Finite Group Databasebeta

← C31×#13602 #21470361

Subgroups — C31×#13602 #21470361

Order 7936.

326 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 90 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×#3131C211yesmaximal, characteristic
2C31×#2132C211yesmaximal, characteristic
3C31×#2126C211yesmaximal, characteristic
4C31×#1414C214yesmaximal, complemented
5C31×#533C411yescharacteristic, complemented
6C31×#532C2211yescharacteristic
7C31×#488C412yescomplemented
8C31×#488C2212yes—
9C31×#487—21no—
10C31×#486—21no—
11C31×#416—21no—
12C31×#416—21no—
13C31×#413—21no—
14C31×#410—21no—
15C2×C4×C8×C31C411yescharacteristic, complemented
16C31×#356—21no—
17C31×#355C2212yes—
18C31×#354C2212yes—
19C31×#353—21no—
20C2×#41×C31D411yescharacteristic
21C2×#41×C31D411yescharacteristic
22C2×M16×C31—41no—
23C22×C8×C31—21no—
24C31×C4⋊Q8—21no—
25C31×C4⋊Q8D411yescharacteristic
26C31×#127D411yescharacteristic
27C31×#125C2×C412yes—
28C31×#124C2×C411yescharacteristic
29C31×#124C2311yescharacteristic
30C31×C22⋊Q8—41no—
31C31×#121—41no—
32C4×Q8×C31—41no—
33C4×D4×C31—41no—
34C31×#117—21no—
35C31×#117—41no—
36C2×C42×C31C2×C411yescharacteristic
37C31×#103—22no—
38C31×#103—22no—
39C31×#102—22no—
40C31×#102—22no—
41C31×#101—22no—
42C31×#100—41no—
43C31×#99—22no—
44C31×#99—41no—
45C4×C8×C31—21no—
46C4×C8×C31C2×C412yes—
47#41×C31—41no—
48#41×C31—41no—
49#41×C31—41no—
50#41×C31—41no—
51C2×Q8×C31—41no—
52C2×Q8×C31C22⋊C411yescharacteristic
53C2×Q8×C31C2×D411yescharacteristic
54C2×D4×C31—21no—
55C2×D4×C31—21no—
56C2×D4×C31—41no—
57C2×D4×C31C22⋊C411yescharacteristic
58C2×D4×C31C2×D411yescharacteristic
59C22×C4×C31—21no—
60C22×C4×C31—21no—
61C22×C4×C31—21no—
62C22×C4×C31C22⋊C411yescharacteristic
63M16×C31—42no—
64M16×C31—81no—
65C2×C8×C31—22no—
66C2×C8×C31—41no—
67C2×C8×C31—41no—
68C2×C8×C31—81no—
69C4⋊C4×C31—22no—
70C4⋊C4×C31—41no—
71C4⋊C4×C31—41no—
72C4⋊C4×C31—42no—
73C4⋊C4×C31—81no—
74C22⋊C4×C31—41no—
75C22⋊C4×C31—41no—
76C22⋊C4×C31—41no—
77C22⋊C4×C31—42no—
78C42×C31—21no—
79C42×C31—41no—
80C42×C31—81no—
81C42×C31C22⋊C411yescharacteristic
82C42×C31C22×C411yescharacteristic
83#13602C3111yesmaximal, characteristic, direct product factor, Sylow
84C23×C31—21no—
85C23×C31—21no—
86C23×C31#9911yescharacteristic
87Q8×C31—41no—
88Q8×C31—41no—
89Q8×C31—82no—
90D4×C31—41no—
91D4×C31—41no—
92D4×C31—82no—
93D4×C31—81no—
94D4×C31—81no—
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—42no—
102C2×C4×C31—41no—
103C2×C4×C31—41no—
104C2×C4×C31—41no—
105C2×C4×C31—41no—
106C2×C4×C31—42no—
107C2×C4×C31—82no—
108C2×C4×C31#9911yescharacteristic
109C2×C4×C31C2×C22⋊C411yescharacteristic
110C8×C31—42no—
111C8×C31—82no—
112#3131C2×C3111yescharacteristic
113#2132C2×C3111yescharacteristic
114#2126C2×C3111yescharacteristic
115#1414C2×C3114yescomplemented
116C22×C31—21no—
117C22×C31—41no—
118C22×C31—41no—
119C22×C31—41no—
120C22×C31—81no—
121C22×C31—81no—
122C22×C31C2×#9911yescharacteristic
123C4×C31—22no—
124C4×C31—21no—
125C4×C31—41no—
126C4×C31—41no—
127C4×C31—41no—
128C4×C31—82no—
129C4×C31—162no—
130C4×C31#35411yescharacteristic
131C4×C31#35411yescharacteristic
132#533C4×C3111yescharacteristic, complemented
133#532C22×C3111yescharacteristic
134#488C4×C3112yescomplemented
135#488C22×C3112yes—
136#487—21no—
137#486—21no—
138#416—21no—
139#416—21no—
140#413—21no—
141#410—21no—
142C2×C4×C8C4×C3111yescharacteristic, complemented
143#356—21no—
144#355C22×C3112yes—
145#354C22×C3112yes—
146#353—21no—
147C2×C31—21no—
148C2×C31—41no—
149C2×C31—81no—
150C2×C31—81no—
151C2×C31C2×#35411yescharacteristic
152C2×#41D4×C3111yescharacteristic
153C2×#41D4×C3111yescharacteristic
154C2×M16—41no—
155C22×C8—21no—
156C4⋊Q8—21no—
157C4⋊Q8D4×C3111yescharacteristic
158#127D4×C3111yescharacteristic
159#125C2×C4×C3112yes—
160#124C2×C4×C3111yescharacteristic
161#124C23×C3111yescharacteristic
162C22⋊Q8—41no—
163#121—41no—
164C4×Q8—41no—
165C4×D4—41no—
166#117—21no—
167#117—41no—
168C2×C42C2×C4×C3111yescharacteristic
169#103—22no—
170#103—22no—
171#102—22no—
172#102—22no—
173#101—22no—
174#100—41no—
175#99—22no—
176#99—41no—
177C4×C8—21no—
178C4×C8C2×C4×C3112yes—
179C31#1360211yescharacteristic, direct product factor, Sylow
180#41—41no—
181#41—41no—
182#41—41no—
183#41—41no—
184C2×Q8—41no—
185C2×Q8C22⋊C4×C3111yescharacteristic
186C2×Q8C2×D4×C3111yescharacteristic
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—41no—
190C2×D4C22⋊C4×C3111yescharacteristic
191C2×D4C2×D4×C3111yescharacteristic
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4C22⋊C4×C3111yescharacteristic
196M16—42no—
197M16—81no—
198C2×C8—22no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—81no—
202C4⋊C4—22no—
203C4⋊C4—41no—
204C4⋊C4—42no—
205C4⋊C4—41no—
206C4⋊C4—81no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—42no—
211C42—21no—
212C42—41no—
213C42—81no—
214C42C22⋊C4×C3111yescharacteristic
215C42C22×C4×C3111yescharacteristic
216C23—21no—
217C23—21no—
218C23C31×#9911yescharacteristic
219Q8—41no—
220Q8—41no—
221Q8—82no—
222D4—41no—
223D4—41no—
224D4—82no—
225D4—81no—
226D4—81no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—82no—
240C2×C4C31×#9911yescharacteristic
241C2×C4C2×C22⋊C4×C3111yescharacteristic
242C8—42no—
243C8—82no—
244C22—21no—
245C22—41no—
246C22—41no—
247C22—41no—
248C22—81no—
249C22—81no—
250C22C2×C31×#9911yescharacteristic
251C4—22no—
252C4—21no—
253C4—41no—
254C4—41no—
255C4—41no—
256C4—82no—
257C4—162no—
258C4C31×#35411yescharacteristic
259C4C31×#35411yescharacteristic
260C2—21no—
261C2—41no—
262C2—81no—
263C2—81no—
264C2C2×C31×#35411yescharacteristic
subgroupG/H
centerC4×C31#354
commutatorC42C22×C4×C31
Frattini, cosocle#124C23×C31
socleC2×C31C2×#354
prime radicalC31×#532C22
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C31 ◁ C2×C4×C31 ◁ C22×C4×C31 ◁ C31×#532 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#1414C2semidirect
5. C31×#533C4semidirect
7. C31×#488C4semidirect
15. C2×C4×C8×C31C4semidirect
83. #13602C31direct (N × H)
115. #1414C2×C31semidirect
132. #533C4×C31semidirect
134. #488C4×C31semidirect
142. C2×C4×C8C4×C31semidirect
179. C31#13602direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31312
C31×#21322
C31×#21262
C31×#14142
#1360231