Finite Group Databasebeta

← #31901960

Subgroups — #31901960

Order 18176.

382 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 97 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#22596615C211yesmaximal, characteristic
2#22588508C212yesmaximal
3#22588473C211yesmaximal, characteristic
4#22586222C212yesmaximal
5C71×#1295C211yesmaximal, characteristic
6C2×#14556062—22no—
7C2×#14556060—22no—
8C2×#14556039C2211yescharacteristic
9#17985933—22no—
10#17985932—22no—
11#17985931—22no—
12#17985869C2211yescharacteristic
13#17985865C2212yes—
14#17985761—22no—
15C2×#97×C71C2212yes—
16#382×C71C2211yescharacteristic
17C22×C71⋊C8—21no—
18C2×C4⋊Dic71—22no—
19C2×C4⋊Dic71—21no—
20C2×C4⋊Dic71—22no—
21C2×C4⋊Dic71D412yes—
22C2×C4⋊Dic71D412yes—
23C2×Dic71⋊C4—22no—
24C2×Dic71⋊C4—22no—
25#14556062—44no—
26#14556060—44no—
27#14556039—21no—
28#14556039Q812yes—
29C71⋊#95—22no—
30C71⋊#95—22no—
31C71⋊#95—22no—
32C22×C8×C71—22no—
33C2×C4⋊C4×C71D412yes—
34C2×C42×C71C2311yescharacteristic
35#97×C71—22no—
36#97×C71Q814yes—
37#95×C71—22no—
38C22×Dic71—22no—
39C22×Dic71—22no—
40C22×Dic71—41no—
41C2×C71⋊C8—22no—
42C2×C71⋊C8—41no—
43C2×C71⋊C8—41no—
44C4⋊Dic71—42no—
45C4⋊Dic71—42no—
46C4⋊Dic71—42no—
47C4⋊Dic71—42no—
48C4⋊Dic71—42no—
49C4⋊Dic71—82no—
50Dic71⋊C4—82no—
51Dic71⋊C4—82no—
52C22×C4×C71—22no—
53C22×C4×C71C2×D412yes—
54C22×C4×C71C2×D411yescharacteristic
55C2×C8×C71—24no—
56C2×C8×C71—24no—
57C2×C8×C71—24no—
58C4⋊C4×C71—42no—
59C42×C71C2×Q811yescharacteristic
60C42×C71C2×Q812yes—
61C42×C71#4111yescharacteristic
62C23×C71#12011yescharacteristic
63C2×C4×C71—22no—
64C2×C4×C71—21no—
65C2×C4×C71—21no—
66C2×C4×C71—22no—
67C2×C4×C71—42no—
68C2×C4×C71—42no—
69C2×C4×C71—42no—
70C2×C4×C71C22⋊Q812yes—
71C2×C4×C71C22⋊Q811yescharacteristic
72C2×C4×C71C4⋊Q812yes—
73C2×C4×C71C4⋊Q811yescharacteristic
74C2×Dic71—42no—
75C2×Dic71—42no—
76C2×Dic71—42no—
77C2×Dic71—42no—
78C2×Dic71—42no—
79C2×Dic71—42no—
80C2×Dic71—82no—
81C2×Dic71—81no—
82C8×C71—28no—
83C71⋊C8—42no—
84C22×C71#39511yescharacteristic
85C22×C71#45312yes—
86C22×C71#45411yescharacteristic
87C22×C71#45511yescharacteristic
88C22×C71#45612yes—
89C4×C71—22no—
90C4×C71—22no—
91C4×C71—21no—
92C4×C71—21no—
93C4×C71—82no—
94Dic71—82no—
95Dic71—82no—
96Dic71—161no—
97#10585—711nomaximal, Sylow
98C2×C71#165611yescharacteristic
99C2×C71#166112yes—
100C2×C71#166311yescharacteristic
101C2×C71#206012yes—
102C2×C71#206111yescharacteristic
103#2845—711no—
104#1847—712no—
105#1836—711no—
106#1295—712no—
107#1295D7111yescharacteristic
108C71#1058511yescharacteristic, complemented, Sylow
109C2×#107—1422no—
110C2×#106—1422no—
111C2×#97—711no—
112C2×#97C2×D7112yes—
113#398—1422no—
114#398—1422no—
115#398—1422no—
116#384—711no—
117#382—712no—
118#382C2×D7111yescharacteristic
119#340—1422no—
120C22×C8—22no—
121C22×C8—1421no—
122C2×C4⋊C4—712no—
123C2×C4⋊C4—712no—
124C2×C4⋊C4—1422no—
125C2×C4⋊C4—1421no—
126C2×C4⋊C4—1422no—
127C2×C4⋊C4—1422no—
128C2×C4⋊C4—1422no—
129C2×C4⋊C4D28412yes—
130C2×C42C22×D7111yescharacteristic
131#107—2844no—
132#106—2844no—
133#97—22no—
134#97—712no—
135#97—1421no—
136#97Dic14214yes—
137#95—22no—
138#95—1422no—
139#95—1422no—
140#95—1422no—
141C22×C4—22no—
142C22×C4—2841no—
143C22×C4—1422no—
144C22×C4—1422no—
145C22×C4D4×D7112yes—
146C22×C4C2×D28411yescharacteristic
147C2×C8—24no—
148C2×C8—24no—
149C2×C8—24no—
150C2×C8—2841no—
151C2×C8—2841no—
152C2×C8—1422no—
153C4⋊C4—42no—
154C4⋊C4—2842no—
155C4⋊C4—2842no—
156C4⋊C4—2842no—
157C4⋊C4—2842no—
158C4⋊C4—2842no—
159C4⋊C4—5682no—
160C4⋊C4—5682no—
161C4⋊C4—5682no—
162C42Q8×D7111yescharacteristic
163C42C2×Dic14212yes—
164C42#1176586411yescharacteristic
165C23#1455609811yescharacteristic
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—2842no—
174C2×C4—2842no—
175C2×C4—2842no—
176C2×C4—2842no—
177C2×C4—2842no—
178C2×C4—2842no—
179C2×C4—5682no—
180C2×C4—5681no—
181C2×C4#1455610512yes—
182C2×C4#1455610811yescharacteristic
183C2×C4#1455612712yes—
184C2×C4#1455612811yescharacteristic
185C8—28no—
186C8—2842no—
187C22#1798592011yescharacteristic
188C22#1798616812yes—
189C22#1798617411yescharacteristic
190C22#1798617911yescharacteristic
191C22#1798618312yes—
192C4—22no—
193C4—22no—
194C4—21no—
195C4—21no—
196C4—82no—
197C4—5682no—
198C4—5682no—
199C4—11361no—
200C2#2258763511yescharacteristic
201C2#2258766012yes—
202C2#2258766911yescharacteristic
203C2#2258968412yes—
204C2#2258969111yescharacteristic
subgroupG/H
centerC23#14556098
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1295C2
socle, prime radicalC23×C71#120
hypercenter#1295D71
hypocenterC71#10585
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
108. C71#10585semidirect

Maximal subgroups

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

subgroupindex
#225966152
#225885082
#225884732
#225862222
C71×#12952
#1058571