Finite Group Databasebeta

← #31884967

Subgroups — #31884967

Order 18176.

252 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 41 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#22589439C211yesmaximal, characteristic
2#22586274C211yesmaximal, characteristic
3C71×#1394C211yesmaximal, characteristic
4C22×#11765855—21no—
5C2×#14556128C411yescharacteristic
6C2×#14556051—21no—
7#17985754C411yescharacteristic
8#17985725—21no—
9C2×#101×C71—21no—
10#391×C71C2211yescharacteristic
11C22×Dic142—21no—
12C2×#11765855—42no—
13C2×#11765855—42no—
14C2×#11765855—41no—
15C2×#11765855—41no—
16C22×C71⋊C8—21no—
17#14556128—42no—
18C2×C4⋊Dic71—41no—
19#14556051—42no—
20C22⋊(C71⋊C8)—41no—
21C22×Q8×C71D411yescharacteristic
22C2×M16×C71—41no—
23C2×C4⋊C4×C71D411yescharacteristic
24C2×C42×C71C2×C411yescharacteristic
25#101×C71—22no—
26#101×C71—41no—
27#95×C71—41no—
28C2×Dic142—22no—
29C2×Dic142—41no—
30C2×Dic142—41no—
31C2×Dic142—41no—
32C2×Dic142—41no—
33C2×Dic142—41no—
34C22×Dic71—41no—
35#11765855—44no—
36#11765855—44no—
37C2×C71⋊C8—22no—
38C2×C71⋊C8—41no—
39C2×C71⋊C8—41no—
40C2×C71⋊C8—81no—
41C4⋊Dic71—42no—
42C4⋊Dic71—42no—
43C2×Q8×C71—21no—
44C2×Q8×C71—41no—
45C2×Q8×C71—41no—
46C2×Q8×C71D811yescharacteristic
47C2×Q8×C71SD1611yescharacteristic
48C22×C4×C71—21no—
49C22×C4×C71—21no—
50C22×C4×C71—21no—
51C22×C4×C71C22⋊C411yescharacteristic
52M16×C71—42no—
53M16×C71—81no—
54C2×C8×C71—81no—
55C4⋊C4×C71—41no—
56C42×C71—41no—
57C23×C71#9911yescharacteristic
58Q8×C71—41no—
59Q8×C71—41no—
60Q8×C71—81no—
61C2×C4×C71—21no—
62C2×C4×C71—21no—
63C2×C4×C71—22no—
64C2×C4×C71—21no—
65C2×C4×C71—21no—
66C2×C4×C71—41no—
67C2×C4×C71—41no—
68C2×C4×C71—41no—
69C2×C4×C71—41no—
70C2×C4×C71—41no—
71C2×C4×C71—81no—
72C2×C4×C71#10211yescharacteristic
73C2×C4×C71#10411yescharacteristic
74C2×Dic71—42no—
75C2×Dic71—81no—
76C2×Dic71—81no—
77Dic142—42no—
78Dic142—42no—
79Dic142—81no—
80Dic142—81no—
81C8×C71—82no—
82C71⋊C8—42no—
83C71⋊C8—161no—
84C22×C71—21no—
85C22×C71—21no—
86C22×C71#32711yescharacteristic
87C22×C71AΣL(1,𝔽16)11yescharacteristic
88C22×C71#35311yescharacteristic
89C4×C71—21no—
90C4×C71—21no—
91C4×C71—42no—
92C4×C71—41no—
93C4×C71—41no—
94C4×C71—81no—
95Dic71—82no—
96#7405—711nomaximal, Sylow
97C2×C71—21no—
98C2×C71—21no—
99C2×C71#134311yescharacteristic
100C2×C71#140711yescharacteristic
101C2×C71#141111yescharacteristic
102#2016—711no—
103#1394D7111yescharacteristic
104#1319—711no—
105C71#740511yescharacteristic, complemented, Sylow
106C22×Q16—1421no—
107C2×C4⋊Q8—711no—
108C2×#103—1421no—
109C2×#101—21no—
110#391C2×D7111yescharacteristic
111#336—711no—
112#324—1421no—
113C22×Q8—1421no—
114C22×Q8C22⋊D7111yescharacteristic
115C2×Q16—2842no—
116C2×Q16—2841no—
117C2×Q16—2841no—
118C2×Q16—2842no—
119C2×M16—41no—
120C22×C8—1421no—
121C4⋊Q8—2842no—
122C2×C4⋊C4—2841no—
123C2×C4⋊C4D28411yescharacteristic
124C2×C42C4×D7111yescharacteristic
125#103—2842no—
126#101—22no—
127#101—41no—
128C22⋊C8—2841no—
129#95—41no—
130C2×Q8—21no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—2841no—
134C2×Q8—2841no—
135C2×Q8—2841no—
136C2×Q8—2841no—
137C2×Q8—2841no—
138C2×Q8—1422no—
139C2×Q8#1176585011yescharacteristic
140C2×Q8#1176585211yescharacteristic
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—2841no—
145C22×C4(C2×D71)⋊C411yescharacteristic
146Q16—2844no—
147Q16—2844no—
148M16—42no—
149M16—81no—
150C2×C8—81no—
151C2×C8—2841no—
152C2×C8—2841no—
153C2×C8—5681no—
154C2×C8—1422no—
155C4⋊C4—41no—
156C4⋊C4—2842no—
157C4⋊C4—2842no—
158C42—41no—
159C23#1455604211yescharacteristic
160Q8—41no—
161Q8—41no—
162Q8—81no—
163Q8—2842no—
164Q8—2842no—
165Q8—5681no—
166Q8—5681no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—81no—
178C2×C4—2842no—
179C2×C4—5681no—
180C2×C4—5681no—
181C2×C4#1455604811yescharacteristic
182C2×C4#1455605611yescharacteristic
183C8—82no—
184C8—2842no—
185C8—11361no—
186C22—21no—
187C22—21no—
188C22#1798573211yescharacteristic
189C22#1798578511yescharacteristic
190C22#1798579111yescharacteristic
191C4—21no—
192C4—21no—
193C4—42no—
194C4—41no—
195C4—41no—
196C4—81no—
197C4—5682no—
198C2—21no—
199C2—21no—
200C2#2258633611yescharacteristic
201C2#2258650111yescharacteristic
202C2#2258650811yescharacteristic
subgroupG/H
centerC22#17985791
commutatorC2×C42×C71C2×C4
Frattini#391C2×D71
FittingC71×#1394C2
socleC22×C71#353
prime radicalC23×C71#99
cosocle#391×C71C22
hypercenter#1394D71
hypocenterC71#7405
LCSG ▷ C2×C42×C71 ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ #1394

Complements

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

NH ≅ G/Ntype
105. C71#7405semidirect

Maximal subgroups

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

subgroupindex
#225894392
#225862742
C71×#13942
#740571