Finite Group Databasebeta

← #13056414

Subgroups — #13056414

Order 1344.

248 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 65 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
1#10621421C211yesmaximal, characteristic
2C2×#66633C211yesmaximal, characteristic, complemented
3C2×C8×D21C211yesmaximal, characteristic, complemented
4#10621380C211yesmaximal, characteristic, complemented
5C3×C7×#102C211yesmaximal, characteristic, complemented
6#10620655C211yesmaximal, characteristic, complemented
7#10620647C211yesmaximal, characteristic, complemented
8#89550—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×#4268—21no—
11C2×C4×D21C2211yescharacteristic
12C2×Dic42C2211yescharacteristic
13C2×C3×C7×C8C2211yescharacteristic
14C3×C7×C4⋊C4C2211yescharacteristic
15#66636—21no—
16#66633—22no—
17#66631—21no—
18C4⋊Dic21—21no—
19C4⋊Dic21C2211yescharacteristic
20C2×C3⋊(C7⋊C8)C2211yescharacteristic
21C8×D21—22no—
22#6501—31no—
23C2×#1142—31no—
24C2×C8×D7—31no—
25#6460—31no—
26C7×#102D311yescharacteristic, complemented
27#6390—31no—
28#6382—31no—
29#4778—71nomaximal, Hall
30C23×C3×C7—21no—
31C22×D21D411yescharacteristic
32C3×C7×D4—21no—
33C3×C7×D4—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36#4268—42no—
37C2×Dic21—21no—
38C2×Dic21—21no—
39C2×Dic21D411yescharacteristic
40C4×D21—21no—
41C4×D21—21no—
42Dic42—21no—
43Dic42—21no—
44C3×C7×C8—21no—
45C3⋊(C7⋊C8)D412yescomplemented
46C2×C7×D4C2×D311yescharacteristic
47C2×C22⋊D7—61no—
48C2×C4×D7—31no—
49C2×Dic14—31no—
50C2×C7×C8C2×D311yescharacteristic
51C7×C4⋊C4C2×D311yescharacteristic
52C22⋊Dic7—61no—
53#1142—62no—
54(C2×D7)⋊C4—61no—
55C4⋊Dic7—31no—
56C4⋊Dic7—61no—
57C2×C7⋊C8—31no—
58C8×D7—62no—
59#938—71no—
60C2×#214—71no—
61C2×D3×C8—71no—
62#897—71no—
63C3×#102D711yescharacteristic, complemented
64#816—71no—
65#808—71no—
66C22×C3×C7—41no—
67C22×C3×C7—41no—
68C22×C3×C7—41no—
69C22×C3×C7C2×D411yescharacteristic
70C2×D21—21no—
71C2×D21—21no—
72C2×D21SD1612yescomplemented
73C3×C4×C7—41no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7#4111yescharacteristic
76Dic21—21no—
77Dic21—41no—
78Dic21—41no—
79#465—211noSylow
80C23×C7—21no—
81C22×D7—31no—
82C7×D4—21no—
83C7×D4—21no—
84C2×C4×C7—21no—
85C2×C4×C7C22×D311yescharacteristic
86C22⋊D7—122no—
87C2×Dic7—31no—
88C2×Dic7—61no—
89C2×Dic7—61no—
90C4×D7—61no—
91C4×D7—61no—
92Dic14—61no—
93Dic14—61no—
94C7×C8—21no—
95C7⋊C8—32no—
96C2×C3×D4C2×D711yescharacteristic
97C2×#67—141no—
98C2×C4×D3—71no—
99C2×Dic6—71no—
100C2×C3×C8C2×D711yescharacteristic
101C3×C4⋊C4C2×D711yescharacteristic
102#217—141no—
103#214—142no—
104(C2×D3)⋊C4—141no—
105C4⋊Dic3—71no—
106C4⋊Dic3—141no—
107C2×C3⋊C8—71no—
108D3×C8—142no—
109C2×C3×C7—81no—
110C2×C3×C7#12111yescharacteristic
111C2×C3×C7C2×SD1611yescharacteristic
112C2×C3×C7#13511yescharacteristic
113D21—22no—
114C2×SD16—211no—
115C22×C8—211no—
116C22⋊Q8—211no—
117#121—211no—
118#106—211no—
119#103—211no—
120#102D2111yescharacteristic, complemented
121C22×C7—41no—
122C22×C7—41no—
123C22×C7—41no—
124C22×C7D3×D411yescharacteristic
125C2×D7—32no—
126C2×D7—61no—
127C2×D7—61no—
128C4×C7—41no—
129C4×C7#23511yescharacteristic
130C4×C7D3×D411yescharacteristic
131Dic7—61no—
132Dic7—121no—
133Dic7—121no—
134C23×C3—21no—
135C22×D3—71no—
136C3×D4—21no—
137C3×D4—21no—
138C2×C3×C4—21no—
139C2×C3×C4C22×D711yescharacteristic
140#67—282no—
141C2×Dic3—71no—
142C2×Dic3—141no—
143C2×Dic3—141no—
144C4×D3—141no—
145C4×D3—141no—
146Dic6—141no—
147Dic6—141no—
148C3×C8—21no—
149C3⋊C8—72no—
150C3×C7#46511yescharacteristic, complemented, Hall
151C2×Q8—211no—
152C2×D4—421no—
153C2×D4C2×D2111yescharacteristic
154C22×C4—211no—
155SD16—422no—
156C2×C8—211no—
157C2×C8—422no—
158C2×C8C2×D2111yescharacteristic
159C4⋊C4—211no—
160C4⋊C4—421no—
161C4⋊C4C2×D2111yescharacteristic
162C22⋊C4—421no—
163C22⋊C4—421no—
164C2×C7—81no—
165C2×C7#88411yescharacteristic
166C2×C7#91211yescharacteristic
167C2×C7D3×SD1611yescharacteristic
168D7—62no—
169C22×C3—41no—
170C22×C3—41no—
171C22×C3—41no—
172C22×C3D4×D711yescharacteristic
173C2×D3—72no—
174C2×D3—141no—
175C2×D3—141no—
176C3×C4—41no—
177C3×C4#115711yescharacteristic
178C3×C4D4×D711yescharacteristic
179Dic3—141no—
180Dic3—281no—
181Dic3—281no—
182C23—21no—
183C23—211no—
184Q8—421no—
185Q8—421no—
186D4—21no—
187D4—21no—
188D4—842no—
189C2×C4—21no—
190C2×C4—211no—
191C2×C4—421no—
192C2×C4—421no—
193C2×C4—421no—
194C2×C4—421no—
195C2×C4C22×D2111yescharacteristic
196C8—21no—
197C8—212no—
198C7#477811yescharacteristic, complemented, Sylow
199C2×C3—81no—
200C2×C3#644711yescharacteristic
201C2×C3#647511yescharacteristic
202C2×C3D7×SD1611yescharacteristic
203D3—142no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—212no—
208C22—421no—
209C22—421no—
210C22D4×D2111yescharacteristic
211C4—41no—
212C4—421no—
213C4—841no—
214C4—841no—
215C4#6672811yescharacteristic
216C4D4×D2111yescharacteristic
217C3#8955011yescharacteristic, complemented, Sylow
218C2—81no—
219C2—422no—
220C2#1062136711yescharacteristic
221C2#1062139511yescharacteristic
222C2SD16×D2111yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#102C2
socleC22×C3×C7C2×D4
prime radical#10621421C2
hypercenter#102D21
hypocenterC3×C7#465
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×#66633C2semidirect
3. C2×C8×D21C2semidirect
4. #10621380C2semidirect
5. C3×C7×#102C2semidirect
6. #10620655C2semidirect
7. #10620647C2semidirect
26. C7×#102D3semidirect
45. C3⋊(C7⋊C8)D4semidirect
63. C3×#102D7semidirect
72. C2×D21SD16semidirect
120. #102D21semidirect
150. C3×C7#465semidirect
198. C7#4778semidirect
217. C3#89550semidirect

Maximal subgroups

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

subgroupindex
#106214212
C2×#666332
C2×C8×D212
#106213802
C3×C7×#1022
#106206552
#106206472
#895503
#47787