Finite Group Databasebeta

← #13054542

Subgroups — #13054542

Order 1344.

208 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 54 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×#66633C211yesmaximal, characteristic, complemented
2C3×#6495C211yesmaximal, characteristic
3#10620875C211yesmaximal, characteristic, complemented
4#10620846C211yesmaximal, characteristic, complemented
5C7×#833C211yesmaximal, characteristic, complemented
6#10620527C211yesmaximal, characteristic, complemented
7#10620498C211yesmaximal, characteristic, complemented
8#89520—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C3×#1157C2211yescharacteristic
12#66633—21no—
13#66633—21no—
14C2×C3⋊(C7⋊C8)C2211yescharacteristic
15C7×#208C2211yescharacteristic
16C7×#208C2211yescharacteristic
17C3×#1142—21no—
18C3×#1141—21no—
19C3×#1136C2211yescharacteristic
20#66570—21no—
21#66565—21no—
22#66557—21no—
23D7×C3⋊C8—21no—
24C2×#1142—31no—
25#6495D311yescharacteristic
26#6472—31no—
27#6470—31no—
28C7×#100—31no—
29#6398—31no—
30#6396—31no—
31#5200—71nomaximal, Hall
32C23×C3×C7—21no—
33C3×C7×D4—21no—
34C3×C7×D4—21no—
35C2×C3×C4×C7C2311yescharacteristic
36C2×Dic21—21no—
37Dic42—21no—
38Dic42—21no—
39C3×C22⋊D7—21no—
40C3×D28D411yescharacteristic
41C3×C4×D7—21no—
42C3×Dic14D411yescharacteristic
43C3⋊(C7⋊C8)D411yescharacteristic
44C3⋊(C7⋊C8)D411yescharacteristic
45C3×C7⋊C8—21no—
46C7×C3⋊C8—21no—
47C7×C3⋊C8—21no—
48C2×C7×D4C2×D311yescharacteristic
49#1157C2×D311yescharacteristic
50C2×Dic14—31no—
51C7×M16—31no—
52C7×M16—31no—
53#1142—21no—
54#1142—61no—
55#1142—61no—
56#1141—21no—
57#1136C2×D311yescharacteristic
58C2×C7⋊C8—31no—
59Dic28—61no—
60#1132—61no—
61#1131—61no—
62C8×D7—61no—
63C3×AGL(1,ℤ/8)—71no—
64#951—71no—
65#948—71no—
66C2×#214—71no—
67#833D711yescharacteristic, complemented
68#824—71no—
69#822—71no—
70C22×C3×C7—41no—
71C22×C3×C7—81no—
72C22×C3×C7#4111yescharacteristic
73C2×C3×D7—21no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7C2×D411yescharacteristic
76Dic21—41no—
77C3×Dic7—21no—
78#471—211noSylow
79C23×C7—21no—
80C7×D4—21no—
81C7×D4—21no—
82C2×C4×C7C22×D311yescharacteristic
83C22⋊D7—21no—
84C2×Dic7—61no—
85D28#6711yescharacteristic
86C4×D7—21no—
87Dic14—61no—
88Dic14—61no—
89Dic14#6711yescharacteristic
90C7×C8—61no—
91C7×C8—61no—
92C7⋊C8—21no—
93C7⋊C8—31no—
94C7⋊C8—31no—
95C3×#41—71no—
96C2×C3×D4C2×D711yescharacteristic
97C2×Dic6—71no—
98C3×SD16—141no—
99C3×D8—141no—
100C3×M16—71no—
101#216—141no—
102#214—141no—
103#214—141no—
104#214—141no—
105#208—141no—
106#208C2×D711yescharacteristic
107#208C2×D711yescharacteristic
108C2×C3⋊C8—71no—
109C2×C3⋊C8—141no—
110C2×C3×C7—21no—
111C2×C3×C7—81no—
112C2×C3×C7#12111yescharacteristic
113C3×D7—41no—
114#137—211no—
115AGL(1,ℤ/8)—71no—
116C2×SD16—211no—
117#131—211no—
118#108—211no—
119#101—211no—
120#100—31no—
121C22×C7—41no—
122C22×C7—81no—
123C22×C7#23711yescharacteristic
124C2×D7—21no—
125C4×C7D3×D411yescharacteristic
126C4×C7C2×#6711yescharacteristic
127Dic7—21no—
128Dic7—121no—
129C23×C3—21no—
130C3×Q8—71no—
131C3×D4—21no—
132C3×D4—21no—
133C3×D4—71no—
134C3×D4—141no—
135C2×C3×C4—141no—
136C2×C3×C4C22×D711yescharacteristic
137C2×Dic3—141no—
138Dic6—141no—
139Dic6—141no—
140C3×C8—141no—
141C3⋊C8—21no—
142C3⋊C8—21no—
143C3⋊C8—71no—
144C3⋊C8—71no—
145C3×C7#47111yescharacteristic, complemented, Hall
146#41—71no—
147C2×Q8—211no—
148C2×D4D3×D711yescharacteristic
149Q16—421no—
150SD16—141no—
151SD16—421no—
152SD16—421no—
153SD16—421no—
154D8—141no—
155M16—31no—
156M16—31no—
157M16—71no—
158M16—421no—
159C2×C8—211no—
160C2×C8—421no—
161C2×C7—21no—
162C2×C7—81no—
163C2×C7#93911yescharacteristic
164D7—41no—
165C22×C3—41no—
166C22×C3—81no—
167C22×C3—141no—
168C22×C3#115711yescharacteristic
169C3×C4—141no—
170C3×C4D4×D711yescharacteristic
171C3×C4D4×D711yescharacteristic
172Dic3—281no—
173C23—21no—
174Q8—71no—
175Q8—421no—
176Q8—421no—
177D4—21no—
178D4—21no—
179D4—71no—
180D4—141no—
181C2×C4—141no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—61no—
185C8—61no—
186C8—141no—
187C8—211no—
188C8—211no—
189C7#520011yescharacteristic, complemented, Sylow
190C2×C3—21no—
191C2×C3—81no—
192C2×C3—281no—
193C2×C3#644711yescharacteristic
194C22—41no—
195C22—81no—
196C22—141no—
197C22#6668311yescharacteristic
198C4—141no—
199C4—841no—
200C4D7×#6711yescharacteristic
201C4#6669411yescharacteristic
202C3#8952011yescharacteristic, complemented, Sylow
203C2—21no—
204C2—81no—
205C2—281no—
206C2#1062111211yescharacteristic
subgroupG/H
centerC2#10621112
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7#121
prime radicalC3×#6495C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#471
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#66633C2semidirect
3. #10620875C2semidirect
4. #10620846C2semidirect
5. C7×#833C2semidirect
6. #10620527C2semidirect
7. #10620498C2semidirect
67. #833D7semidirect
145. C3×C7#471semidirect
189. C7#5200semidirect
202. C3#89520semidirect

Maximal subgroups

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

subgroupindex
C2×#666332
C3×#64952
#106208752
#106208462
C7×#8332
#106205272
#106204982
#895203
#52007