Finite Group Databasebeta

← #31901546

Subgroups — #31901546

Order 18176.

364 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 79 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#22599685C211yesmaximal, characteristic
2#22587038C212yesmaximal
3#22586851C212yesmaximal
4#22586210C211yesmaximal, characteristic
5C71×#1843C211yesmaximal, characteristic
6#17986556C2212yes—
7#17986295—22no—
8#17986240—22no—
9#17986210—22no—
10#17986056C2212yes—
11#17985746—22no—
12#17985744—22no—
13#17985735—24no—
14C2×C4⋊C8×C71—22no—
15C2×C4⋊C8×C71C2211yescharacteristic
16C2×C4⋊C8×C71C2211yescharacteristic
17#376×C71C2211yescharacteristic
18C2×Dic284—22no—
19#14556128D412yes—
20#14556128D412yes—
21#14556120—22no—
22#14556107—22no—
23#14556107—22no—
24C4×Dic142—22no—
25C4×Dic142—22no—
26#14556087—22no—
27#14556062—22no—
28C4⋊(C71⋊C8)—22no—
29#14556053—24no—
30C22⋊(C71⋊C8)—22no—
31#14556039—22no—
32C22×C8×C71—21no—
33C22×C8×C71D412yes—
34C2×C42×C71—21no—
35C2×C42×C71C2311yescharacteristic
36C4⋊C8×C71—22no—
37C4⋊C8×C71—22no—
38C4⋊C8×C71—44no—
39#95×C71—21no—
40#95×C71D412yes—
41C2×Dic142—22no—
42C2×Dic142—22no—
43Dic284—82no—
44C2×C71⋊C8—42no—
45C4⋊Dic71—22no—
46C4⋊Dic71—22no—
47C4⋊Dic71—22no—
48C4⋊Dic71—22no—
49Dic71⋊C4—42no—
50Dic71⋊C4—42no—
51C22⋊Dic71—42no—
52C4×Dic71—42no—
53C22×C4×C71—22no—
54C22×C4×C71C2×D411yescharacteristic
55C22×C4×C71C2×D411yescharacteristic
56C22×C4×C71#4111yescharacteristic
57C2×C8×C71—22no—
58C2×C8×C71—42no—
59C2×C8×C71—42no—
60C2×C8×C71—41no—
61C42×C71—22no—
62C42×C71—22no—
63C42×C71C2×D412yes—
64C42×C71#4112yes—
65C23×C71#12111yescharacteristic
66C2×C4×C71—22no—
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—24no—
70C2×C4×C71—41no—
71C2×C4×C71—42no—
72C2×C4×C71—42no—
73C2×C4×C71#12011yescharacteristic
74C2×C4×C71#12112yes—
75C2×C4×C71#12312yes—
76C2×C4×C71#12411yescharacteristic
77C2×Dic71—42no—
78C2×Dic71—42no—
79Dic142—42no—
80Dic142—42no—
81Dic142—82no—
82C8×C71—42no—
83C8×C71—81no—
84C71⋊C8—82no—
85C22×C71—21no—
86C22×C71—21no—
87C22×C71#39411yescharacteristic
88C22×C71#47312yes—
89C4×C71—22no—
90C4×C71—21no—
91C4×C71—22no—
92C4×C71—44no—
93C4×C71#45311yescharacteristic
94C4×C71#45611yescharacteristic
95Dic71—82no—
96Dic71—82no—
97#10516—711nomaximal, Sylow
98C2×C71—22no—
99C2×C71#203911yescharacteristic
100C2×C71#204711yescharacteristic
101C2×C71#205311yescharacteristic
102#3249—711no—
103#1843D7111yescharacteristic
104#1552—712no—
105#1506—712no—
106#1290—711no—
107C71#1051611yescharacteristic, complemented, Sylow
108#533—712no—
109#477—1422no—
110#467—1422no—
111#462—1422no—
112#432—712no—
113C2×C4⋊C8—22no—
114C2×C4⋊C8C2×D7111yescharacteristic
115C2×C4⋊C8C2×D7111yescharacteristic
116#376C2×D7111yescharacteristic
117#332—1422no—
118#332—1422no—
119#328—1424no—
120C2×Q16—1422no—
121C22×C8—21no—
122C22×C8C22⋊D7112yes—
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×C42—21no—
132C2×C42C22×D7111yescharacteristic
133#107—1422no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—44no—
137C4⋊C8—1422no—
138#103—1424no—
139C22⋊C8—1422no—
140#97—1422no—
141#95—21no—
142#95D28412yes—
143C2×Q8—1422no—
144C2×Q8—1422no—
145C22×C4—22no—
146C22×C4C2×C22⋊D7111yescharacteristic
147C22×C4C2×D28411yescharacteristic
148C22×C4#1176586511yescharacteristic
149Q16—5682no—
150C2×C8—22no—
151C2×C8—42no—
152C2×C8—42no—
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—22no—
163C42—22no—
164C42—2842no—
165C42D4×D7112yes—
166C42#1176586612yes—
167C23#1455610211yescharacteristic
168Q8—2842no—
169Q8—2842no—
170Q8—5682no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—2842no—
179C2×C4—2842no—
180C2×C4#1455609611yescharacteristic
181C2×C4#1455610312yes—
182C2×C4#1455611312yes—
183C2×C4#1455611711yescharacteristic
184C8—42no—
185C8—81no—
186C8—5682no—
187C22—21no—
188C22—21no—
189C22#1798591611yescharacteristic
190C22#1798627412yes—
191C4—21no—
192C4—22no—
193C4—22no—
194C4—44no—
195C4—5682no—
196C4—5682no—
197C4#1798616611yescharacteristic
198C4#1798618111yescharacteristic
199C2—22no—
200C2#2258956411yescharacteristic
201C2#2258960811yescharacteristic
202C2#2258964611yescharacteristic
subgroupG/H
centerC22#17985916
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1843C2
socleC22×C71#394
prime radicalC23×C71#121
hypercenter#1843D71
hypocenterC71#10516
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1843

Complements

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

NH ≅ G/Ntype
107. C71#10516semidirect

Maximal subgroups

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

subgroupindex
#225996852
#225870382
#225868512
#225862102
C71×#18432
#1051671