Finite Group Databasebeta

← C31×#12218 #21468977

Subgroups — C31×#12218 #21468977

Order 7936.

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

normal subgroups only

This group has 456 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×#2842C211yesmaximal, characteristic
2C31×#2029C211yesmaximal, characteristic
3C31×#1614C212yesmaximal, complemented
4C31×#1352C212yesmaximal, complemented
5C31×#1316C211yesmaximal, characteristic
6C31×#543C2211yescharacteristic
7C31×#484—22no—
8C31×#399—21no—
9C31×#399C2212yes—
10C31×#396—22no—
11C31×#396—22no—
12C31×#379—21no—
13C31×#379—21no—
14C31×#379C2211yescharacteristic
15C31×#342—21no—
16C31×#342C2211yescharacteristic
17C31×#328—22no—
18C31×#323C2212yes—
19C25×C31D411yescharacteristic
20C31×C22⋊Q8—22no—
21C31×C22⋊Q8—42no—
22C2×C4⋊C4×C31—22no—
23C2×C4⋊C4×C31—21no—
24C2×C4⋊C4×C31—22no—
25C2×C4⋊C4×C31D412yes—
26C2×C22⋊C4×C31—21no—
27C2×C22⋊C4×C31—21no—
28C2×C22⋊C4×C31—21no—
29C2×C22⋊C4×C31—22no—
30C2×C22⋊C4×C31—21no—
31C2×C22⋊C4×C31—22no—
32C2×C22⋊C4×C31—22no—
33C2×C22⋊C4×C31—42no—
34C2×C22⋊C4×C31D411yescharacteristic
35C2×C22⋊C4×C31C2311yescharacteristic
36C31×#107—42no—
37C31×#103—42no—
38C31×C22⋊C8—22no—
39C31×#95—21no—
40C31×#95D412yes—
41C24×C31—21no—
42C24×C31—21no—
43C24×C31—41no—
44C24×C31—41no—
45C24×C31—82no—
46C24×C31Q1612yes—
47C24×C31C2×D411yescharacteristic
48C2×Q8×C31—22no—
49C22×C4×C31—22no—
50C22×C4×C31—21no—
51C22×C4×C31—22no—
52C22×C4×C31—22no—
53C22×C4×C31—41no—
54C22×C4×C31C2×D412yes—
55C2×C8×C31—42no—
56C4⋊C4×C31—22no—
57C4⋊C4×C31—42no—
58C4⋊C4×C31—42no—
59C4⋊C4×C31—82no—
60C4⋊C4×C31—82no—
61C4⋊C4×C31—81no—
62C22⋊C4×C31—41no—
63C22⋊C4×C31—41no—
64C22⋊C4×C31—41no—
65C22⋊C4×C31—41no—
66C22⋊C4×C31—41no—
67C22⋊C4×C31—41no—
68C22⋊C4×C31—82no—
69C22⋊C4×C31—82no—
70C22⋊C4×C31—82no—
71C22⋊C4×C31—81no—
72C22⋊C4×C31—81no—
73C22⋊C4×C31—84no—
74#12218C3111yesmaximal, characteristic, direct product factor, Sylow
75C23×C31—21no—
76C23×C31—21no—
77C23×C31—22no—
78C23×C31—42no—
79C23×C31—42no—
80C23×C31—42no—
81C23×C31—42no—
82C23×C31—44no—
83C23×C31—41no—
84C23×C31—41no—
85C23×C31—81no—
86C23×C31—82no—
87C23×C31—82no—
88C23×C31—82no—
89C23×C31—84no—
90C23×C31#12011yescharacteristic
91C23×C31C2×Q1611yescharacteristic
92C23×C31#13711yescharacteristic
93Q8×C31—82no—
94C2×C4×C31—22no—
95C2×C4×C31—42no—
96C2×C4×C31—42no—
97C2×C4×C31—42no—
98C2×C4×C31—42no—
99C2×C4×C31—42no—
100C2×C4×C31—42no—
101C2×C4×C31—41no—
102C2×C4×C31—42no—
103C2×C4×C31—42no—
104C2×C4×C31—42no—
105C2×C4×C31—82no—
106C2×C4×C31—81no—
107C2×C4×C31—82no—
108C8×C31—82no—
109#2842C2×C3111yescharacteristic
110#2029C2×C3111yescharacteristic
111#1614C2×C3112yescomplemented
112#1352C2×C3112yescomplemented
113#1316C2×C3111yescharacteristic
114C22×C31—22no—
115C22×C31—21no—
116C22×C31—21no—
117C22×C31—41no—
118C22×C31—41no—
119C22×C31—42no—
120C22×C31—42no—
121C22×C31—42no—
122C22×C31—44no—
123C22×C31—42no—
124C22×C31—81no—
125C22×C31—82no—
126C22×C31—82no—
127C22×C31—84no—
128C22×C31—82no—
129C22×C31#45111yescharacteristic
130C22×C31#45511yescharacteristic
131C22×C31B(4,𝔽2)11yescharacteristic
132C4×C31—42no—
133C4×C31—81no—
134C4×C31—82no—
135C4×C31—82no—
136C4×C31—82no—
137C4×C31—161no—
138#543C22×C3111yescharacteristic
139#484—22no—
140#399—21no—
141#399C22×C3112yes—
142#396—22no—
143#396—22no—
144#379—21no—
145#379—21no—
146#379C22×C3111yescharacteristic
147#342—21no—
148#342C22×C3111yescharacteristic
149#328—22no—
150#323C22×C3112yes—
151C2×C31—21no—
152C2×C31—21no—
153C2×C31—41no—
154C2×C31—41no—
155C2×C31—82no—
156C2×C31#160211yescharacteristic
157C2×C31#220012yes—
158C25D4×C3111yescharacteristic
159C22⋊Q8—22no—
160C22⋊Q8—42no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4D4×C3112yes—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4D4×C3111yescharacteristic
174C2×C22⋊C4C23×C3111yescharacteristic
175#107—42no—
176#103—42no—
177C22⋊C8—22no—
178#95—21no—
179#95D4×C3112yes—
180C31#1221811yescharacteristic, direct product factor, Sylow
181C24—21no—
182C24—21no—
183C24—41no—
184C24—41no—
185C24—82no—
186C24Q16×C3112yes—
187C24C2×D4×C3111yescharacteristic
188C2×Q8—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—41no—
194C22×C4C2×D4×C3112yes—
195C2×C8—42no—
196C4⋊C4—22no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—82no—
200C4⋊C4—82no—
201C4⋊C4—81no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—82no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4—81no—
212C22⋊C4—81no—
213C22⋊C4—84no—
214C23—21no—
215C23—21no—
216C23—22no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—44no—
222C23—41no—
223C23—41no—
224C23—81no—
225C23—82no—
226C23—82no—
227C23—82no—
228C23—84no—
229C23C31×#12011yescharacteristic
230C23C2×Q16×C3111yescharacteristic
231C23C31×#13711yescharacteristic
232Q8—82no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—82no—
245C2×C4—81no—
246C2×C4—82no—
247C8—82no—
248C22—22no—
249C22—21no—
250C22—21no—
251C22—41no—
252C22—41no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—44no—
257C22—42no—
258C22—81no—
259C22—82no—
260C22—82no—
261C22—84no—
262C22—82no—
263C22C31×#45111yescharacteristic
264C22C31×#45511yescharacteristic
265C22C31×B(4,𝔽2)11yescharacteristic
266C4—42no—
267C4—81no—
268C4—82no—
269C4—82no—
270C4—82no—
271C4—161no—
272C2—21no—
273C2—21no—
274C2—41no—
275C2—41no—
276C2—82no—
277C2C31×#160211yescharacteristic
278C2C31×#220012yes—
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×#1614C2semidirect
4. C31×#1352C2semidirect
74. #12218C31direct (N × H)
111. #1614C2×C31semidirect
112. #1352C2×C31semidirect
180. C31#12218direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28422
C31×#20292
C31×#16142
C31×#13522
C31×#13162
#1221831