Finite Group Databasebeta

← C31×#7826 #21464585

Subgroups — C31×#7826 #21464585

Order 7936.

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

normal subgroups only

This group has 462 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
2C31×#2100C212yesmaximal, complemented
3C31×#2094C211yesmaximal, characteristic
4C31×#1271C212yesmaximal, complemented
5C31×#1270C211yesmaximal, characteristic
6C2×C31×C22⋊Q8—22no—
7C2×C31×#106—21no—
8C2×C31×#102—21no—
9C31×#396—22no—
10C31×#395—21no—
11C31×#390C2211yescharacteristic
12C31×#389C2212yes—
13C31×#386—21no—
14C31×#384C2211yescharacteristic
15C31×#340—22no—
16C31×#340—22no—
17C31×#336C2211yescharacteristic
18C31×#336C2212yes—
19C22×Q8×C31—21no—
20C22×D4×C31D411yescharacteristic
21C23×C4×C31—21no—
22C22×C8×C31—21no—
23C22×C8×C31—22no—
24C31×C22⋊Q8—44no—
25C31×C22⋊Q8—44no—
26C2×C4⋊C4×C31—22no—
27C2×C4⋊C4×C31—22no—
28C2×C4⋊C4×C31—21no—
29C2×C4⋊C4×C31—22no—
30C2×C4⋊C4×C31D412yes—
31C2×C4⋊C4×C31D411yescharacteristic
32C2×C4⋊C4×C31D412yes—
33C2×C22⋊C4×C31—22no—
34C2×C22⋊C4×C31—22no—
35C2×C22⋊C4×C31—21no—
36C2×C42×C31C2311yescharacteristic
37C31×#106—42no—
38C31×#102—42no—
39C31×#95—21no—
40C31×#95—22no—
41C24×C31—21no—
42C2×Q8×C31—44no—
43C2×Q8×C31—81no—
44C2×D4×C31—21no—
45C2×D4×C31—41no—
46C2×D4×C31—41no—
47C2×D4×C31Q1612yes—
48C22×C4×C31—22no—
49C22×C4×C31—22no—
50C22×C4×C31—21no—
51C22×C4×C31—44no—
52C22×C4×C31—41no—
53C22×C4×C31—81no—
54C22×C4×C31C2×D412yes—
55C22×C4×C31C2×D411yescharacteristic
56C2×C8×C31—22no—
57C2×C8×C31—41no—
58C2×C8×C31—41no—
59C2×C8×C31—42no—
60C2×C8×C31—42no—
61C2×C8×C31—42no—
62C4⋊C4×C31—22no—
63C4⋊C4×C31—44no—
64C4⋊C4×C31—41no—
65C4⋊C4×C31—42no—
66C4⋊C4×C31—41no—
67C4⋊C4×C31—41no—
68C4⋊C4×C31—44no—
69C4⋊C4×C31—42no—
70C4⋊C4×C31—42no—
71C4⋊C4×C31SD1614yes—
72C22⋊C4×C31—44no—
73C22⋊C4×C31—42no—
74C22⋊C4×C31—42no—
75C22⋊C4×C31—81no—
76C42×C31—41no—
77#7826C3111yesmaximal, characteristic, direct product factor, Sylow
78C23×C31—22no—
79C23×C31—42no—
80C23×C31—41no—
81C23×C31—42no—
82C23×C31—41no—
83C23×C31#12011yescharacteristic
84Q8×C31—84no—
85D4×C31—42no—
86D4×C31—81no—
87C2×C4×C31—21no—
88C2×C4×C31—22no—
89C2×C4×C31—22no—
90C2×C4×C31—21no—
91C2×C4×C31—24no—
92C2×C4×C31—42no—
93C2×C4×C31—42no—
94C2×C4×C31—42no—
95C2×C4×C31—42no—
96C2×C4×C31—41no—
97C2×C4×C31—42no—
98C2×C4×C31—42no—
99C2×C4×C31—84no—
100C2×C4×C31—81no—
101C2×C4×C31—82no—
102C2×C4×C31C2×SD1612yes—
103C2×C4×C31C2×Q1611yescharacteristic
104C2×C4×C31AGL(1,ℤ/8)12yes—
105C2×C4×C31#13711yescharacteristic
106C8×C31—42no—
107C8×C31—82no—
108#2683C2×C3111yescharacteristic
109#2100C2×C3112yescomplemented
110#2094C2×C3111yescharacteristic
111#1271C2×C3112yescomplemented
112#1270C2×C3111yescharacteristic
113C22×C31—42no—
114C22×C31—44no—
115C22×C31—82no—
116C22×C31—81no—
117C22×C31—81no—
118C22×C31#45012yes—
119C22×C31#45111yescharacteristic
120C22×C31#45311yescharacteristic
121C22×C31#45412yes—
122C22×C31#45811yescharacteristic
123C4×C31—21no—
124C4×C31—21no—
125C4×C31—22no—
126C4×C31—22no—
127C4×C31—44no—
128C4×C31—82no—
129C4×C31—81no—
130C4×C31—161no—
131C2×C22⋊Q8—22no—
132C2×#106—21no—
133C2×#102—21no—
134#396—22no—
135#395—21no—
136#390C22×C3111yescharacteristic
137#389C22×C3112yes—
138#386—21no—
139#384C22×C3111yescharacteristic
140#340—22no—
141#340—22no—
142#336C22×C3111yescharacteristic
143#336C22×C3112yes—
144C2×C31—82no—
145C2×C31#161411yescharacteristic
146C2×C31#161612yes—
147C2×C31#162911yescharacteristic
148C2×C31#163712yes—
149C2×C31#165711yescharacteristic
150C22×Q8—21no—
151C22×D4D4×C3111yescharacteristic
152C23×C4—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22⋊Q8—44no—
156C22⋊Q8—44no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4D4×C3112yes—
162C2×C4⋊C4D4×C3112yes—
163C2×C4⋊C4D4×C3111yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—21no—
167C2×C42C23×C3111yescharacteristic
168#106—42no—
169#102—42no—
170#95—21no—
171#95—22no—
172C31#782611yescharacteristic, direct product factor, Sylow
173C24—21no—
174C2×Q8—44no—
175C2×Q8—81no—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4Q16×C3112yes—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—44no—
184C22×C4—41no—
185C22×C4—81no—
186C22×C4C2×D4×C3112yes—
187C22×C4C2×D4×C3111yescharacteristic
188C2×C8—22no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—44no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4SD16×C3114yes—
204C22⋊C4—44no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—81no—
208C42—41no—
209C23—22no—
210C23—42no—
211C23—41no—
212C23—42no—
213C23—41no—
214C23C31×#12011yescharacteristic
215Q8—84no—
216D4—42no—
217D4—81no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—24no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—84no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4C2×SD16×C3112yes—
234C2×C4C2×Q16×C3111yescharacteristic
235C2×C4C31×AGL(1,ℤ/8)12yes—
236C2×C4C31×#13711yescharacteristic
237C8—42no—
238C8—82no—
239C22—42no—
240C22—44no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22C31×#45012yes—
245C22C31×#45111yescharacteristic
246C22C31×#45311yescharacteristic
247C22C31×#45412yes—
248C22C31×#45811yescharacteristic
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—44no—
254C4—82no—
255C4—81no—
256C4—161no—
257C2—82no—
258C2C31×#161411yescharacteristic
259C2C31×#161612yes—
260C2C31×#162911yescharacteristic
261C2C31×#163712yes—
262C2C31×#165711yescharacteristic
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
2. C31×#2100C2semidirect
4. C31×#1271C2semidirect
77. #7826C31direct (N × H)
109. #2100C2×C31semidirect
111. #1271C2×C31semidirect
172. C31#7826direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26832
C31×#21002
C31×#20942
C31×#12712
C31×#12702
#782631