Finite Group Databasebeta

← #31900542

Subgroups — #31900542

Order 18176.

416 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 101 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#22589317C211yesmaximal, characteristic
2#22586903C212yesmaximal
3#22586198C212yesmaximal
4#22586173C211yesmaximal, characteristic
5C71×#2862C211yesmaximal, characteristic
6#17986556C2211yescharacteristic
7#17986052C2212yes—
8#17986023C2211yescharacteristic
9#17985759—21no—
10#17985759—21no—
11#17985735—24no—
12C2×C4⋊Q8×C71—21no—
13#391×C71—24no—
14#382×C71—21no—
15#382×C71C2212yes—
16C43×C71C2211yescharacteristic
17C2×#11765849—21no—
18#14556128Q812yescomplemented
19#14556120—21no—
20#14556107—22no—
21C4×Dic142—22no—
22#14556087—21no—
23C4⋊(C71⋊C8)—22no—
24C4⋊(C71⋊C8)—22no—
25C22⋊(C71⋊C8)—22no—
26C4×C71⋊C8—22no—
27C22×Q8×C71—22no—
28C4⋊Q8×C71—42no—
29C4⋊Q8×C71—42no—
30C2×C4⋊C4×C71—22no—
31C2×C4⋊C4×C71—22no—
32C2×C42×C71—22no—
33C2×C42×C71D411yescharacteristic
34C2×C42×C71D411yescharacteristic
35C2×C42×C71C2311yescharacteristic
36#95×C71—24no—
37#95×C71—22no—
38#95×C71D414yes—
39C2×Dic142—22no—
40#11765849—41no—
41#11765849—41no—
42C2×C71⋊C8—42no—
43C2×C71⋊C8—41no—
44C4⋊Dic71—22no—
45C4⋊Dic71—22no—
46Dic71⋊C4—42no—
47C22⋊Dic71—41no—
48C4×Dic71—41no—
49C2×Q8×C71—44no—
50C2×Q8×C71—48no—
51C22×C4×C71—21no—
52C22×C4×C71—21no—
53C22×C4×C71—24no—
54C22×C4×C71C2×D412yes—
55C22×C4×C71C2×D411yescharacteristic
56C4⋊C4×C71—44no—
57C4⋊C4×C71—42no—
58C4⋊C4×C71—42no—
59C42×C71—21no—
60C42×C71—21no—
61C42×C71—44no—
62C42×C71SD1612yes—
63C42×C71Q1612yes—
64C42×C71C2×Q811yescharacteristic
65C42×C71#4111yescharacteristic
66C42×C71#4112yes—
67C23×C71#12011yescharacteristic
68Q8×C71—88no—
69C2×C4×C71—24no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71—42no—
73C2×C4×C71—42no—
74C2×C4×C71—48no—
75C2×C4×C71—44no—
76C2×C4×C71C22⋊Q812yes—
77C2×C4×C71C22⋊Q811yescharacteristic
78C2×C4×C71#12412yes—
79C2×C4×C71#12411yescharacteristic
80C2×C4×C71C2×SD1611yescharacteristic
81C2×C4×C71C2×Q1611yescharacteristic
82C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
83C2×C4×C71#13711yescharacteristic
84C2×Dic71—42no—
85Dic142—82no—
86C71⋊C8—82no—
87C71⋊C8—81no—
88C22×C71—22no—
89C22×C71#39311yescharacteristic
90C22×C71#45011yescharacteristic
91C22×C71#45111yescharacteristic
92C4×C71—24no—
93C4×C71—22no—
94C4×C71—22no—
95C4×C71—42no—
96C4×C71—84no—
97C4×C71#47511yescharacteristic
98C4×C71#47711yescharacteristic
99C4×C71#48711yescharacteristic
100C4×C71#48911yescharacteristic
101Dic71—82no—
102#10363—711nomaximal, Sylow
103C2×C71—21no—
104C2×C71—21no—
105C2×C71#202411yescharacteristic
106C2×C71#203112yes—
107#2862D7111yescharacteristic
108#1994—711no—
109#1518—712no—
110#1286—712no—
111#1276—711no—
112C71#1036311yescharacteristic, complemented, Sylow
113#533—711no—
114C2×C4⋊Q8—21no—
115#431—712no—
116#423—711no—
117#391—24no—
118#382—21no—
119#382C2×D7112yes—
120C43C2×D7111yescharacteristic
121#339—1421no—
122#339—1421no—
123#328—1424no—
124C22×Q8—22no—
125C2×M16—1421no—
126C4⋊Q8—42no—
127C4⋊Q8—42no—
128C4⋊Q8—712no—
129#125—1421no—
130C22⋊Q8—1422no—
131C4×Q8—1422no—
132#117—1421no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C42—22no—
136C2×C42C22⋊D7111yescharacteristic
137C2×C42C22⋊D7111yescharacteristic
138C2×C42C22×D7111yescharacteristic
139C4⋊C8—1422no—
140C4⋊C8—1422no—
141C22⋊C8—1422no—
142C4×C8—1422no—
143#95—24no—
144#95—22no—
145#95C22⋊D7114yes—
146C2×Q8—44no—
147C2×Q8—48no—
148C2×Q8—1422no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—24no—
152C22×C4C2×C22⋊D7112yes—
153C22×C4C2×C22⋊D7111yescharacteristic
154M16—2841no—
155M16—2841no—
156C2×C8—2842no—
157C2×C8—2841no—
158C4⋊C4—44no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—2842no—
162C4⋊C4—1422no—
163C4⋊C4—1422no—
164C22⋊C4—2841no—
165C42—21no—
166C42—21no—
167C42—44no—
168C42—2841no—
169C42#1176585312yes—
170C42#1176585512yes—
171C42Q8×D7111yescharacteristic
172C42#1176586611yescharacteristic
173C42#1176586612yes—
174C23#1455609711yescharacteristic
175Q8—88no—
176Q8—5682no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—48no—
183C2×C4—44no—
184C2×C4—2842no—
185C2×C4#1455610412yes—
186C2×C4#1455610411yescharacteristic
187C2×C4#1455611712yes—
188C2×C4#1455611711yescharacteristic
189C2×C4C2×#1176585311yescharacteristic
190C2×C4C2×#1176585511yescharacteristic
191C2×C4#1455615411yescharacteristic
192C2×C4#1455615911yescharacteristic
193C8—5682no—
194C8—5681no—
195C22—22no—
196C22#1798591011yescharacteristic
197C22#1798615011yescharacteristic
198C22#1798615511yescharacteristic
199C4—24no—
200C4—22no—
201C4—22no—
202C4—42no—
203C4—84no—
204C4—5682no—
205C4#1798628111yescharacteristic
206C4#1798629111yescharacteristic
207C4#1798635111yescharacteristic
208C4#1798636111yescharacteristic
209C2—21no—
210C2—21no—
211C2#2258947811yescharacteristic
212C2#2258950912yes—
subgroupG/H
centerC22#17985910
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2862C2
socleC22×C71#393
prime radicalC23×C71#120
hypercenter#2862D71
hypocenterC71#10363
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
18. #14556128Q8semidirect
112. C71#10363semidirect

Maximal subgroups

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

subgroupindex
#225893172
#225869032
#225861982
#225861732
C71×#28622
#1036371