Finite Group Databasebeta

← C31×#23565 #21480324

Subgroups — C31×#23565 #21480324

Order 7936.

1334 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 366 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×#543C211yesmaximal, characteristic
2C31×#2315C213yesmaximal
3C31×#2062C218yesmaximal, complemented
4C2×C31×#342C213yesmaximal
5C31×#543—26no—
6C31×#543C2214yes—
7C2×C31×C22⋊Q8—26no—
8C22×C22⋊C4×C31C2211yescharacteristic
9C22×C22⋊C4×C31C2213yes—
10C22×C22⋊C4×C31C2213yes—
11C31×#385—26no—
12C31×#385C22112yes—
13C31×#380—26no—
14C31×#342C22112yes—
15C25×C31C2311yescharacteristic
16C22×Q8×C31—22no—
17C23×C4×C31—23no—
18C23×C4×C31D416yes—
19C23×C4×C31C2312yes—
20C31×C22⋊Q8—224no—
21C31×C22⋊Q8—224no—
22C2×C4⋊C4×C31—26no—
23C2×C4⋊C4×C31—212no—
24C2×C22⋊C4×C31—26no—
25C2×C22⋊C4×C31—26no—
26C2×C22⋊C4×C31—23no—
27C2×C22⋊C4×C31—26no—
28C2×C22⋊C4×C31—23no—
29C2×C22⋊C4×C31—23no—
30C2×C22⋊C4×C31D416yes—
31C2×C22⋊C4×C31Q8112yescomplemented
32C2×C22⋊C4×C31C2316yes—
33C2×C22⋊C4×C31C2316yes—
34C31×#95—212no—
35C24×C31—82no—
36C24×C31C2×D413yes—
37C24×C31C2×D413yes—
38C24×C31C2×Q816yes—
39C24×C31#4112yes—
40C24×C31C2411yescharacteristic
41C2×Q8×C31—28no—
42C2×Q8×C31—84no—
43C22×C4×C31—26no—
44C22×C4×C31—26no—
45C22×C4×C31—26no—
46C22×C4×C31—212no—
47C22×C4×C31—26no—
48C22×C4×C31—82no—
49C22×C4×C31—86no—
50C22×C4×C31—86no—
51C22×C4×C31C2×D4112yes—
52C4⋊C4×C31—224no—
53C4⋊C4×C31—812no—
54C22⋊C4×C31—212no—
55C22⋊C4×C31—212no—
56C22⋊C4×C31—212no—
57C22⋊C4×C31—26no—
58C22⋊C4×C31—43no—
59C22⋊C4×C31—41no—
60C22⋊C4×C31C2×Q8112yescomplemented
61#23565C3111yesmaximal, characteristic, direct product factor, Sylow
62C23×C31—46no—
63C23×C31—46no—
64C23×C31—42no—
65C23×C31—86no—
66C23×C31—82no—
67C23×C31#12013yes—
68C23×C31#12011yescharacteristic
69C23×C31C22⋊Q816yes—
70C23×C31C22⋊Q816yes—
71C23×C31C4⋊Q816yes—
72C23×C31C4⋊Q816yes—
73C23×C31C22×D413yes—
74C23×C31C22×Q813yes—
75C23×C31C2×#4111yescharacteristic
76Q8×C31—88no—
77C2×C4×C31—212no—
78C2×C4×C31—212no—
79C2×C4×C31—412no—
80C2×C4×C31—412no—
81C2×C4×C31—412no—
82C2×C4×C31—412no—
83C2×C4×C31—42no—
84C2×C4×C31—86no—
85C2×C4×C31—812no—
86C2×#543C2×C3111yescharacteristic
87#2315C2×C3113yes—
88#2062C2×C3118yescomplemented
89C2×#342C2×C3113yes—
90C22×C31—22no—
91C22×C31—26no—
92C22×C31—26no—
93C22×C31—46no—
94C22×C31—46no—
95C22×C31—46no—
96C22×C31—46no—
97C22×C31—43no—
98C22×C31—41no—
99C22×C31#39516yes—
100C22×C31#39512yes—
101C22×C31C2×#12011yescharacteristic
102C22×C31C2×C22⋊Q813yes—
103C22×C31C2×C4⋊Q813yes—
104C4×C31—412no—
105C4×C31—412no—
106C4×C31—82no—
107#543—26no—
108#543C22×C3114yes—
109C2×C22⋊Q8—26no—
110C22×C22⋊C4C22×C3111yescharacteristic
111C22×C22⋊C4C22×C3113yes—
112C22×C22⋊C4C22×C3113yes—
113#385—26no—
114#385C22×C31112yes—
115#380—26no—
116#342C22×C31112yes—
117C2×C31—23no—
118C2×C31—23no—
119C2×C31—22no—
120C2×C31—26no—
121C2×C31C2×#39511yescharacteristic
122C2×C31#302612yes—
123C25C23×C3111yescharacteristic
124C22×Q8—22no—
125C23×C4—23no—
126C23×C4D4×C3116yes—
127C23×C4C23×C3112yes—
128C22⋊Q8—224no—
129C22⋊Q8—224no—
130C2×C4⋊C4—26no—
131C2×C4⋊C4—212no—
132C2×C22⋊C4—26no—
133C2×C22⋊C4—26no—
134C2×C22⋊C4—23no—
135C2×C22⋊C4—26no—
136C2×C22⋊C4—23no—
137C2×C22⋊C4—23no—
138C2×C22⋊C4D4×C3116yes—
139C2×C22⋊C4Q8×C31112yescomplemented
140C2×C22⋊C4C23×C3116yes—
141C2×C22⋊C4C23×C3116yes—
142#95—212no—
143C31#2356511yescharacteristic, direct product factor, Sylow
144C24—82no—
145C24C2×D4×C3113yes—
146C24C2×D4×C3113yes—
147C24C2×Q8×C3116yes—
148C24#41×C3112yes—
149C24C24×C3111yescharacteristic
150C2×Q8—28no—
151C2×Q8—84no—
152C22×C4—26no—
153C22×C4—26no—
154C22×C4—26no—
155C22×C4—212no—
156C22×C4—26no—
157C22×C4—82no—
158C22×C4—86no—
159C22×C4—86no—
160C22×C4C2×D4×C31112yes—
161C4⋊C4—224no—
162C4⋊C4—812no—
163C22⋊C4—212no—
164C22⋊C4—212no—
165C22⋊C4—212no—
166C22⋊C4—26no—
167C22⋊C4—43no—
168C22⋊C4—41no—
169C22⋊C4C2×Q8×C31112yescomplemented
170C23—46no—
171C23—46no—
172C23—42no—
173C23—86no—
174C23—82no—
175C23C31×#12013yes—
176C23C31×#12011yescharacteristic
177C23C31×C22⋊Q816yes—
178C23C31×C22⋊Q816yes—
179C23C31×C4⋊Q816yes—
180C23C31×C4⋊Q816yes—
181C23C22×D4×C3113yes—
182C23C22×Q8×C3113yes—
183C23C2×#41×C3111yescharacteristic
184Q8—88no—
185C2×C4—212no—
186C2×C4—212no—
187C2×C4—412no—
188C2×C4—412no—
189C2×C4—412no—
190C2×C4—412no—
191C2×C4—42no—
192C2×C4—86no—
193C2×C4—812no—
194C22—22no—
195C22—26no—
196C22—26no—
197C22—46no—
198C22—46no—
199C22—46no—
200C22—46no—
201C22—43no—
202C22—41no—
203C22C31×#39516yes—
204C22C31×#39512yes—
205C22C2×C31×#12011yescharacteristic
206C22C2×C31×C22⋊Q813yes—
207C22C2×C31×C4⋊Q813yes—
208C4—412no—
209C4—412no—
210C4—82no—
211C2—23no—
212C2—22no—
213C2—26no—
214C2—23no—
215C2C2×C31×#39511yescharacteristic
216C2C31×#302612yes—
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC24C24×C31
prime radicalC25×C31C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C25×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#2062C2semidirect
31. C2×C22⋊C4×C31Q8semidirect
60. C22⋊C4×C31C2×Q8semidirect
61. #23565C31direct (N × H)
88. #2062C2×C31semidirect
139. C2×C22⋊C4Q8×C31semidirect
143. C31#23565direct (N × H)
169. C22⋊C4C2×Q8×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5432
C31×#23152
C31×#20622
C2×C31×#3422
#2356531