Finite Group Databasebeta

← C31×#12306 #21469065

Subgroups — C31×#12306 #21469065

Order 7936.

274 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3263C211yesmaximal, characteristic
2C31×#2239C211yesmaximal, characteristic, complemented
3C31×#2238C211yesmaximal, characteristic
4C31×#2103C211yesmaximal, characteristic, complemented
5C31×#1342C211yesmaximal, characteristic, complemented
6C31×#1341C211yesmaximal, characteristic
7C31×#1328C211yesmaximal, characteristic, complemented
8C31×#542—21no—
9C31×#476—21no—
10C31×#475—21no—
11C31×#470C2211yescharacteristic
12C31×#469C2211yescharacteristic
13C31×#465—21no—
14C31×#464—21no—
15C31×#463—21no—
16C31×#430C2211yescharacteristic
17C31×#424C2211yescharacteristic
18C31×#365—21no—
19C31×#365—21no—
20C31×#362—21no—
21C31×#361—21no—
22C31×#349C2211yescharacteristic
23C31×#349C2211yescharacteristic
24C31×#344—21no—
25C31×#341—21no—
26C31×#338C2211yescharacteristic
27C2×Q16×C31D411yescharacteristic
28C2×SD16×C31—21no—
29C2×D8×C31D411yescharacteristic
30C2×M16×C31—21no—
31C2×M16×C31—21no—
32C2×M16×C31C2311yescharacteristic
33C22×C8×C31D411yescharacteristic
34C31×C4⋊Q8—21no—
35C31×#126—41no—
36C31×#124—21no—
37C31×#124—41no—
38C31×#123—41no—
39C31×C22⋊Q8—21no—
40C31×C22⋊Q8—41no—
41C31×#121—21no—
42C4×Q8×C31—21no—
43C4×D4×C31—21no—
44C31×#117D411yescharacteristic
45C31×M32—21no—
46C31×M32—21no—
47C31×#108—21no—
48C31×#108—21no—
49C31×#106—21no—
50C31×#106D411yescharacteristic
51C31×#106D411yescharacteristic
52C31×C4⋊C8—21no—
53C31×C4⋊C8—21no—
54C31×#103—21no—
55C31×#103—21no—
56C31×#102—21no—
57C31×#102—21no—
58C2×Q8×C31—21no—
59C2×Q8×C31—41no—
60C2×D4×C31—21no—
61C2×D4×C31—41no—
62C22×C4×C31—41no—
63C22×C4×C31C2×D411yescharacteristic
64Q16×C31—41no—
65SD16×C31—81no—
66D8×C31—41no—
67M16×C31—41no—
68M16×C31—41no—
69M16×C31—41no—
70M16×C31—41no—
71M16×C31—41no—
72C2×C8×C31—21no—
73C2×C8×C31—41no—
74C2×C8×C31—41no—
75C2×C8×C31—41no—
76C2×C8×C31C2×D411yescharacteristic
77C2×C8×C31C2×D411yescharacteristic
78C4⋊C4×C31—21no—
79C4⋊C4×C31—21no—
80C4⋊C4×C31—21no—
81C4⋊C4×C31—41no—
82C4⋊C4×C31—41no—
83C4⋊C4×C31—41no—
84C22⋊C4×C31—41no—
85C22⋊C4×C31—41no—
86C22⋊C4×C31—41no—
87C22⋊C4×C31—41no—
88C22⋊C4×C31—41no—
89C22⋊C4×C31SD1611yescharacteristic
90C22⋊C4×C31SD1611yescharacteristic
91C42×C31—21no—
92C42×C31—41no—
93C16×C31—41no—
94C16×C31—41no—
95#12306C3111yesmaximal, characteristic, direct product factor, Sylow
96C23×C31—41no—
97C23×C31C2×SD1611yescharacteristic
98Q8×C31—41no—
99Q8×C31—41no—
100Q8×C31—161no—
101D4×C31—41no—
102D4×C31—41no—
103D4×C31—161no—
104C2×C4×C31—21no—
105C2×C4×C31—21no—
106C2×C4×C31—21no—
107C2×C4×C31—21no—
108C2×C4×C31—41no—
109C2×C4×C31—41no—
110C2×C4×C31—41no—
111C2×C4×C31—161no—
112C2×C4×C31#12011yescharacteristic
113C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
114C8×C31—21no—
115C8×C31—21no—
116C8×C31—41no—
117C8×C31—81no—
118C8×C31—81no—
119#3263C2×C3111yescharacteristic
120#2239C2×C3111yescharacteristic, complemented
121#2238C2×C3111yescharacteristic
122#2103C2×C3111yescharacteristic, complemented
123#1342C2×C3111yescharacteristic, complemented
124#1341C2×C3111yescharacteristic
125#1328C2×C3111yescharacteristic, complemented
126C22×C31—21no—
127C22×C31—41no—
128C22×C31—81no—
129C22×C31—81no—
130C22×C31—81no—
131C22×C31#45011yescharacteristic
132C4×C31—21no—
133C4×C31—41no—
134C4×C31—41no—
135C4×C31—81no—
136C4×C31—81no—
137C4×C31—81no—
138C4×C31#45411yescharacteristic
139C4×C31#45811yescharacteristic
140#542—21no—
141#476—21no—
142#475—21no—
143#470C22×C3111yescharacteristic
144#469C22×C3111yescharacteristic
145#465—21no—
146#464—21no—
147#463—21no—
148#430C22×C3111yescharacteristic
149#424C22×C3111yescharacteristic
150#365—21no—
151#365—21no—
152#362—21no—
153#361—21no—
154#349C22×C3111yescharacteristic
155#349C22×C3111yescharacteristic
156#344—21no—
157#341—21no—
158#338C22×C3111yescharacteristic
159C2×C31—21no—
160C2×C31—41no—
161C2×C31—161no—
162C2×C31#161611yescharacteristic
163C2×Q16D4×C3111yescharacteristic
164C2×SD16—21no—
165C2×D8D4×C3111yescharacteristic
166C2×M16—21no—
167C2×M16—21no—
168C2×M16C23×C3111yescharacteristic
169C22×C8D4×C3111yescharacteristic
170C4⋊Q8—21no—
171#126—41no—
172#124—21no—
173#124—41no—
174#123—41no—
175C22⋊Q8—21no—
176C22⋊Q8—41no—
177#121—21no—
178C4×Q8—21no—
179C4×D4—21no—
180#117D4×C3111yescharacteristic
181M32—21no—
182M32—21no—
183#108—21no—
184#108—21no—
185#106—21no—
186#106D4×C3111yescharacteristic
187#106D4×C3111yescharacteristic
188C4⋊C8—21no—
189C4⋊C8—21no—
190#103—21no—
191#103—21no—
192#102—21no—
193#102—21no—
194C31#1230611yescharacteristic, direct product factor, Sylow
195C2×Q8—21no—
196C2×Q8—41no—
197C2×D4—21no—
198C2×D4—41no—
199C22×C4—41no—
200C22×C4C2×D4×C3111yescharacteristic
201Q16—41no—
202SD16—81no—
203D8—41no—
204M16—41no—
205M16—41no—
206M16—41no—
207M16—41no—
208M16—41no—
209C2×C8—21no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8C2×D4×C3111yescharacteristic
214C2×C8C2×D4×C3111yescharacteristic
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4SD16×C3111yescharacteristic
227C22⋊C4SD16×C3111yescharacteristic
228C42—21no—
229C42—41no—
230C16—41no—
231C16—41no—
232C23—41no—
233C23C2×SD16×C3111yescharacteristic
234Q8—41no—
235Q8—41no—
236Q8—161no—
237D4—41no—
238D4—41no—
239D4—161no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—161no—
248C2×C4C31×#12011yescharacteristic
249C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
250C8—21no—
251C8—21no—
252C8—41no—
253C8—81no—
254C8—81no—
255C22—21no—
256C22—41no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22C31×#45011yescharacteristic
261C4—21no—
262C4—41no—
263C4—41no—
264C4—81no—
265C4—81no—
266C4—81no—
267C4C31×#45411yescharacteristic
268C4C31×#45811yescharacteristic
269C2—21no—
270C2—41no—
271C2—161no—
272C2C31×#161611yescharacteristic
subgroupG/H
center, socleC2×C31#1616
commutator, Frattini, cosocleC2×M16C23×C31
prime radicalC31×#469C22
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×M16×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2239C2semidirect
4. C31×#2103C2semidirect
5. C31×#1342C2semidirect
7. C31×#1328C2semidirect
95. #12306C31direct (N × H)
120. #2239C2×C31semidirect
122. #2103C2×C31semidirect
123. #1342C2×C31semidirect
125. #1328C2×C31semidirect
194. C31#12306direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32632
C31×#22392
C31×#22382
C31×#21032
C31×#13422
C31×#13412
C31×#13282
#1230631