Finite Group Databasebeta

← C31×#10227 #21466986

Subgroups — C31×#10227 #21466986

Order 7936.

466 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3229C212yesmaximal
2C31×#3162C211yesmaximal, characteristic
3C31×#1960C211yesmaximal, characteristic
4C31×#1518C212yesmaximal
5C31×#1281C211yesmaximal, characteristic
6C31×#533C2212yes—
7C2×C31×C4⋊Q8C2211yescharacteristic
8C31×#501—22no—
9C31×#501—22no—
10C31×#498—22no—
11C31×#487—22no—
12C31×#483—24no—
13C31×#475—24no—
14C31×#431C2212yes—
15C31×#427—22no—
16C2×C31×#103—22no—
17C2×C31×#97C2211yescharacteristic
18C31×#382C2211yescharacteristic
19C31×#328—24no—
20C22×Q8×C31—21no—
21C22×C8×C31D412yes—
22C31×C4⋊Q8—22no—
23C31×C4⋊Q8—21no—
24C31×C4⋊Q8D412yes—
25C31×C4⋊Q8D412yes—
26C31×C4⋊Q8D412yes—
27C31×#125—22no—
28C31×C22⋊Q8—22no—
29C31×C22⋊Q8—22no—
30C4×Q8×C31—22no—
31C4×Q8×C31—22no—
32C31×#117—22no—
33C2×C4⋊C4×C31—22no—
34C2×C4⋊C4×C31—21no—
35C2×C42×C31C2311yescharacteristic
36C31×#107—24no—
37C31×#106—22no—
38C31×#106—22no—
39C31×#106—28no—
40C31×C4⋊C8—22no—
41C31×#103—28no—
42C31×#103—44no—
43C31×C22⋊C8D414yes—
44C31×#97—22no—
45C4×C8×C31—22no—
46C31×#95—22no—
47C2×Q8×C31—22no—
48C2×Q8×C31—22no—
49C2×Q8×C31—22no—
50C2×Q8×C31—41no—
51C2×Q8×C31—42no—
52C2×Q8×C31—42no—
53C22×C4×C31—41no—
54C22×C4×C31—41no—
55C22×C4×C31C2×D412yes—
56C22×C4×C31C2×D411yescharacteristic
57C2×C8×C31—22no—
58C2×C8×C31—22no—
59C2×C8×C31—24no—
60C2×C8×C31SD1614yes—
61C4⋊C4×C31—24no—
62C4⋊C4×C31—22no—
63C4⋊C4×C31—22no—
64C4⋊C4×C31—22no—
65C4⋊C4×C31—22no—
66C4⋊C4×C31—42no—
67C4⋊C4×C31—42no—
68C4⋊C4×C31—42no—
69C4⋊C4×C31—42no—
70C22⋊C4×C31—42no—
71C42×C31—42no—
72C42×C31C2×D412yes—
73C42×C31C2×D411yescharacteristic
74C42×C31#4111yescharacteristic
75#10227C3111yesmaximal, characteristic, direct product factor, Sylow
76C23×C31#12711yescharacteristic
77Q8×C31—44no—
78Q8×C31—44no—
79Q8×C31—82no—
80Q8×C31—82no—
81C2×C4×C31—42no—
82C2×C4×C31—42no—
83C2×C4×C31—42no—
84C2×C4×C31—42no—
85C2×C4×C31—81no—
86C2×C4×C31—81no—
87C2×C4×C31—82no—
88C2×C4×C31—81no—
89C2×C4×C31#12012yes—
90C2×C4×C31#12011yescharacteristic
91C2×C4×C31#12112yes—
92C2×C4×C31#12111yescharacteristic
93C2×C4×C31C2×SD1612yes—
94C2×C4×C31#13712yes—
95C8×C31—24no—
96C8×C31—44no—
97#3229C2×C3112yes—
98#3162C2×C3111yescharacteristic
99#1960C2×C3111yescharacteristic
100#1518C2×C3112yes—
101#1281C2×C3111yescharacteristic
102C22×C31—22no—
103C22×C31#39211yescharacteristic
104C22×C31#49211yescharacteristic
105C22×C31#49711yescharacteristic
106C4×C31—22no—
107C4×C31—22no—
108C4×C31—82no—
109C4×C31—82no—
110C4×C31—82no—
111C4×C31—161no—
112C4×C31#44812yes—
113C4×C31#46112yes—
114#533C22×C3112yes—
115C2×C4⋊Q8C22×C3111yescharacteristic
116#501—22no—
117#501—22no—
118#498—22no—
119#487—22no—
120#483—24no—
121#475—24no—
122#431C22×C3112yes—
123#427—22no—
124C2×#103—22no—
125C2×#97C22×C3111yescharacteristic
126#382C22×C3111yescharacteristic
127#328—24no—
128C2×C31—21no—
129C2×C31—21no—
130C2×C31#200912yes—
131C2×C31#201311yescharacteristic
132C22×Q8—21no—
133C22×C8D4×C3112yes—
134C4⋊Q8—22no—
135C4⋊Q8—21no—
136C4⋊Q8D4×C3112yes—
137C4⋊Q8D4×C3112yes—
138C4⋊Q8D4×C3112yes—
139#125—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144#117—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C42C23×C3111yescharacteristic
148#107—24no—
149#106—22no—
150#106—22no—
151#106—28no—
152C4⋊C8—22no—
153#103—28no—
154#103—44no—
155C22⋊C8D4×C3114yes—
156#97—22no—
157C4×C8—22no—
158#95—22no—
159C31#1022711yescharacteristic, direct product factor, Sylow
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—42no—
165C2×Q8—42no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4C2×D4×C3112yes—
169C22×C4C2×D4×C3111yescharacteristic
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—24no—
173C2×C8SD16×C3114yes—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C22⋊C4—42no—
184C42—42no—
185C42C2×D4×C3112yes—
186C42C2×D4×C3111yescharacteristic
187C42#41×C3111yescharacteristic
188C23C31×#12711yescharacteristic
189Q8—44no—
190Q8—44no—
191Q8—82no—
192Q8—82no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—82no—
200C2×C4—81no—
201C2×C4C31×#12012yes—
202C2×C4C31×#12011yescharacteristic
203C2×C4C31×#12112yes—
204C2×C4C31×#12111yescharacteristic
205C2×C4C2×SD16×C3112yes—
206C2×C4C31×#13712yes—
207C8—24no—
208C8—44no—
209C22—22no—
210C22C31×#39211yescharacteristic
211C22C31×#49211yescharacteristic
212C22C31×#49711yescharacteristic
213C4—22no—
214C4—22no—
215C4—82no—
216C4—82no—
217C4—82no—
218C4—161no—
219C4C31×#44812yes—
220C4C31×#46112yes—
221C2—21no—
222C2—21no—
223C2C31×#200912yes—
224C2C31×#201311yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC23×C31#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
75. #10227C31direct (N × H)
159. C31#10227direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32292
C31×#31622
C31×#19602
C31×#15182
C31×#12812
#1022731