Finite Group Databasebeta

← C37×#9070 #22630323

Subgroups — C37×#9070 #22630323

Order 9472.

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
1C37×#3131C211yesmaximal, characteristic
2C37×#2986C211yesmaximal, characteristic, complemented
3C37×#1526C211yesmaximal, characteristic, complemented
4C37×#1484C212yesmaximal
5C37×#1278C212yesmaximal, complemented
6C37×#533C2211yescharacteristic
7C37×#532C2211yescharacteristic
8C37×#488—22no—
9C37×#487C412yes—
10C37×#486C412yes—
11C37×#446—22no—
12C8×Q8×C37—22no—
13C37×#433—22no—
14C37×#431C2212yes—
15C37×#416—22no—
16C4×M16×C37C2212yes—
17C2×C4×C8×C37C2211yescharacteristic
18C37×#330—22no—
19C37×D4⋊C8—22no—
20C37×D4⋊C8—22no—
21C2×#41×C37—21no—
22C2×M16×C37—22no—
23C22×C8×C37—21no—
24C4⋊Q8×C37C2×C411yescharacteristic
25C4⋊Q8×C37C2×C412yes—
26#127×C37C2×C411yescharacteristic
27#125×C37D412yes—
28#124×C37—21no—
29C22⋊Q8×C37—22no—
30#121×C37—22no—
31C4×Q8×C37—22no—
32C4×D4×C37—22no—
33#117×C37D412yes—
34C2×C42×C37C2311yescharacteristic
35C4⋊C8×C37—24no—
36C4⋊C8×C37D412yes—
37C4⋊C8×C37D412yes—
38#103×C37—24no—
39#102×C37—24no—
40C22⋊C8×C37—22no—
41#97×C37—22no—
42#97×C37—22no—
43C4×C8×C37—22no—
44C4×C8×C37—21no—
45C4×C8×C37—22no—
46C4×C8×C37—22no—
47C4×C8×C37C2×C412yes—
48#41×C37—44no—
49C2×Q8×C37—22no—
50C2×Q8×C37—21no—
51C2×D4×C37—21no—
52C2×D4×C37—22no—
53C2×D4×C37—41no—
54C22×C4×C37—41no—
55C22×C4×C37C2×D411yescharacteristic
56C22×C4×C37#4112yes—
57M16×C37—24no—
58M16×C37—42no—
59C2×C8×C37—22no—
60C2×C8×C37—22no—
61C2×C8×C37—22no—
62C2×C8×C37—42no—
63C2×C8×C37—42no—
64C4⋊C4×C37—22no—
65C4⋊C4×C37—24no—
66C4⋊C4×C37—22no—
67C4⋊C4×C37C22⋊C414yes—
68C22⋊C4×C37—22no—
69C22⋊C4×C37—42no—
70C42×C37—22no—
71C42×C37C22×C411yescharacteristic
72C42×C37C2×D411yescharacteristic
73C42×C37C2×D412yes—
74C23×C37—41no—
75C23×C37#12311yescharacteristic
76Q8×C37—42no—
77Q8×C37—42no—
78Q8×C37—42no—
79D4×C37—42no—
80D4×C37—42no—
81D4×C37—42no—
82D4×C37—82no—
83C2×C4×C37—22no—
84C2×C4×C37—22no—
85C2×C4×C37—22no—
86C2×C4×C37—24no—
87C2×C4×C37—41no—
88C2×C4×C37—82no—
89C2×C4×C37C2×C22⋊C411yescharacteristic
90C2×C4×C37C4×D412yes—
91C2×C4×C37#12011yescharacteristic
92C2×C4×C37#12112yes—
93C8×C37—42no—
94C8×C37—44no—
95C8×C37—42no—
96C8×C37—42no—
97#9070C3711yesmaximal, characteristic, direct product factor, Sylow
98C22×C37—22no—
99C22×C37—21no—
100C22×C37—82no—
101C22×C37—81no—
102C22×C37#38611yescharacteristic
103C4×C37—22no—
104C4×C37—21no—
105C4×C37—44no—
106C4×C37—81no—
107C4×C37#44312yes—
108C4×C37#44412yes—
109C4×C37#45412yes—
110#3131C2×C3711yescharacteristic
111#2986C2×C3711yescharacteristic, complemented
112#1526C2×C3711yescharacteristic, complemented
113#1484C2×C3712yes—
114#1278C2×C3712yescomplemented
115C2×C37—41no—
116C2×C37—161no—
117C2×C37#187612yes—
118C2×C37#188811yescharacteristic
119#533C22×C3711yescharacteristic
120#532C22×C3711yescharacteristic
121#488—22no—
122#487C4×C3712yes—
123#486C4×C3712yes—
124#446—22no—
125C8×Q8—22no—
126#433—22no—
127#431C22×C3712yes—
128#416—22no—
129C4×M16C22×C3712yes—
130C2×C4×C8C22×C3711yescharacteristic
131#330—22no—
132D4⋊C8—22no—
133D4⋊C8—22no—
134C37#907011yescharacteristic, direct product factor, Sylow
135C2×#41—21no—
136C2×M16—22no—
137C22×C8—21no—
138C4⋊Q8C2×C4×C3711yescharacteristic
139C4⋊Q8C2×C4×C3712yes—
140#127C2×C4×C3711yescharacteristic
141#125D4×C3712yes—
142#124—21no—
143C22⋊Q8—22no—
144#121—22no—
145C4×Q8—22no—
146C4×D4—22no—
147#117D4×C3712yes—
148C2×C42C23×C3711yescharacteristic
149C4⋊C8—24no—
150C4⋊C8D4×C3712yes—
151C4⋊C8D4×C3712yes—
152#103—24no—
153#102—24no—
154C22⋊C8—22no—
155#97—22no—
156#97—22no—
157C4×C8—22no—
158C4×C8—21no—
159C4×C8—22no—
160C4×C8—22no—
161C4×C8C2×C4×C3712yes—
162#41—44no—
163C2×Q8—22no—
164C2×Q8—21no—
165C2×D4—21no—
166C2×D4—22no—
167C2×D4—41no—
168C22×C4—41no—
169C22×C4C2×D4×C3711yescharacteristic
170C22×C4#41×C3712yes—
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×C3714yes—
182C22⋊C4—22no—
183C22⋊C4—42no—
184C42—22no—
185C42C22×C4×C3711yescharacteristic
186C42C2×D4×C3711yescharacteristic
187C42C2×D4×C3712yes—
188C23—41no—
189C23#123×C3711yescharacteristic
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×C3711yescharacteristic
204C2×C4C4×D4×C3712yes—
205C2×C4#120×C3711yescharacteristic
206C2×C4#121×C3712yes—
207C8—42no—
208C8—44no—
209C8—42no—
210C8—42no—
211C22—22no—
212C22—21no—
213C22—82no—
214C22—81no—
215C22C37×#38611yescharacteristic
216C4—22no—
217C4—21no—
218C4—44no—
219C4—81no—
220C4C37×#44312yes—
221C4C37×#44412yes—
222C4C37×#45412yes—
223C2—41no—
224C2—161no—
225C2C37×#187612yes—
226C2C37×#188811yescharacteristic
subgroupG/H
centerC2×C4×C37#120
commutatorC42C22×C4×C37
Frattini, cosocleC2×C42C23×C37
socleC22×C37#386
prime radicalC37×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C37 ◁ C2×C42×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C37×#2986C2semidirect
3. C37×#1526C2semidirect
5. C37×#1278C2semidirect
97. #9070C37direct (N × H)
111. #2986C2×C37semidirect
112. #1526C2×C37semidirect
114. #1278C2×C37semidirect
134. C37#9070direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#31312
C37×#29862
C37×#15262
C37×#14842
C37×#12782
#907037