Finite Group Databasebeta

← C31×#12365 #21469124

Subgroups — C31×#12365 #21469124

Order 7936.

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

normal subgroups only

This group has 346 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×#3291C211yesmaximal, characteristic
2C31×#2230C212yesmaximal, complemented
3C31×#1708C211yesmaximal, characteristic, complemented
4C31×#1359C211yesmaximal, characteristic, complemented
5C31×#1335C212yesmaximal, complemented
6C31×#577—22no—
7C31×#498C2211yescharacteristic, complemented
8C31×#492C2211yescharacteristic
9C31×#487—21no—
10C31×#486—21no—
11C31×#471—22no—
12C31×#456—21no—
13C31×#453—21no—
14C31×#443C2212yes—
15C31×#365—22no—
16C31×#364C2212yescomplemented
17C31×#360—22no—
18C82×C31C2211yescharacteristic, complemented
19C31×2−1+4—21no—
20C31×H(5,𝔽2)—21no—
21C31×#137—22no—
22C31×#137—42no—
23C31×AGL(1,ℤ/8)—22no—
24C31×AGL(1,ℤ/8)—42no—
25C31×#135—42no—
26C31×#135D412yes—
27C2×SD16×C31—22no—
28C2×SD16×C31—42no—
29C31×#131—22no—
30C31×C4⋊Q8—21no—
31C31×C4⋊Q8D411yescharacteristic
32C31×#127D411yescharacteristic
33C31×M32—22no—
34C31×#108D412yes—
35C31×#106—22no—
36C31×#104—22no—
37C31×#103—42no—
38C31×#102—42no—
39C31×#101—21no—
40C31×#100—21no—
41C4×C8×C31—21no—
42C4×C8×C31C2311yescharacteristic
43#41×C31—22no—
44#41×C31—42no—
45#41×C31—42no—
46#41×C31—81no—
47C2×Q8×C31—21no—
48C2×Q8×C31—41no—
49C2×Q8×C31—42no—
50C2×D4×C31—21no—
51C2×D4×C31—41no—
52C2×D4×C31—42no—
53C2×D4×C31—81no—
54Q16×C31—42no—
55Q16×C31SD1612yes—
56SD16×C31—22no—
57SD16×C31—42no—
58SD16×C31—42no—
59SD16×C31—42no—
60SD16×C31—42no—
61D8×C31—42no—
62D8×C31SD1612yes—
63M16×C31—22no—
64M16×C31—42no—
65C2×C8×C31—22no—
66C2×C8×C31—42no—
67C2×C8×C31C2×D412yes—
68C4⋊C4×C31—41no—
69C4⋊C4×C31—41no—
70C4⋊C4×C31—41no—
71C42×C31C2×D411yescharacteristic
72C16×C31—42no—
73#12365C3111yesmaximal, characteristic, direct product factor, Sylow
74C23×C31—41no—
75C23×C31—81no—
76Q8×C31—22no—
77Q8×C31—42no—
78Q8×C31—82no—
79Q8×C31—81no—
80D4×C31—22no—
81D4×C31—42no—
82D4×C31—42no—
83D4×C31—82no—
84D4×C31—82no—
85D4×C31—81no—
86C2×C4×C31—21no—
87C2×C4×C31—42no—
88C2×C4×C31—41no—
89C2×C4×C31—81no—
90C2×C4×C31—82no—
91C2×C4×C31—81no—
92C2×C4×C31#12011yescharacteristic
93C8×C31—42no—
94C8×C31—42no—
95C8×C31C2×SD1612yes—
96C8×C31AGL(1,ℤ/8)12yes—
97#3291C2×C3111yescharacteristic
98#2230C2×C3112yescomplemented
99#1708C2×C3111yescharacteristic, complemented
100#1359C2×C3111yescharacteristic, complemented
101#1335C2×C3112yescomplemented
102C22×C31—42no—
103C22×C31—81no—
104C22×C31—161no—
105C22×C31—321no—
106C22×C31#45311yescharacteristic
107C4×C31—22no—
108C4×C31—42no—
109C4×C31—81no—
110C4×C31—161no—
111C4×C31#45012yes—
112#577—22no—
113#498C22×C3111yescharacteristic, complemented
114#492C22×C3111yescharacteristic
115#487—21no—
116#486—21no—
117#471—22no—
118#456—21no—
119#453—21no—
120#443C22×C3112yes—
121#365—22no—
122#364C22×C3112yescomplemented
123#360—22no—
124C82C22×C3111yescharacteristic, complemented
125C2×C31—21no—
126C2×C31—82no—
127C2×C31—161no—
128C2×C31#162911yescharacteristic
1292−1+4—21no—
130H(5,𝔽2)—21no—
131#137—22no—
132#137—42no—
133AGL(1,ℤ/8)—22no—
134AGL(1,ℤ/8)—42no—
135#135—42no—
136#135D4×C3112yes—
137C2×SD16—22no—
138C2×SD16—42no—
139#131—22no—
140C4⋊Q8—21no—
141C4⋊Q8D4×C3111yescharacteristic
142#127D4×C3111yescharacteristic
143M32—22no—
144#108D4×C3112yes—
145#106—22no—
146#104—22no—
147#103—42no—
148#102—42no—
149#101—21no—
150#100—21no—
151C4×C8—21no—
152C4×C8C23×C3111yescharacteristic
153C31#1236511yescharacteristic, direct product factor, Sylow
154#41—22no—
155#41—42no—
156#41—42no—
157#41—81no—
158C2×Q8—21no—
159C2×Q8—41no—
160C2×Q8—42no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4—42no—
164C2×D4—81no—
165Q16—42no—
166Q16SD16×C3112yes—
167SD16—22no—
168SD16—42no—
169SD16—42no—
170SD16—42no—
171SD16—42no—
172D8—42no—
173D8SD16×C3112yes—
174M16—22no—
175M16—42no—
176C2×C8—22no—
177C2×C8—42no—
178C2×C8C2×D4×C3112yes—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C42C2×D4×C3111yescharacteristic
183C16—42no—
184C23—41no—
185C23—81no—
186Q8—22no—
187Q8—42no—
188Q8—82no—
189Q8—81no—
190D4—22no—
191D4—42no—
192D4—42no—
193D4—82no—
194D4—82no—
195D4—81no—
196C2×C4—21no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—81no—
200C2×C4—82no—
201C2×C4—81no—
202C2×C4C31×#12011yescharacteristic
203C8—42no—
204C8—42no—
205C8C2×SD16×C3112yes—
206C8C31×AGL(1,ℤ/8)12yes—
207C22—42no—
208C22—81no—
209C22—161no—
210C22—321no—
211C22C31×#45311yescharacteristic
212C4—22no—
213C4—42no—
214C4—81no—
215C4—161no—
216C4C31×#45012yes—
217C2—21no—
218C2—82no—
219C2—161no—
220C2C31×#162911yescharacteristic
subgroupG/H
center, socleC2×C31#1629
commutator, Frattini, cosocleC4×C8C23×C31
prime radicalC31×#3291C2
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
2. C31×#2230C2semidirect
3. C31×#1708C2semidirect
4. C31×#1359C2semidirect
5. C31×#1335C2semidirect
7. C31×#498C22semidirect
16. C31×#364C22semidirect
18. C82×C31C22semidirect
73. #12365C31direct (N × H)
98. #2230C2×C31semidirect
99. #1708C2×C31semidirect
100. #1359C2×C31semidirect
101. #1335C2×C31semidirect
113. #498C22×C31semidirect
122. #364C22×C31semidirect
124. C82C22×C31semidirect
153. C31#12365direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32912
C31×#22302
C31×#17082
C31×#13592
C31×#13352
#1236531