Finite Group Databasebeta

← C31×#8527 #21465286

Subgroups — C31×#8527 #21465286

Order 7936.

656 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C2×C31×#541C211yesmaximal, characteristic
2C31×#1558C212yesmaximal, complemented
3C31×#1534C212yesmaximal, complemented
4C31×#1464C212yesmaximal
5C31×#541—24no—
6C31×#541—24no—
7C2×C31×C4⋊Q8C2211yescharacteristic
8C2×C31×#126—22no—
9C2×C31×#125—22no—
10C2×C31×#125C2211yescharacteristic
11C2×C31×C22⋊Q8C412yes—
12C2×C4×Q8×C31C412yes—
13C2×C31×#117C2211yescharacteristic
14C31×#432C2214yes—
15C31×#333—24no—
16C31×#330—24no—
17C31×#324—22no—
18C31×#323—22no—
19C22×Q8×C31C2×C412yes—
20C23×C4×C31C2×C411yescharacteristic
21C31×C4⋊Q8—22no—
22C31×C4⋊Q8D414yes—
23C31×#126—216no—
24C31×#125—216no—
25C31×#125—22no—
26C31×#125D414yes—
27C31×C22⋊Q8—24no—
28C31×C22⋊Q8—24no—
29C4×Q8×C31—24no—
30C4×Q8×C31—24no—
31C31×#117—22no—
32C31×#117—22no—
33C2×C4⋊C4×C31—22no—
34C2×C4⋊C4×C31—22no—
35C2×C4⋊C4×C31—22no—
36C2×C4⋊C4×C31—22no—
37C2×C4⋊C4×C31C2×C412yes—
38C2×C4⋊C4×C31D412yes—
39C2×C22⋊C4×C31—22no—
40C2×C22⋊C4×C31D412yes—
41C2×C42×C31—22no—
42C2×C42×C31C2×C411yescharacteristic
43C2×C42×C31C2311yescharacteristic
44C31×C4⋊C8—24no—
45C31×C22⋊C8—24no—
46C31×#97—24no—
47C24×C31C22⋊C411yescharacteristic
48C2×Q8×C31—22no—
49C2×Q8×C31—44no—
50C2×Q8×C31C22⋊C414yes—
51C22×C4×C31—22no—
52C22×C4×C31—22no—
53C22×C4×C31—21no—
54C22×C4×C31—21no—
55C22×C4×C31—42no—
56C22×C4×C31C22⋊C411yescharacteristic
57C22×C4×C31C22⋊C412yes—
58C22×C4×C31C22×C411yescharacteristic
59C22×C4×C31C2×D412yes—
60C2×C8×C31—44no—
61C4⋊C4×C31—28no—
62C4⋊C4×C31—28no—
63C4⋊C4×C31—22no—
64C4⋊C4×C31—28no—
65C4⋊C4×C31—28no—
66C4⋊C4×C31—24no—
67C4⋊C4×C31C22⋊C414yes—
68C22⋊C4×C31—28no—
69C22⋊C4×C31—24no—
70C42×C31—28no—
71C42×C31—21no—
72C42×C31—21no—
73C42×C31C2×D412yes—
74C42×C31C2×D412yes—
75#8527C3111yesmaximal, characteristic, direct product factor, Sylow
76C23×C31—21no—
77C23×C31—22no—
78C23×C31—42no—
79C23×C31C2×C22⋊C411yescharacteristic
80Q8×C31—44no—
81Q8×C31—44no—
82C2×C4×C31—22no—
83C2×C4×C31—22no—
84C2×C4×C31—24no—
85C2×C4×C31—24no—
86C2×C4×C31—21no—
87C2×C4×C31—22no—
88C2×C4×C31—42no—
89C2×C4×C31—42no—
90C2×C4×C31—42no—
91C2×C4×C31—44no—
92C2×C4×C31—44no—
93C2×C4×C31—42no—
94C2×C4×C31C2×C22⋊C411yescharacteristic
95C2×C4×C31C2×C22⋊C411yescharacteristic
96C2×C4×C31#12014yes—
97C8×C31—84no—
98C2×#541C2×C3111yescharacteristic
99#1558C2×C3112yescomplemented
100#1534C2×C3112yescomplemented
101#1464C2×C3112yes—
102C22×C31—22no—
103C22×C31—42no—
104C22×C31—41no—
105C22×C31—82no—
106C22×C31#37911yescharacteristic
107C22×C31#41312yes—
108C4×C31—24no—
109C4×C31—21no—
110C4×C31—21no—
111C4×C31—42no—
112C4×C31—44no—
113C4×C31—41no—
114C4×C31—44no—
115#541—24no—
116#541—24no—
117C2×C4⋊Q8C22×C3111yescharacteristic
118C2×#126—22no—
119C2×#125—22no—
120C2×#125C22×C3111yescharacteristic
121C2×C22⋊Q8C4×C3112yes—
122C2×C4×Q8C4×C3112yes—
123C2×#117C22×C3111yescharacteristic
124#432C22×C3114yes—
125#333—24no—
126#330—24no—
127#324—22no—
128#323—22no—
129C2×C31—22no—
130C2×C31—81no—
131C2×C31#178511yescharacteristic
132C2×C31#179612yes—
133C22×Q8C2×C4×C3112yes—
134C23×C4C2×C4×C3111yescharacteristic
135C4⋊Q8—22no—
136C4⋊Q8D4×C3114yes—
137#126—216no—
138#125—216no—
139#125—22no—
140#125D4×C3114yes—
141C22⋊Q8—24no—
142C22⋊Q8—24no—
143C4×Q8—24no—
144C4×Q8—24no—
145#117—22no—
146#117—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4C2×C4×C3112yes—
152C2×C4⋊C4D4×C3112yes—
153C2×C22⋊C4—22no—
154C2×C22⋊C4D4×C3112yes—
155C2×C42—22no—
156C2×C42C2×C4×C3111yescharacteristic
157C2×C42C23×C3111yescharacteristic
158C4⋊C8—24no—
159C22⋊C8—24no—
160#97—24no—
161C31#852711yescharacteristic, direct product factor, Sylow
162C24C22⋊C4×C3111yescharacteristic
163C2×Q8—22no—
164C2×Q8—44no—
165C2×Q8C22⋊C4×C3114yes—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—42no—
171C22×C4C22⋊C4×C3111yescharacteristic
172C22×C4C22⋊C4×C3112yes—
173C22×C4C22×C4×C3111yescharacteristic
174C22×C4C2×D4×C3112yes—
175C2×C8—44no—
176C4⋊C4—28no—
177C4⋊C4—28no—
178C4⋊C4—22no—
179C4⋊C4—28no—
180C4⋊C4—28no—
181C4⋊C4—24no—
182C4⋊C4C22⋊C4×C3114yes—
183C22⋊C4—28no—
184C22⋊C4—24no—
185C42—28no—
186C42—21no—
187C42—21no—
188C42C2×D4×C3112yes—
189C42C2×D4×C3112yes—
190C23—21no—
191C23—22no—
192C23—42no—
193C23C2×C22⋊C4×C3111yescharacteristic
194Q8—44no—
195Q8—44no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—21no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4C2×C22⋊C4×C3111yescharacteristic
209C2×C4C2×C22⋊C4×C3111yescharacteristic
210C2×C4C31×#12014yes—
211C8—84no—
212C22—22no—
213C22—42no—
214C22—41no—
215C22—82no—
216C22C31×#37911yescharacteristic
217C22C31×#41312yes—
218C4—24no—
219C4—21no—
220C4—21no—
221C4—44no—
222C4—42no—
223C4—41no—
224C4—44no—
225C2—22no—
226C2—81no—
227C2C31×#178511yescharacteristic
228C2C31×#179612yes—
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC24×C31C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1558C2semidirect
3. C31×#1534C2semidirect
75. #8527C31direct (N × H)
99. #1558C2×C31semidirect
100. #1534C2×C31semidirect
161. C31#8527direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5412
C31×#15582
C31×#15342
C31×#14642
#852731