Finite Group Databasebeta

← #13056431

Subgroups — #13056431

Order 1344.

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

normal subgroups only

This group has 216 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#10621427C211yesmaximal, characteristic
2C2×#66635C211yesmaximal, characteristic
3C2×Dic84C211yesmaximal, characteristic
4#10621370C211yesmaximal, characteristic
5C3×C7×#103C211yesmaximal, characteristic
6#10620653C211yesmaximal, characteristic
7#10620649C211yesmaximal, characteristic
8#89565—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×Dic42C2211yescharacteristic
12C2×C3×C7×C8C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14#66635—22no—
15#66629—21no—
16#66629—21no—
17#66629—21no—
18C4×Dic21—21no—
19C4×Dic21C2211yescharacteristic
20C2×C3⋊(C7⋊C8)C2211yescharacteristic
21Dic84—22no—
22#6507—31no—
23C2×#1144—31no—
24C2×Dic28—31no—
25#6450—31no—
26C7×#103D311yescharacteristic
27Dic7⋊C8—31no—
28#6384—31no—
29#4795—71nomaximal, Hall
30C3×C7×Q8—21no—
31C3×C7×Q8—21no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36C2×Dic21—21no—
37C2×Dic21D411yescharacteristic
38C2×Dic21D411yescharacteristic
39Dic42—21no—
40Dic42—21no—
41Dic42—21no—
42Dic42D412yes—
43C3×C7×C8—21no—
44C3⋊(C7⋊C8)—21no—
45C2×C7×Q8C2×D311yescharacteristic
46C2×Dic14—31no—
47C2×Dic14—31no—
48C2×C7×C8C2×D311yescharacteristic
49C7×C4⋊C4C2×D311yescharacteristic
50#1144—62no—
51Dic7⋊C4—61no—
52Dic7⋊C4—61no—
53Dic7⋊C4—61no—
54C4×Dic7—31no—
55C4×Dic7—61no—
56C2×C7⋊C8—31no—
57Dic28—62no—
58#944—71no—
59C2×#216—71no—
60C2×Dic12—71no—
61#887—71no—
62C3×#103D711yescharacteristic
63Dic3⋊C8—71no—
64#810—71no—
65C22×C3×C7C2×D411yescharacteristic
66C3×C4×C7—41no—
67C3×C4×C7—41no—
68C3×C4×C7C2×D411yescharacteristic
69C3×C4×C7#4111yescharacteristic
70Dic21—22no—
71Dic21—21no—
72Dic21—41no—
73Dic21Q1612yescomplemented
74#462—211noSylow
75C7×Q8—21no—
76C7×Q8—21no—
77C2×C4×C7—21no—
78C2×C4×C7—21no—
79C2×C4×C7C22×D311yescharacteristic
80C2×Dic7—31no—
81C2×Dic7—31no—
82C2×Dic7—61no—
83C2×Dic7—61no—
84Dic14—32no—
85Dic14—61no—
86Dic14—61no—
87Dic14—61no—
88C7×C8—21no—
89C7⋊C8—61no—
90C2×C3×Q8C2×D711yescharacteristic
91C2×Dic6—71no—
92C2×Dic6—71no—
93C2×C3×C8C2×D711yescharacteristic
94C3×C4⋊C4C2×D711yescharacteristic
95#216—142no—
96Dic3⋊C4—141no—
97Dic3⋊C4—141no—
98Dic3⋊C4—141no—
99C4×Dic3—71no—
100C4×Dic3—141no—
101C2×C3⋊C8—71no—
102Dic12—142no—
103C2×C3×C7#12111yescharacteristic
104C2×C3×C7C2×Q1611yescharacteristic
105C2×C3×C7#13711yescharacteristic
106C2×Q16—211no—
107C2×Q16—211no—
108C4⋊Q8—211no—
109C4×Q8—211no—
110C4⋊C8—211no—
111#103—211no—
112#103D2111yescharacteristic
113C22×C7D3×D411yescharacteristic
114C4×C7—41no—
115C4×C7—41no—
116C4×C7#23511yescharacteristic
117C4×C7D3×D411yescharacteristic
118Dic7—32no—
119Dic7—62no—
120Dic7—61no—
121Dic7—121no—
122C3×Q8—21no—
123C3×Q8—21no—
124C2×C3×C4—21no—
125C2×C3×C4—21no—
126C2×C3×C4C22×D711yescharacteristic
127C2×Dic3—71no—
128C2×Dic3—71no—
129C2×Dic3—141no—
130C2×Dic3—141no—
131Dic6—72no—
132Dic6—141no—
133Dic6—141no—
134Dic6—141no—
135C3×C8—21no—
136C3⋊C8—141no—
137C3×C7#46211yescharacteristic, complemented, Hall
138C2×Q8—211no—
139C2×Q8—211no—
140C2×Q8C2×D2111yescharacteristic
141Q16—422no—
142Q16—422no—
143C2×C8—211no—
144C2×C8C2×D2111yescharacteristic
145C4⋊C4—421no—
146C4⋊C4—421no—
147C4⋊C4—421no—
148C4⋊C4C2×D2111yescharacteristic
149C42—211no—
150C42—421no—
151C2×C7#88411yescharacteristic
152C2×C7#91511yescharacteristic
153C2×C7D3×Q1611yescharacteristic
154C22×C3D4×D711yescharacteristic
155C3×C4—41no—
156C3×C4—41no—
157C3×C4#115711yescharacteristic
158C3×C4D4×D711yescharacteristic
159Dic3—72no—
160Dic3—142no—
161Dic3—141no—
162Dic3—281no—
163Q8—21no—
164Q8—21no—
165Q8—212no—
166Q8—421no—
167Q8—421no—
168Q8—421no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—211no—
172C2×C4—211no—
173C2×C4—421no—
174C2×C4—421no—
175C2×C4C22×D2111yescharacteristic
176C8—21no—
177C8—421no—
178C7#479511yescharacteristic, complemented, Sylow
179C2×C3#644711yescharacteristic
180C2×C3#647811yescharacteristic
181C2×C3D7×Q1611yescharacteristic
182C22D4×D2111yescharacteristic
183C4—41no—
184C4—41no—
185C4—212no—
186C4—422no—
187C4—421no—
188C4—841no—
189C4#6672811yescharacteristic
190C4D4×D2111yescharacteristic
191C3#8956511yescharacteristic, complemented, Sylow
192C2#1062136711yescharacteristic
193C2#1062139811yescharacteristic
194C2Q16×D2111yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#103C2
socle, prime radicalC22×C3×C7C2×D4
hypercenter#103D21
hypocenterC3×C7#462
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
73. Dic21Q16semidirect
137. C3×C7#462semidirect
178. C7#4795semidirect
191. C3#89565semidirect

Maximal subgroups

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

subgroupindex
#106214272
C2×#666352
C2×Dic842
#106213702
C3×C7×#1032
#106206532
#106206492
#895653
#47957