Finite Group Databasebeta

← #31901648

Subgroups — #31901648

Order 18176.

358 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 83 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#22599734C211yesmaximal, characteristic
2#22587077C212yesmaximal
3#22586796C212yesmaximal
4C71⋊#1309C211yesmaximal, characteristic
5C71×#1848C211yesmaximal, characteristic
6#17986556C2212yes—
7#17986285—22no—
8#17986257—22no—
9#17986210—22no—
10#17986023C2212yes—
11#17985746—22no—
12#17985744—22no—
13#17985731—24no—
14C2×#108×C71—22no—
15#423×C71C2211yescharacteristic
16#423×C71C2211yescharacteristic
17C2×C4×C8×C71C2211yescharacteristic
18C2×Dic284—22no—
19C2×#11765849—22no—
20#14556128D412yes—
21#14556128D412yes—
22#14556120—22no—
23#14556107—22no—
24#14556107—22no—
25C4×Dic142—22no—
26C4×Dic142—22no—
27#14556087—22no—
28#14556060—22no—
29C4⋊(C71⋊C8)—22no—
30C4⋊(C71⋊C8)—22no—
31#14556053—24no—
32C2×M16×C71—21no—
33C2×M16×C71D412yes—
34C22×C8×C71—21no—
35C2×C42×C71C2311yescharacteristic
36#108×C71—44no—
37C4⋊C8×C71—22no—
38C4⋊C8×C71—22no—
39C4×C8×C71—21no—
40C4×C8×C71D412yes—
41C2×Dic142—22no—
42C2×Dic142—22no—
43Dic284—82no—
44#11765849—82no—
45C2×C71⋊C8—42no—
46C4⋊Dic71—22no—
47C4⋊Dic71—22no—
48C4⋊Dic71—22no—
49C4⋊Dic71—22no—
50Dic71⋊C4—42no—
51Dic71⋊C4—42no—
52C22⋊Dic71—42no—
53C4×Dic71—42no—
54C22×C4×C71C2×D411yescharacteristic
55C22×C4×C71#4112yes—
56M16×C71—42no—
57M16×C71—42no—
58C2×C8×C71—22no—
59C2×C8×C71—22no—
60C2×C8×C71—22no—
61C2×C8×C71—22no—
62C2×C8×C71—41no—
63C42×C71C2×D412yes—
64C42×C71C2×D411yescharacteristic
65C42×C71#4111yescharacteristic
66C23×C71#12411yescharacteristic
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71#12011yescharacteristic
72C2×C4×C71#12111yescharacteristic
73C2×C4×C71#12112yes—
74C2×C4×C71#12312yes—
75C2×Dic71—42no—
76C2×Dic71—42no—
77Dic142—42no—
78Dic142—42no—
79Dic142—42no—
80Dic142—42no—
81C8×C71—24no—
82C8×C71—42no—
83C8×C71—81no—
84C71⋊C8—82no—
85C22×C71—22no—
86C22×C71—21no—
87C22×C71#39411yescharacteristic
88C4×C71—22no—
89C4×C71—21no—
90C4×C71#45311yescharacteristic
91C4×C71#45611yescharacteristic
92C4×C71#47112yes—
93C4×C71#47312yes—
94Dic71—82no—
95Dic71—82no—
96#10534—711nomaximal, Sylow
97C2×C71—41no—
98C2×C71#203911yescharacteristic
99C2×C71#205212yes—
100#3255—711no—
101#1848D7111yescharacteristic
102#1561—712no—
103#1493—712no—
104#1309—711no—
105C71#1053411yescharacteristic, complemented, Sylow
106#533—712no—
107#475—1422no—
108#470—1422no—
109#462—1422no—
110C2×#108—22no—
111#423—712no—
112#423C2×D7111yescharacteristic
113#423C2×D7111yescharacteristic
114C2×C4×C8C2×D7111yescharacteristic
115#332—1422no—
116#332—1422no—
117Q8⋊C8—1424no—
118C2×Q16—1422no—
119C2×M16—21no—
120C2×M16—1422no—
121C2×M16C22⋊D7112yes—
122C22×C8—21no—
123C4⋊Q8—712no—
124C4⋊Q8—712no—
125#125—1422no—
126C22⋊Q8—1422no—
127C22⋊Q8—1422no—
128C4×Q8—1422no—
129C4×Q8—1422no—
130#117—1422no—
131C2×C42C22×D7111yescharacteristic
132#108—44no—
133#106—1422no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—1422no—
137C4⋊C8—1422no—
138#103—1424no—
139C4×C8—21no—
140C4×C8D28412yes—
141C2×Q8—1422no—
142C2×Q8—1422no—
143C22×C4C2×C22⋊D7111yescharacteristic
144C22×C4#1176586612yes—
145Q16—5682no—
146M16—42no—
147M16—42no—
148M16—5682no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—41no—
154C2×C8—2842no—
155C4⋊C4—2842no—
156C4⋊C4—2842no—
157C4⋊C4—1422no—
158C4⋊C4—1422no—
159C4⋊C4—1422no—
160C4⋊C4—1422no—
161C22⋊C4—2842no—
162C42—2842no—
163C42D4×D7112yes—
164C42C2×D28411yescharacteristic
165C42#1176586511yescharacteristic
166C23#1455611711yescharacteristic
167Q8—2842no—
168Q8—2842no—
169Q8—2842no—
170Q8—2842no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—2842no—
176C2×C4—2842no—
177C2×C4#1455609611yescharacteristic
178C2×C4#1455610211yescharacteristic
179C2×C4#1455610312yes—
180C2×C4#1455611312yes—
181C8—24no—
182C8—42no—
183C8—81no—
184C8—5682no—
185C22—21no—
186C22—22no—
187C22#1798591611yescharacteristic
188C4—21no—
189C4—22no—
190C4—5682no—
191C4—5682no—
192C4#1798616611yescharacteristic
193C4#1798618111yescharacteristic
194C4#1798626312yes—
195C4#1798627512yes—
196C2—41no—
197C2#2258956411yescharacteristic
198C2#2258964112yes—
subgroupG/H
centerC22#17985916
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1848C2
socleC22×C71#394
prime radicalC23×C71#124
hypercenter#1848D71
hypocenterC71#10534
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1848

Complements

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

NH ≅ G/Ntype
105. C71#10534semidirect

Maximal subgroups

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

subgroupindex
#225997342
#225870772
#225867962
C71⋊#13092
C71×#18482
#1053471