Finite Group Databasebeta

← #31901650

Subgroups — #31901650

Order 18176.

342 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 85 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#22588477C211yesmaximal, characteristic
2#22587078C212yesmaximal
3#22586795C212yesmaximal
4C71⋊#1309C211yesmaximal, characteristic
5C71×#3255C211yesmaximal, characteristic
6#17986023C2211yescharacteristic
7#17986023C2212yes—
8C2×C4×C71⋊C8C2211yescharacteristic
9#17985763—22no—
10#17985745—22no—
11#17985745—22no—
12Q8⋊(C71⋊C8)—24no—
13#533×C71C2212yes—
14#475×C71—22no—
15#470×C71—22no—
16#462×C71—22no—
17#423×C71C2211yescharacteristic
18C2×#11765849—21no—
19C2×#11765849—22no—
20C22×C71⋊C8—21no—
21C4⋊(C71⋊C8)—22no—
22C4⋊(C71⋊C8)—22no—
23C4⋊(C71⋊C8)—22no—
24C4×C71⋊C8—21no—
25C4×C71⋊C8D412yes—
26C2×Q16×C71—22no—
27C2×M16×C71D412yes—
28C4⋊Q8×C71D412yes—
29C4⋊Q8×C71D412yes—
30#125×C71—22no—
31C22⋊Q8×C71—22no—
32C22⋊Q8×C71—22no—
33C4×Q8×C71—22no—
34C4×Q8×C71—22no—
35#117×C71—22no—
36C2×C42×C71C2311yescharacteristic
37#106×C71—22no—
38C4⋊C8×C71—22no—
39#103×C71—24no—
40#11765849—42no—
41#11765849—82no—
42C2×C71⋊C8—22no—
43C2×C71⋊C8—41no—
44C2×C71⋊C8—41no—
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—42no—
69Q8×C71—42no—
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71—42no—
75C2×C4×C71—42no—
76C2×C4×C71#12011yescharacteristic
77C2×C4×C71#12111yescharacteristic
78C2×C4×C71#12112yes—
79C2×C4×C71#12312yes—
80C8×C71—42no—
81C71⋊C8—42no—
82C71⋊C8—81no—
83C71⋊C8—82no—
84C22×C71—22no—
85C22×C71—21no—
86C22×C71#39411yescharacteristic
87C4×C71—22no—
88C4×C71—21no—
89C4×C71—82no—
90C4×C71—82no—
91C4×C71#45411yescharacteristic
92C4×C71#45511yescharacteristic
93C4×C71#47112yes—
94C4×C71#47312yes—
95#10535—711nomaximal, Sylow
96C2×C71—41no—
97C2×C71#204011yescharacteristic
98C2×C71#205212yes—
99#3255D7111yescharacteristic
100#1837—711no—
101#1561—712no—
102#1493—712no—
103#1309—711no—
104C71#1053511yescharacteristic, complemented, Sylow
105#533C2×D7112yes—
106#475—22no—
107#470—22no—
108#462—22no—
109#423—711no—
110#423—712no—
111#423C2×D7111yescharacteristic
112C2×C4×C8—711no—
113#341—1422no—
114#332—1422no—
115#332—1422no—
116Q8⋊C8—1424no—
117C2×Q16—22no—
118C2×M16—1421no—
119C2×M16—1422no—
120C2×M16C22⋊D7112yes—
121C22×C8—1421no—
122C4⋊Q8C22⋊D7112yes—
123C4⋊Q8C22⋊D7112yes—
124#125—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129#117—22no—
130C2×C42C22×D7111yescharacteristic
131#106—22no—
132C4⋊C8—22no—
133C4⋊C8—1422no—
134C4⋊C8—1422no—
135C4⋊C8—1422no—
136#103—24no—
137C4×C8—712no—
138C4×C8—1421no—
139C2×Q8—22no—
140C2×Q8—22no—
141C22×C4C2×C22⋊D7111yescharacteristic
142C22×C4#1176586412yes—
143Q16—82no—
144M16—42no—
145M16—2842no—
146M16—5682no—
147C2×C8—22no—
148C2×C8—2841no—
149C2×C8—2841no—
150C2×C8—2841no—
151C2×C8—2842no—
152C2×C8—1422no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C22⋊C4—42no—
160C42—42no—
161C42D4×D7111yescharacteristic
162C42C2×C22⋊D7112yes—
163C42#1176586411yescharacteristic
164C23#1455611511yescharacteristic
165Q8—42no—
166Q8—42no—
167Q8—42no—
168Q8—42no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4#1455609711yescharacteristic
176C2×C4#1455609911yescharacteristic
177C2×C4#1455610012yes—
178C2×C4#1455610912yes—
179C8—42no—
180C8—2842no—
181C8—5681no—
182C8—5682no—
183C22—21no—
184C22—22no—
185C22#1798591311yescharacteristic
186C4—21no—
187C4—22no—
188C4—82no—
189C4—82no—
190C4#1798617211yescharacteristic
191C4#1798617711yescharacteristic
192C4#1798626412yes—
193C4#1798627212yes—
194C2—41no—
195C2#2258956611yescharacteristic
196C2#2258963612yes—
subgroupG/H
centerC22#17985913
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#3255C2
socleC22×C71#394
prime radicalC23×C71#124
hypercenter#3255D71
hypocenterC71#10535
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
104. C71#10535semidirect

Maximal subgroups

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

subgroupindex
#225884772
#225870782
#225867952
C71⋊#13092
C71×#32552
#1053571