Finite Group Databasebeta

← #13054701

Subgroups — #13054701

Order 1344.

216 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 68 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
1Q8×Dic21C211yesmaximal, characteristic
2C2×C7×#216C211yesmaximal, characteristic
3C2×C3×#1144C211yesmaximal, characteristic
4#10620947C211yesmaximal, characteristic
5#10620518C211yesmaximal, characteristic
6#10620508C211yesmaximal, characteristic
7#10620506C211yesmaximal, characteristic
8#89954—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12C4⋊Dic21—21no—
13C4⋊Dic21C2211yescharacteristic
14C4×Dic21—21no—
15C4×Dic21C2211yescharacteristic
16C7×#216—22no—
17C2×C7×C3⋊C8C2211yescharacteristic
18C3×#1144—22no—
19C2×C3×C7⋊C8C2211yescharacteristic
20#66577—21no—
21#66577—21no—
22C2×C7×Q16—31no—
23Q8×Dic7—31no—
24#6507—31no—
25C2×#1144D311yescharacteristic
26#6409—31no—
27#6390—31no—
28Dic7⋊C8—31no—
29#5182—71nomaximal, Hall
30C3×C7×Q8—21no—
31C3×C7×Q8D412yes—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7C2311yescharacteristic
34C2×Dic21—21no—
35C2×Dic21D411yescharacteristic
36C2×Dic21D411yescharacteristic
37C2×C7×Dic3—21no—
38C7×Dic6—21no—
39C7×Dic6—21no—
40C2×C3×Dic7—21no—
41C3×Dic14—21no—
42C3×Dic14—21no—
43C3×C7⋊C8—21no—
44C7×C3⋊C8—21no—
45C2×C7×Q8—31no—
46C2×C7×Q8C2×D311yescharacteristic
47C2×Dic14C2×D311yescharacteristic
48C7×Q16—62no—
49C2×C7×C8—31no—
50#1144—22no—
51C4⋊Dic7—31no—
52C4⋊Dic7—61no—
53Dic7⋊C4—61no—
54Dic7⋊C4—61no—
55C4×Dic7—31no—
56C4×Dic7—61no—
57C2×C7⋊C8C2×D311yescharacteristic
58C2×C3×Q16—71no—
59Q8×Dic3—71no—
60#944—71no—
61C2×#216D711yescharacteristic
62#835—71no—
63#816—71no—
64Dic3⋊C8—71no—
65C22×C3×C7C2×D411yescharacteristic
66C3×C4×C7—22no—
67C3×C4×C7C2×D411yescharacteristic
68C3×C4×C7#4111yescharacteristic
69Dic21—21no—
70Dic21—41no—
71Dic21Q1612yescomplemented
72C3×Dic7—41no—
73C7×Dic3—41no—
74#462—211noSylow
75C7×Q8—21no—
76C7×Q8—61no—
77C7×Q8—61no—
78C7×Q8#6712yes—
79C2×C4×C7—21no—
80C2×C4×C7—61no—
81C2×C4×C7C22×D311yescharacteristic
82C2×Dic7—21no—
83C2×Dic7—31no—
84C2×Dic7—31no—
85C2×Dic7—61no—
86Dic14—21no—
87Dic14—21no—
88C7×C8—61no—
89C7⋊C8—21no—
90C2×C3×Q8—71no—
91C2×C3×Q8C2×D711yescharacteristic
92C2×Dic6C2×D711yescharacteristic
93C3×Q16—142no—
94C2×C3×C8—71no—
95#216—22no—
96C4⋊Dic3—71no—
97C4⋊Dic3—141no—
98Dic3⋊C4—141no—
99Dic3⋊C4—141no—
100C4×Dic3—71no—
101C4×Dic3—141no—
102C2×C3⋊C8C2×D711yescharacteristic
103C2×C3×C7#12111yescharacteristic
104C2×C3×C7C2×Q1611yescharacteristic
105C2×C3×C7#13711yescharacteristic
106C2×Q16—31no—
107C2×Q16—71no—
108C4⋊Q8—211no—
109C4×Q8—211no—
110C4⋊C8—211no—
111#103—211no—
112#103—211no—
113C22×C7D3×D411yescharacteristic
114C4×C7—22no—
115C4×C7—121no—
116C4×C7#23711yescharacteristic
117C4×C7C2×#6711yescharacteristic
118Dic7—32no—
119Dic7—41no—
120Dic7—61no—
121Dic7—121no—
122C3×Q8—21no—
123C3×Q8—141no—
124C3×Q8—141no—
125C3×Q8C22⋊D712yes—
126C2×C3×C4—21no—
127C2×C3×C4—141no—
128C2×C3×C4C22×D711yescharacteristic
129C2×Dic3—21no—
130C2×Dic3—71no—
131C2×Dic3—71no—
132C2×Dic3—141no—
133Dic6—21no—
134Dic6—21no—
135C3×C8—141no—
136C3⋊C8—21no—
137C3×C7#46211yescharacteristic, complemented, Hall
138C2×Q8—31no—
139C2×Q8—71no—
140C2×Q8D3×D711yescharacteristic
141Q16—62no—
142Q16—142no—
143C2×C8—31no—
144C2×C8—71no—
145C4⋊C4—211no—
146C4⋊C4—421no—
147C4⋊C4—421no—
148C4⋊C4—421no—
149C42—211no—
150C42—421no—
151C2×C7D3×Q1611yescharacteristic
152C2×C7#91811yescharacteristic
153C2×C7#93911yescharacteristic
154C22×C3D4×D711yescharacteristic
155C3×C4—22no—
156C3×C4—281no—
157C3×C4#115911yescharacteristic
158C3×C4C2×C22⋊D711yescharacteristic
159Dic3—41no—
160Dic3—72no—
161Dic3—141no—
162Dic3—281no—
163Q8—21no—
164Q8—61no—
165Q8—61no—
166Q8—141no—
167Q8—141no—
168Q8#424512yes—
169C2×C4—21no—
170C2×C4—61no—
171C2×C4—141no—
172C2×C4—211no—
173C2×C4—211no—
174C2×C4—421no—
175C2×C4C2×D3×D711yescharacteristic
176C8—61no—
177C8—141no—
178C7#518211yescharacteristic, complemented, Sylow
179C2×C3D7×Q1611yescharacteristic
180C2×C3#648111yescharacteristic
181C2×C3#650211yescharacteristic
182C22#6669411yescharacteristic
183C4—22no—
184C4—121no—
185C4—212no—
186C4—281no—
187C4—421no—
188C4—841no—
189C4#6668411yescharacteristic
190C4C2×#424511yescharacteristic
191C3#8995411yescharacteristic, complemented, Sylow
192C2#1062107311yescharacteristic
193C2#1062107411yescharacteristic
194C2#1062113311yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socle, prime radicalC22×C3×C7C2×D4
hypercenterC2×Q8D3×D7
hypocenterC3×C7#462
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
71. Dic21Q16semidirect
137. C3×C7#462semidirect
178. C7#5182semidirect
191. C3#89954semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic212
C2×C7×#2162
C2×C3×#11442
#106209472
#106205182
#106205082
#106205062
#899543
#51827