Finite Group Databasebeta

← C73×#7214 #32916618

Subgroups — C73×#7214 #32916618

Order 18688.

290 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 74 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×#1820C211yesmaximal, characteristic
2C73×#1750C211yesmaximal, characteristic
3C73×#1464C211yesmaximal, characteristic
4C2×#117×C73C2211yescharacteristic
5#432×C73—21no—
6C2×C22⋊C8×C73—21no—
7C2×#97×C73C411yescharacteristic, complemented
8C2×C4×C8×C73C411yescharacteristic, complemented
9#381×C73—21no—
10#342×C73—21no—
11#342×C73C412yes—
12#336×C73—21no—
13#324×C73C412yescomplemented
14#323×C73—21no—
15C23×C4×C73D411yescharacteristic
16C22×C8×C73—21no—
17C22×C8×C73—21no—
18#117×C73—41no—
19#117×C73—41no—
20C2×C4⋊C4×C73C2×C411yescharacteristic
21C2×C4⋊C4×C73Q811yescharacteristic
22C2×C22⋊C4×C73—22no—
23C2×C22⋊C4×C73C2×C411yescharacteristic
24C2×C22⋊C4×C73D411yescharacteristic
25C2×C42×C73C2×C411yescharacteristic
26C2×C42×C73D411yescharacteristic
27C4⋊C8×C73—41no—
28C22⋊C8×C73—24no—
29C22⋊C8×C73—22no—
30#97×C73—41no—
31#97×C73—41no—
32C4×C8×C73—41no—
33#95×C73—41no—
34#95×C73—41no—
35C24×C73C22⋊C411yescharacteristic
36C22×C4×C73—22no—
37C22×C4×C73—21no—
38C22×C4×C73—21no—
39C22×C4×C73—41no—
40C22×C4×C73—42no—
41C22×C4×C73C4211yescharacteristic
42C22×C4×C73C22⋊C411yescharacteristic
43C22×C4×C73C22⋊C411yescharacteristic
44C22×C4×C73C4⋊C411yescharacteristic
45C22×C4×C73C4⋊C411yescharacteristic
46C22×C4×C73C4⋊C411yescharacteristic
47C2×C8×C73—22no—
48C2×C8×C73—22no—
49C2×C8×C73—41no—
50C2×C8×C73—41no—
51C2×C8×C73—41no—
52C2×C8×C73—41no—
53C2×C8×C73—42no—
54C4⋊C4×C73—41no—
55C4⋊C4×C73—41no—
56C22⋊C4×C73—41no—
57C22⋊C4×C73—41no—
58C22⋊C4×C73—42no—
59C22⋊C4×C73—42no—
60C42×C73—21no—
61C42×C73—21no—
62C42×C73—21no—
63C42×C73—21no—
64C23×C73—21no—
65C23×C73—21no—
66C23×C73—41no—
67C23×C73—41no—
68C23×C73#9511yescharacteristic
69C23×C73#10412yes—
70C2×C4×C73—21no—
71C2×C4×C73—21no—
72C2×C4×C73—21no—
73C2×C4×C73—21no—
74C2×C4×C73—21no—
75C2×C4×C73—21no—
76C2×C4×C73—21no—
77C2×C4×C73—21no—
78C2×C4×C73—21no—
79C2×C4×C73—42no—
80C2×C4×C73—42no—
81C2×C4×C73—41no—
82C2×C4×C73—41no—
83C2×C4×C73—41no—
84C2×C4×C73—41no—
85C2×C4×C73—82no—
86C2×C4×C73—82no—
87C2×C4×C73—82no—
88C2×C4×C73#9911yescharacteristic
89C2×C4×C73#9911yescharacteristic
90C8×C73—42no—
91C8×C73—42no—
92C8×C73—82no—
93C22×C73—21no—
94C22×C73—21no—
95C22×C73—22no—
96C22×C73—42no—
97C22×C73—41no—
98C22×C73—41no—
99C22×C73—81no—
100C22×C73H(3,ℤ/4)11yescharacteristic
101C22×C73#33911yescharacteristic
102C22×C73#34211yescharacteristic
103C4×C73—21no—
104C4×C73—21no—
105C4×C73—41no—
106C4×C73—41no—
107C4×C73—41no—
108C4×C73—41no—
109C4×C73—41no—
110C4×C73—162no—
111#7214C7311yesmaximal, characteristic, direct product factor, Sylow
112C2×C73—21no—
113C2×C73—21no—
114C2×C73—42no—
115C2×C73#128411yescharacteristic
116C2×C73#139111yescharacteristic
117C2×C73#139511yescharacteristic
118#1820C2×C7311yescharacteristic
119#1750C2×C7311yescharacteristic
120#1464C2×C7311yescharacteristic
121C73#721411yescharacteristic, direct product factor, Sylow
122C2×#117C22×C7311yescharacteristic
123#432—21no—
124C2×C22⋊C8—21no—
125C2×#97C4×C7311yescharacteristic, complemented
126C2×C4×C8C4×C7311yescharacteristic, complemented
127#381—21no—
128#342—21no—
129#342C4×C7312yes—
130#336—21no—
131#324C4×C7312yescomplemented
132#323—21no—
133C23×C4D4×C7311yescharacteristic
134C22×C8—21no—
135C22×C8—21no—
136#117—41no—
137#117—41no—
138C2×C4⋊C4C2×C4×C7311yescharacteristic
139C2×C4⋊C4Q8×C7311yescharacteristic
140C2×C22⋊C4—22no—
141C2×C22⋊C4C2×C4×C7311yescharacteristic
142C2×C22⋊C4D4×C7311yescharacteristic
143C2×C42C2×C4×C7311yescharacteristic
144C2×C42D4×C7311yescharacteristic
145C4⋊C8—41no—
146C22⋊C8—24no—
147C22⋊C8—22no—
148#97—41no—
149#97—41no—
150C4×C8—41no—
151#95—41no—
152#95—41no—
153C24C22⋊C4×C7311yescharacteristic
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4C42×C7311yescharacteristic
160C22×C4C22⋊C4×C7311yescharacteristic
161C22×C4C22⋊C4×C7311yescharacteristic
162C22×C4C4⋊C4×C7311yescharacteristic
163C22×C4C4⋊C4×C7311yescharacteristic
164C22×C4C4⋊C4×C7311yescharacteristic
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—42no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C22⋊C4—41no—
175C22⋊C4—41no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C23—21no—
183C23—21no—
184C23—41no—
185C23—41no—
186C23#95×C7311yescharacteristic
187C23#104×C7312yes—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4#99×C7311yescharacteristic
207C2×C4#99×C7311yescharacteristic
208C8—42no—
209C8—42no—
210C8—82no—
211C22—21no—
212C22—21no—
213C22—22no—
214C22—42no—
215C22—41no—
216C22—41no—
217C22—81no—
218C22H(3,ℤ/4)×C7311yescharacteristic
219C22#339×C7311yescharacteristic
220C22#342×C7311yescharacteristic
221C4—21no—
222C4—21no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—162no—
229C2—21no—
230C2—21no—
231C2—42no—
232C2C73×#128411yescharacteristic
233C2C73×#139111yescharacteristic
234C2C73×#139511yescharacteristic
subgroupG/H
center, socleC22×C73#342
commutatorC22×C4C42×C73
Frattini, cosocleC2×#117C22×C73
prime radicalC24×C73C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C22×C4×C73 ◁ C2×#117×C73 ◁ G

Complements

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

NH ≅ G/Ntype
7. C2×#97×C73C4semidirect
8. C2×C4×C8×C73C4semidirect
13. #324×C73C4semidirect
111. #7214C73direct (N × H)
121. C73#7214direct (N × H)
125. C2×#97C4×C73semidirect
126. C2×C4×C8C4×C73semidirect
131. #324C4×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#18202
C73×#17502
C73×#14642
#721473