Finite Group Databasebeta

← C31×#12364 #21469123

Subgroups — C31×#12364 #21469123

Order 7936.

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

normal subgroups only

This group has 358 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×#1705C211yesmaximal, characteristic, complemented
4C31×#1359C211yesmaximal, characteristic, complemented
5C31×#1335C212yesmaximal, complemented
6C31×#577—22no—
7C31×#498—21no—
8C31×#498C2211yescharacteristic, complemented
9C31×#492—21no—
10C31×#492C2211yescharacteristic
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—
29C2×SD16×C31—42no—
30C31×#131—22no—
31C31×C4⋊Q8—21no—
32C31×C4⋊Q8D411yescharacteristic
33C31×#127D411yescharacteristic
34C31×M32—22no—
35C31×#108D412yes—
36C31×#106—22no—
37C31×#106—42no—
38C31×#104—22no—
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×C31SD1612yescomplemented
56SD16×C31—22no—
57SD16×C31—42no—
58SD16×C31—42no—
59SD16×C31—42no—
60SD16×C31—42no—
61SD16×C31—84no—
62D8×C31—42no—
63D8×C31SD1612yescomplemented
64M16×C31—22no—
65M16×C31—42no—
66C2×C8×C31—22no—
67C2×C8×C31—42no—
68C2×C8×C31C2×D412yes—
69C4⋊C4×C31—41no—
70C4⋊C4×C31—41no—
71C4⋊C4×C31—41no—
72C42×C31C2×D411yescharacteristic
73C16×C31—42no—
74#12364C3111yesmaximal, characteristic, direct product factor, Sylow
75C23×C31—41no—
76C23×C31—81no—
77Q8×C31—22no—
78Q8×C31—42no—
79Q8×C31—82no—
80Q8×C31—81no—
81D4×C31—22no—
82D4×C31—42no—
83D4×C31—42no—
84D4×C31—82no—
85D4×C31—82no—
86D4×C31—81no—
87C2×C4×C31—21no—
88C2×C4×C31—42no—
89C2×C4×C31—41no—
90C2×C4×C31—81no—
91C2×C4×C31—82no—
92C2×C4×C31—81no—
93C2×C4×C31#12011yescharacteristic
94C8×C31—24no—
95C8×C31—42no—
96C8×C31C2×SD1612yes—
97C8×C31AGL(1,ℤ/8)12yes—
98#3291C2×C3111yescharacteristic
99#2230C2×C3112yescomplemented
100#1705C2×C3111yescharacteristic, complemented
101#1359C2×C3111yescharacteristic, complemented
102#1335C2×C3112yescomplemented
103C22×C31—42no—
104C22×C31—81no—
105C22×C31—161no—
106C22×C31—321no—
107C22×C31#45311yescharacteristic
108C4×C31—22no—
109C4×C31—42no—
110C4×C31—81no—
111C4×C31—161no—
112C4×C31#45012yes—
113#577—22no—
114#498—21no—
115#498C22×C3111yescharacteristic, complemented
116#492—21no—
117#492C22×C3111yescharacteristic
118#471—22no—
119#456—21no—
120#453—21no—
121#443C22×C3112yes—
122#365—22no—
123#364C22×C3112yescomplemented
124#360—22no—
125C82C22×C3111yescharacteristic, complemented
126C2×C31—21no—
127C2×C31—82no—
128C2×C31—161no—
129C2×C31#162911yescharacteristic
1302−1+4—21no—
131H(5,𝔽2)—21no—
132#137—22no—
133#137—42no—
134AGL(1,ℤ/8)—22no—
135AGL(1,ℤ/8)—42no—
136#135—42no—
137#135D4×C3112yes—
138C2×SD16—22no—
139C2×SD16—42no—
140C2×SD16—42no—
141#131—22no—
142C4⋊Q8—21no—
143C4⋊Q8D4×C3111yescharacteristic
144#127D4×C3111yescharacteristic
145M32—22no—
146#108D4×C3112yes—
147#106—22no—
148#106—42no—
149#104—22no—
150#101—21no—
151#100—21no—
152C4×C8—21no—
153C4×C8C23×C3111yescharacteristic
154C31#1236411yescharacteristic, direct product factor, Sylow
155#41—22no—
156#41—42no—
157#41—42no—
158#41—81no—
159C2×Q8—21no—
160C2×Q8—41no—
161C2×Q8—42no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4—42no—
165C2×D4—81no—
166Q16—42no—
167Q16SD16×C3112yescomplemented
168SD16—22no—
169SD16—42no—
170SD16—42no—
171SD16—42no—
172SD16—42no—
173SD16—84no—
174D8—42no—
175D8SD16×C3112yescomplemented
176M16—22no—
177M16—42no—
178C2×C8—22no—
179C2×C8—42no—
180C2×C8C2×D4×C3112yes—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C42C2×D4×C3111yescharacteristic
185C16—42no—
186C23—41no—
187C23—81no—
188Q8—22no—
189Q8—42no—
190Q8—82no—
191Q8—81no—
192D4—22no—
193D4—42no—
194D4—42no—
195D4—82no—
196D4—82no—
197D4—81no—
198C2×C4—21no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—81no—
202C2×C4—82no—
203C2×C4—81no—
204C2×C4C31×#12011yescharacteristic
205C8—24no—
206C8—42no—
207C8C2×SD16×C3112yes—
208C8C31×AGL(1,ℤ/8)12yes—
209C22—42no—
210C22—81no—
211C22—161no—
212C22—321no—
213C22C31×#45311yescharacteristic
214C4—22no—
215C4—42no—
216C4—81no—
217C4—161no—
218C4C31×#45012yes—
219C2—21no—
220C2—82no—
221C2—161no—
222C2C31×#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×#1705C2semidirect
4. C31×#1359C2semidirect
5. C31×#1335C2semidirect
8. C31×#498C22semidirect
16. C31×#364C22semidirect
18. C82×C31C22semidirect
55. Q16×C31SD16semidirect
63. D8×C31SD16semidirect
74. #12364C31direct (N × H)
99. #2230C2×C31semidirect
100. #1705C2×C31semidirect
101. #1359C2×C31semidirect
102. #1335C2×C31semidirect
115. #498C22×C31semidirect
123. #364C22×C31semidirect
125. C82C22×C31semidirect
154. C31#12364direct (N × H)
167. Q16SD16×C31semidirect
175. D8SD16×C31semidirect

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×#17052
C31×#13592
C31×#13352
#1236431