Finite Group Databasebeta

← C31×#12217 #21468976

Subgroups — C31×#12217 #21468976

Order 7936.

484 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2840C211yesmaximal, characteristic
2C31×#2029C211yesmaximal, characteristic
3C31×#1602C212yesmaximal, complemented
4C31×#1351C212yesmaximal, complemented
5C31×#1316C211yesmaximal, characteristic
6C31×#543C2211yescharacteristic
7C31×#451—22no—
8C31×#393—22no—
9C31×#393—21no—
10C31×#393—22no—
11C31×#393C2212yes—
12C31×#379—21no—
13C31×#379—21no—
14C31×#379C2211yescharacteristic
15C31×#342—21no—
16C31×#342C2211yescharacteristic
17C31×#328—22no—
18C31×#323C2212yes—
19C25×C31D411yescharacteristic
20C22×Q8×C31—21no—
21C22×Q8×C31D412yes—
22C2×Q16×C31—42no—
23C31×C22⋊Q8—22no—
24C31×C22⋊Q8—42no—
25C2×C22⋊C4×C31—21no—
26C2×C22⋊C4×C31—21no—
27C2×C22⋊C4×C31—21no—
28C2×C22⋊C4×C31—22no—
29C2×C22⋊C4×C31—21no—
30C2×C22⋊C4×C31—22no—
31C2×C22⋊C4×C31—22no—
32C2×C22⋊C4×C31—42no—
33C2×C22⋊C4×C31D411yescharacteristic
34C2×C22⋊C4×C31C2311yescharacteristic
35C31×#103—42no—
36C31×C22⋊C8—22no—
37C31×#95—21no—
38C31×#95—22no—
39C31×#95—22no—
40C31×#95D412yes—
41C24×C31—21no—
42C24×C31—21no—
43C24×C31—41no—
44C24×C31—41no—
45C24×C31—82no—
46C24×C31Q1612yescomplemented
47C24×C31C2×D411yescharacteristic
48C2×Q8×C31—22no—
49C2×Q8×C31—42no—
50C2×Q8×C31—82no—
51C2×Q8×C31—82no—
52C2×Q8×C31—81no—
53C22×C4×C31—22no—
54C22×C4×C31—21no—
55C22×C4×C31—22no—
56C22×C4×C31—22no—
57C22×C4×C31—41no—
58C22×C4×C31C2×D412yes—
59Q16×C31—84no—
60C2×C8×C31—42no—
61C4⋊C4×C31—22no—
62C4⋊C4×C31—42no—
63C22⋊C4×C31—41no—
64C22⋊C4×C31—41no—
65C22⋊C4×C31—41no—
66C22⋊C4×C31—41no—
67C22⋊C4×C31—41no—
68C22⋊C4×C31—41no—
69C22⋊C4×C31—82no—
70C22⋊C4×C31—82no—
71C22⋊C4×C31—82no—
72C22⋊C4×C31—81no—
73C22⋊C4×C31—81no—
74C22⋊C4×C31—84no—
75#12217C3111yesmaximal, characteristic, direct product factor, Sylow
76C23×C31—21no—
77C23×C31—21no—
78C23×C31—22no—
79C23×C31—42no—
80C23×C31—42no—
81C23×C31—42no—
82C23×C31—42no—
83C23×C31—44no—
84C23×C31—41no—
85C23×C31—41no—
86C23×C31—81no—
87C23×C31—82no—
88C23×C31—82no—
89C23×C31—82no—
90C23×C31—84no—
91C23×C31#12011yescharacteristic
92C23×C31C2×Q1611yescharacteristic
93C23×C31#13711yescharacteristic
94Q8×C31—44no—
95Q8×C31—82no—
96Q8×C31—82no—
97Q8×C31—84no—
98C2×C4×C31—22no—
99C2×C4×C31—42no—
100C2×C4×C31—42no—
101C2×C4×C31—42no—
102C2×C4×C31—42no—
103C2×C4×C31—42no—
104C2×C4×C31—42no—
105C2×C4×C31—41no—
106C2×C4×C31—42no—
107C2×C4×C31—42no—
108C2×C4×C31—42no—
109C2×C4×C31—82no—
110C2×C4×C31—81no—
111C2×C4×C31—82no—
112C8×C31—82no—
113#2840C2×C3111yescharacteristic
114#2029C2×C3111yescharacteristic
115#1602C2×C3112yescomplemented
116#1351C2×C3112yescomplemented
117#1316C2×C3111yescharacteristic
118C22×C31—22no—
119C22×C31—21no—
120C22×C31—21no—
121C22×C31—41no—
122C22×C31—41no—
123C22×C31—42no—
124C22×C31—42no—
125C22×C31—42no—
126C22×C31—44no—
127C22×C31—42no—
128C22×C31—81no—
129C22×C31—82no—
130C22×C31—82no—
131C22×C31—84no—
132C22×C31—82no—
133C22×C31#45111yescharacteristic
134C22×C31#45511yescharacteristic
135C22×C31B(4,𝔽2)11yescharacteristic
136C4×C31—42no—
137C4×C31—81no—
138C4×C31—82no—
139C4×C31—82no—
140C4×C31—82no—
141C4×C31—161no—
142#543C22×C3111yescharacteristic
143#451—22no—
144#393—22no—
145#393—21no—
146#393—22no—
147#393C22×C3112yes—
148#379—21no—
149#379—21no—
150#379C22×C3111yescharacteristic
151#342—21no—
152#342C22×C3111yescharacteristic
153#328—22no—
154#323C22×C3112yes—
155C2×C31—21no—
156C2×C31—21no—
157C2×C31—41no—
158C2×C31—41no—
159C2×C31—82no—
160C2×C31#160211yescharacteristic
161C2×C31#219912yes—
162C25D4×C3111yescharacteristic
163C22×Q8—21no—
164C22×Q8D4×C3112yes—
165C2×Q16—42no—
166C22⋊Q8—22no—
167C22⋊Q8—42no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4D4×C3111yescharacteristic
177C2×C22⋊C4C23×C3111yescharacteristic
178#103—42no—
179C22⋊C8—22no—
180#95—21no—
181#95—22no—
182#95—22no—
183#95D4×C3112yes—
184C31#1221711yescharacteristic, direct product factor, Sylow
185C24—21no—
186C24—21no—
187C24—41no—
188C24—41no—
189C24—82no—
190C24Q16×C3112yescomplemented
191C24C2×D4×C3111yescharacteristic
192C2×Q8—22no—
193C2×Q8—42no—
194C2×Q8—82no—
195C2×Q8—82no—
196C2×Q8—81no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—41no—
202C22×C4C2×D4×C3112yes—
203Q16—84no—
204C2×C8—42no—
205C4⋊C4—22no—
206C4⋊C4—42no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—82no—
214C22⋊C4—82no—
215C22⋊C4—82no—
216C22⋊C4—81no—
217C22⋊C4—81no—
218C22⋊C4—84no—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—44no—
227C23—41no—
228C23—41no—
229C23—81no—
230C23—82no—
231C23—82no—
232C23—82no—
233C23—84no—
234C23C31×#12011yescharacteristic
235C23C2×Q16×C3111yescharacteristic
236C23C31×#13711yescharacteristic
237Q8—44no—
238Q8—82no—
239Q8—82no—
240Q8—84no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—82no—
253C2×C4—81no—
254C2×C4—82no—
255C8—82no—
256C22—22no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—44no—
265C22—42no—
266C22—81no—
267C22—82no—
268C22—82no—
269C22—84no—
270C22—82no—
271C22C31×#45111yescharacteristic
272C22C31×#45511yescharacteristic
273C22C31×B(4,𝔽2)11yescharacteristic
274C4—42no—
275C4—81no—
276C4—82no—
277C4—82no—
278C4—82no—
279C4—161no—
280C2—21no—
281C2—21no—
282C2—41no—
283C2—41no—
284C2—82no—
285C2C31×#160211yescharacteristic
286C2C31×#219912yes—
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC25×C31D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C22⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#1602C2semidirect
4. C31×#1351C2semidirect
46. C24×C31Q16semidirect
75. #12217C31direct (N × H)
115. #1602C2×C31semidirect
116. #1351C2×C31semidirect
184. C31#12217direct (N × H)
190. C24Q16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#28402
C31×#20292
C31×#16022
C31×#13512
C31×#13162
#1221731