Finite Group Databasebeta

← C31×#9070 #21465829

Subgroups — C31×#9070 #21465829

Order 7936.

426 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2986C211yesmaximal, characteristic, complemented
3C31×#1526C211yesmaximal, characteristic, complemented
4C31×#1484C212yesmaximal
5C31×#1278C212yesmaximal, complemented
6C31×#533C2211yescharacteristic
7C31×#532C2211yescharacteristic
8C31×#488—22no—
9C31×#487C412yes—
10C31×#486C412yes—
11C31×#446—22no—
12C8×Q8×C31—22no—
13C31×#433—22no—
14C31×#431C2212yes—
15C31×#416—22no—
16C4×M16×C31C2212yes—
17C2×C4×C8×C31C2211yescharacteristic
18C31×#330—22no—
19C31×D4⋊C8—22no—
20C31×D4⋊C8—22no—
21C2×#41×C31—21no—
22C2×M16×C31—22no—
23C22×C8×C31—21no—
24C31×C4⋊Q8C2×C411yescharacteristic
25C31×C4⋊Q8C2×C412yes—
26C31×#127C2×C411yescharacteristic
27C31×#125D412yes—
28C31×#124—21no—
29C31×C22⋊Q8—22no—
30C31×#121—22no—
31C4×Q8×C31—22no—
32C4×D4×C31—22no—
33C31×#117D412yes—
34C2×C42×C31C2311yescharacteristic
35C31×C4⋊C8—24no—
36C31×C4⋊C8D412yes—
37C31×C4⋊C8D412yes—
38C31×#103—24no—
39C31×#102—24no—
40C31×C22⋊C8—22no—
41C31×#97—22no—
42C31×#97—22no—
43C4×C8×C31—22no—
44C4×C8×C31—21no—
45C4×C8×C31—22no—
46C4×C8×C31—22no—
47C4×C8×C31C2×C412yes—
48#41×C31—44no—
49C2×Q8×C31—22no—
50C2×Q8×C31—21no—
51C2×D4×C31—21no—
52C2×D4×C31—22no—
53C2×D4×C31—41no—
54C22×C4×C31—41no—
55C22×C4×C31C2×D411yescharacteristic
56C22×C4×C31#4112yes—
57M16×C31—24no—
58M16×C31—42no—
59C2×C8×C31—22no—
60C2×C8×C31—22no—
61C2×C8×C31—22no—
62C2×C8×C31—42no—
63C2×C8×C31—42no—
64C4⋊C4×C31—22no—
65C4⋊C4×C31—24no—
66C4⋊C4×C31—22no—
67C4⋊C4×C31C22⋊C414yes—
68C22⋊C4×C31—22no—
69C22⋊C4×C31—42no—
70C42×C31—22no—
71C42×C31C22×C411yescharacteristic
72C42×C31C2×D411yescharacteristic
73C42×C31C2×D412yes—
74#9070C3111yesmaximal, characteristic, direct product factor, Sylow
75C23×C31—41no—
76C23×C31#12311yescharacteristic
77Q8×C31—42no—
78Q8×C31—42no—
79Q8×C31—42no—
80D4×C31—42no—
81D4×C31—42no—
82D4×C31—42no—
83D4×C31—82no—
84C2×C4×C31—22no—
85C2×C4×C31—22no—
86C2×C4×C31—22no—
87C2×C4×C31—24no—
88C2×C4×C31—41no—
89C2×C4×C31—82no—
90C2×C4×C31C2×C22⋊C411yescharacteristic
91C2×C4×C31C4×D412yes—
92C2×C4×C31#12011yescharacteristic
93C2×C4×C31#12112yes—
94C8×C31—42no—
95C8×C31—44no—
96C8×C31—42no—
97C8×C31—42no—
98#3131C2×C3111yescharacteristic
99#2986C2×C3111yescharacteristic, complemented
100#1526C2×C3111yescharacteristic, complemented
101#1484C2×C3112yes—
102#1278C2×C3112yescomplemented
103C22×C31—22no—
104C22×C31—21no—
105C22×C31—82no—
106C22×C31—81no—
107C22×C31#38611yescharacteristic
108C4×C31—22no—
109C4×C31—21no—
110C4×C31—44no—
111C4×C31—81no—
112C4×C31#44312yes—
113C4×C31#44412yes—
114C4×C31#45412yes—
115#533C22×C3111yescharacteristic
116#532C22×C3111yescharacteristic
117#488—22no—
118#487C4×C3112yes—
119#486C4×C3112yes—
120#446—22no—
121C8×Q8—22no—
122#433—22no—
123#431C22×C3112yes—
124#416—22no—
125C4×M16C22×C3112yes—
126C2×C4×C8C22×C3111yescharacteristic
127#330—22no—
128D4⋊C8—22no—
129D4⋊C8—22no—
130C2×C31—41no—
131C2×C31—161no—
132C2×C31#187612yes—
133C2×C31#188811yescharacteristic
134C2×#41—21no—
135C2×M16—22no—
136C22×C8—21no—
137C4⋊Q8C2×C4×C3111yescharacteristic
138C4⋊Q8C2×C4×C3112yes—
139#127C2×C4×C3111yescharacteristic
140#125D4×C3112yes—
141#124—21no—
142C22⋊Q8—22no—
143#121—22no—
144C4×Q8—22no—
145C4×D4—22no—
146#117D4×C3112yes—
147C2×C42C23×C3111yescharacteristic
148C4⋊C8—24no—
149C4⋊C8D4×C3112yes—
150C4⋊C8D4×C3112yes—
151#103—24no—
152#102—24no—
153C22⋊C8—22no—
154#97—22no—
155#97—22no—
156C4×C8—22no—
157C4×C8—21no—
158C4×C8—22no—
159C4×C8—22no—
160C4×C8C2×C4×C3112yes—
161C31#907011yescharacteristic, direct product factor, Sylow
162#41—44no—
163C2×Q8—22no—
164C2×Q8—21no—
165C2×D4—21no—
166C2×D4—22no—
167C2×D4—41no—
168C22×C4—41no—
169C22×C4C2×D4×C3111yescharacteristic
170C22×C4#41×C3112yes—
171M16—24no—
172M16—42no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—42no—
177C2×C8—42no—
178C4⋊C4—22no—
179C4⋊C4—24no—
180C4⋊C4—22no—
181C4⋊C4C22⋊C4×C3114yes—
182C22⋊C4—22no—
183C22⋊C4—42no—
184C42—22no—
185C42C22×C4×C3111yescharacteristic
186C42C2×D4×C3111yescharacteristic
187C42C2×D4×C3112yes—
188C23—41no—
189C23C31×#12311yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—42no—
193D4—42no—
194D4—42no—
195D4—42no—
196D4—82no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—41no—
202C2×C4—82no—
203C2×C4C2×C22⋊C4×C3111yescharacteristic
204C2×C4C4×D4×C3112yes—
205C2×C4C31×#12011yescharacteristic
206C2×C4C31×#12112yes—
207C8—42no—
208C8—44no—
209C8—42no—
210C8—42no—
211C22—22no—
212C22—21no—
213C22—82no—
214C22—81no—
215C22C31×#38611yescharacteristic
216C4—22no—
217C4—21no—
218C4—44no—
219C4—81no—
220C4C31×#44312yes—
221C4C31×#44412yes—
222C4C31×#45412yes—
223C2—41no—
224C2—161no—
225C2C31×#187612yes—
226C2C31×#188811yescharacteristic
subgroupG/H
centerC2×C4×C31#120
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
socleC22×C31#386
prime radicalC31×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2986C2semidirect
3. C31×#1526C2semidirect
5. C31×#1278C2semidirect
74. #9070C31direct (N × H)
99. #2986C2×C31semidirect
100. #1526C2×C31semidirect
102. #1278C2×C31semidirect
161. C31#9070direct (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×#29862
C31×#15262
C31×#14842
C31×#12782
#907031