Finite Group Databasebeta

← #31885047

Subgroups — #31885047

Order 18176.

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

normal subgroups only

This group has 206 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#22589501C211yesmaximal, characteristic
2#22586467C211yesmaximal, characteristic
3C71×#1272C211yesmaximal, characteristic
4#17986630C411yescharacteristic
5#17986008—21no—
6C2×#14556047—21no—
7#17985768C411yescharacteristic
8#17985759—21no—
9#391×C71C2211yescharacteristic
10#336×C71—21no—
11C2×#11765849—21no—
12C2×#11765849—41no—
13#14556127—41no—
14#14556119—41no—
15#14556107—41no—
16#14556087—21no—
17#14556052—41no—
18#14556052—41no—
19#14556047—22no—
20#14556047—41no—
21C22×Q8×C71D411yescharacteristic
22C22×C8×C71—41no—
23C2×C4⋊C4×C71C2×C411yescharacteristic
24C2×C42×C71D411yescharacteristic
25#95×C71—41no—
26C2×Dic142—41no—
27#11765849—42no—
28#11765849—41no—
29#11765849—41no—
30#11765849—81no—
31C2×C71⋊C8—41no—
32C2×C71⋊C8—81no—
33C4⋊Dic71—21no—
34C4⋊Dic71—21no—
35C4⋊Dic71—41no—
36Dic71⋊C4—42no—
37C22⋊Dic71—41no—
38C4×Dic71—41no—
39C2×Q8×C71—21no—
40C2×Q8×C71—41no—
41C2×Q8×C71—41no—
42C2×Q8×C71SD1611yescharacteristic
43C2×Q8×C71Q1611yescharacteristic
44C22×C4×C71—21no—
45C22×C4×C71—21no—
46C22×C4×C71—21no—
47C22×C4×C71C22⋊C411yescharacteristic
48C2×C8×C71—42no—
49C2×C8×C71—81no—
50C2×C8×C71—81no—
51C4⋊C4×C71—41no—
52C42×C71—41no—
53C23×C71#9911yescharacteristic
54Q8×C71—41no—
55Q8×C71—41no—
56Q8×C71—81no—
57C2×C4×C71—21no—
58C2×C4×C71—21no—
59C2×C4×C71—21no—
60C2×C4×C71—21no—
61C2×C4×C71—21no—
62C2×C4×C71—21no—
63C2×C4×C71—41no—
64C2×C4×C71—41no—
65C2×C4×C71—41no—
66C2×C4×C71—41no—
67C2×C4×C71—41no—
68C2×C4×C71—81no—
69C2×C4×C71#10311yescharacteristic
70C2×C4×C71#10411yescharacteristic
71C2×Dic71—41no—
72C2×Dic71—41no—
73Dic142—161no—
74C8×C71—82no—
75C71⋊C8—81no—
76C71⋊C8—82no—
77C22×C71—21no—
78C22×C71—21no—
79C22×C71#32811yescharacteristic
80C22×C71AΣL(1,𝔽16)11yescharacteristic
81C22×C71#35411yescharacteristic
82C4×C71—21no—
83C4×C71—21no—
84C4×C71—41no—
85C4×C71—41no—
86C4×C71—41no—
87C4×C71—41no—
88C4×C71—81no—
89Dic71—81no—
90Dic71—81no—
91#7431—711nomaximal, Sylow
92C2×C71—21no—
93C2×C71—21no—
94C2×C71#135111yescharacteristic
95C2×C71#140311yescharacteristic
96C2×C71#140511yescharacteristic
97#2028—711no—
98#1394—711no—
99#1272D7111yescharacteristic
100C71#743111yescharacteristic, complemented, Sylow
101#544—711no—
102#419—1421no—
103C2×#101—1421no—
104#391C2×D7111yescharacteristic
105#343—711no—
106#339—1421no—
107#336—21no—
108C22×Q8D28411yescharacteristic
109C2×M16—2841no—
110C2×M16—1421no—
111C22×C8—41no—
112C4⋊Q8—2841no—
113#125—2841no—
114C22⋊Q8—2841no—
115#117—1421no—
116C2×C4⋊C4C4×D7111yescharacteristic
117C2×C42C22⋊D7111yescharacteristic
118#103—2841no—
119#103—2841no—
120#101—2841no—
121#101—1422no—
122#95—41no—
123C2×Q8—21no—
124C2×Q8—41no—
125C2×Q8—41no—
126C2×Q8—2841no—
127C2×Q8#1176585411yescharacteristic
128C2×Q8Dic28411yescharacteristic
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4(C2×D71)⋊C411yescharacteristic
133M16—2842no—
134M16—2841no—
135M16—2841no—
136M16—5681no—
137C2×C8—42no—
138C2×C8—81no—
139C2×C8—81no—
140C2×C8—2841no—
141C2×C8—5681no—
142C4⋊C4—41no—
143C4⋊C4—2841no—
144C4⋊C4—2842no—
145C4⋊C4—1421no—
146C4⋊C4—1421no—
147C22⋊C4—2841no—
148C42—41no—
149C42—2841no—
150C23#1455604211yescharacteristic
151Q8—41no—
152Q8—41no—
153Q8—81no—
154Q8—11361no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—81no—
167C2×C4—2841no—
168C2×C4—2841no—
169C2×C4#1455605311yescharacteristic
170C2×C4#1455605411yescharacteristic
171C8—82no—
172C8—5681no—
173C8—5682no—
174C22—21no—
175C22—21no—
176C22#1798573711yescharacteristic
177C22#1798578711yescharacteristic
178C22#1798579311yescharacteristic
179C4—21no—
180C4—21no—
181C4—41no—
182C4—41no—
183C4—41no—
184C4—41no—
185C4—81no—
186C4—5681no—
187C4—5681no—
188C2—21no—
189C2—21no—
190C2#2258636211yescharacteristic
191C2#2258648911yescharacteristic
192C2#2258649511yescharacteristic
subgroupG/H
centerC22#17985787
commutatorC2×C4⋊C4×C71C2×C4
Frattini#391C2×D71
FittingC71×#1272C2
socleC22×C71AΣL(1,𝔽16)
prime radicalC23×C71#99
cosocle#391×C71C22
hypercenter#1272D71
hypocenterC71#7431
LCSG ▷ C2×C4⋊C4×C71 ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ #1272

Complements

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

NH ≅ G/Ntype
100. C71#7431semidirect

Maximal subgroups

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

subgroupindex
#225895012
#225864672
C71×#12722
#743171