Finite Group Databasebeta

← #31901956

Subgroups — #31901956

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#22588509C212yesmaximal
2#22588473C211yesmaximal, characteristic
3#22586221C212yesmaximal
4#22586221C211yesmaximal, characteristic
5C71×#2845C211yesmaximal, characteristic
6C2×#14556061—22no—
7C2×#14556059—22no—
8C2×#14556039C2211yescharacteristic
9C2×#14556039C2212yes—
10#17985869C2211yescharacteristic
11#17985761—22no—
12#398×C71—22no—
13#398×C71—22no—
14#398×C71—22no—
15#382×C71C2211yescharacteristic
16#382×C71C2212yes—
17C22×C71⋊C8—22no—
18C22×C71⋊C8—21no—
19C2×C4⋊Dic71—21no—
20C2×C4⋊Dic71—22no—
21#14556061—44no—
22#14556059—44no—
23#14556039—22no—
24#14556039—21no—
25#14556039Q814yes—
26#14556039Q812yes—
27C2×C4⋊C4×C71—22no—
28C2×C4⋊C4×C71—22no—
29C2×C4⋊C4×C71—22no—
30C2×C4⋊C4×C71D412yes—
31C2×C4⋊C4×C71D412yes—
32C2×C4⋊C4×C71D412yes—
33C2×C42×C71C2311yescharacteristic
34#95×C71—22no—
35#95×C71—22no—
36#95×C71—22no—
37#95×C71—22no—
38C22×Dic71—41no—
39C2×C71⋊C8—24no—
40C2×C71⋊C8—24no—
41C2×C71⋊C8—24no—
42C2×C71⋊C8—22no—
43C2×C71⋊C8—41no—
44C2×C71⋊C8—41no—
45C4⋊Dic71—42no—
46C4⋊Dic71—42no—
47C4⋊Dic71—42no—
48C22×C4×C71—22no—
49C22×C4×C71—22no—
50C22×C4×C71—22no—
51C22×C4×C71C2×D412yes—
52C22×C4×C71C2×D411yescharacteristic
53C4⋊C4×C71—42no—
54C4⋊C4×C71—42no—
55C4⋊C4×C71—42no—
56C4⋊C4×C71—82no—
57C4⋊C4×C71—82no—
58C4⋊C4×C71—82no—
59C42×C71C2×Q812yes—
60C42×C71C2×Q811yescharacteristic
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×C71—42no—
71C2×C4×C71—42no—
72C2×C4×C71—42no—
73C2×C4×C71—42no—
74C2×C4×C71—42no—
75C2×C4×C71—42no—
76C2×C4×C71C22⋊Q812yes—
77C2×C4×C71C22⋊Q811yescharacteristic
78C2×C4×C71C4⋊Q812yes—
79C2×C4×C71C4⋊Q811yescharacteristic
80C2×Dic71—82no—
81C2×Dic71—81no—
82C71⋊C8—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—
94C4×C71—82no—
95C4×C71—82no—
96Dic71—161no—
97#10585—711nomaximal, Sylow
98C2×C71#165611yescharacteristic
99C2×C71#166112yes—
100C2×C71#166311yescharacteristic
101C2×C71#206012yes—
102C2×C71#206111yescharacteristic
103#2845D7111yescharacteristic
104#1847—712no—
105#1836—711no—
106#1295—711no—
107#1295—712no—
108C71#1058511yescharacteristic, complemented, Sylow
109C2×#107—1422no—
110C2×#106—1422no—
111C2×#97—711no—
112C2×#97—712no—
113#398—22no—
114#398—22no—
115#398—22no—
116#384—711no—
117#382C2×D7111yescharacteristic
118#382C2×D7112yes—
119#340—1422no—
120C22×C8—1421no—
121C22×C8—1422no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—1421no—
126C2×C4⋊C4—1422no—
127C2×C4⋊C4C22⋊D7112yes—
128C2×C4⋊C4C22⋊D7112yes—
129C2×C4⋊C4C22⋊D7112yes—
130C2×C42C22×D7111yescharacteristic
131#107—2844no—
132#106—2844no—
133#97—712no—
134#97—714no—
135#97—1421no—
136#97—1422no—
137#95—22no—
138#95—22no—
139#95—22no—
140#95—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—2841no—
145C22×C4C2×C22⋊D7112yes—
146C22×C4C2×C22⋊D7111yescharacteristic
147C2×C8—2841no—
148C2×C8—2841no—
149C2×C8—1422no—
150C2×C8—1424no—
151C2×C8—1424no—
152C2×C8—1424no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—82no—
157C4⋊C4—82no—
158C4⋊C4—82no—
159C4⋊C4—2842no—
160C4⋊C4—2842no—
161C4⋊C4—2842no—
162C42Q8×D7112yes—
163C42Q8×D7111yescharacteristic
164C42#1176586611yescharacteristic
165C23#1455609711yescharacteristic
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—5682no—
180C2×C4—5681no—
181C2×C4#1455610412yes—
182C2×C4#1455610411yescharacteristic
183C2×C4#1455612612yes—
184C2×C4#1455612611yescharacteristic
185C8—2842no—
186C8—1428no—
187C22#1798591811yescharacteristic
188C22#1798616712yes—
189C22#1798617211yescharacteristic
190C22#1798617711yescharacteristic
191C22#1798618212yes—
192C4—22no—
193C4—22no—
194C4—21no—
195C4—21no—
196C4—82no—
197C4—82no—
198C4—82no—
199C4—11361no—
200C2#2258763311yescharacteristic
201C2#2258765712yes—
202C2#2258766811yescharacteristic
203C2#2258968212yes—
204C2#2258968711yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2845C2
socle, prime radicalC23×C71#120
hypercenter#2845D71
hypocenterC71#10585
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2845

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
#225885092
#225884732
#225862212
C71×#28452
#1058571