C4⋊C4×C31D412yes—
37C2×C4⋊C4×C31D411yescharacteristic
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31—22no—
40C2×C22⋊C4×C31—22no—
41C2×C22⋊C4×C31—22no—
42C2×C42×C31—21no—
43C2×C42×C31C2311yescharacteristic
44C31×#102—44no—
45C31×#102—42no—
46C31×#95—21no—
47C31×#95—22no—
48C24×C31—22no—
49C24×C31—41no—
50C2×Q8×C31—21no—
51C2×Q8×C31—41no—
52C2×Q8×C31—41no—
53C2×Q8×C31D812yescomplemented
54C2×D4×C31—24no—
55C2×D4×C31—22no—
56C2×D4×C31—22no—
57C2×D4×C31—22no—
58C2×D4×C31—42no—
59C2×D4×C31—42no—
60C2×D4×C31—42no—
61C2×D4×C31—41no—
62C2×D4×C31—42no—
63C2×D4×C31—41no—
64C2×D4×C31—42no—
65C2×D4×C31—42no—
66C2×D4×C31—41no—
67C2×D4×C31—41no—
68C2×D4×C31—42no—
69C2×D4×C31—42no—
70C2×D4×C31—41no—
71C2×D4×C31—81no—
72C2×D4×C31—81no—
73C2×D4×C31—82no—
74C2×D4×C31—81no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—22no—
78C22×C4×C31C2×D411yescharacteristic
79C22×C4×C31C2×D412yes—
80SD16×C31—48no—
81D8×C31—416no—
82C2×C8×C31—24no—
83C2×C8×C31—22no—
84C2×C8×C31—42no—
85C2×C8×C31—41no—
86C2×C8×C31—41no—
87C2×C8×C31—42no—
88C4⋊C4×C31—42no—
89C4⋊C4×C31—41no—
90C22⋊C4×C31—44no—
91C22⋊C4×C31—42no—
92C22⋊C4×C31—42no—
93C22⋊C4×C31—42no—
94C22⋊C4×C31—42no—
95C22⋊C4×C31—82no—
96C42×C31—41no—
97C42×C31—42no—
98#7742C3111yesmaximal, characteristic, direct product factor, Sylow
99C23×C31—42no—
100C23×C31—42no—
101C23×C31—42no—
102C23×C31—42no—
103C23×C31—42no—
104C23×C31—42no—
105C23×C31—42no—
106C23×C31—42no—
107C23×C31—81no—
108C23×C31—82no—
109C23×C31—81no—
110C23×C31—82no—
111C23×C31#12011yescharacteristic
112Q8×C31—42no—
113Q8×C31—81no—
114D4×C31—44no—
115D4×C31—44no—
116D4×C31—42no—
117D4×C31—42no—
118D4×C31—44no—
119D4×C31—82no—
120D4×C31—81no—
121D4×C31—81no—
122D4×C31—82no—
123D4×C31—82no—
124D4×C31—84no—
125D4×C31—84no—
126C2×C4×C31—22no—
127C2×C4×C31—21no—
128C2×C4×C31—22no—
129C2×C4×C31—21no—
130C2×C4×C31—22no—
131C2×C4×C31—41no—
132C2×C4×C31—41no—
133C2×C4×C31—42no—
134C2×C4×C31—42no—
135C2×C4×C31—42no—
136C2×C4×C31—42no—
137C2×C4×C31—41no—
138C2×C4×C31C2×D811yescharacteristic
139C2×C4×C31#13512yes—
140C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C31AGL(1,ℤ/8)12yes—
142C8×C31—44no—
143C8×C31—42no—
144#2733C2×C3111yescharacteristic, complemented
145#2014C2×C3111yescharacteristic, complemented
146#2011C2×C3112yescomplemented
147#1272C2×C3111yescharacteristic, complemented
148#1270C2×C3112yescomplemented
149C22×C31—82no—
150C22×C31—82no—
151C22×C31—84no—
152C22×C31—82no—
153C22×C31—82no—
154C22×C31—82no—
155C22×C31—82no—
156C22×C31—82no—
157C22×C31—82no—
158C22×C31—162no—
159C22×C31—161no—
160C22×C31—161no—
161C22×C31#44711yescharacteristic
162C22×C31#44912yes—
163C22×C31#45311yescharacteristic
164C22×C31#45312yes—
165C22×C31B(4,𝔽2)11yescharacteristic
166C4×C31—21no—
167C4×C31—21no—
168C4×C31—22no—
169C4×C31—22no—
170C4×C31—42no—
171C4×C31—82no—
172C4×C31—81no—
173C22×SD16—21no—
174C22×D8—22no—
175C2×#127C22×C3111yescharacteristic
176C2×#124—21no—
177C2×#102—22no—
178C2×#102—21no—
179#394—21no—
180#392—22no—
181#391C22×C3111yescharacteristic
182#390C22×C3112yes—
183#388—22no—
184#336C22×C3112yes—
185#336C22×C3111yescharacteristic
186C2×C31—162no—
187C2×C31—162no—
188C2×C31#159511yescharacteristic
189C2×C31#160012yes—
190C2×C31#162112yes—
191C2×C31#162411yescharacteristic
192C2×C31#165511yescharacteristic
193C22×Q8D4×C3111yescharacteristic
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—22no—
197C22×D4—21no—
198C22×D4D4×C3112yes—
199C2×SD16—42no—
200C2×SD16—42no—
201C2×SD16—42no—
202C2×D8—44no—
203C2×D8—44no—
204C2×D8—44no—
205C22×C8—22no—
206C22×C8—21no—
207#127—42no—
208#124—42no—
209#124—42no—
210C2×C4⋊C4D4×C3112yes—
211C2×C4⋊C4D4×C3111yescharacteristic
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C42—21no—
217C2×C42C23×C3111yescharacteristic
218#102—44no—
219#102—42no—
220#95—21no—
221#95—22no—
222C31#774211yescharacteristic, direct product factor, Sylow
223C24—22no—
224C24—41no—
225C2×Q8—21no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8D8×C3112yescomplemented
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—41no—
237C2×D4—42no—
238C2×D4—41no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—41no—
246C2×D4—81no—
247C2×D4—81no—
248C2×D4—82no—
249C2×D4—81no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4C2×D4×C3111yescharacteristic
254C22×C4C2×D4×C3112yes—
255SD16—48no—
256D8—416no—
257C2×C8—24no—
258C2×C8—22no—
259C2×C8—42no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—42no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C22⋊C4—44no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—82no—
271C42—41no—
272C42—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—81no—
282C23—82no—
283C23—81no—
284C23—82no—
285C23C31×#12011yescharacteristic
286Q8—42no—
287Q8—81no—
288D4—44no—
289D4—44no—
290D4—42no—
291D4—42no—
292D4—44no—
293D4—82no—
294D4—81no—
295D4—81no—
296D4—82no—
297D4—82no—
298D4—84no—
299D4—84no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4C2×D8×C3111yescharacteristic
313C2×C4C31×#13512yes—
314C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
315C2×C4C31×AGL(1,ℤ/8)12yes—
316C8—44no—
317C8—42no—
318C22—82no—
319C22—82no—
320C22—84no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—162no—
328C22—161no—
329C22—161no—
330C22C31×#44711yescharacteristic
331C22C31×#44912yes—
332C22C31×#45311yescharacteristic
333C22C31×#45312yes—
334C22C31×B(4,𝔽2)11yescharacteristic
335C4—21no—
336C4—21no—
337C4—22no—
338C4—22no—
339C4—42no—
340C4—82no—
341C4—81no—
342C2—162no—
343C2—162no—
344C2C31×#159511yescharacteristic
345C2C31×#160012yes—
346C2C31×#162112yes—
347C2C31×#162411yescharacteristic
348C2C31×#165511yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2733C2semidirect
2. C31×#2014C2semidirect
3. C31×#2011C2semidirect
4. C31×#1272C2semidirect
5. C31×#1270C2semidirect
53. C2×Q8×C31D8semidirect
98. #7742C31direct (N × H)
144. #2733C2×C31semidirect
145. #2014C2×C31semidirect
146. #2011C2×C31semidirect
147. #1272C2×C31semidirect
148. #1270C2×C31semidirect
222. C31#7742direct (N × H)
228. C2×Q8D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#27332
C31×#20142
C31×#20112
C31×#12722
C31×#12702
#774231