Finite Group Databasebeta

← #13053592

Subgroups — #13053592

Order 1344.

216 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 56 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
1C3×#6471C211yesmaximal, characteristic, complemented
2#10620863C211yesmaximal, characteristic, complemented
3C2×#66561C211yesmaximal, characteristic, complemented
4#10620845C211yesmaximal, characteristic, complemented
5#10620662C211yesmaximal, characteristic, complemented
6C7×#822C211yesmaximal, characteristic, complemented
7#10620493C211yesmaximal, characteristic, complemented
8#89669—31nomaximal, Hall
9C2×D84C2211yescharacteristic
10C3×#1157C2211yescharacteristic
11C2×C3×D28C2211yescharacteristic
12C3×C7×M16C2211yescharacteristic
13#66627C2211yescharacteristic
14C7×#208C2211yescharacteristic
15C2×C7×C3⋊C8C2211yescharacteristic
16C3×D56—21no—
17C3×#1132—21no—
18#66566—21no—
19#66561—21no—
20#66561—21no—
21#66561—21no—
22#66557—21no—
23D7×C3⋊C8—21no—
24#6471—31no—
25#6471D311yescharacteristic, complemented
26C2×D56—31no—
27#6464—31no—
28C7×#108—31no—
29#6397—31no—
30#6397—31no—
31#5199—71nomaximal, Hall
32C22×D21—21no—
33C22×C3×D7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35D84—21no—
36D84—21no—
37C3×C22⋊D7—21no—
38C3×D28—21no—
39C3×D28—21no—
40C3×D28D411yescharacteristic
41C3×C4×D7—21no—
42C3×Dic14D411yescharacteristic
43C3×C7×C8—21no—
44C3⋊(C7⋊C8)—21no—
45C7×C3⋊C8—21no—
46C7×C3⋊C8D411yescharacteristic
47C7×C3⋊C8D411yescharacteristic
48#1157C2×D311yescharacteristic
49C2×D28—31no—
50C2×D28C2×D311yescharacteristic
51C7×M16—31no—
52C7×M16C2×D311yescharacteristic
53C2×C7×C8—31no—
54#1136—31no—
55D56—21no—
56D56—61no—
57D56—61no—
58D56—61no—
59#1132—21no—
60#1132—61no—
61#1131—61no—
62C8×D7—61no—
63C3×AGL(1,ℤ/8)—71no—
64#949—71no—
65#948—71no—
66C2×#213—71no—
67#833—71no—
68#823—71no—
69#822D711yescharacteristic, complemented
70C22×C3×C7#4111yescharacteristic
71C2×D21—41no—
72C2×D21—81no—
73C2×C3×D7—21no—
74C2×C3×D7—41no—
75C2×C3×D7—81no—
76C3×C4×C7C2×D411yescharacteristic
77C3×C4×C7C2×D411yescharacteristic
78C3×Dic7—21no—
79#472—211noSylow
80C22×D7—21no—
81C22×D7—61no—
82C2×C4×C7C22×D311yescharacteristic
83C22⋊D7—21no—
84D28—21no—
85D28—21no—
86D28—61no—
87D28—61no—
88D28#6711yescharacteristic
89C4×D7—21no—
90Dic14#6711yescharacteristic
91C7×C8—21no—
92C7×C8—31no—
93C7×C8—31no—
94C7×C8—61no—
95C7⋊C8—61no—
96C3×#41—71no—
97C2×C3×D4—71no—
98C2×D12—71no—
99C3×SD16—141no—
100C3×D8—141no—
101C3×M16C2×D711yescharacteristic
102#215—141no—
103#213—141no—
104#213—141no—
105#213—141no—
106#208—71no—
107#208—141no—
108#208C2×D711yescharacteristic
109C2×C3⋊C8—141no—
110C2×C3⋊C8C2×D711yescharacteristic
111C2×C3×C7—21no—
112C2×C3×C7#12111yescharacteristic
113D21—81no—
114C3×D7—41no—
115C3×D7—81no—
116AGL(1,ℤ/8)—71no—
117AGL(1,ℤ/8)—211no—
118C2×D8—211no—
119#131—211no—
120#108—31no—
121#100—211no—
122#100—211no—
123C22×C7#23711yescharacteristic
124C2×D7—21no—
125C2×D7—41no—
126C2×D7—81no—
127C2×D7—121no—
128C2×D7—241no—
129C4×C7D3×D411yescharacteristic
130C4×C7C2×#6711yescharacteristic
131Dic7—21no—
132C23×C3—141no—
133C22×D3—141no—
134C3×Q8—71no—
135C3×D4—71no—
136C3×D4—141no—
137C3×D4—141no—
138C3×D4—141no—
139C2×C3×C4—141no—
140C2×C3×C4C22×D711yescharacteristic
141D12—141no—
142D12—141no—
143C3×C8—21no—
144C3⋊C8—21no—
145C3⋊C8—141no—
146C3⋊C8D2811yescharacteristic
147C3⋊C8D2811yescharacteristic
148C3×C7#47211yescharacteristic, complemented, Hall
149#41—71no—
150C2×D4—71no—
151C2×D4—211no—
152SD16—141no—
153SD16—421no—
154D8—141no—
155D8—421no—
156D8—421no—
157D8—421no—
158M16—31no—
159M16—211no—
160M16—421no—
161M16D3×D711yescharacteristic
162C2×C8—31no—
163C2×C8—421no—
164C2×C7—21no—
165C2×C7#93911yescharacteristic
166D7—41no—
167D7—81no—
168D7—241no—
169C22×C3—141no—
170C22×C3—281no—
171C22×C3—561no—
172C22×C3#116111yescharacteristic
173C2×D3—281no—
174C2×D3—561no—
175C3×C4—141no—
176C3×C4C2×D2811yescharacteristic
177C3×C4D4×D711yescharacteristic
178C23—141no—
179C23—421no—
180Q8—71no—
181D4—71no—
182D4—141no—
183D4—141no—
184D4—141no—
185D4—421no—
186D4—421no—
187C2×C4—141no—
188C2×C4C2×D3×D711yescharacteristic
189C8—21no—
190C8—31no—
191C8—31no—
192C8—61no—
193C8—421no—
194C7#519911yescharacteristic, complemented, Sylow
195C2×C3—21no—
196C2×C3—281no—
197C2×C3—561no—
198C2×C3#645811yescharacteristic
199D3—561no—
200C22—141no—
201C22—281no—
202C22—561no—
203C22—841no—
204C22—1681no—
205C22#6667011yescharacteristic
206C4—141no—
207C4D3×D2811yescharacteristic
208C4D7×#6711yescharacteristic
209C3#8966911yescharacteristic, complemented, Sylow
210C2—21no—
211C2—281no—
212C2—561no—
213C2—1681no—
214C2#1062097111yescharacteristic
subgroupG/H
centerC2#10620971
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×M16C22
socleC2×C3×C7#121
hypercenterM16D3×D7
hypocenterC3×C7#472
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. C3×#6471C2semidirect
2. #10620863C2semidirect
3. C2×#66561C2semidirect
4. #10620845C2semidirect
5. #10620662C2semidirect
6. C7×#822C2semidirect
7. #10620493C2semidirect
25. #6471D3semidirect
69. #822D7semidirect
148. C3×C7#472semidirect
194. C7#5199semidirect
209. C3#89669semidirect

Maximal subgroups

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

subgroupindex
C3×#64712
#106208632
C2×#665612
#106208452
#106206622
C7×#8222
#106204932
#896693
#51997