Finite Group Databasebeta

← C31×#7821 #21464580

Subgroups — C31×#7821 #21464580

Order 7936.

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

normal subgroups only

This group has 422 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×#2804C211yesmaximal, characteristic
2C31×#2093C211yesmaximal, characteristic
3C31×#2072C212yesmaximal, complemented
4C31×#1272C212yesmaximal, complemented
5C31×#1270C211yesmaximal, characteristic
6C2×C31×#107—21no—
7C2×C31×#103—22no—
8C2×C31×#102—21no—
9C31×#400—21no—
10C31×#393—22no—
11C31×#391C2212yes—
12C31×#390C2211yescharacteristic
13C31×#386—21no—
14C31×#385—22no—
15C31×#384C2211yescharacteristic
16C31×#340—22no—
17C31×#336C2212yes—
18C31×#336C2211yescharacteristic
19C22×Q8×C31D412yes—
20C22×D4×C31D411yescharacteristic
21C23×C4×C31—21no—
22C22×C8×C31—21no—
23C22×C8×C31—22no—
24C2×C4⋊C4×C31—21no—
25C2×C4⋊C4×C31—22no—
26C2×C4⋊C4×C31—21no—
27C2×C4⋊C4×C31D411yescharacteristic
28C2×C4⋊C4×C31D412yes—
29C2×C22⋊C4×C31—22no—
30C2×C22⋊C4×C31—22no—
31C2×C22⋊C4×C31—21no—
32C2×C42×C31C2311yescharacteristic
33C31×#107—42no—
34C31×#103—44no—
35C31×#102—42no—
36C31×#95—22no—
37C31×#95—21no—
38C31×#95—22no—
39C31×#95—22no—
40C24×C31—21no—
41C2×Q8×C31—22no—
42C2×Q8×C31—22no—
43C2×Q8×C31—42no—
44C2×Q8×C31—42no—
45C2×D4×C31—21no—
46C2×D4×C31—41no—
47C2×D4×C31—41no—
48C2×D4×C31Q1612yes—
49C22×C4×C31—22no—
50C22×C4×C31—21no—
51C22×C4×C31—22no—
52C22×C4×C31—41no—
53C22×C4×C31—81no—
54C22×C4×C31—82no—
55C22×C4×C31C2×D411yescharacteristic
56C22×C4×C31C2×D412yes—
57C2×C8×C31—22no—
58C2×C8×C31—41no—
59C2×C8×C31—41no—
60C2×C8×C31—42no—
61C2×C8×C31—42no—
62C2×C8×C31—42no—
63C4⋊C4×C31—41no—
64C4⋊C4×C31—42no—
65C4⋊C4×C31—41no—
66C4⋊C4×C31—41no—
67C4⋊C4×C31—42no—
68C4⋊C4×C31—42no—
69C4⋊C4×C31—81no—
70C22⋊C4×C31—42no—
71C22⋊C4×C31—42no—
72C22⋊C4×C31—82no—
73C22⋊C4×C31—81no—
74C42×C31—41no—
75#7821C3111yesmaximal, characteristic, direct product factor, Sylow
76C23×C31—22no—
77C23×C31—42no—
78C23×C31—41no—
79C23×C31—42no—
80C23×C31—41no—
81C23×C31#12011yescharacteristic
82Q8×C31—44no—
83Q8×C31—82no—
84D4×C31—42no—
85D4×C31—81no—
86C2×C4×C31—21no—
87C2×C4×C31—22no—
88C2×C4×C31—21no—
89C2×C4×C31—22no—
90C2×C4×C31—42no—
91C2×C4×C31—42no—
92C2×C4×C31—41no—
93C2×C4×C31—42no—
94C2×C4×C31—42no—
95C2×C4×C31—42no—
96C2×C4×C31—42no—
97C2×C4×C31—42no—
98C2×C4×C31—84no—
99C2×C4×C31—82no—
100C2×C4×C31—81no—
101C2×C4×C31C2×Q1611yescharacteristic
102C2×C4×C31#13512yes—
103C2×C4×C31AGL(1,ℤ/8)12yes—
104C2×C4×C31#13711yescharacteristic
105C8×C31—42no—
106C8×C31—82no—
107#2804C2×C3111yescharacteristic
108#2093C2×C3111yescharacteristic
109#2072C2×C3112yescomplemented
110#1272C2×C3112yescomplemented
111#1270C2×C3111yescharacteristic
112C22×C31—42no—
113C22×C31—44no—
114C22×C31—82no—
115C22×C31—81no—
116C22×C31—81no—
117C22×C31#44912yes—
118C22×C31#45111yescharacteristic
119C22×C31#45311yescharacteristic
120C22×C31#45512yes—
121C22×C31#45811yescharacteristic
122C4×C31—22no—
123C4×C31—22no—
124C4×C31—21no—
125C4×C31—21no—
126C4×C31—82no—
127C4×C31—81no—
128C4×C31—82no—
129C4×C31—161no—
130C2×#107—21no—
131C2×#103—22no—
132C2×#102—21no—
133#400—21no—
134#393—22no—
135#391C22×C3112yes—
136#390C22×C3111yescharacteristic
137#386—21no—
138#385—22no—
139#384C22×C3111yescharacteristic
140#340—22no—
141#336C22×C3112yes—
142#336C22×C3111yescharacteristic
143C2×C31—82no—
144C2×C31#161212yes—
145C2×C31#161411yescharacteristic
146C2×C31#162011yescharacteristic
147C2×C31#164412yes—
148C2×C31#165911yescharacteristic
149C22×Q8D4×C3112yes—
150C22×D4D4×C3111yescharacteristic
151C23×C4—21no—
152C22×C8—21no—
153C22×C8—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4D4×C3112yes—
158C2×C4⋊C4D4×C3111yescharacteristic
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—21no—
162C2×C42C23×C3111yescharacteristic
163#107—42no—
164#103—44no—
165#102—42no—
166#95—22no—
167#95—21no—
168#95—22no—
169#95—22no—
170C31#782111yescharacteristic, direct product factor, Sylow
171C24—21no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4Q16×C3112yes—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—41no—
184C22×C4—81no—
185C22×C4—82no—
186C22×C4C2×D4×C3111yescharacteristic
187C22×C4C2×D4×C3112yes—
188C2×C8—22no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—81no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—82no—
204C22⋊C4—81no—
205C42—41no—
206C23—22no—
207C23—42no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23C31×#12011yescharacteristic
212Q8—44no—
213Q8—82no—
214D4—42no—
215D4—81no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—84no—
229C2×C4—82no—
230C2×C4—81no—
231C2×C4C2×Q16×C3111yescharacteristic
232C2×C4C31×#13512yes—
233C2×C4C31×AGL(1,ℤ/8)12yes—
234C2×C4C31×#13711yescharacteristic
235C8—42no—
236C8—82no—
237C22—42no—
238C22—44no—
239C22—82no—
240C22—81no—
241C22—81no—
242C22C31×#44912yes—
243C22C31×#45111yescharacteristic
244C22C31×#45311yescharacteristic
245C22C31×#45512yes—
246C22C31×#45811yescharacteristic
247C4—22no—
248C4—22no—
249C4—21no—
250C4—21no—
251C4—82no—
252C4—81no—
253C4—82no—
254C4—161no—
255C2—82no—
256C2C31×#161212yes—
257C2C31×#161411yescharacteristic
258C2C31×#162011yescharacteristic
259C2C31×#164412yes—
260C2C31×#165911yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
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
3. C31×#2072C2semidirect
4. C31×#1272C2semidirect
75. #7821C31direct (N × H)
109. #2072C2×C31semidirect
110. #1272C2×C31semidirect
170. C31#7821direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28042
C31×#20932
C31×#20722
C31×#12722
C31×#12702
#782131