Finite Group Databasebeta

← #31885726

Subgroups — #31885726

Order 18176.

468 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 127 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#22588089C212yesmaximal
2#22586166C214yesmaximal
3C4×C4⋊Q8×C71C211yesmaximal, characteristic
4C2×#14556039—22no—
5#17985754—22no—
6#17985754C2214yes—
7C2×C4⋊Q8×C71C411yescharacteristic
8C2×C4×Q8×C71C412yes—
9#391×C71C2212yes—
10#389×C71—22no—
11#384×C71C411yescharacteristic
12C4×C4⋊C4×C71—22no—
13C43×C71C2211yescharacteristic
14C22×C71⋊C8—24no—
15#14556039—28no—
16C22×Q8×C71C2×C412yes—
17C4⋊Q8×C71—24no—
18C4×Q8×C71—44no—
19C2×C4⋊C4×C71—22no—
20C2×C4⋊C4×C71—22no—
21C2×C4⋊C4×C71—22no—
22C2×C4⋊C4×C71C2×C412yes—
23C2×C42×C71—22no—
24C2×C42×C71C2×C411yescharacteristic
25C2×C42×C71C2×C411yescharacteristic
26C2×C42×C71D412yes—
27C2×C42×C71D412yes—
28C2×C42×C71C2311yescharacteristic
29#95×C71—24no—
30C2×C71⋊C8—28no—
31C2×C71⋊C8—44no—
32C2×C71⋊C8—44no—
33C2×Q8×C71—22no—
34C2×Q8×C71—42no—
35C2×Q8×C71—42no—
36C2×Q8×C71M1614yes—
37C22×C4×C71—22no—
38C22×C4×C71—22no—
39C22×C4×C71C22⋊C411yescharacteristic
40C22×C4×C71C22⋊C411yescharacteristic
41C22×C4×C71C22⋊C412yes—
42C22×C4×C71C22×C411yescharacteristic
43C22×C4×C71C2×D412yes—
44C4⋊C4×C71—28no—
45C4⋊C4×C71—42no—
46C4⋊C4×C71—44no—
47C4⋊C4×C71—44no—
48C42×C71—22no—
49C42×C71—24no—
50C42×C71—22no—
51C42×C71—24no—
52C42×C71—41no—
53C42×C71—44no—
54C23×C71C2×C22⋊C411yescharacteristic
55Q8×C71—44no—
56Q8×C71—44no—
57C2×C4×C71—22no—
58C2×C4×C71—21no—
59C2×C4×C71—22no—
60C2×C4×C71—21no—
61C2×C4×C71—22no—
62C2×C4×C71—22no—
63C2×C4×C71—22no—
64C2×C4×C71—24no—
65C2×C4×C71—21no—
66C2×C4×C71—22no—
67C2×C4×C71—42no—
68C2×C4×C71—42no—
69C2×C4×C71—42no—
70C2×C4×C71—44no—
71C2×C4×C71#10012yes—
72C2×C4×C71#10414yes—
73C2×C4×C71C2×M1612yes—
74C2×C4×C71#13714yes—
75C71⋊C8—48no—
76C22×C71#40711yescharacteristic
77C22×C71#41011yescharacteristic
78C22×C71C2×#10011yescharacteristic
79C22×C71#41912yes—
80C22×C71C2×#10412yes—
81C4×C71—22no—
82C4×C71—22no—
83C4×C71—24no—
84C4×C71—24no—
85C4×C71—44no—
86C4×C71—41no—
87C4×C71—82no—
88#7639—711nomaximal, Sylow
89C2×C71#146611yescharacteristic
90C2×C71#149112yes—
91C2×C71#150612yes—
92C2×C71#152511yescharacteristic
93C2×C71#155811yescharacteristic
94C4×C4⋊Q8D7111yescharacteristic
95#1745—712no—
96#1272—714no—
97C71#763911yescharacteristic, complemented, Sylow
98C2×C4⋊Q8Dic7111yescharacteristic
99C2×C4×Q8Dic7112yes—
100C2×#97—1422no—
101#391C2×D7112yes—
102#389—22no—
103#384Dic7111yescharacteristic
104C4×C4⋊C4—22no—
105C43C2×D7111yescharacteristic
106#336—714no—
107#336—1422no—
108C22×Q8C2×Dic7112yes—
109C22×C8—1424no—
110C4⋊Q8—24no—
111C4×Q8—44no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4C2×Dic7112yes—
116C2×C42—22no—
117C2×C42C2×Dic7111yescharacteristic
118C2×C42C2×Dic7111yescharacteristic
119C2×C42C22⋊D7112yes—
120C2×C42C22⋊D7112yes—
121C2×C42C22×D7111yescharacteristic
122#97—1428no—
123#95—24no—
124C2×Q8—22no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8#1176584914yes—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4C22⋊Dic7111yescharacteristic
131C22×C4C22⋊Dic7111yescharacteristic
132C22×C4C22⋊Dic7112yes—
133C22×C4C22×Dic7111yescharacteristic
134C22×C4C2×C22⋊D7112yes—
135C2×C8—2844no—
136C2×C8—2844no—
137C2×C8—1428no—
138C4⋊C4—28no—
139C4⋊C4—42no—
140C4⋊C4—44no—
141C4⋊C4—44no—
142C42—22no—
143C42—24no—
144C42—22no—
145C42—24no—
146C42—41no—
147C42—44no—
148C23C2×C22⋊Dic7111yescharacteristic
149Q8—44no—
150Q8—44no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—44no—
165C2×C4#1455604512yes—
166C2×C4#1455605514yes—
167C2×C4C2×#1176584912yes—
168C2×C4#1455615914yes—
169C8—2848no—
170C22#1798596311yescharacteristic
171C22#1798597311yescharacteristic
172C22C2×#1455604511yescharacteristic
173C22#1798600812yes—
174C22C2×#1455605512yes—
175C4—22no—
176C4—24no—
177C4—22no—
178C4—24no—
179C4—41no—
180C4—44no—
181C4—82no—
182C2#2258668111yescharacteristic
183C2#2258678712yes—
184C2#2258685212yes—
185C2#2258693211yescharacteristic
186C2#2258706611yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC4×C4⋊Q8×C71C2
socle, prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenterC4×C4⋊Q8D71
hypocenterC71#7639
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C43 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
97. C71#7639semidirect

Maximal subgroups

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

subgroupindex
#225880892
#225861662
C4×C4⋊Q8×C712
#763971