Finite Group Databasebeta

← C31×#7309 #21464068

Subgroups — C31×#7309 #21464068

Order 7936.

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

normal subgroups only

This group has 470 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×#1785C211yesmaximal, characteristic
2C31×#1782C211yesmaximal, characteristic
3C31×#1780C211yesmaximal, characteristic
4C22×#41×C31C2211yescharacteristic
5C2×C31×#101C412yescomplemented
6C2×C31×#100—21no—
7C2×C31×#99—21no—
8C2×C31×#99C412yes—
9C31×#407—21no—
10C2×C31×C22⋊C8—21no—
11C2×C31×C22⋊C8—21no—
12C2×C31×C22⋊C8C412yescomplemented
13C31×#381—21no—
14C2×#41×C31—42no—
15C2×#41×C31—42no—
16C2×#41×C31—42no—
17C22×Q8×C31C2×C411yescharacteristic
18C22×D4×C31C2×C411yescharacteristic
19C22×D4×C31D411yescharacteristic
20C22×D4×C31Q811yescharacteristic
21C23×C4×C31C2×C411yescharacteristic
22C23×C4×C31D411yescharacteristic
23C23×C4×C31D411yescharacteristic
24C2×M16×C31—22no—
25C22×C8×C31—22no—
26C2×C22⋊C4×C31—22no—
27C31×#101—42no—
28C31×#100—42no—
29C31×#99—42no—
30C31×#99—42no—
31C31×C22⋊C8—42no—
32C31×C22⋊C8—42no—
33C31×C22⋊C8—42no—
34C31×C22⋊C8—42no—
35C31×#95—42no—
36C24×C31C22⋊C411yescharacteristic
37C24×C31C4⋊C411yescharacteristic
38C24×C31C4⋊C411yescharacteristic
39#41×C31—416no—
40C2×Q8×C31—22no—
41C2×Q8×C31—41no—
42C2×Q8×C31—41no—
43C2×D4×C31—22no—
44C2×D4×C31—22no—
45C2×D4×C31—22no—
46C2×D4×C31—41no—
47C2×D4×C31—41no—
48C2×D4×C31—41no—
49C2×D4×C31—41no—
50C2×D4×C31—41no—
51C2×D4×C31—41no—
52C22×C4×C31—22no—
53C22×C4×C31—22no—
54C22×C4×C31—22no—
55C22×C4×C31—22no—
56C22×C4×C31—22no—
57C22×C4×C31—41no—
58C22×C4×C31—41no—
59C22×C4×C31—41no—
60C22×C4×C31—41no—
61C22×C4×C31—42no—
62C22×C4×C31C4211yescharacteristic
63C22×C4×C31C22⋊C411yescharacteristic
64C22×C4×C31C22⋊C411yescharacteristic
65C22×C4×C31C4⋊C411yescharacteristic
66M16×C31—42no—
67M16×C31—42no—
68C2×C8×C31—42no—
69C2×C8×C31—42no—
70C2×C8×C31—42no—
71C2×C8×C31—42no—
72C22⋊C4×C31—42no—
73C22⋊C4×C31—42no—
74#7309C3111yesmaximal, characteristic, direct product factor, Sylow
75C23×C31—21no—
76C23×C31—21no—
77C23×C31—21no—
78C23×C31—21no—
79C23×C31—21no—
80C23×C31—21no—
81C23×C31—21no—
82C23×C31—21no—
83C23×C31—42no—
84C23×C31—42no—
85C23×C31—42no—
86C23×C31#9511yescharacteristic
87C23×C31#9912yes—
88Q8×C31—44no—
89D4×C31—44no—
90D4×C31—44no—
91D4×C31—44no—
92C2×C4×C31—21no—
93C2×C4×C31—21no—
94C2×C4×C31—21no—
95C2×C4×C31—21no—
96C2×C4×C31—21no—
97C2×C4×C31—21no—
98C2×C4×C31—21no—
99C2×C4×C31—21no—
100C2×C4×C31—21no—
101C2×C4×C31—21no—
102C2×C4×C31—44no—
103C2×C4×C31—44no—
104C2×C4×C31—44no—
105C2×C4×C31—82no—
106C2×C4×C31—84no—
107C2×C4×C31#9912yes—
108C2×C4×C31#9912yes—
109C8×C31—82no—
110C8×C31—82no—
111#1785C2×C3111yescharacteristic
112#1782C2×C3111yescharacteristic
113#1780C2×C3111yescharacteristic
114C22×C31—22no—
115C22×C31—22no—
116C22×C31—42no—
117C22×C31—42no—
118C22×C31—42no—
119C22×C31—41no—
120C22×C31—41no—
121C22×C31—42no—
122C22×C31—42no—
123C22×C31—81no—
124C22×C31—81no—
125C22×C31—81no—
126C22×C31—81no—
127C22×C31#34211yescharacteristic
128C22×C31#34211yescharacteristic
129C22×C31#34211yescharacteristic
130C4×C31—22no—
131C4×C31—22no—
132C4×C31—41no—
133C4×C31—41no—
134C4×C31—162no—
135C22×#41C22×C3111yescharacteristic
136C2×#101C4×C3112yescomplemented
137C2×#100—21no—
138C2×#99—21no—
139C2×#99C4×C3112yes—
140#407—21no—
141C2×C22⋊C8—21no—
142C2×C22⋊C8—21no—
143C2×C22⋊C8C4×C3112yescomplemented
144#381—21no—
145C2×C31—21no—
146C2×C31—21no—
147C2×C31—42no—
148C2×C31—81no—
149C2×C31—81no—
150C2×C31#130411yescharacteristic
151C2×C31#139512yes—
152C2×#41—42no—
153C2×#41—42no—
154C2×#41—42no—
155C22×Q8C2×C4×C3111yescharacteristic
156C22×D4C2×C4×C3111yescharacteristic
157C22×D4D4×C3111yescharacteristic
158C22×D4Q8×C3111yescharacteristic
159C23×C4C2×C4×C3111yescharacteristic
160C23×C4D4×C3111yescharacteristic
161C23×C4D4×C3111yescharacteristic
162C2×M16—22no—
163C22×C8—22no—
164C2×C22⋊C4—22no—
165#101—42no—
166#100—42no—
167#99—42no—
168#99—42no—
169C22⋊C8—42no—
170C22⋊C8—42no—
171C22⋊C8—42no—
172C22⋊C8—42no—
173#95—42no—
174C31#730911yescharacteristic, direct product factor, Sylow
175C24C22⋊C4×C3111yescharacteristic
176C24C4⋊C4×C3111yescharacteristic
177C24C4⋊C4×C3111yescharacteristic
178#41—416no—
179C2×Q8—22no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—41no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4C42×C3111yescharacteristic
202C22×C4C22⋊C4×C3111yescharacteristic
203C22×C4C22⋊C4×C3111yescharacteristic
204C22×C4C4⋊C4×C3111yescharacteristic
205M16—42no—
206M16—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23C31×#9511yescharacteristic
225C23C31×#9912yes—
226Q8—44no—
227D4—44no—
228D4—44no—
229D4—44no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—82no—
244C2×C4—84no—
245C2×C4C31×#9912yes—
246C2×C4C31×#9912yes—
247C8—82no—
248C8—82no—
249C22—22no—
250C22—22no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—41no—
255C22—41no—
256C22—42no—
257C22—42no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22C31×#34211yescharacteristic
263C22C31×#34211yescharacteristic
264C22C31×#34211yescharacteristic
265C4—22no—
266C4—22no—
267C4—41no—
268C4—41no—
269C4—162no—
270C2—21no—
271C2—21no—
272C2—42no—
273C2—81no—
274C2—81no—
275C2C31×#130411yescharacteristic
276C2C31×#139512yes—
subgroupG/H
center, socleC22×C31#342
commutatorC22×C4C42×C31
Frattini, cosocleC22×#41C22×C31
prime radicalC22×#41×C31C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C22×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
5. C2×C31×#101C4semidirect
12. C2×C31×C22⋊C8C4semidirect
74. #7309C31direct (N × H)
136. C2×#101C4×C31semidirect
143. C2×C22⋊C8C4×C31semidirect
174. C31#7309direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#17852
C31×#17822
C31×#17802
#730931