Finite Group Databasebeta

← #31901592

Subgroups — #31901592

Order 18176.

370 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 77 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#22588480C211yesmaximal, characteristic, complemented
2#22587062C212yesmaximal, complemented
3#22586951C212yesmaximal, complemented
4#22586251C211yesmaximal, characteristic, complemented
5C71×#3253C211yesmaximal, characteristic
6#17986056C2212yes—
7#17986023C2211yescharacteristic
8C2×#14556039C2211yescharacteristic
9#17985763—22no—
10#17985745—22no—
11#17985743—22no—
12#17985739—24no—
13#532×C71C2212yes—
14#476×C71—22no—
15#469×C71—22no—
16#459×C71—22no—
17#423×C71C2211yescharacteristic
18C2×#11765849—21no—
19C22×C71⋊C8—21no—
20C4⋊(C71⋊C8)—22no—
21C4⋊(C71⋊C8)—22no—
22C22⋊(C71⋊C8)—22no—
23#14556039—22no—
24#14556039—21no—
25#14556039D412yes—
26C2×#41×C71—22no—
27C2×D8×C71—22no—
28C2×M16×C71D412yes—
29C4⋊Q8×C71D412yes—
30#127×C71D412yes—
31#124×C71—22no—
32#121×C71—22no—
33#121×C71—22no—
34C4×D4×C71—22no—
35C4×D4×C71—22no—
36C2×C42×C71C2311yescharacteristic
37#106×C71—22no—
38C4⋊C8×C71—22no—
39#102×C71—24no—
40#11765849—42no—
41C2×C71⋊C8—22no—
42C2×C71⋊C8—41no—
43C2×C71⋊C8—41no—
44C2×C71⋊C8—41no—
45C2×C71⋊C8—42no—
46#41×C71—84no—
47C2×Q8×C71—22no—
48C2×D4×C71—22no—
49C2×D4×C71—22no—
50C2×D4×C71—22no—
51C2×D4×C71—42no—
52C22×C4×C71—42no—
53C22×C4×C71C2×D411yescharacteristic
54C22×C4×C71#4112yes—
55D8×C71—82no—
56M16×C71—42no—
57C2×C8×C71—22no—
58C4⋊C4×C71—22no—
59C4⋊C4×C71—22no—
60C22⋊C4×C71—42no—
61C22⋊C4×C71—42no—
62C42×C71C2×D411yescharacteristic
63C42×C71C2×D412yes—
64C42×C71#4111yescharacteristic
65C23×C71—42no—
66C23×C71#12411yescharacteristic
67Q8×C71—82no—
68D4×C71—42no—
69D4×C71—42no—
70D4×C71—82no—
71D4×C71—82no—
72D4×C71—84no—
73C2×C4×C71—21no—
74C2×C4×C71—21no—
75C2×C4×C71—42no—
76C2×C4×C71—42no—
77C2×C4×C71—84no—
78C2×C4×C71#12011yescharacteristic
79C2×C4×C71#12111yescharacteristic
80C2×C4×C71#12112yes—
81C2×C4×C71#12312yes—
82C8×C71—42no—
83C71⋊C8—42no—
84C71⋊C8—82no—
85C71⋊C8—81no—
86C22×C71—22no—
87C22×C71—21no—
88C22×C71—82no—
89C22×C71—82no—
90C22×C71—82no—
91C22×C71#39411yescharacteristic
92C4×C71—22no—
93C4×C71—22no—
94C4×C71—21no—
95C4×C71—82no—
96C4×C71#45411yescharacteristic
97C4×C71#45511yescharacteristic
98#10525—711nomaximal, Sylow
99C2×C71—41no—
100C2×C71—162no—
101C2×C71#204011yescharacteristic
102C2×C71#204812yes—
103#3253D7111yescharacteristic
104#1838—711no—
105#1557—712no—
106#1530—712no—
107#1307—711no—
108C71#1052511yescharacteristic, complemented, Sylow
109#532C2×D7112yes—
110#476—22no—
111#469—22no—
112#459—22no—
113#432—712no—
114#423—711no—
115#423C2×D7111yescharacteristic
116C2×#97—711no—
117#341—1422no—
118#332—1422no—
119#331—1422no—
120#329—1424no—
121C2×#41—22no—
122C2×D8—22no—
123C2×M16—1421no—
124C2×M16C22⋊D7112yes—
125C22×C8—1421no—
126C4⋊Q8C22⋊D7112yes—
127#127C22⋊D7112yes—
128#124—22no—
129#121—22no—
130#121—22no—
131C4×D4—22no—
132C4×D4—22no—
133C2×C42C22×D7111yescharacteristic
134#106—22no—
135C4⋊C8—22no—
136C4⋊C8—1422no—
137C4⋊C8—1422no—
138#102—24no—
139C22⋊C8—1422no—
140#97—712no—
141#97—1422no—
142#97—1421no—
143#41—84no—
144C2×Q8—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—42no—
149C22×C4—42no—
150C22×C4C2×C22⋊D7111yescharacteristic
151C22×C4#1176586412yes—
152D8—82no—
153M16—42no—
154M16—2842no—
155C2×C8—22no—
156C2×C8—2841no—
157C2×C8—2841no—
158C2×C8—2841no—
159C2×C8—2842no—
160C2×C8—1422no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C22⋊C4—42no—
164C22⋊C4—42no—
165C42D4×D7111yescharacteristic
166C42C2×C22⋊D7112yes—
167C42#1176586411yescharacteristic
168C23—42no—
169C23#1455611511yescharacteristic
170Q8—82no—
171D4—42no—
172D4—42no—
173D4—82no—
174D4—82no—
175D4—84no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—84no—
181C2×C4#1455609711yescharacteristic
182C2×C4#1455609911yescharacteristic
183C2×C4#1455610012yes—
184C2×C4#1455610912yes—
185C8—42no—
186C8—2842no—
187C8—5682no—
188C8—5681no—
189C22—21no—
190C22—22no—
191C22—82no—
192C22—82no—
193C22—82no—
194C22#1798591311yescharacteristic
195C4—21no—
196C4—22no—
197C4—22no—
198C4—82no—
199C4#1798617211yescharacteristic
200C4#1798617711yescharacteristic
201C2—41no—
202C2—162no—
203C2#2258956611yescharacteristic
204C2#2258961012yes—
subgroupG/H
centerC22#17985913
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
Fitting, prime radicalC71×#3253C2
socleC22×C71#394
hypercenter#3253D71
hypocenterC71#10525
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3253

Complements

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

NH ≅ G/Ntype
1. #22588480C2semidirect
2. #22587062C2semidirect
3. #22586951C2semidirect
4. #22586251C2semidirect
108. C71#10525semidirect

Maximal subgroups

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

subgroupindex
#225884802
#225870622
#225869512
#225862512
C71×#32532
#1052571