Finite Group Databasebeta

← #31901611

Subgroups — #31901611

Order 18176.

342 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 77 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#22588514C211yesmaximal, characteristic
2#22587070C212yesmaximal
3#22586955C212yesmaximal
4#22586251C211yesmaximal, characteristic
5C71×#3255C211yesmaximal, characteristic
6#17986056C2212yes—
7C2×#14556063—22no—
8#17986023C2211yescharacteristic
9C2×#14556039C2211yescharacteristic
10#17985748—22no—
11#17985745—22no—
12#17985741—24no—
13#533×C71C2212yes—
14#475×C71—22no—
15#470×C71—22no—
16#462×C71—22no—
17#423×C71C2211yescharacteristic
18C2×#11765849—21no—
19C22×C71⋊C8—21no—
20#14556063—44no—
21C4⋊(C71⋊C8)—22no—
22C4⋊(C71⋊C8)—22no—
23C22⋊(C71⋊C8)—22no—
24#14556039—22no—
25#14556039—21no—
26#14556039D412yes—
27C2×Q16×C71—22no—
28C2×M16×C71D412yes—
29C4⋊Q8×C71D412yes—
30C4⋊Q8×C71D412yes—
31#125×C71—22no—
32C22⋊Q8×C71—22no—
33C22⋊Q8×C71—22no—
34C4×Q8×C71—22no—
35C4×Q8×C71—22no—
36#117×C71—22no—
37C2×C42×C71C2311yescharacteristic
38#106×C71—22no—
39C4⋊C8×C71—22no—
40#103×C71—24no—
41#11765849—42no—
42C2×C71⋊C8—22no—
43C2×C71⋊C8—22no—
44C2×C71⋊C8—22no—
45C2×C71⋊C8—41no—
46C2×C71⋊C8—42no—
47C2×Q8×C71—22no—
48C2×Q8×C71—22no—
49C22×C4×C71C2×D411yescharacteristic
50C22×C4×C71#4112yes—
51Q16×C71—82no—
52M16×C71—42no—
53C2×C8×C71—22no—
54C4⋊C4×C71—22no—
55C4⋊C4×C71—22no—
56C4⋊C4×C71—22no—
57C4⋊C4×C71—22no—
58C4⋊C4×C71—42no—
59C4⋊C4×C71—42no—
60C22⋊C4×C71—42no—
61C42×C71—42no—
62C42×C71C2×D411yescharacteristic
63C42×C71C2×D412yes—
64C42×C71#4111yescharacteristic
65C23×C71#12411yescharacteristic
66Q8×C71—42no—
67Q8×C71—42no—
68Q8×C71—82no—
69C2×C4×C71—21no—
70C2×C4×C71—21no—
71C2×C4×C71—42no—
72C2×C4×C71—42no—
73C2×C4×C71—42no—
74C2×C4×C71#12011yescharacteristic
75C2×C4×C71#12111yescharacteristic
76C2×C4×C71#12112yes—
77C2×C4×C71#12312yes—
78C8×C71—42no—
79C71⋊C8—24no—
80C71⋊C8—82no—
81C71⋊C8—81no—
82C22×C71—22no—
83C22×C71—21no—
84C22×C71#39411yescharacteristic
85C4×C71—22no—
86C4×C71—22no—
87C4×C71—21no—
88C4×C71—82no—
89C4×C71—82no—
90C4×C71#45311yescharacteristic
91C4×C71#45611yescharacteristic
92#10528—711nomaximal, Sylow
93C2×C71—41no—
94C2×C71#203911yescharacteristic
95C2×C71#204912yes—
96#3255D7111yescharacteristic
97#1849—711no—
98#1559—712no—
99#1531—712no—
100#1307—711no—
101C71#1052811yescharacteristic, complemented, Sylow
102#533C2×D7112yes—
103#475—22no—
104#470—22no—
105#462—22no—
106#432—712no—
107C2×#108—1422no—
108#423—711no—
109#423C2×D7111yescharacteristic
110C2×#97—711no—
111#333—1422no—
112#332—1422no—
113#330—1424no—
114C2×Q16—22no—
115C2×M16—1421no—
116C2×M16C22⋊D7112yes—
117C22×C8—1421no—
118C4⋊Q8C22⋊D7112yes—
119C4⋊Q8C22⋊D7112yes—
120#125—22no—
121C22⋊Q8—22no—
122C22⋊Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125#117—22no—
126C2×C42C22×D7111yescharacteristic
127#108—2844no—
128#106—22no—
129C4⋊C8—22no—
130C4⋊C8—1422no—
131C4⋊C8—1422no—
132#103—24no—
133C22⋊C8—1422no—
134#97—712no—
135#97—1422no—
136#97—1421no—
137C2×Q8—22no—
138C2×Q8—22no—
139C22×C4C2×C22⋊D7111yescharacteristic
140C22×C4#1176586412yes—
141Q16—82no—
142M16—42no—
143M16—2842no—
144C2×C8—22no—
145C2×C8—2841no—
146C2×C8—2842no—
147C2×C8—1422no—
148C2×C8—1422no—
149C2×C8—1422no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C22⋊C4—42no—
157C42—42no—
158C42D4×D7111yescharacteristic
159C42C2×C22⋊D7112yes—
160C42#1176586411yescharacteristic
161C23#1455611511yescharacteristic
162Q8—42no—
163Q8—42no—
164Q8—82no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4#1455609711yescharacteristic
171C2×C4#1455609911yescharacteristic
172C2×C4#1455610012yes—
173C2×C4#1455610912yes—
174C8—42no—
175C8—5682no—
176C8—5681no—
177C8—1424no—
178C22—21no—
179C22—22no—
180C22#1798591311yescharacteristic
181C4—21no—
182C4—22no—
183C4—22no—
184C4—82no—
185C4—82no—
186C4#1798616711yescharacteristic
187C4#1798618211yescharacteristic
188C2—41no—
189C2#2258956111yescharacteristic
190C2#2258961712yes—
subgroupG/H
centerC22#17985913
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#3255C2
socleC22×C71#394
prime radicalC23×C71#124
hypercenter#3255D71
hypocenterC71#10528
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3255

Complements

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

NH ≅ G/Ntype
101. C71#10528semidirect

Maximal subgroups

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

subgroupindex
#225885142
#225870702
#225869552
#225862512
C71×#32552
#1052871