Finite Group Databasebeta

← #31917818

Subgroups — #31917818

Order 18176.

318 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 79 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#22590025C212yesmaximal
2#22586513C214yesmaximal, complemented
3C71×#3142C211yesmaximal, characteristic
4#17985983—22no—
5#17985798C2214yes—
6#17985796—22no—
7#532×C71C2211yescharacteristic
8C2×#125×C71C411yescharacteristic
9#491×C71C2212yes—
10#490×C71C412yes—
11#488×C71—22no—
12#417×C71—22no—
13C4×M16×C71C411yescharacteristic
14C2×#11765849—42no—
15#14556047—24no—
16#14556045—42no—
17C2×#41×C71D412yes—
18C2×M16×C71—21no—
19C4⋊Q8×C71C2311yescharacteristic
20#127×C71C2×C411yescharacteristic
21#125×C71—21no—
22#125×C71—41no—
23#125×C71C2×C412yes—
24#124×C71D412yes—
25#121×C71—41no—
26C4×D4×C71—41no—
27C2×C4⋊C4×C71—21no—
28C2×C4⋊C4×C71—41no—
29C2×C42×C71C2×C411yescharacteristic
30#103×C71—24no—
31#102×C71—24no—
32#97×C71C2×C412yes—
33C4×C8×C71—21no—
34#11765849—44no—
35#11765849—82no—
36C2×C71⋊C8—82no—
37#41×C71—42no—
38#41×C71—42no—
39C2×Q8×C71C2×D412yes—
40C2×D4×C71—22no—
41C2×D4×C71—41no—
42C2×D4×C71C22⋊C412yes—
43C22×C4×C71—21no—
44C22×C4×C71—22no—
45C22×C4×C71—41no—
46C22×C4×C71C22⋊C411yescharacteristic
47M16×C71—42no—
48C2×C8×C71—22no—
49C4⋊C4×C71—22no—
50C4⋊C4×C71—41no—
51C4⋊C4×C71—42no—
52C4⋊C4×C71—42no—
53C4⋊C4×C71—41no—
54C22⋊C4×C71—42no—
55C42×C71—21no—
56C42×C71C22⋊C411yescharacteristic
57C42×C71C22×C411yescharacteristic
58C23×C71—22no—
59C23×C71#9911yescharacteristic
60Q8×C71—42no—
61D4×C71—42no—
62D4×C71—82no—
63D4×C71—82no—
64C2×C4×C71—22no—
65C2×C4×C71—21no—
66C2×C4×C71—21no—
67C2×C4×C71—22no—
68C2×C4×C71—42no—
69C2×C4×C71—42no—
70C2×C4×C71—42no—
71C2×C4×C71—81no—
72C2×C4×C71—82no—
73C2×C4×C71#9911yescharacteristic
74C2×C4×C71C2×C22⋊C411yescharacteristic
75C8×C71—42no—
76C71⋊C8—84no—
77C22×C71—21no—
78C22×C71—41no—
79C22×C71—42no—
80C22×C71—82no—
81C22×C71C2×#9911yescharacteristic
82C4×C71—21no—
83C4×C71—41no—
84C4×C71—41no—
85C4×C71—42no—
86C4×C71—82no—
87C4×C71#35412yes—
88#13600—711nomaximal, Sylow
89C2×C71—21no—
90C2×C71—41no—
91C2×C71—82no—
92C2×C71C2×#35411yescharacteristic
93#3142D7111yescharacteristic
94#2132—712no—
95#1413—714no—
96C71#1360011yescharacteristic, complemented, Sylow
97#532C2×D7111yescharacteristic
98C2×#125Dic7111yescharacteristic
99#491C2×D7112yes—
100#490Dic7112yes—
101#488—22no—
102#417—22no—
103#413—1422no—
104C4×M16Dic7111yescharacteristic
105#356—714no—
106#355—1422no—
107C2×#41C22⋊D7112yes—
108C2×M16—21no—
109C2×M16—2842no—
110C4⋊Q8C22×D7111yescharacteristic
111#127C2×Dic7111yescharacteristic
112#125—21no—
113#125—41no—
114#125C2×Dic7112yes—
115#124C22⋊D7112yes—
116#121—41no—
117C4×D4—41no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—41no—
120C2×C42C2×Dic7111yescharacteristic
121#103—24no—
122#102—24no—
123#101—1424no—
124#100—2842no—
125#97C2×Dic7112yes—
126C4×C8—21no—
127#41—42no—
128#41—42no—
129C2×Q8C2×C22⋊D7112yes—
130C2×D4—22no—
131C2×D4—41no—
132C2×D4C22⋊Dic7112yes—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—41no—
136C22×C4C22⋊Dic7111yescharacteristic
137M16—42no—
138M16—2844no—
139M16—5682no—
140C2×C8—22no—
141C2×C8—5682no—
142C4⋊C4—22no—
143C4⋊C4—41no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—41no—
147C22⋊C4—42no—
148C42—21no—
149C42C22⋊Dic7111yescharacteristic
150C42C22×Dic7111yescharacteristic
151C23—22no—
152C23#1455604311yescharacteristic
153Q8—42no—
154D4—42no—
155D4—82no—
156D4—82no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—81no—
165C2×C4—82no—
166C2×C4#1455604311yescharacteristic
167C2×C4C2×C22⋊Dic7111yescharacteristic
168C8—42no—
169C8—5684no—
170C22—21no—
171C22—41no—
172C22—42no—
173C22—82no—
174C22C2×#1455604311yescharacteristic
175C4—21no—
176C4—41no—
177C4—41no—
178C4—42no—
179C4—82no—
180C4#1798579412yes—
181C2—21no—
182C2—41no—
183C2—82no—
184C2C2×#1798579411yescharacteristic
subgroupG/H
centerC2C2×#17985794
commutatorC42×C71C22×C4
FrattiniC4⋊Q8C22×D71
FittingC71×#3142C2
socleC2×C71C2×#354
prime radical#532×C71C22
cosocleC4⋊Q8×C71C23
hypercenter#3142D71
hypocenterC71#13600
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3142

Complements

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

NH ≅ G/Ntype
2. #22586513C2semidirect
96. C71#13600semidirect

Maximal subgroups

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

subgroupindex
#225900252
#225865132
C71×#31422
#1360071