Finite Group Databasebeta

← C31×#12174 #21468933

Subgroups — C31×#12174 #21468933

Order 7936.

340 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3255C211yesmaximal, characteristic, complemented
2C31×#2191C212yesmaximal
3C31×#2050C211yesmaximal, characteristic
4C31×#1342C212yesmaximal, complemented
5C31×#1327C211yesmaximal, characteristic, complemented
6C31×#575C2211yescharacteristic
7C31×#533—21no—
8C31×#510—22no—
9C31×#475—22no—
10C31×#470—21no—
11C31×#470C2212yes—
12C31×#462—22no—
13C31×#423C2211yescharacteristic
14C2×C31×#104—21no—
15C31×#413—21no—
16C31×#360—22no—
17C31×#349C2212yes—
18C31×#338C2211yescharacteristic
19C2×#41×C31—21no—
20C31×#137—41no—
21C31×AGL(1,ℤ/8)—41no—
22C31×#135—22no—
23C2×Q16×C31—21no—
24C2×Q16×C31D412yes—
25C2×M16×C31—21no—
26C2×M16×C31D411yescharacteristic
27C2×M16×C31C2311yescharacteristic
28C31×C4⋊Q8—21no—
29C31×C4⋊Q8—21no—
30C31×#125—41no—
31C31×C22⋊Q8—22no—
32C31×C22⋊Q8—41no—
33C4×Q8×C31—22no—
34C4×Q8×C31—41no—
35C31×#117—41no—
36C2×C42×C31D411yescharacteristic
37C31×Q32—42no—
38C31×SD32—42no—
39C31×M32—22no—
40C31×#106—21no—
41C31×#106D412yes—
42C31×C4⋊C8—22no—
43C31×#104—41no—
44C31×#104—41no—
45C31×#103—22no—
46C31×#103—22no—
47C31×#101—41no—
48C31×#100—41no—
49#41×C31—41no—
50#41×C31—41no—
51#41×C31—41no—
52#41×C31—41no—
53C2×Q8×C31—21no—
54C2×Q8×C31—22no—
55C2×Q8×C31—41no—
56C2×D4×C31—21no—
57C2×D4×C31—41no—
58C22×C4×C31—21no—
59C22×C4×C31—41no—
60C22×C4×C31C2×D411yescharacteristic
61Q16×C31—22no—
62Q16×C31—42no—
63Q16×C31—81no—
64SD16×C31—42no—
65D8×C31—22no—
66M16×C31—41no—
67M16×C31—41no—
68M16×C31—41no—
69M16×C31—41no—
70C2×C8×C31—21no—
71C2×C8×C31—41no—
72C2×C8×C31C2×D412yes—
73C4⋊C4×C31—22no—
74C4⋊C4×C31—41no—
75C4⋊C4×C31—41no—
76C4⋊C4×C31—42no—
77C4⋊C4×C31—41no—
78C4⋊C4×C31—42no—
79C22⋊C4×C31—42no—
80C42×C31—21no—
81C42×C31—42no—
82C42×C31D811yescharacteristic
83C42×C31D811yescharacteristic
84C16×C31—42no—
85#12174C3111yesmaximal, characteristic, direct product factor, Sylow
86C23×C31—41no—
87C23×C31AGL(1,ℤ/8)11yescharacteristic
88Q8×C31—41no—
89Q8×C31—42no—
90Q8×C31—41no—
91Q8×C31—42no—
92Q8×C31—82no—
93D4×C31—41no—
94D4×C31—41no—
95D4×C31—81no—
96D4×C31—81no—
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—42no—
104C2×C4×C31—42no—
105C2×C4×C31—81no—
106C2×C4×C31—81no—
107C2×C4×C31#12011yescharacteristic
108C2×C4×C31C2×D811yescharacteristic
109C8×C31—22no—
110C8×C31—41no—
111C8×C31—81no—
112#3255C2×C3111yescharacteristic, complemented
113#2191C2×C3112yes—
114#2050C2×C3111yescharacteristic
115#1342C2×C3112yescomplemented
116#1327C2×C3111yescharacteristic, complemented
117C22×C31—21no—
118C22×C31—41no—
119C22×C31—81no—
120C22×C31—161no—
121C22×C31#44711yescharacteristic
122C4×C31—22no—
123C4×C31—21no—
124C4×C31—41no—
125C4×C31—82no—
126C4×C31—81no—
127C4×C31—82no—
128C4×C31#45411yescharacteristic
129C4×C31B(4,𝔽2)11yescharacteristic
130#575C22×C3111yescharacteristic
131#533—21no—
132#510—22no—
133#475—22no—
134#470—21no—
135#470C22×C3112yes—
136#462—22no—
137#423C22×C3111yescharacteristic
138C2×#104—21no—
139#413—21no—
140#360—22no—
141#349C22×C3112yes—
142#338C22×C3111yescharacteristic
143C2×C31—21no—
144C2×C31—41no—
145C2×C31—161no—
146C2×C31#159811yescharacteristic
147C2×#41—21no—
148#137—41no—
149AGL(1,ℤ/8)—41no—
150#135—22no—
151C2×Q16—21no—
152C2×Q16D4×C3112yes—
153C2×M16—21no—
154C2×M16D4×C3111yescharacteristic
155C2×M16C23×C3111yescharacteristic
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158#125—41no—
159C22⋊Q8—22no—
160C22⋊Q8—41no—
161C4×Q8—22no—
162C4×Q8—41no—
163#117—41no—
164C2×C42D4×C3111yescharacteristic
165Q32—42no—
166SD32—42no—
167M32—22no—
168#106—21no—
169#106D4×C3112yes—
170C4⋊C8—22no—
171#104—41no—
172#104—41no—
173#103—22no—
174#103—22no—
175#101—41no—
176#100—41no—
177C31#1217411yescharacteristic, direct product factor, Sylow
178#41—41no—
179#41—41no—
180#41—41no—
181#41—41no—
182C2×Q8—21no—
183C2×Q8—22no—
184C2×Q8—41no—
185C2×D4—21no—
186C2×D4—41no—
187C22×C4—21no—
188C22×C4—41no—
189C22×C4C2×D4×C3111yescharacteristic
190Q16—22no—
191Q16—42no—
192Q16—81no—
193SD16—42no—
194D8—22no—
195M16—41no—
196M16—41no—
197M16—41no—
198M16—41no—
199C2×C8—21no—
200C2×C8—41no—
201C2×C8C2×D4×C3112yes—
202C4⋊C4—22no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C22⋊C4—42no—
209C42—21no—
210C42—42no—
211C42D8×C3111yescharacteristic
212C42D8×C3111yescharacteristic
213C16—42no—
214C23—41no—
215C23C31×AGL(1,ℤ/8)11yescharacteristic
216Q8—41no—
217Q8—42no—
218Q8—41no—
219Q8—42no—
220Q8—82no—
221D4—41no—
222D4—41no—
223D4—81no—
224D4—81no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4C31×#12011yescharacteristic
236C2×C4C2×D8×C3111yescharacteristic
237C8—22no—
238C8—41no—
239C8—81no—
240C22—21no—
241C22—41no—
242C22—81no—
243C22—161no—
244C22C31×#44711yescharacteristic
245C4—22no—
246C4—21no—
247C4—41no—
248C4—82no—
249C4—81no—
250C4—82no—
251C4C31×#45411yescharacteristic
252C4C31×B(4,𝔽2)11yescharacteristic
253C2—21no—
254C2—41no—
255C2—161no—
256C2C31×#159811yescharacteristic
subgroupG/H
center, socleC2×C31#1598
commutator, Frattini, cosocleC2×M16C23×C31
prime radicalC31×#575C22
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×M16×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3255C2semidirect
4. C31×#1342C2semidirect
5. C31×#1327C2semidirect
85. #12174C31direct (N × H)
112. #3255C2×C31semidirect
115. #1342C2×C31semidirect
116. #1327C2×C31semidirect
177. C31#12174direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32552
C31×#21912
C31×#20502
C31×#13422
C31×#13272
#1217431