Finite Group Databasebeta

← C31×#9757 #21466516

Subgroups — C31×#9757 #21466516

Order 7936.

400 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 134 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×#3131C211yesmaximal, characteristic
2C31×#1835C211yesmaximal, characteristic, complemented
3C31×#1518C212yesmaximal, complemented
4C31×#1517C212yesmaximal
5C31×#1450C211yesmaximal, characteristic, complemented
6C31×#533C2211yescharacteristic
7C31×#532C2211yescharacteristic
8C31×#488—21no—
9C31×#488—21no—
10C31×#487C412yescomplemented
11C31×#486C412yescomplemented
12C31×#431C2212yes—
13C31×#416—22no—
14C2×C4×C8×C31C2211yescharacteristic
15C31×#382C2212yes—
16C31×#340—22no—
17C31×#328—24no—
18C31×#327—24no—
19C31×#322—21no—
20C82×C31—21no—
21C2×#41×C31—21no—
22C22×C8×C31D412yes—
23C31×C4⋊Q8C2×C412yes—
24C31×C4⋊Q8C2×C411yescharacteristic
25C31×#127C2×C411yescharacteristic
26C31×#125—21no—
27C31×#124—21no—
28C31×C22⋊Q8—22no—
29C31×#121—22no—
30C4×Q8×C31—22no—
31C4×D4×C31—22no—
32C31×#117—21no—
33C2×C4⋊C4×C31—22no—
34C2×C42×C31C2311yescharacteristic
35C31×C4⋊C8—22no—
36C31×#103—22no—
37C31×#103—22no—
38C31×#102—22no—
39C31×#102—22no—
40C31×C22⋊C8D414yes—
41C4×C8×C31—22no—
42C4×C8×C31C2×C412yes—
43C4×C8×C31D411yescharacteristic
44C4×C8×C31D411yescharacteristic
45C31×#95—22no—
46C31×#95—22no—
47#41×C31—44no—
48C2×Q8×C31—21no—
49C2×Q8×C31—22no—
50C2×D4×C31—21no—
51C2×D4×C31—22no—
52C2×D4×C31—41no—
53C22×C4×C31—41no—
54C22×C4×C31—42no—
55C22×C4×C31C2×D412yes—
56C22×C4×C31C2×D411yescharacteristic
57C2×C8×C31—24no—
58C2×C8×C31—24no—
59C2×C8×C31C22⋊C412yes—
60C2×C8×C31C22⋊C412yes—
61C4⋊C4×C31—22no—
62C4⋊C4×C31—22no—
63C4⋊C4×C31—22no—
64C4⋊C4×C31—42no—
65C4⋊C4×C31—44no—
66C22⋊C4×C31—41no—
67C22⋊C4×C31—42no—
68C42×C31—41no—
69C42×C31C22×C411yescharacteristic
70C42×C31C2×D411yescharacteristic
71C42×C31#4112yes—
72#9757C3111yesmaximal, characteristic, direct product factor, Sylow
73C23×C31—41no—
74C23×C31#12711yescharacteristic
75Q8×C31—42no—
76Q8×C31—82no—
77D4×C31—42no—
78D4×C31—82no—
79D4×C31—82no—
80C2×C4×C31—42no—
81C2×C4×C31—42no—
82C2×C4×C31—41no—
83C2×C4×C31—82no—
84C2×C4×C31—84no—
85C2×C4×C31—82no—
86C2×C4×C31C2×C22⋊C411yescharacteristic
87C2×C4×C31C4×D412yes—
88C2×C4×C31#12112yes—
89C2×C4×C31#12411yescharacteristic
90C2×C4×C31#13514yes—
91C8×C31—22no—
92C8×C31—22no—
93C8×C31—44no—
94#3131C2×C3111yescharacteristic
95#1835C2×C3111yescharacteristic, complemented
96#1518C2×C3112yescomplemented
97#1517C2×C3112yes—
98#1450C2×C3111yescharacteristic, complemented
99C22×C31—22no—
100C22×C31—81no—
101C22×C31—82no—
102C22×C31#39011yescharacteristic
103C22×C31#49512yes—
104C4×C31—22no—
105C4×C31—22no—
106C4×C31—81no—
107C4×C31—82no—
108C4×C31—162no—
109C4×C31#41612yes—
110C4×C31#48812yes—
111#533C22×C3111yescharacteristic
112#532C22×C3111yescharacteristic
113#488—21no—
114#488—21no—
115#487C4×C3112yescomplemented
116#486C4×C3112yescomplemented
117#431C22×C3112yes—
118#416—22no—
119C2×C4×C8C22×C3111yescharacteristic
120#382C22×C3112yes—
121#340—22no—
122#328—24no—
123#327—24no—
124#322—21no—
125C82—21no—
126C2×C31—22no—
127C2×C31—161no—
128C2×C31#195811yescharacteristic
129C2×C31#197612yes—
130C2×#41—21no—
131C22×C8D4×C3112yes—
132C4⋊Q8C2×C4×C3111yescharacteristic
133C4⋊Q8C2×C4×C3112yes—
134#127C2×C4×C3111yescharacteristic
135#125—21no—
136#124—21no—
137C22⋊Q8—22no—
138#121—22no—
139C4×Q8—22no—
140C4×D4—22no—
141#117—21no—
142C2×C4⋊C4—22no—
143C2×C42C23×C3111yescharacteristic
144C4⋊C8—22no—
145#103—22no—
146#103—22no—
147#102—22no—
148#102—22no—
149C22⋊C8D4×C3114yes—
150C4×C8—22no—
151C4×C8C2×C4×C3112yes—
152C4×C8D4×C3111yescharacteristic
153C4×C8D4×C3111yescharacteristic
154#95—22no—
155#95—22no—
156C31#975711yescharacteristic, direct product factor, Sylow
157#41—44no—
158C2×Q8—21no—
159C2×Q8—22no—
160C2×D4—21no—
161C2×D4—22no—
162C2×D4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4C2×D4×C3112yes—
166C22×C4C2×D4×C3111yescharacteristic
167C2×C8—24no—
168C2×C8—24no—
169C2×C8C22⋊C4×C3112yes—
170C2×C8C22⋊C4×C3112yes—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—44no—
176C22⋊C4—41no—
177C22⋊C4—42no—
178C42—41no—
179C42C22×C4×C3111yescharacteristic
180C42C2×D4×C3111yescharacteristic
181C42#41×C3112yes—
182C23—41no—
183C23C31×#12711yescharacteristic
184Q8—42no—
185Q8—82no—
186D4—42no—
187D4—82no—
188D4—82no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—82no—
193C2×C4—84no—
194C2×C4—82no—
195C2×C4C2×C22⋊C4×C3111yescharacteristic
196C2×C4C4×D4×C3112yes—
197C2×C4C31×#12112yes—
198C2×C4C31×#12411yescharacteristic
199C2×C4C31×#13514yes—
200C8—22no—
201C8—22no—
202C8—44no—
203C22—22no—
204C22—81no—
205C22—82no—
206C22C31×#39011yescharacteristic
207C22C31×#49512yes—
208C4—22no—
209C4—22no—
210C4—81no—
211C4—82no—
212C4—162no—
213C4C31×#41612yes—
214C4C31×#48812yes—
215C2—22no—
216C2—161no—
217C2C31×#195811yescharacteristic
218C2C31×#197612yes—
subgroupG/H
center, socleC22×C31#390
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1835C2semidirect
3. C31×#1518C2semidirect
5. C31×#1450C2semidirect
10. C31×#487C4semidirect
11. C31×#486C4semidirect
72. #9757C31direct (N × H)
95. #1835C2×C31semidirect
96. #1518C2×C31semidirect
98. #1450C2×C31semidirect
115. #487C4×C31semidirect
116. #486C4×C31semidirect
156. C31#9757direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31312
C31×#18352
C31×#15182
C31×#15172
C31×#14502
#975731