Finite Group Databasebeta

← C31×#7738 #21464497

Subgroups — C31×#7738 #21464497

Order 7936.

604 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2683C211yesmaximal, characteristic, complemented
2C31×#2041C212yesmaximal, complemented
3C31×#2011C211yesmaximal, characteristic
4C31×#1271C212yesmaximal, complemented
5C31×#1270C211yesmaximal, characteristic, complemented
6C22×D8×C31—21no—
7C2×C31×#127C2211yescharacteristic
8C2×C31×C22⋊Q8—22no—
9C2×C31×#102—21no—
10C2×C31×#102—22no—
11C2×C31×#102—22no—
12C31×#396—22no—
13C31×#395—21no—
14C31×#390C2211yescharacteristic
15C31×#389C2212yes—
16C31×#386—21no—
17C31×#336C2211yescharacteristic
18C31×#336C2212yes—
19C22×Q8×C31—21no—
20C22×D4×C31—22no—
21C22×D4×C31—21no—
22C22×D4×C31D411yescharacteristic
23C23×C4×C31—21no—
24C2×D8×C31—42no—
25C2×D8×C31—42no—
26C2×D8×C31—42no—
27C22×C8×C31—21no—
28C22×C8×C31—22no—
29C31×#127—42no—
30C31×C22⋊Q8—44no—
31C31×C22⋊Q8—44no—
32C2×C4⋊C4×C31—22no—
33C2×C4⋊C4×C31—22no—
34C2×C4⋊C4×C31D412yes—
35C2×C4⋊C4×C31D411yescharacteristic
36C2×C4⋊C4×C31D412yes—
37C2×C22⋊C4×C31—22no—
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31—21no—
40C2×C42×C31C2311yescharacteristic
41C31×#102—42no—
42C31×#102—44no—
43C31×#102—44no—
44C31×#95—21no—
45C31×#95—22no—
46C24×C31—21no—
47C24×C31—41no—
48C2×Q8×C31—44no—
49C2×Q8×C31—81no—
50C2×D4×C31—24no—
51C2×D4×C31—22no—
52C2×D4×C31—21no—
53C2×D4×C31—41no—
54C2×D4×C31—41no—
55C2×D4×C31—41no—
56C2×D4×C31—41no—
57C2×D4×C31—42no—
58C2×D4×C31—42no—
59C2×D4×C31—42no—
60C2×D4×C31—41no—
61C2×D4×C31—42no—
62C2×D4×C31—41no—
63C2×D4×C31—42no—
64C2×D4×C31—41no—
65C2×D4×C31SD1612yes—
66C22×C4×C31—22no—
67C22×C4×C31—22no—
68C22×C4×C31—21no—
69C22×C4×C31—44no—
70C22×C4×C31—81no—
71C22×C4×C31C2×D412yes—
72C22×C4×C31C2×D411yescharacteristic
73D8×C31—48no—
74C2×C8×C31—22no—
75C2×C8×C31—41no—
76C2×C8×C31—41no—
77C2×C8×C31—42no—
78C2×C8×C31—42no—
79C2×C8×C31—42no—
80C4⋊C4×C31—22no—
81C4⋊C4×C31—44no—
82C4⋊C4×C31—41no—
83C4⋊C4×C31—42no—
84C4⋊C4×C31—42no—
85C4⋊C4×C31—42no—
86C4⋊C4×C31D814yescomplemented
87C22⋊C4×C31—44no—
88C22⋊C4×C31—42no—
89C22⋊C4×C31—42no—
90C22⋊C4×C31—81no—
91C42×C31—41no—
92#7738C3111yesmaximal, characteristic, direct product factor, Sylow
93C23×C31—22no—
94C23×C31—42no—
95C23×C31—41no—
96C23×C31—42no—
97C23×C31—41no—
98C23×C31—42no—
99C23×C31—82no—
100C23×C31—81no—
101C23×C31—81no—
102C23×C31—82no—
103C23×C31#12011yescharacteristic
104Q8×C31—84no—
105D4×C31—42no—
106D4×C31—42no—
107D4×C31—42no—
108D4×C31—44no—
109D4×C31—44no—
110D4×C31—81no—
111D4×C31—81no—
112D4×C31—81no—
113D4×C31—82no—
114D4×C31—82no—
115C2×C4×C31—21no—
116C2×C4×C31—22no—
117C2×C4×C31—22no—
118C2×C4×C31—21no—
119C2×C4×C31—24no—
120C2×C4×C31—42no—
121C2×C4×C31—42no—
122C2×C4×C31—42no—
123C2×C4×C31—42no—
124C2×C4×C31—41no—
125C2×C4×C31—42no—
126C2×C4×C31—42no—
127C2×C4×C31—84no—
128C2×C4×C31C2×D812yes—
129C2×C4×C31C2×SD1611yescharacteristic
130C2×C4×C31AGL(1,ℤ/8)12yes—
131C2×C4×C31#13711yescharacteristic
132C8×C31—42no—
133C8×C31—82no—
134#2683C2×C3111yescharacteristic, complemented
135#2041C2×C3112yescomplemented
136#2011C2×C3111yescharacteristic
137#1271C2×C3112yescomplemented
138#1270C2×C3111yescharacteristic, complemented
139C22×C31—42no—
140C22×C31—44no—
141C22×C31—82no—
142C22×C31—81no—
143C22×C31—81no—
144C22×C31—82no—
145C22×C31—84no—
146C22×C31—161no—
147C22×C31—162no—
148C22×C31—161no—
149C22×C31#44712yes—
150C22×C31#44811yescharacteristic
151C22×C31#45311yescharacteristic
152C22×C31#45412yes—
153C22×C31B(4,𝔽2)11yescharacteristic
154C4×C31—21no—
155C4×C31—21no—
156C4×C31—22no—
157C4×C31—22no—
158C4×C31—44no—
159C4×C31—82no—
160C4×C31—81no—
161C22×D8—21no—
162C2×#127C22×C3111yescharacteristic
163C2×C22⋊Q8—22no—
164C2×#102—21no—
165C2×#102—22no—
166C2×#102—22no—
167#396—22no—
168#395—21no—
169#390C22×C3111yescharacteristic
170#389C22×C3112yes—
171#386—21no—
172#336C22×C3111yescharacteristic
173#336C22×C3112yes—
174C2×C31—82no—
175C2×C31—162no—
176C2×C31#159611yescharacteristic
177C2×C31#159812yes—
178C2×C31#161911yescharacteristic
179C2×C31#162712yes—
180C2×C31#165711yescharacteristic
181C22×Q8—21no—
182C22×D4—21no—
183C22×D4—22no—
184C22×D4D4×C3111yescharacteristic
185C23×C4—21no—
186C2×D8—42no—
187C2×D8—42no—
188C2×D8—42no—
189C22×C8—21no—
190C22×C8—22no—
191#127—42no—
192C22⋊Q8—44no—
193C22⋊Q8—44no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4D4×C3112yes—
197C2×C4⋊C4D4×C3112yes—
198C2×C4⋊C4D4×C3111yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C42C23×C3111yescharacteristic
203#102—42no—
204#102—44no—
205#102—44no—
206#95—21no—
207#95—22no—
208C31#773811yescharacteristic, direct product factor, Sylow
209C24—21no—
210C24—41no—
211C2×Q8—44no—
212C2×Q8—81no—
213C2×D4—24no—
214C2×D4—22no—
215C2×D4—21no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—41no—
224C2×D4—42no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—41no—
228C2×D4SD16×C3112yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—44no—
233C22×C4—81no—
234C22×C4C2×D4×C3112yes—
235C22×C4C2×D4×C3111yescharacteristic
236D8—48no—
237C2×C8—22no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4D8×C3114yescomplemented
250C22⋊C4—44no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—81no—
254C42—41no—
255C23—22no—
256C23—42no—
257C23—41no—
258C23—42no—
259C23—41no—
260C23—42no—
261C23—82no—
262C23—81no—
263C23—81no—
264C23—82no—
265C23C31×#12011yescharacteristic
266Q8—84no—
267D4—42no—
268D4—42no—
269D4—42no—
270D4—44no—
271D4—44no—
272D4—81no—
273D4—81no—
274D4—81no—
275D4—82no—
276D4—82no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—84no—
290C2×C4C2×D8×C3112yes—
291C2×C4C2×SD16×C3111yescharacteristic
292C2×C4C31×AGL(1,ℤ/8)12yes—
293C2×C4C31×#13711yescharacteristic
294C8—42no—
295C8—82no—
296C22—42no—
297C22—44no—
298C22—82no—
299C22—81no—
300C22—81no—
301C22—82no—
302C22—84no—
303C22—161no—
304C22—162no—
305C22—161no—
306C22C31×#44712yes—
307C22C31×#44811yescharacteristic
308C22C31×#45311yescharacteristic
309C22C31×#45412yes—
310C22C31×B(4,𝔽2)11yescharacteristic
311C4—21no—
312C4—21no—
313C4—22no—
314C4—22no—
315C4—44no—
316C4—82no—
317C4—81no—
318C2—82no—
319C2—162no—
320C2C31×#159611yescharacteristic
321C2C31×#159812yes—
322C2C31×#161911yescharacteristic
323C2C31×#162712yes—
324C2C31×#165711yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#2011C2
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×#2683C2semidirect
2. C31×#2041C2semidirect
4. C31×#1271C2semidirect
5. C31×#1270C2semidirect
86. C4⋊C4×C31D8semidirect
92. #7738C31direct (N × H)
134. #2683C2×C31semidirect
135. #2041C2×C31semidirect
137. #1271C2×C31semidirect
138. #1270C2×C31semidirect
208. C31#7738direct (N × H)
249. C4⋊C4D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#26832
C31×#20412
C31×#20112
C31×#12712
C31×#12702
#773831