Finite Group Databasebeta

← #89446⋊C3 #13051575

Subgroups — #89446⋊C3 #13051575

Order 1344.

340 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 92 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×(C2×AGL(1,𝔽7))⋊C4C211yesmaximal, characteristic
2C2×(C2×AGL(1,𝔽7))⋊C4C212yesmaximal
3C2×C4⋊(ΓL(1,𝔽8)⋊C4)C211yesmaximal, characteristic, complemented
4ΓL(1,𝔽8)×#95C211yesmaximal, characteristic, complemented
5ΓL(1,𝔽8)⋊#95C212yesmaximal, complemented
6#89446C311yesmaximal, characteristic, complemented, Hall
7C23×AGL(1,𝔽7)C2211yescharacteristic
8C22×C4×ΓL(1,𝔽8)C2212yes—
9C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
10C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
11C22×ΓL(1,𝔽8)⋊C4C2212yes—
12(C2×AGL(1,𝔽7))⋊C4—24no—
13(C2×AGL(1,𝔽7))⋊C4—22no—
14C4⋊(ΓL(1,𝔽8)⋊C4)—22no—
15C2×(C2×D7)⋊C4C2×C312yes—
16C2×(C2×D7)⋊C4C2×C311yescharacteristic
17C2×C4⋊Dic7C2×C311yescharacteristic, complemented
18C7×#95C2×C311yescharacteristic, complemented
19C7⋊#95C2×C312yescomplemented
20C3×#397—71nomaximal, Hall
21C23×ΓL(1,𝔽8)C2311yescharacteristic
22C22×AGL(1,𝔽7)—21no—
23C22×AGL(1,𝔽7)—21no—
24C22×AGL(1,𝔽7)—22no—
25C22×AGL(1,𝔽7)—22no—
26C22×AGL(1,𝔽7)D412yes—
27C2×C4×ΓL(1,𝔽8)—22no—
28C2×C4×ΓL(1,𝔽8)—22no—
29C2×C4×ΓL(1,𝔽8)—22no—
30C2×C4×ΓL(1,𝔽8)—22no—
31C2×C4×ΓL(1,𝔽8)D412yes—
32C2×ΓL(1,𝔽8)⋊C4—22no—
33C2×ΓL(1,𝔽8)⋊C4—22no—
34C2×ΓL(1,𝔽8)⋊C4—21no—
35C2×ΓL(1,𝔽8)⋊C4—22no—
36C2×ΓL(1,𝔽8)⋊C4—22no—
37C23×D7C22×C311yescharacteristic
38C22×C4×C7C22×C312yes—
39C22×C4×C7C22×C311yescharacteristic
40C22×Dic7C22×C311yescharacteristic
41C22×Dic7C22×C312yes—
42(C2×D7)⋊C4—24no—
43(C2×D7)⋊C4—22no—
44C4⋊Dic7—22no—
45C2×C3×C4⋊C4—71no—
46C2×C3×C22⋊C4—71no—
47C2×C3×C22⋊C4—72no—
48C3×#95—72no—
49C3×#95—71no—
50C22×ΓL(1,𝔽8)C2×D411yescharacteristic
51C22×ΓL(1,𝔽8)C2×D411yescharacteristic
52C22×ΓL(1,𝔽8)#4111yescharacteristic
53C22×ΓL(1,𝔽8)#4112yes—
54C22×ΓL(1,𝔽8)#4112yes—
55C2×AGL(1,𝔽7)—22no—
56C2×AGL(1,𝔽7)—24no—
57C2×AGL(1,𝔽7)—41no—
58C2×AGL(1,𝔽7)—41no—
59C2×AGL(1,𝔽7)—42no—
60C4×ΓL(1,𝔽8)—22no—
61C4×ΓL(1,𝔽8)—42no—
62ΓL(1,𝔽8)⋊C4—42no—
63ΓL(1,𝔽8)⋊C4—41no—
64#397—71noSylow
65C23×C7C23×C311yescharacteristic
66C22×D7—21no—
67C22×D7—22no—
68C22×D7—22no—
69C22×D7—21no—
70C22×D7C3×D412yes—
71C2×C4×C7—22no—
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7C3×D412yes—
76C2×Dic7—22no—
77C2×Dic7—21no—
78C2×Dic7—22no—
79C2×Dic7—22no—
80C2×Dic7—22no—
81C24×C3—71no—
82C22×C3×C4—71no—
83C22×C3×C4—72no—
84C22×C3×C4—71no—
85C22×C3×C4—72no—
86C3×C4⋊C4—142no—
87C3×C22⋊C4—142no—
88C3×C22⋊C4—144no—
89C2×ΓL(1,𝔽8)#12111yescharacteristic
90C2×ΓL(1,𝔽8)#12311yescharacteristic
91C2×ΓL(1,𝔽8)#12312yes—
92C2×ΓL(1,𝔽8)#12411yescharacteristic
93C2×ΓL(1,𝔽8)#12612yes—
94AGL(1,𝔽7)—42no—
95C2×C4⋊C4—71no—
96C2×C22⋊C4—72no—
97C2×C22⋊C4—71no—
98#95—72no—
99#95AGL(1,𝔽7)11yescharacteristic, complemented
100C22×C7C2×C3×D411yescharacteristic
101C22×C7C2×C3×D411yescharacteristic
102C22×C7C3×#4112yes—
103C22×C7C3×#4112yes—
104C22×C7C3×#4111yescharacteristic
105C2×D7—22no—
106C2×D7—24no—
107C2×D7—41no—
108C2×D7—41no—
109C2×D7—42no—
110C4×C7—22no—
111C4×C7—42no—
112Dic7—42no—
113Dic7—41no—
114C23×C3—72no—
115C23×C3—71no—
116C23×C3—141no—
117C23×C3—142no—
118C23×C3—142no—
119C23×C3—141no—
120C2×C3×C4—72no—
121C2×C3×C4—142no—
122C2×C3×C4—142no—
123C2×C3×C4—142no—
124C2×C3×C4—142no—
125C2×C3×C4—141no—
126C2×C3×C4—142no—
127C2×C3×C4—142no—
128C2×C3×C4—142no—
129C2×C3×C4—142no—
130ΓL(1,𝔽8)#39711yescharacteristic, complemented, Hall
131C24—71no—
132C22×C4—71no—
133C22×C4—72no—
134C22×C4C2×AGL(1,𝔽7)12yes—
135C22×C4C2×AGL(1,𝔽7)11yescharacteristic
136C4⋊C4—142no—
137C22⋊C4—144no—
138C22⋊C4—142no—
139C2×C7C3×#12111yescharacteristic
140C2×C7C3×#12312yes—
141C2×C7C3×#12311yescharacteristic
142C2×C7C3×#12411yescharacteristic
143C2×C7C3×#12612yes—
144D7—42no—
145C22×C3—71no—
146C22×C3—72no—
147C22×C3—71no—
148C22×C3—72no—
149C22×C3—71no—
150C22×C3—144no—
151C22×C3—142no—
152C22×C3—281no—
153C22×C3—281no—
154C22×C3—282no—
155C3×C4—142no—
156C3×C4—282no—
157C3×C4—281no—
158C3×C4—282no—
159C23—72no—
160C23—141no—
161C23—142no—
162C23—142no—
163C23—141no—
164C23C22×AGL(1,𝔽7)11yescharacteristic
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—141no—
170C2×C4—142no—
171C2×C4—142no—
172C2×C4—142no—
173C2×C4—142no—
174C2×C4C4⋊AGL(1,𝔽7)12yes—
175C7C3×#39711yescharacteristic, complemented, Sylow
176C2×C3—71no—
177C2×C3—72no—
178C2×C3—71no—
179C2×C3—71no—
180C2×C3—72no—
181C2×C3—282no—
182C22—144no—
183C22—142no—
184C22—281no—
185C22—281no—
186C22—282no—
187C22C2×C4⋊AGL(1,𝔽7)11yescharacteristic
188C22#1157⋊C312yes—
189C22D4×AGL(1,𝔽7)11yescharacteristic
190C22#1159⋊C312yes—
191C22#6665911yescharacteristic
192C4—22no—
193C4—42no—
194C4—281no—
195C4—282no—
196C3—71noSylow
197C2—282no—
198C2#6438⋊C311yescharacteristic
199C2#6446⋊C312yes—
200C2#6449⋊C311yescharacteristic
201C2#6458⋊C311yescharacteristic
202C2#6461⋊C312yes—
subgroupG/H
center, FrattiniC23C22×AGL(1,𝔽7)
commutator, socle, cosocleC23×C7C23×C3
FittingC7×#95C2×C3
prime radicalC23×AGL(1,𝔽7)C22
hypercenter#95AGL(1,𝔽7)
hypocenterC7C3×#397
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊(ΓL(1,𝔽8)⋊C4)C2semidirect
4. ΓL(1,𝔽8)×#95C2semidirect
5. ΓL(1,𝔽8)⋊#95C2semidirect
6. #89446C3semidirect
17. C2×C4⋊Dic7C2×C3semidirect
18. C7×#95C2×C3semidirect
19. C7⋊#95C2×C3semidirect
99. #95AGL(1,𝔽7)semidirect
130. ΓL(1,𝔽8)#397semidirect
175. C7C3×#397semidirect

Maximal subgroups

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

subgroupindex
C2×(C2×AGL(1,𝔽7))⋊C42
C2×C4⋊(ΓL(1,𝔽8)⋊C4)2
ΓL(1,𝔽8)×#952
ΓL(1,𝔽8)⋊#952
#894463
C3×#3977