Finite Group Databasebeta

← #37524628

Subgroups — #37524628

Order 22932.

360 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 76 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
1C2×C13⋊C3×#3861C211yesmaximal, characteristic
2#21439560C311yesmaximal, characteristic, complemented
3#21439526C311yesmaximal, characteristic, complemented
4#21439523C311yesmaximal, characteristic, complemented
5#21439522C311yesmaximal, characteristic, complemented
6C13⋊C3×#3861C411yescharacteristic, complemented, Hall
7C2×#14274262C2×C311yescharacteristic
8C2×#14274263C2×C311yescharacteristic
9C2×C13×#3861C2×C311yescharacteristic
10C2×C72×C13⋊C3C2×C311yescharacteristic
11ΓL(1,𝔽8)⋊(Dic13⋊C3)—71nomaximal
12ΓL(1,𝔽8)⋊(Dic13⋊C3)—72nomaximal
13ΓL(1,𝔽8)⋊(Dic13⋊C3)—74nomaximal
14ΓL(1,𝔽8)⋊(Dic13⋊C3)—71nomaximal
15C7⋊Dic91C3211yescharacteristic, complemented, Hall
16#14274263C3×C411yescharacteristic, complemented
17#14274262C3×C411yescharacteristic, complemented
18C13×#3861C3×C411yescharacteristic, complemented
19C72×C13⋊C3C3×C411yescharacteristic, complemented
20C3×#10596874—131nomaximal, Hall
21C2×ΓL(1,𝔽8)×C13⋊C3—71no—
22C2×ΓL(1,𝔽8)×C13⋊C3—72no—
23C2×ΓL(1,𝔽8)×C13⋊C3—74no—
24C2×ΓL(1,𝔽8)×C13⋊C3—71no—
25C2×C72×C13C2×C3211yescharacteristic
26#11763964—71no—
27#11763964—72no—
28#11763964—71no—
29#11763964—74no—
30#11763963—71no—
31#11763963—72no—
32#11763963—71no—
33#11763963—74no—
34ΓL(1,𝔽8)⋊Dic13—71no—
35ΓL(1,𝔽8)⋊Dic13—72no—
36ΓL(1,𝔽8)⋊Dic13—71no—
37ΓL(1,𝔽8)⋊Dic13—74no—
38#11763955—71no—
39#11763955—212no—
40#11763955—214no—
41#11763955—211no—
42C2×C3×#3861—131no—
43ΓL(1,𝔽8)×C13⋊C3—71no—
44ΓL(1,𝔽8)×C13⋊C3—72no—
45ΓL(1,𝔽8)×C13⋊C3—74no—
46ΓL(1,𝔽8)×C13⋊C3—71no—
47C72×C13C32×C411yescharacteristic, complemented, Hall
48C3×C7⋊Dic7—131no—
49#10596874—131no—
50#10596874—131no—
51#10596874—391no—
52C2×#63271—71no—
53C2×#63271—72no—
54C2×#63271—71no—
55C2×#63271—74no—
56C2×#63270—71no—
57C2×#63270—72no—
58C2×#63270—71no—
59C2×#63270—74no—
60C2×C13×ΓL(1,𝔽8)—71no—
61C2×C13×ΓL(1,𝔽8)—72no—
62C2×C13×ΓL(1,𝔽8)—71no—
63C2×C13×ΓL(1,𝔽8)—74no—
64C2×C7×C13⋊C3AGL(1,𝔽7)11yescharacteristic
65C2×C7×C13⋊C3AGL(1,𝔽7)12yes—
66C2×C7×C13⋊C3AGL(1,𝔽7)14yes—
67C2×C7×C13⋊C3AGL(1,𝔽7)11yescharacteristic
68C3×Dic13⋊C3—491noHall
69C3×#3861—131noHall
70Dic91—71no—
71Dic91—212no—
72Dic91—211no—
73Dic91—214no—
74C2×C3×C72—131no—
75C2×#3861—131no—
76C2×#3861—131no—
77C2×#3861D13⋊C311yescharacteristic
78#63271—71no—
79#63271—72no—
80#63271—71no—
81#63271—74no—
82#63270—71no—
83#63270—72no—
84#63270—71no—
85#63270—74no—
86C7×C13⋊C3ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
87C7×C13⋊C3ΓL(1,𝔽8)⋊C412yescomplemented
88C7×C13⋊C3ΓL(1,𝔽8)⋊C414yescomplemented
89C7×C13⋊C3ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
90C13×ΓL(1,𝔽8)—71no—
91C13×ΓL(1,𝔽8)—72no—
92C13×ΓL(1,𝔽8)—71no—
93C13×ΓL(1,𝔽8)—74no—
94C3×ΓL(1,𝔽8)⋊C4—912no—
95C3×ΓL(1,𝔽8)⋊C4—914no—
96C3×ΓL(1,𝔽8)⋊C4—911no—
97C3×ΓL(1,𝔽8)⋊C4—911no—
98C2×C3×C13⋊C3—491no—
99C7⋊Dic7—391noHall
100C2×C7×C13C3×AGL(1,𝔽7)11yescharacteristic
101C2×C7×C13C3×AGL(1,𝔽7)12yes—
102C2×C7×C13C3×AGL(1,𝔽7)11yescharacteristic
103C2×C7×C13C3×AGL(1,𝔽7)14yes—
104C3×Dic13—491no—
105Dic13⋊C3—491no—
106Dic13⋊C3—491no—
107Dic13⋊C3—1471no—
108C3×C72—131no—
109#3861—131no—
110#3861—131no—
111#3861Dic13⋊C311yescharacteristic, complemented
112C2×C3×ΓL(1,𝔽8)—912no—
113C2×C3×ΓL(1,𝔽8)—914no—
114C2×C3×ΓL(1,𝔽8)—911no—
115C2×C3×ΓL(1,𝔽8)—911no—
116C3×C13⋊C3—491noHall
117C2×C72C3×D13⋊C311yescharacteristic
118C7×C13C3×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
119C7×C13C3×ΓL(1,𝔽8)⋊C412yescomplemented
120C7×C13C3×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
121C7×C13C3×ΓL(1,𝔽8)⋊C414yescomplemented
122C3×Dic7—2732no—
123C3×Dic7—2734no—
124C3×Dic7—911no—
125C3×Dic7—911no—
126ΓL(1,𝔽8)⋊C4—2732no—
127ΓL(1,𝔽8)⋊C4—2731no—
128ΓL(1,𝔽8)⋊C4—2734no—
129ΓL(1,𝔽8)⋊C4—911no—
130ΓL(1,𝔽8)⋊C4—912no—
131ΓL(1,𝔽8)⋊C4—911no—
132ΓL(1,𝔽8)⋊C4—912no—
133ΓL(1,𝔽8)⋊C4—911no—
134ΓL(1,𝔽8)⋊C4—911no—
135ΓL(1,𝔽8)⋊C4—914no—
136ΓL(1,𝔽8)⋊C4—911no—
137ΓL(1,𝔽8)⋊C4—914no—
138C2×C3×C13—491no—
139C2×C13⋊C3—491no—
140C2×C13⋊C3—491no—
141C2×C13⋊C3#6442411yescharacteristic
142C3×ΓL(1,𝔽8)—912no—
143C3×ΓL(1,𝔽8)—914no—
144C3×ΓL(1,𝔽8)—911no—
145C3×ΓL(1,𝔽8)—911no—
146Dic13—1471noHall
147C72C3×Dic13⋊C311yescharacteristic, complemented, Sylow
148C2×C3×C7—132no—
149C2×C3×C7—134no—
150C2×C3×C7—131no—
151C2×C3×C7—131no—
152C2×ΓL(1,𝔽8)—72no—
153C2×ΓL(1,𝔽8)—71no—
154C2×ΓL(1,𝔽8)—74no—
155C2×ΓL(1,𝔽8)—71no—
156C2×ΓL(1,𝔽8)—912no—
157C2×ΓL(1,𝔽8)—911no—
158C2×ΓL(1,𝔽8)—911no—
159C2×ΓL(1,𝔽8)—914no—
160C2×ΓL(1,𝔽8)—912no—
161C2×ΓL(1,𝔽8)—911no—
162C2×ΓL(1,𝔽8)—911no—
163C2×ΓL(1,𝔽8)—914no—
164C3×C13—491no—
165C13⋊C3—491no—
166C13⋊C3—491no—
167C13⋊C3#1059687411yescharacteristic, complemented
168C32×C4—6371noHall
169Dic7—2732no—
170Dic7—2731no—
171Dic7—8191no—
172Dic7—8194no—
173C2×C13C3×#6442411yescharacteristic
174C3×C7—132no—
175C3×C7—134no—
176C3×C7—131no—
177C3×C7—131no—
178ΓL(1,𝔽8)—72no—
179ΓL(1,𝔽8)—71no—
180ΓL(1,𝔽8)—74no—
181ΓL(1,𝔽8)—71no—
182ΓL(1,𝔽8)—912no—
183ΓL(1,𝔽8)—911no—
184ΓL(1,𝔽8)—911no—
185ΓL(1,𝔽8)—914no—
186ΓL(1,𝔽8)—912no—
187ΓL(1,𝔽8)—911no—
188ΓL(1,𝔽8)—911no—
189ΓL(1,𝔽8)—914no—
190C2×C32—6371no—
191C2×C7ΓL(1,𝔽8)⋊(D13⋊C3)12yes—
192C2×C7ΓL(1,𝔽8)⋊(D13⋊C3)11yescharacteristic
193C2×C7ΓL(1,𝔽8)⋊(D13⋊C3)11yescharacteristic
194C2×C7ΓL(1,𝔽8)⋊(D13⋊C3)14yes—
195C13C3×#1059687411yescharacteristic, complemented, Sylow
196C3×C4—19111no—
197C3×C4—19111no—
198C3×C4—6371no—
199C3×C4—6371no—
200C32—6371noSylow
201C7ΓL(1,𝔽8)⋊(Dic13⋊C3)12yescomplemented
202C7ΓL(1,𝔽8)⋊(Dic13⋊C3)11yescharacteristic, complemented
203C7ΓL(1,𝔽8)⋊(Dic13⋊C3)11yescharacteristic, complemented
204C7ΓL(1,𝔽8)⋊(Dic13⋊C3)14yescomplemented
205C2×C3—131no—
206C2×C3—491no—
207C2×C3—6371no—
208C2×C3—6371no—
209C4—57331noSylow
210C3—131no—
211C3—491no—
212C3—6371no—
213C3—6371no—
214C2(C13⋊C3)⋊#6442411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2(C13⋊C3)⋊#64424
commutator, hypocenterC72×C13C32×C4
Fitting, socle, cosocleC2×C72×C13C2×C32
prime radicalC2×C13⋊C3×#3861C2
LCSG ▷ C72×C13
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. #21439560C3semidirect
3. #21439526C3semidirect
4. #21439523C3semidirect
5. #21439522C3semidirect
6. C13⋊C3×#3861C4semidirect
15. C7⋊Dic91C32semidirect
16. #14274263C3×C4semidirect
17. #14274262C3×C4semidirect
18. C13×#3861C3×C4semidirect
19. C72×C13⋊C3C3×C4semidirect
47. C72×C13C32×C4semidirect
86. C7×C13⋊C3ΓL(1,𝔽8)⋊C4semidirect
87. C7×C13⋊C3ΓL(1,𝔽8)⋊C4semidirect
88. C7×C13⋊C3ΓL(1,𝔽8)⋊C4semidirect
89. C7×C13⋊C3ΓL(1,𝔽8)⋊C4semidirect
111. #3861Dic13⋊C3semidirect
118. C7×C13C3×ΓL(1,𝔽8)⋊C4semidirect
119. C7×C13C3×ΓL(1,𝔽8)⋊C4semidirect
120. C7×C13C3×ΓL(1,𝔽8)⋊C4semidirect
121. C7×C13C3×ΓL(1,𝔽8)⋊C4semidirect
147. C72C3×Dic13⋊C3semidirect
167. C13⋊C3#10596874semidirect
195. C13C3×#10596874semidirect
201. C7ΓL(1,𝔽8)⋊(Dic13⋊C3)semidirect
202. C7ΓL(1,𝔽8)⋊(Dic13⋊C3)semidirect
203. C7ΓL(1,𝔽8)⋊(Dic13⋊C3)semidirect
204. C7ΓL(1,𝔽8)⋊(Dic13⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C2×C13⋊C3×#38612
#214395603
#214395263
#214395233
#214395223
ΓL(1,𝔽8)⋊(Dic13⋊C3)7
C3×#1059687413