Finite Group Databasebeta

← D21×#10587723 #37525213

Subgroups — D21×#10587723 #37525213

Order 22932.

184 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 37 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
1D21×#63271C211yesmaximal, characteristic, complemented
2C3⋊#16838469C211yesmaximal, characteristic, complemented
3C3×C7×#10587723C211yesmaximal, characteristic, complemented
4D21×D91C311yesmaximal, characteristic, complemented
5D7×#10587723—31nomaximal
6C3×C7×#63271C2211yescharacteristic, complemented, Hall
7C3⋊(C7⋊D91)C2×C311yescharacteristic, complemented
8C7×C13×D21C2×C311yescharacteristic, complemented
9C3×C7×D91C2×C311yescharacteristic, complemented
10D7×#63271—31no—
11#16838469—31no—
12C7×#10587723—21no—
13C7×#10587723D311yescharacteristic, complemented
14D21×D13⋊C3—71nomaximal
15D3×#10587723—71nomaximal
16D7×D91—31noHall
17C3×C72×C13C22×C311yescharacteristic, complemented
18C7×#63271—21no—
19C7×#63271C2×D311yescharacteristic, complemented
20AGL(1,𝔽7)×D21—131nomaximal, Hall
21C3⋊#10587723—71no—
22D3×#63271—71no—
23C3⋊((C13⋊C3)⋊D7)—71no—
24C13⋊C3×D21—141no—
25C3×#10587723D711yescharacteristic, complemented
26C3×C7×D13⋊C3—71no—
27C7⋊D91—31no—
28C7×D91C3×D311yescharacteristic, complemented
29C7×C13×D7—31no—
30D3×D91—71no—
31D13×D21—71no—
32C2×#10587723—211no—
33D7×D13⋊C3—211no—
34C3⋊(ΓL(1,𝔽8)⋊D7)—131no—
35ΓL(1,𝔽8)×D21—261no—
36C3×C7×AGL(1,𝔽7)—131no—
37C3×#63271C2×D711yescharacteristic, complemented
38C3×C7×C13⋊C3—141no—
39C72×C13C2×C3×D311yescharacteristic, complemented, Hall
40D7×D21—131no—
41D7×AGL(1,𝔽7)—391no—
42D273—71no—
43D273—71no—
44D273—421no—
45C13×D21AGL(1,𝔽7)11yescharacteristic, complemented
46D3×C7×C13—71no—
47C3×D91C3×D711yescharacteristic, complemented
48C3×C7×D13—71no—
49C2×#63271—211no—
50#10587723—21no—
51#10587723—211no—
52#10587723D2111yescharacteristic, direct product factor
53(C13⋊C3)⋊D7—211no—
54D7×C13⋊C3—421no—
55C7×D13⋊C3—71no—
56C7×D13⋊C3—141no—
57D3×D13⋊C3—491noHall
58C3×C7×ΓL(1,𝔽8)—261noHall
59C2×D91—211no—
60D7×D13—211no—
61C3⋊(C7⋊D7)—131no—
62C7×D21D13⋊C311yescharacteristic, complemented
63C3×C7×D7—131no—
64ΓL(1,𝔽8)⋊D7—391no—
65D7×ΓL(1,𝔽8)—781no—
66C7×AGL(1,𝔽7)—131no—
67C7×AGL(1,𝔽7)—261no—
68C3×C7×C13—61no—
69C3×C7×C13C2×AGL(1,𝔽7)11yescharacteristic, complemented
70C3×C7×C13C2×C3×D711yescharacteristic, complemented
71#63271—21no—
72#63271C2×D2111yescharacteristic, complemented
73C7×C13⋊C3—141no—
74C7×C13⋊C3—281no—
75C2×C3×D21—911no—
76D3×AGL(1,𝔽7)—911no—
77C3⋊(D13⋊C3)—491no—
78D3×C13⋊C3—981no—
79C3×D13⋊C3—71no—
80D72—391noHall
81C2×C7×C13—211no—
82D91—211no—
83D91—211no—
84D91—1261no—
85D91C3×D2111yescharacteristic, complemented
86C7×D13—71no—
87C13×D7—31no—
88D3×D13—491no—
89C2×D13⋊C3—1471no—
90C3×C72C2×D13⋊C311yescharacteristic, complemented
91C7×ΓL(1,𝔽8)—261no—
92C7×ΓL(1,𝔽8)—521no—
93C2×C32×C7—911no—
94C3×D21—911no—
95C3×D21—1821no—
96C3⋊AGL(1,𝔽7)—911no—
97D3×ΓL(1,𝔽8)—1821no—
98C3×AGL(1,𝔽7)—131no—
99C3×C13⋊C3—141noHall
100C7⋊D7—391no—
101C7×D7—31no—
102C7×D7—131no—
103C7×C13—61no—
104C7×C13D3×AGL(1,𝔽7)11yescharacteristic, complemented
105C7×C13C2×C3×D2111yescharacteristic, complemented
106C2×D21—911no—
107C2×C3×D7—2731no—
108D3×D7—911no—
109C2×AGL(1,𝔽7)—2731no—
110D39—491no—
111C3×D13—71no—
112D3×C13—71no—
113C2×C13⋊C3—1471no—
114D13⋊C3—71no—
115D13⋊C3—141no—
116D13⋊C3—1471no—
117C32×C7—1821no—
118C3×ΓL(1,𝔽8)—261no—
119C2×D13—1471noHall
120C72D3×D13⋊C311yescharacteristic, complemented, Sylow
121C2×C3×C7—911no—
122C2×C3×C7—911no—
123C2×C3×C7—1821no—
124D21—5461no—
125D21—911no—
126D21—911no—
127D21#1058772311yescharacteristic, direct product factor
128C3×D7—131no—
129C3×D7—2731no—
130C3×D7—5461no—
131D3×C7—71no—
132C2×ΓL(1,𝔽8)—5461no—
133AGL(1,𝔽7)—131no—
134AGL(1,𝔽7)—2731no—
135AGL(1,𝔽7)—261no—
136C3×C13D7×AGL(1,𝔽7)11yescharacteristic, complemented
137C13⋊C3—71no—
138C13⋊C3—281no—
139C2×C3×D3—6371noHall
140C2×D7—2731no—
141C2×D7—2731no—
142C2×C13—211no—
143D13—71no—
144D13—1471no—
145C3×C7—61no—
146C3×C7—3641no—
147C3×C7—1821no—
148C3×C7D7×D13⋊C311yescharacteristic, complemented
149C3×C7C2×#1058772311yescharacteristic, complemented
150ΓL(1,𝔽8)—261no—
151ΓL(1,𝔽8)—521no—
152C2×C32—911no—
153C3×D3—6371no—
154C3×D3—12741no—
155C2×C7—211no—
156C2×C7—911no—
157D7—31no—
158D7—131no—
159D7—2731no—
160D7—2731no—
161D7—16381no—
162C13AGL(1,𝔽7)×D2111yescharacteristic, complemented, Sylow
163C22×C3—19111no—
164C2×D3—6371no—
165C32—1821noSylow
166C7—61no—
167C7D3×#1058772311yescharacteristic, complemented
168C7D21×D13⋊C311yescharacteristic, complemented
169C2×C3—911no—
170C2×C3—911no—
171C2×C3—19111no—
172C2×C3—1821no—
173C2×C3—38221no—
174D3—71no—
175D3—6371no—
176C22—19111noSylow
177C3—3641no—
178C3—1821no—
179C3D7×#1058772311yescharacteristic, complemented
180C2—211no—
181C2—911no—
182C2—19111no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C72×C13C22×C3
LCSG ▷ C3×C72×C13
UCSC1

Complements

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

NH ≅ G/Ntype
1. D21×#63271C2semidirect
2. C3⋊#16838469C2semidirect
3. C3×C7×#10587723C2semidirect
4. D21×D91C3semidirect
6. C3×C7×#63271C22semidirect
7. C3⋊(C7⋊D91)C2×C3semidirect
8. C7×C13×D21C2×C3semidirect
9. C3×C7×D91C2×C3semidirect
13. C7×#10587723D3semidirect
17. C3×C72×C13C22×C3semidirect
19. C7×#63271C2×D3semidirect
25. C3×#10587723D7semidirect
28. C7×D91C3×D3semidirect
37. C3×#63271C2×D7semidirect
39. C72×C13C2×C3×D3semidirect
45. C13×D21AGL(1,𝔽7)semidirect
47. C3×D91C3×D7semidirect
52. #10587723D21direct (N × H)
62. C7×D21D13⋊C3semidirect
69. C3×C7×C13C2×AGL(1,𝔽7)semidirect
70. C3×C7×C13C2×C3×D7semidirect
72. #63271C2×D21semidirect
85. D91C3×D21semidirect
90. C3×C72C2×D13⋊C3semidirect
104. C7×C13D3×AGL(1,𝔽7)semidirect
105. C7×C13C2×C3×D21semidirect
120. C72D3×D13⋊C3semidirect
127. D21#10587723direct (N × H)
136. C3×C13D7×AGL(1,𝔽7)semidirect
148. C3×C7D7×D13⋊C3semidirect
149. C3×C7C2×#10587723semidirect
162. C13AGL(1,𝔽7)×D21semidirect
167. C7D3×#10587723semidirect
168. C7D21×D13⋊C3semidirect
179. C3D7×#10587723semidirect

Maximal subgroups

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

subgroupindex
D21×#632712
C3⋊#168384692
C3×C7×#105877232
D21×D913
D7×#105877233
D21×D13⋊C37
D3×#105877237
AGL(1,𝔽7)×D2113