Finite Group Databasebeta

← C31×#12310 #21469069

Subgroups — C31×#12310 #21469069

Order 7936.

406 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 78 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×#3292C211yesmaximal, characteristic, complemented
2C31×#2213C212yesmaximal, complemented
3C31×#1712C211yesmaximal, characteristic, complemented
4C31×#1357C211yesmaximal, characteristic, complemented
5C31×#1335C212yesmaximal, complemented
6C31×#576—22no—
7C31×#509—22no—
8C31×#493—22no—
9C31×#493C2211yescharacteristic
10C31×#493C2211yescharacteristic, complemented
11C31×#472—22no—
12C31×#453—22no—
13C31×#443C2212yes—
14C31×#364C2212yescomplemented
15C31×#361—22no—
16C82×C31C2211yescharacteristic, complemented
17C31×H(5,𝔽2)—22no—
18C31×AGL(1,ℤ/8)—24no—
19C31×AGL(1,ℤ/8)—44no—
20C31×#135—42no—
21C31×#135D412yes—
22C2×D8×C31—22no—
23C2×D8×C31—22no—
24C2×D8×C31—44no—
25C2×D8×C31—42no—
26C31×#131—22no—
27C31×#127—21no—
28C31×#127D412yes—
29C31×SD32—42no—
30C31×D16—42no—
31C31×M32—22no—
32C31×#108D412yes—
33C31×#104—22no—
34C31×#100—22no—
35C4×C8×C31—21no—
36C4×C8×C31C2311yescharacteristic
37#41×C31—22no—
38#41×C31—44no—
39C2×D4×C31—22no—
40C2×D4×C31—42no—
41C2×D4×C31—42no—
42C2×D4×C31—44no—
43C2×D4×C31—41no—
44C2×D4×C31—82no—
45Q16×C31D812yescomplemented
46SD16×C31—22no—
47SD16×C31—44no—
48D8×C31—44no—
49D8×C31—42no—
50D8×C31—42no—
51D8×C31—42no—
52D8×C31—42no—
53D8×C31—88no—
54D8×C31D812yescomplemented
55M16×C31—22no—
56M16×C31—42no—
57C2×C8×C31—22no—
58C2×C8×C31—42no—
59C2×C8×C31C2×D412yes—
60C42×C31C2×D411yescharacteristic
61C16×C31—42no—
62#12310C3111yesmaximal, characteristic, direct product factor, Sylow
63C23×C31—42no—
64C23×C31—82no—
65C23×C31—81no—
66Q8×C31—22no—
67D4×C31—22no—
68D4×C31—42no—
69D4×C31—44no—
70D4×C31—84no—
71D4×C31—84no—
72D4×C31—84no—
73D4×C31—82no—
74C2×C4×C31—21no—
75C2×C4×C31—42no—
76C2×C4×C31—82no—
77C2×C4×C31#12011yescharacteristic
78C8×C31—24no—
79C8×C31—42no—
80C8×C31C2×D812yes—
81C8×C31AGL(1,ℤ/8)12yes—
82#3292C2×C3111yescharacteristic, complemented
83#2213C2×C3112yescomplemented
84#1712C2×C3111yescharacteristic, complemented
85#1357C2×C3111yescharacteristic, complemented
86#1335C2×C3112yescomplemented
87C22×C31—42no—
88C22×C31—82no—
89C22×C31—162no—
90C22×C31—162no—
91C22×C31—161no—
92C22×C31—322no—
93C22×C31#45311yescharacteristic
94C4×C31—22no—
95C4×C31—42no—
96C4×C31#44712yes—
97#576—22no—
98#509—22no—
99#493—22no—
100#493C22×C3111yescharacteristic
101#493C22×C3111yescharacteristic, complemented
102#472—22no—
103#453—22no—
104#443C22×C3112yes—
105#364C22×C3112yescomplemented
106#361—22no—
107C82C22×C3111yescharacteristic, complemented
108C2×C31—21no—
109C2×C31—82no—
110C2×C31—162no—
111C2×C31—321no—
112C2×C31#161911yescharacteristic
113H(5,𝔽2)—22no—
114AGL(1,ℤ/8)—24no—
115AGL(1,ℤ/8)—44no—
116#135—42no—
117#135D4×C3112yes—
118C2×D8—22no—
119C2×D8—22no—
120C2×D8—44no—
121C2×D8—42no—
122#131—22no—
123#127—21no—
124#127D4×C3112yes—
125SD32—42no—
126D16—42no—
127M32—22no—
128#108D4×C3112yes—
129#104—22no—
130#100—22no—
131C4×C8—21no—
132C4×C8C23×C3111yescharacteristic
133C31#1231011yescharacteristic, direct product factor, Sylow
134#41—22no—
135#41—44no—
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—44no—
140C2×D4—41no—
141C2×D4—82no—
142Q16D8×C3112yescomplemented
143SD16—22no—
144SD16—44no—
145D8—44no—
146D8—42no—
147D8—42no—
148D8—42no—
149D8—42no—
150D8—88no—
151D8D8×C3112yescomplemented
152M16—22no—
153M16—42no—
154C2×C8—22no—
155C2×C8—42no—
156C2×C8C2×D4×C3112yes—
157C42C2×D4×C3111yescharacteristic
158C16—42no—
159C23—42no—
160C23—82no—
161C23—81no—
162Q8—22no—
163D4—22no—
164D4—42no—
165D4—44no—
166D4—84no—
167D4—84no—
168D4—84no—
169D4—82no—
170C2×C4—21no—
171C2×C4—42no—
172C2×C4—82no—
173C2×C4C31×#12011yescharacteristic
174C8—24no—
175C8—42no—
176C8C2×D8×C3112yes—
177C8C31×AGL(1,ℤ/8)12yes—
178C22—42no—
179C22—82no—
180C22—162no—
181C22—162no—
182C22—161no—
183C22—322no—
184C22C31×#45311yescharacteristic
185C4—22no—
186C4—42no—
187C4C31×#44712yes—
188C2—21no—
189C2—82no—
190C2—162no—
191C2—321no—
192C2C31×#161911yescharacteristic
subgroupG/H
center, socleC2×C31#1619
commutator, Frattini, cosocleC4×C8C23×C31
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C4×C8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3292C2semidirect
2. C31×#2213C2semidirect
3. C31×#1712C2semidirect
4. C31×#1357C2semidirect
5. C31×#1335C2semidirect
10. C31×#493C22semidirect
14. C31×#364C22semidirect
16. C82×C31C22semidirect
45. Q16×C31D8semidirect
54. D8×C31D8semidirect
62. #12310C31direct (N × H)
82. #3292C2×C31semidirect
83. #2213C2×C31semidirect
84. #1712C2×C31semidirect
85. #1357C2×C31semidirect
86. #1335C2×C31semidirect
101. #493C22×C31semidirect
105. #364C22×C31semidirect
107. C82C22×C31semidirect
133. C31#12310direct (N × H)
142. Q16D8×C31semidirect
151. D8D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#32922
C31×#22132
C31×#17122
C31×#13572
C31×#13352
#1231031