Finite Group Databasebeta

← C22×C31×#356 #21490123

Subgroups — C22×C31×#356 #21490123

Order 7936.

1254 conjugacy classes of subgroups; only the 382 normal subgroups are shown.

normal subgroups only

This group has 1254 conjugacy classes of subgroups — too many to draw. Only its 382 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
1C22×C31×C4⋊Q8C211yesmaximal, characteristic
2C22×C31×#101C212yesmaximal
3C2×C31×#356C2112yesmaximal, direct product factor
4C23×Q8×C31C411yescharacteristic
5C23×Q8×C31C2211yescharacteristic
6C22×M16×C31—22no—
7C2×C31×C4⋊Q8—26no—
8C2×C31×C4⋊Q8C416yes—
9C2×C31×C4⋊Q8C2216yes—
10C22×C4⋊C4×C31—22no—
11C22×C42×C31C411yescharacteristic
12C2×C31×#101—26no—
13C2×C31×#101C22112yes—
14C31×#356C22116yesdirect product factor
15C22×Q8×C31—23no—
16C22×Q8×C31—43no—
17C22×Q8×C31—41no—
18C22×Q8×C31—43no—
19C22×Q8×C31—41no—
20C22×Q8×C31C2×C416yes—
21C22×Q8×C31C2×C413yes—
22C22×Q8×C31D416yes—
23C22×Q8×C31C2313yes—
24C23×C4×C31—21no—
25C23×C4×C31—21no—
26C23×C4×C31C2×C411yescharacteristic
27C23×C4×C31D412yes—
28C2×M16×C31—212no—
29C2×M16×C31—46no—
30C22×C8×C31—42no—
31C31×C4⋊Q8—212no—
32C31×C4⋊Q8—212no—
33C31×C4⋊Q8C2×C4112yes—
34C31×C4⋊Q8C2314yes—
35C2×C4⋊C4×C31—212no—
36C2×C4⋊C4×C31—212no—
37C2×C42×C31—23no—
38C2×C42×C31C2×C416yes—
39C31×#101—212no—
40C31×#101C2318yes—
41C24×C31C22⋊C411yescharacteristic
42C2×Q8×C31—26no—
43C2×Q8×C31—43no—
44C2×Q8×C31—43no—
45C2×Q8×C31—46no—
46C2×Q8×C31—46no—
47C2×Q8×C31—43no—
48C2×Q8×C31—43no—
49C2×Q8×C31—46no—
50C2×Q8×C31—46no—
51C2×Q8×C31—46no—
52C2×Q8×C31—46no—
53C2×Q8×C31C22⋊C416yes—
54C2×Q8×C31C22×C414yes—
55C2×Q8×C31C22×C413yes—
56C2×Q8×C31C2×D416yes—
57C2×Q8×C31C2411yescharacteristic
58C22×C4×C31—28no—
59C22×C4×C31—26no—
60C22×C4×C31—23no—
61C22×C4×C31—22no—
62C22×C4×C31—21no—
63C22×C4×C31—26no—
64C22×C4×C31—26no—
65C22×C4×C31—43no—
66C22×C4×C31—41no—
67C22×C4×C31C22⋊C413yes—
68C22×C4×C31C22⋊C416yes—
69C22×C4×C31C22×C413yes—
70C22×C4×C31C2×D416yes—
71M16×C31—28no—
72M16×C31—412no—
73C2×C8×C31—412no—
74C4⋊C4×C31—224no—
75C4⋊C4×C31—28no—
76C42×C31—26no—
77C42×C31C22×C414yes—
78C22×#356C3111yesmaximal, characteristic, direct product factor, Sylow
79C23×C31—24no—
80C23×C31#9913yes—
81C23×C31#9911yescharacteristic
82C23×C31C2×C22⋊C413yes—
83Q8×C31—46no—
84Q8×C31—43no—
85Q8×C31—46no—
86Q8×C31—41no—
87Q8×C31—44no—
88Q8×C31—412no—
89C2×C4×C31—224no—
90C2×C4×C31—24no—
91C2×C4×C31—26no—
92C2×C4×C31—26no—
93C2×C4×C31—26no—
94C2×C4×C31—26no—
95C2×C4×C31—23no—
96C2×C4×C31—23no—
97C2×C4×C31—23no—
98C2×C4×C31—23no—
99C2×C4×C31—24no—
100C2×C4×C31—46no—
101C2×C4×C31—46no—
102C2×C4×C31C2×C22⋊C413yes—
103C2×C4×C31C2×C22⋊C416yes—
104C2×C4×C31C23×C411yescharacteristic
105C2×C4×C31C22×D412yes—
106C8×C31—48no—
107C22×C4⋊Q8C2×C3111yescharacteristic
108C22×#101C2×C3112yes—
109C2×#356C2×C31112yesdirect product factor
110C22×C31—26no—
111C22×C31—26no—
112C22×C31#35614yesdirect product factor
113C22×C31C2×#9913yes—
114C22×C31C2×#9913yes—
115C22×C31C22×C22⋊C411yescharacteristic
116C4×C31—28no—
117C4×C31—22no—
118C4×C31—26no—
119C4×C31—23no—
120C4×C31—21no—
121C4×C31—44no—
122C23×Q8C4×C3111yescharacteristic
123C23×Q8C22×C3111yescharacteristic
124C22×M16—22no—
125C2×C4⋊Q8—26no—
126C2×C4⋊Q8C4×C3116yes—
127C2×C4⋊Q8C22×C3116yes—
128C22×C4⋊C4—22no—
129C22×C42C4×C3111yescharacteristic
130C2×#101—26no—
131C2×#101C22×C31112yes—
132#356C22×C31116yesdirect product factor
133C2×C31—21no—
134C2×C31—23no—
135C2×C31C2×#35616yesdirect product factor
136C2×C31C22×#9911yescharacteristic
137C22×Q8—23no—
138C22×Q8—41no—
139C22×Q8—43no—
140C22×Q8—43no—
141C22×Q8—41no—
142C22×Q8C2×C4×C3116yes—
143C22×Q8C2×C4×C3113yes—
144C22×Q8D4×C3116yes—
145C22×Q8C23×C3113yes—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4C2×C4×C3111yescharacteristic
149C23×C4D4×C3112yes—
150C2×M16—212no—
151C2×M16—46no—
152C22×C8—42no—
153C4⋊Q8—212no—
154C4⋊Q8—212no—
155C4⋊Q8C2×C4×C31112yes—
156C4⋊Q8C23×C3114yes—
157C2×C4⋊C4—212no—
158C2×C4⋊C4—212no—
159C2×C42—23no—
160C2×C42C2×C4×C3116yes—
161#101—212no—
162#101C23×C3118yes—
163C31C22×#35611yescharacteristic, direct product factor, Sylow
164C24C22⋊C4×C3111yescharacteristic
165C2×Q8—26no—
166C2×Q8—43no—
167C2×Q8—43no—
168C2×Q8—46no—
169C2×Q8—46no—
170C2×Q8—43no—
171C2×Q8—43no—
172C2×Q8—46no—
173C2×Q8—46no—
174C2×Q8—46no—
175C2×Q8—46no—
176C2×Q8C22⋊C4×C3116yes—
177C2×Q8C22×C4×C3114yes—
178C2×Q8C22×C4×C3113yes—
179C2×Q8C2×D4×C3116yes—
180C2×Q8C24×C3111yescharacteristic
181C22×C4—28no—
182C22×C4—22no—
183C22×C4—26no—
184C22×C4—23no—
185C22×C4—21no—
186C22×C4—26no—
187C22×C4—26no—
188C22×C4—43no—
189C22×C4—41no—
190C22×C4C22⋊C4×C3113yes—
191C22×C4C22⋊C4×C3116yes—
192C22×C4C22×C4×C3113yes—
193C22×C4C2×D4×C3116yes—
194M16—28no—
195M16—412no—
196C2×C8—412no—
197C4⋊C4—224no—
198C4⋊C4—28no—
199C42—26no—
200C42C22×C4×C3114yes—
201C23—24no—
202C23C31×#9913yes—
203C23C31×#9911yescharacteristic
204C23C2×C22⋊C4×C3113yes—
205Q8—46no—
206Q8—43no—
207Q8—46no—
208Q8—41no—
209Q8—44no—
210Q8—412no—
211C2×C4—224no—
212C2×C4—26no—
213C2×C4—26no—
214C2×C4—26no—
215C2×C4—26no—
216C2×C4—23no—
217C2×C4—23no—
218C2×C4—23no—
219C2×C4—23no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—46no—
223C2×C4—46no—
224C2×C4C2×C22⋊C4×C3113yes—
225C2×C4C2×C22⋊C4×C3116yes—
226C2×C4C23×C4×C3111yescharacteristic
227C2×C4C22×D4×C3112yes—
228C8—48no—
229C22—26no—
230C22—26no—
231C22C31×#35614yesdirect product factor
232C22C2×C31×#9913yes—
233C22C2×C31×#9913yes—
234C22C22×C22⋊C4×C3111yescharacteristic
235C4—28no—
236C4—22no—
237C4—23no—
238C4—26no—
239C4—21no—
240C4—44no—
241C2—21no—
242C2—23no—
243C2C2×C31×#35616yesdirect product factor
244C2C22×C31×#9911yescharacteristic
subgroupG/H
center, socleC23×C31#99
commutatorC2×C4C23×C4×C31
Frattini, cosocleC2×Q8C24×C31
prime radicalC24×C31C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C24×C31 ◁ C23×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C31×#356C2direct (N × H)
14. C31×#356C22direct (N × H)
78. C22×#356C31direct (N × H)
109. C2×#356C2×C31direct (N × H)
112. C22×C31#356direct (N × H)
132. #356C22×C31direct (N × H)
135. C2×C31C2×#356direct (N × H)
163. C31C22×#356direct (N × H)
231. C22C31×#356direct (N × H)
243. C2C2×C31×#356direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C31×C4⋊Q82
C22×C31×#1012
C2×C31×#3562
C22×#35631