Finite Group Databasebeta

← #31900557

Subgroups — #31900557

Order 18176.

402 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 93 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#22589168C211yesmaximal, characteristic
2#22586899C212yesmaximal
3#22586198C212yesmaximal, complemented
4#22586173C211yesmaximal, characteristic, complemented
5C71×#2864C211yesmaximal, characteristic, complemented
6#17986549C2211yescharacteristic
7#17986052C2212yes—
8#17986023C2211yescharacteristic
9C2×#14556056—22no—
10#17985732—24no—
11#389×C71—22no—
12#387×C71—24no—
13#382×C71C2212yes—
14C43×C71C2211yescharacteristic
15C2×#11765865—21no—
16C2×#11765849—21no—
17#14556128Q811yescharacteristic
18#14556125Q811yescharacteristic
19#14556116—21no—
20#14556102—22no—
21C4×D284—22no—
22C4⋊(C71⋊C8)—22no—
23C4⋊(C71⋊C8)—22no—
24#14556056—44no—
25C22⋊(C71⋊C8)—22no—
26C4×C71⋊C8—22no—
27C2×C4⋊C4×C71—24no—
28C2×C4⋊C4×C71—22no—
29C2×C42×C71—22no—
30C2×C42×C71D412yes—
31C2×C42×C71C2311yescharacteristic
32#95×C71—24no—
33#95×C71—22no—
34#95×C71D414yes—
35#11765865—42no—
36#11765865—42no—
37C2×Dic142—21no—
38C2×D284—21no—
39C2×D284—22no—
40C2×C22⋊D71—41no—
41C2×C4×D71—41no—
42#11765849—42no—
43C2×C71⋊C8—42no—
44C2×C71⋊C8—41no—
45C4⋊Dic71—22no—
46(C2×D71)⋊C4—42no—
47C22×C4×C71—22no—
48C22×C4×C71—24no—
49C22×C4×C71C2×D412yes—
50C22×C4×C71C2×D411yescharacteristic
51C4⋊C4×C71—48no—
52C4⋊C4×C71—42no—
53C4⋊C4×C71—42no—
54C42×C71—22no—
55C42×C71—22no—
56C42×C71—44no—
57C42×C71C2×Q811yescharacteristic
58C42×C71#4111yescharacteristic
59C42×C71#4112yes—
60C23×C71#12011yescharacteristic
61C22×D71—41no—
62C2×C4×C71—24no—
63C2×C4×C71—42no—
64C2×C4×C71—42no—
65C2×C4×C71—42no—
66C2×C4×C71—48no—
67C2×C4×C71—44no—
68C2×C4×C71C22⋊Q812yes—
69C2×C4×C71C22⋊Q811yescharacteristic
70C2×C4×C71#12412yes—
71C2×C4×C71#12411yescharacteristic
72C2×C4×C71#13512yes—
73C2×C4×C71AGL(1,ℤ/8)12yes—
74C22⋊D71—82no—
75C2×Dic71—41no—
76D284—42no—
77D284—82no—
78C4×D71—81no—
79C4×D71—81no—
80Dic142—42no—
81C71⋊C8—82no—
82C71⋊C8—81no—
83C22×C71—22no—
84C22×C71#39311yescharacteristic
85C22×C71#44912yes—
86C2×D71—81no—
87C2×D71—82no—
88C4×C71—22no—
89C4×C71—22no—
90C4×C71—44no—
91C4×C71—84no—
92C4×C71#47812yes—
93C4×C71#48811yescharacteristic
94C4×C71#49011yescharacteristic
95Dic71—81no—
96#10366—711nomaximal, Sylow
97C2×C71—22no—
98C2×C71#202511yescharacteristic
99C2×C71#203112yes—
100D71—161no—
101#2864D7111yescharacteristic, complemented
102#1962—711no—
103#1517—712no—
104#1286—712no—
105#1276—711no—
106C71#1036611yescharacteristic, complemented, Sylow
107#532—711no—
108#431—712no—
109#423—711no—
110C2×#104—1422no—
111#389—22no—
112#387—24no—
113#382C2×D7112yes—
114C43C2×D7111yescharacteristic
115#327—1424no—
116C2×#41—1421no—
117C2×M16—1421no—
118C4⋊Q8—711no—
119#127—711no—
120#124—1421no—
121#121—1422no—
122C4×D4—1422no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C42—22no—
126C2×C42C22⋊D7112yes—
127C2×C42C22×D7111yescharacteristic
128C4⋊C8—1422no—
129C4⋊C8—1422no—
130#104—2844no—
131C22⋊C8—1422no—
132C4×C8—1422no—
133#95—24no—
134#95—22no—
135#95C22⋊D7114yes—
136#41—2842no—
137#41—2842no—
138C2×Q8—1421no—
139C2×D4—2841no—
140C2×D4—1421no—
141C2×D4—1422no—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—2841no—
145C22×C4C2×C22⋊D7112yes—
146C22×C4C2×C22⋊D7111yescharacteristic
147M16—2842no—
148C2×C8—2842no—
149C2×C8—2841no—
150C4⋊C4—48no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—1422no—
154C22⋊C4—2842no—
155C42—22no—
156C42—22no—
157C42—44no—
158C42Q8×D7111yescharacteristic
159C42#1176586611yescharacteristic
160C42#1176586612yes—
161C23—2841no—
162C23#1455609711yescharacteristic
163Q8—2842no—
164D4—2842no—
165D4—5682no—
166D4—5682no—
167C2×C4—24no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—48no—
172C2×C4—44no—
173C2×C4—2841no—
174C2×C4—5681no—
175C2×C4—5681no—
176C2×C4#1455610412yes—
177C2×C4#1455610411yescharacteristic
178C2×C4#1455611712yes—
179C2×C4#1455611711yescharacteristic
180C2×C4#1455614812yes—
181C2×C4#1455615312yes—
182C8—5682no—
183C8—5681no—
184C22—22no—
185C22—5681no—
186C22—5682no—
187C22#1798591011yescharacteristic
188C22#1798614412yes—
189C4—22no—
190C4—22no—
191C4—44no—
192C4—84no—
193C4—5681no—
194C4#1798629612yes—
195C4#1798635611yescharacteristic
196C4#1798636511yescharacteristic
197C2—22no—
198C2—11361no—
199C2#2258948311yescharacteristic
200C2#2258950912yes—
subgroupG/H
centerC22#17985910
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2864C2
socleC22×C71#393
prime radical#17986549C22
hypercenter#2864D71
hypocenterC71#10366
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2864

Complements

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

NH ≅ G/Ntype
3. #22586198C2semidirect
4. #22586173C2semidirect
5. C71×#2864C2semidirect
101. #2864D71semidirect
106. C71#10366semidirect

Maximal subgroups

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

subgroupindex
#225891682
#225868992
#225861982
#225861732
C71×#28642
#1036671