Finite Group Databasebeta

← #31892417

Subgroups — #31892417

Order 18176.

398 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 103 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#22597446C211yesmaximal, characteristic, complemented
2#22587005C211yesmaximal, characteristic, complemented
3#22586809C212yesmaximal
4#22586239C212yesmaximal, complemented
5C71×#3135C211yesmaximal, characteristic
6#17986130—24no—
7#17986060—22no—
8#17986056C2212yes—
9#17986052C2212yes—
10#17985745—22no—
11D4⋊(C71⋊C8)—24no—
12#533×C71C2211yescharacteristic
13#532×C71C2211yescharacteristic
14#491×C71C412yes—
15#490×C71C412yes—
16#489×C71—22no—
17#416×C71—22no—
18C2×#97×C71C2211yescharacteristic
19C4⋊(C71⋊C8)—24no—
20C4⋊(C71⋊C8)—22no—
21C4⋊(C71⋊C8)D414yes—
22C22⋊(C71⋊C8)—22no—
23C22⋊(C71⋊C8)—22no—
24#14556039—22no—
25#14556039—22no—
26C4×C71⋊C8—22no—
27C4×C71⋊C8—22no—
28C2×#41×C71—21no—
29C22×C8×C71—21no—
30C4⋊Q8×C71C2×C411yescharacteristic
31C4⋊Q8×C71D412yes—
32#127×C71C2×C411yescharacteristic
33#125×C71C2×C412yes—
34#124×C71—21no—
35C22⋊Q8×C71—22no—
36#121×C71—22no—
37C4×Q8×C71—22no—
38C4×D4×C71—22no—
39#117×C71D412yes—
40C2×C42×C71C2311yescharacteristic
41#103×C71—24no—
42#102×C71—24no—
43#97×C71—21no—
44#97×C71C2×C412yes—
45C2×C71⋊C8—24no—
46C2×C71⋊C8—42no—
47#41×C71—44no—
48C2×Q8×C71—22no—
49C2×Q8×C71—21no—
50C2×D4×C71—21no—
51C2×D4×C71—22no—
52C2×D4×C71—41no—
53C22×C4×C71—41no—
54C22×C4×C71C2×D411yescharacteristic
55C22×C4×C71#4112yes—
56C2×C8×C71—22no—
57C2×C8×C71—42no—
58C4⋊C4×C71—24no—
59C4⋊C4×C71—22no—
60C4⋊C4×C71—22no—
61C4⋊C4×C71C22⋊C414yes—
62C22⋊C4×C71—22no—
63C22⋊C4×C71—42no—
64C42×C71—22no—
65C42×C71C22×C411yescharacteristic
66C42×C71C2×D412yes—
67C42×C71C2×D411yescharacteristic
68C23×C71—41no—
69C23×C71#12311yescharacteristic
70Q8×C71—42no—
71Q8×C71—82no—
72D4×C71—44no—
73D4×C71—42no—
74D4×C71—82no—
75C2×C4×C71—22no—
76C2×C4×C71—24no—
77C2×C4×C71—42no—
78C2×C4×C71—41no—
79C2×C4×C71—82no—
80C2×C4×C71C2×C22⋊C411yescharacteristic
81C2×C4×C71C4×D412yes—
82C2×C4×C71#12011yescharacteristic
83C2×C4×C71#12112yes—
84C8×C71—42no—
85C71⋊C8—44no—
86C71⋊C8—82no—
87C22×C71—22no—
88C22×C71—21no—
89C22×C71—82no—
90C22×C71—81no—
91C22×C71#38611yescharacteristic
92C4×C71—22no—
93C4×C71—22no—
94C4×C71—21no—
95C4×C71—44no—
96C4×C71—81no—
97C4×C71#45412yes—
98#9008—711nomaximal, Sylow
99C2×C71—41no—
100C2×C71—161no—
101C2×C71#188112yes—
102C2×C71#188811yescharacteristic
103#3135D7111yescharacteristic
104#2987—711no—
105#1544—711no—
106#1496—712no—
107#1302—712no—
108C71#900811yescharacteristic, complemented, Sylow
109#533C2×D7111yescharacteristic
110#532C2×D7111yescharacteristic
111#491Dic7112yes—
112#490Dic7112yes—
113#489—22no—
114#446—1424no—
115#433—1422no—
116#432—712no—
117#431—712no—
118#416—22no—
119C2×#97C2×D7111yescharacteristic
120#332—1422no—
121D4⋊C8—1424no—
122C2×#41—21no—
123C22×C8—21no—
124C4⋊Q8C2×Dic7111yescharacteristic
125C4⋊Q8C22⋊D7112yes—
126#127C2×Dic7111yescharacteristic
127#125C2×Dic7112yes—
128#124—21no—
129C22⋊Q8—22no—
130#121—22no—
131C4×Q8—22no—
132C4×D4—22no—
133#117C22⋊D7112yes—
134C2×C42C22×D7111yescharacteristic
135C4⋊C8—714no—
136C4⋊C8—1424no—
137C4⋊C8—1422no—
138#103—24no—
139#102—24no—
140C22⋊C8—1422no—
141C22⋊C8—1422no—
142#97—21no—
143#97—1422no—
144#97—1422no—
145#97C2×Dic7112yes—
146C4×C8—1422no—
147C4×C8—1422no—
148#41—44no—
149C2×Q8—22no—
150C2×Q8—21no—
151C2×D4—21no—
152C2×D4—22no—
153C2×D4—41no—
154C22×C4—41no—
155C22×C4C2×C22⋊D7111yescharacteristic
156C22×C4#1176586412yes—
157C2×C8—22no—
158C2×C8—42no—
159C2×C8—2842no—
160C2×C8—1424no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4C22⋊Dic7114yes—
165C22⋊C4—22no—
166C22⋊C4—42no—
167C42—22no—
168C42C22×Dic7111yescharacteristic
169C42D4×D7112yes—
170C42C2×C22⋊D7111yescharacteristic
171C23—41no—
172C23#1455610911yescharacteristic
173Q8—42no—
174Q8—82no—
175D4—42no—
176D4—44no—
177D4—82no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—82no—
183C2×C4C2×C22⋊Dic7111yescharacteristic
184C2×C4D4×Dic7112yes—
185C2×C4#1455609611yescharacteristic
186C2×C4#1455610012yes—
187C8—42no—
188C8—2844no—
189C8—5682no—
190C22—22no—
191C22—21no—
192C22—82no—
193C22—81no—
194C22#1798587911yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—44no—
199C4—81no—
200C4#1798617112yes—
201C2—41no—
202C2—161no—
203C2#2258870412yes—
204C2#2258874411yescharacteristic
subgroupG/H
centerC2×C4#14556096
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#3135C2
socleC22×C71#386
prime radical#532×C71C22
cosocleC2×C42×C71C23
hypercenter#3135D71
hypocenterC71#9008
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3135

Complements

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

NH ≅ G/Ntype
1. #22597446C2semidirect
2. #22587005C2semidirect
4. #22586239C2semidirect
108. C71#9008semidirect

Maximal subgroups

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

subgroupindex
#225974462
#225870052
#225868092
#225862392
C71×#31352
#900871