Finite Group Databasebeta

← C73×#7347 #32916751

Subgroups — C73×#7347 #32916751

Order 18688.

318 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 46 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
1C73×#1794C211yesmaximal, characteristic
2C73×#1321C211yesmaximal, characteristic, complemented
3C73×#1315C211yesmaximal, characteristic, complemented
4#585×C73C411yescharacteristic
5#421×C73—21no—
6#416×C73—21no—
7#410×C73C411yescharacteristic
8#408×C73C2211yescharacteristic
9#349×C73—21no—
10C22⋊C16×C73—21no—
112−1+4×C73—41no—
122−1+4×C73C812yes—
13H(5,𝔽2)×C73—21no—
14H(5,𝔽2)×C73—41no—
15H(5,𝔽2)×C73—41no—
16C2×#41×C73—21no—
17C2×#41×C73—41no—
18C2×#41×C73—42no—
19C2×#41×C73C2×C411yescharacteristic
20C2×M16×C73D411yescharacteristic
21C22×C8×C73D411yescharacteristic
22#117×C73—21no—
23M32×C73—41no—
24C2×C16×C73—41no—
25#104×C73—42no—
26#103×C73—41no—
27#102×C73—41no—
28#99×C73—41no—
29C22⋊C8×C73—41no—
30#41×C73—22no—
31#41×C73—22no—
32#41×C73—41no—
33#41×C73—42no—
34#41×C73—41no—
35#41×C73—41no—
36#41×C73—42no—
37#41×C73—42no—
38#41×C73—44no—
39#41×C73—42no—
40#41×C73—41no—
41#41×C73—42no—
42C2×Q8×C73—21no—
43C2×Q8×C73—42no—
44C2×Q8×C73—41no—
45C2×Q8×C73C2×C811yescharacteristic
46C2×D4×C73—21no—
47C2×D4×C73—21no—
48C2×D4×C73—42no—
49C2×D4×C73—42no—
50C2×D4×C73—44no—
51C2×D4×C73—41no—
52C2×D4×C73—41no—
53C2×D4×C73M1611yescharacteristic
54C22×C4×C73—21no—
55C22×C4×C73—41no—
56C22×C4×C73—42no—
57C22×C4×C73C22⋊C411yescharacteristic
58M16×C73—41no—
59C2×C8×C73—21no—
60C2×C8×C73—21no—
61C2×C8×C73—41no—
62C4⋊C4×C73—41no—
63C22⋊C4×C73—41no—
64C42×C73—41no—
65C16×C73—81no—
66C16×C73—81no—
67C23×C73—21no—
68C23×C73—41no—
69C23×C73—42no—
70C23×C73#10011yescharacteristic
71Q8×C73—22no—
72Q8×C73—41no—
73Q8×C73—42no—
74Q8×C73—41no—
75D4×C73—22no—
76D4×C73—41no—
77D4×C73—42no—
78D4×C73—42no—
79D4×C73—42no—
80D4×C73—44no—
81D4×C73—42no—
82D4×C73—41no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—21no—
86C2×C4×C73—42no—
87C2×C4×C73—41no—
88C2×C4×C73—44no—
89C2×C4×C73—42no—
90C2×C4×C73—41no—
91C2×C4×C73—41no—
92C2×C4×C73—42no—
93C2×C4×C73—81no—
94C2×C4×C73C22⋊C811yescharacteristic
95C2×C4×C73#9911yescharacteristic
96C8×C73—41no—
97C8×C73—41no—
98C22×C73—21no—
99C22×C73—41no—
100C22×C73—42no—
101C22×C73—41no—
102C22×C73—81no—
103C22×C73—161no—
104C22×C73—162no—
105C22×C73#32311yescharacteristic
106C4×C73—21no—
107C4×C73—41no—
108C4×C73—42no—
109C4×C73—161no—
110C4×C73AΣL(1,𝔽16)11yescharacteristic
111C4×C73AΣL(1,𝔽16)11yescharacteristic
112#7347C7311yesmaximal, characteristic, direct product factor, Sylow
113C2×C73—21no—
114C2×C73—41no—
115C2×C73—81no—
116C2×C73—82no—
117C2×C73#131611yescharacteristic
118#1794C2×C7311yescharacteristic
119#1321C2×C7311yescharacteristic, complemented
120#1315C2×C7311yescharacteristic, complemented
121C73#734711yescharacteristic, direct product factor, Sylow
122#585C4×C7311yescharacteristic
123#421—21no—
124#416—21no—
125#410C4×C7311yescharacteristic
126#408C22×C7311yescharacteristic
127#349—21no—
128C22⋊C16—21no—
1292−1+4—41no—
1302−1+4C8×C7312yes—
131H(5,𝔽2)—21no—
132H(5,𝔽2)—41no—
133H(5,𝔽2)—41no—
134C2×#41—21no—
135C2×#41—41no—
136C2×#41—42no—
137C2×#41C2×C4×C7311yescharacteristic
138C2×M16D4×C7311yescharacteristic
139C22×C8D4×C7311yescharacteristic
140#117—21no—
141M32—41no—
142C2×C16—41no—
143#104—42no—
144#103—41no—
145#102—41no—
146#99—41no—
147C22⋊C8—41no—
148#41—22no—
149#41—22no—
150#41—41no—
151#41—42no—
152#41—41no—
153#41—41no—
154#41—42no—
155#41—42no—
156#41—44no—
157#41—42no—
158#41—41no—
159#41—42no—
160C2×Q8—21no—
161C2×Q8—42no—
162C2×Q8—41no—
163C2×Q8C2×C8×C7311yescharacteristic
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—44no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4M16×C7311yescharacteristic
172C22×C4—21no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4C22⋊C4×C7311yescharacteristic
176M16—41no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—41no—
180C4⋊C4—41no—
181C22⋊C4—41no—
182C42—41no—
183C16—81no—
184C16—81no—
185C23—21no—
186C23—41no—
187C23—42no—
188C23#100×C7311yescharacteristic
189Q8—22no—
190Q8—41no—
191Q8—42no—
192Q8—41no—
193D4—22no—
194D4—41no—
195D4—42no—
196D4—42no—
197D4—42no—
198D4—44no—
199D4—42no—
200D4—41no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—81no—
212C2×C4C22⋊C8×C7311yescharacteristic
213C2×C4#99×C7311yescharacteristic
214C8—41no—
215C8—41no—
216C22—21no—
217C22—41no—
218C22—42no—
219C22—41no—
220C22—81no—
221C22—161no—
222C22—162no—
223C22#323×C7311yescharacteristic
224C4—21no—
225C4—41no—
226C4—42no—
227C4—161no—
228C4AΣL(1,𝔽16)×C7311yescharacteristic
229C4AΣL(1,𝔽16)×C7311yescharacteristic
230C2—21no—
231C2—41no—
232C2—81no—
233C2—82no—
234C2C73×#131611yescharacteristic
subgroupG/H
centerC4×C73AΣL(1,𝔽16)
commutatorC2×Q8C2×C8×C73
Frattini, cosocle#408C22×C73
socleC2×C73#1316
prime radical#585×C73C4
LCSG ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C73 ◁ C2×C4×C73 ◁ C22×C4×C73 ◁ #408×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#1321C2semidirect
3. C73×#1315C2semidirect
112. #7347C73direct (N × H)
119. #1321C2×C73semidirect
120. #1315C2×C73semidirect
121. C73#7347direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#17942
C73×#13212
C73×#13152
#734773