Finite Group Databasebeta

← C31×#8345 #21465104

Subgroups — C31×#8345 #21465104

Order 7936.

802 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 198 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
1C2×C31×#533C211yesmaximal, characteristic
2C31×#1797C212yesmaximal
3C2×C31×#332C214yesmaximal, complemented
4C31×#533—28no—
5C2×C31×C4⋊Q8C2212yes—
6C2×C31×#125C412yes—
7C2×C31×C22⋊Q8—22no—
8C2×C4×Q8×C31—22no—
9C2×C31×#117C412yes—
10C22×C42×C31C2211yescharacteristic
11C2×C31×C4⋊C8C2214yes—
12C2×C31×C22⋊C8—22no—
13C31×#336—22no—
14C31×#332—28no—
15C22×Q8×C31—22no—
16C23×C4×C31C2×C411yescharacteristic
17C23×C4×C31D412yes—
18C22×C8×C31—24no—
19C31×C4⋊Q8—24no—
20C31×C4⋊Q8D418yes—
21C31×#125—28no—
22C31×C22⋊Q8—216no—
23C4×Q8×C31—216no—
24C31×#117—28no—
25C2×C4⋊C4×C31—24no—
26C2×C4⋊C4×C31—22no—
27C2×C4⋊C4×C31C2×C414yes—
28C2×C22⋊C4×C31—22no—
29C2×C42×C31—22no—
30C2×C42×C31—24no—
31C2×C42×C31C2×C411yescharacteristic
32C2×C42×C31D412yes—
33C2×C42×C31C2311yescharacteristic
34C31×C4⋊C8—28no—
35C31×C22⋊C8—44no—
36C24×C31C22⋊C411yescharacteristic
37C2×Q8×C31—28no—
38C2×Q8×C31—44no—
39C2×Q8×C31—42no—
40C2×Q8×C31—42no—
41C22×C4×C31—22no—
42C22×C4×C31—22no—
43C22×C4×C31—22no—
44C22×C4×C31—24no—
45C22×C4×C31—24no—
46C22×C4×C31—24no—
47C22×C4×C31—22no—
48C22×C4×C31—22no—
49C22×C4×C31C22⋊C412yes—
50C22×C4×C31C22⋊C411yescharacteristic
51C22×C4×C31C22×C411yescharacteristic
52C22×C4×C31C2×D412yes—
53C2×C8×C31—44no—
54C2×C8×C31—44no—
55C2×C8×C31—44no—
56C4⋊C4×C31—216no—
57C4⋊C4×C31—24no—
58C4⋊C4×C31—28no—
59C4⋊C4×C31C22⋊C418yes—
60C22⋊C4×C31—28no—
61C42×C31—28no—
62C42×C31—22no—
63C42×C31—24no—
64C42×C31C2×D414yes—
65#8345C3111yesmaximal, characteristic, direct product factor, Sylow
66C23×C31—21no—
67C23×C31—21no—
68C23×C31—22no—
69C23×C31—22no—
70C23×C31#10112yes—
71C23×C31C2×C22⋊C411yescharacteristic
72Q8×C31—48no—
73Q8×C31—48no—
74C2×C4×C31—28no—
75C2×C4×C31—21no—
76C2×C4×C31—21no—
77C2×C4×C31—21no—
78C2×C4×C31—21no—
79C2×C4×C31—22no—
80C2×C4×C31—28no—
81C2×C4×C31—22no—
82C2×C4×C31—22no—
83C2×C4×C31—44no—
84C2×C4×C31—44no—
85C2×C4×C31—44no—
86C2×C4×C31—44no—
87C2×C4×C31#10112yes—
88C2×C4×C31C2×C22⋊C412yes—
89C2×C4×C31#12014yes—
90C2×C4×C31#13514yes—
91C2×C4×C31AGL(1,ℤ/8)12yes—
92C2×C4×C31#13712yes—
93C8×C31—84no—
94C2×#533C2×C3111yescharacteristic
95#1797C2×C3112yes—
96C2×#332C2×C3114yescomplemented
97C22×C31—24no—
98C22×C31—22no—
99C22×C31—41no—
100C22×C31—41no—
101C22×C31—42no—
102C22×C31#37911yescharacteristic
103C22×C31C2×#10111yescharacteristic
104C22×C31C2×#10111yescharacteristic
105C22×C31#41612yes—
106C22×C31#41712yes—
107C4×C31—22no—
108C4×C31—22no—
109C4×C31—42no—
110C4×C31—48no—
111C4×C31#44914yes—
112C4×C31#45214yes—
113#533—28no—
114C2×C4⋊Q8C22×C3112yes—
115C2×#125C4×C3112yes—
116C2×C22⋊Q8—22no—
117C2×C4×Q8—22no—
118C2×#117C4×C3112yes—
119C22×C42C22×C3111yescharacteristic
120C2×C4⋊C8C22×C3114yes—
121C2×C22⋊C8—22no—
122#336—22no—
123#332—28no—
124C2×C31—42no—
125C2×C31#154411yescharacteristic
126C2×C31#155212yes—
127C2×C31#156111yescharacteristic
128C2×C31#178411yescharacteristic
129C2×C31#178812yes—
130C22×Q8—22no—
131C23×C4C2×C4×C3111yescharacteristic
132C23×C4D4×C3112yes—
133C22×C8—24no—
134C4⋊Q8—24no—
135C4⋊Q8D4×C3118yes—
136#125—28no—
137C22⋊Q8—216no—
138C4×Q8—216no—
139#117—28no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4C2×C4×C3114yes—
143C2×C22⋊C4—22no—
144C2×C42—22no—
145C2×C42—24no—
146C2×C42C2×C4×C3111yescharacteristic
147C2×C42D4×C3112yes—
148C2×C42C23×C3111yescharacteristic
149C4⋊C8—28no—
150C22⋊C8—44no—
151C31#834511yescharacteristic, direct product factor, Sylow
152C24C22⋊C4×C3111yescharacteristic
153C2×Q8—28no—
154C2×Q8—44no—
155C2×Q8—42no—
156C2×Q8—42no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4C22⋊C4×C3112yes—
166C22×C4C22⋊C4×C3111yescharacteristic
167C22×C4C22×C4×C3111yescharacteristic
168C22×C4C2×D4×C3112yes—
169C2×C8—44no—
170C2×C8—44no—
171C2×C8—44no—
172C4⋊C4—216no—
173C4⋊C4—24no—
174C4⋊C4—28no—
175C4⋊C4C22⋊C4×C3118yes—
176C22⋊C4—28no—
177C42—28no—
178C42—22no—
179C42—24no—
180C42C2×D4×C3114yes—
181C23—21no—
182C23—22no—
183C23—22no—
184C23—21no—
185C23C31×#10112yes—
186C23C2×C22⋊C4×C3111yescharacteristic
187Q8—48no—
188Q8—48no—
189C2×C4—28no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—28no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4C31×#10112yes—
203C2×C4C2×C22⋊C4×C3112yes—
204C2×C4C31×#12014yes—
205C2×C4C31×#13514yes—
206C2×C4C31×AGL(1,ℤ/8)12yes—
207C2×C4C31×#13712yes—
208C8—84no—
209C22—24no—
210C22—22no—
211C22—41no—
212C22—41no—
213C22—42no—
214C22C31×#37911yescharacteristic
215C22C2×C31×#10111yescharacteristic
216C22C2×C31×#10111yescharacteristic
217C22C31×#41612yes—
218C22C31×#41712yes—
219C4—22no—
220C4—22no—
221C4—42no—
222C4—48no—
223C4C31×#44914yes—
224C4C31×#45214yes—
225C2—42no—
226C2C31×#154411yescharacteristic
227C2C31×#155212yes—
228C2C31×#156111yescharacteristic
229C2C31×#178411yescharacteristic
230C2C31×#178812yes—
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC24×C31C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C31×#332C2semidirect
65. #8345C31direct (N × H)
96. C2×#332C2×C31semidirect
151. C31#8345direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5332
C31×#17972
C2×C31×#3322
#834531