Finite Group Databasebeta

← #31900546

Subgroups — #31900546

Order 18176.

416 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 97 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#22596718C211yesmaximal, characteristic
2#22589317C211yesmaximal, characteristic
3#22586905C212yesmaximal
4#22586199C212yesmaximal
5C71×#1276C211yesmaximal, characteristic
6#17986556C2211yescharacteristic
7C2×#14556128—21no—
8#17986023C2211yescharacteristic
9#17985905—24no—
10#17985865—21no—
11#17985865C2212yes—
12#17985759—21no—
13#17985759—21no—
14#17985737—24no—
15#431×C71C2212yes—
16C43×C71C2211yescharacteristic
17C22×Dic142—22no—
18C2×#11765849—21no—
19#14556128—42no—
20#14556128—42no—
21#14556128Q812yescomplemented
22#14556120—21no—
23#14556107—22no—
24C4×Dic142—22no—
25#14556087—21no—
26C2×C4⋊Dic71—22no—
27C2×C4⋊Dic71—22no—
28C4⋊(C71⋊C8)—22no—
29C71⋊#95—24no—
30C71⋊#95—22no—
31C71⋊#95D414yes—
32C2×C42×C71—22no—
33C2×C42×C71D411yescharacteristic
34C2×C42×C71D411yescharacteristic
35C2×C42×C71C2311yescharacteristic
36C4⋊C8×C71—22no—
37C22⋊C8×C71—22no—
38C4×C8×C71—22no—
39C2×Dic142—22no—
40C2×Dic142—48no—
41C2×Dic142—44no—
42C22×Dic71—24no—
43#11765849—41no—
44#11765849—41no—
45C2×C71⋊C8—41no—
46C4⋊Dic71—22no—
47C4⋊Dic71—22no—
48C4⋊Dic71—44no—
49C4⋊Dic71—42no—
50C4⋊Dic71—42no—
51Dic71⋊C4—42no—
52C22⋊Dic71—41no—
53C4×Dic71—41no—
54C22×C4×C71—21no—
55C22×C4×C71—21no—
56C22×C4×C71C2×D412yes—
57C22×C4×C71C2×D411yescharacteristic
58C2×C8×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
68C2×C4×C71—24no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71—42no—
72C2×C4×C71—42no—
73C2×C4×C71C22⋊Q812yes—
74C2×C4×C71C22⋊Q811yescharacteristic
75C2×C4×C71#12412yes—
76C2×C4×C71#12411yescharacteristic
77C2×C4×C71C2×SD1611yescharacteristic
78C2×C4×C71C2×Q1611yescharacteristic
79C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
80C2×C4×C71#13711yescharacteristic
81C2×Dic71—48no—
82C2×Dic71—42no—
83C2×Dic71—44no—
84Dic142—88no—
85Dic142—82no—
86C8×C71—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#47511yescharacteristic
97C4×C71#47711yescharacteristic
98C4×C71#48711yescharacteristic
99C4×C71#48911yescharacteristic
100Dic71—84no—
101Dic71—82no—
102#10363—711nomaximal, Sylow
103C2×C71—21no—
104C2×C71—21no—
105C2×C71#202411yescharacteristic
106C2×C71#203112yes—
107#2862—711no—
108#1994—711no—
109#1518—712no—
110#1286—712no—
111#1276D7111yescharacteristic
112C71#1036311yescharacteristic, complemented, Sylow
113#533—711no—
114C2×C4⋊Q8—1421no—
115#431C2×D7112yes—
116#423—711no—
117#391—1424no—
118#382—712no—
119#382—1421no—
120C43C2×D7111yescharacteristic
121#339—1421no—
122#339—1421no—
123#328—1424no—
124C22×Q8—1422no—
125C2×M16—1421no—
126C4⋊Q8—2842no—
127C4⋊Q8—2842no—
128C4⋊Q8—712no—
129#125—1421no—
130C22⋊Q8—1422no—
131C4×Q8—1422no—
132#117—1421no—
133C2×C4⋊C4—1422no—
134C2×C4⋊C4—1422no—
135C2×C42—22no—
136C2×C42D28411yescharacteristic
137C2×C42D28411yescharacteristic
138C2×C42C22×D7111yescharacteristic
139C4⋊C8—22no—
140C4⋊C8—1422no—
141C22⋊C8—22no—
142C4×C8—22no—
143#95—714no—
144#95—1424no—
145#95—1422no—
146C2×Q8—2844no—
147C2×Q8—2848no—
148C2×Q8—1422no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—1424no—
152C22×C4D4×D7112yes—
153C22×C4C2×D28411yescharacteristic
154M16—2841no—
155M16—2841no—
156C2×C8—42no—
157C2×C8—2841no—
158C4⋊C4—2844no—
159C4⋊C4—2842no—
160C4⋊C4—2842no—
161C4⋊C4—2842no—
162C4⋊C4—1422no—
163C4⋊C4—1422no—
164C22⋊C4—2841no—
165C42—21no—
166C42—21no—
167C42—44no—
168C42—2841no—
169C42#1176585412yes—
170C42Dic28412yes—
171C42Q8×D7111yescharacteristic
172C42#1176586411yescharacteristic
173C42#1176586512yes—
174C23#1455609811yescharacteristic
175Q8—5682no—
176Q8—5688no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—2848no—
183C2×C4—2844no—
184C2×C4—2842no—
185C2×C4#1455610612yes—
186C2×C4#1455610811yescharacteristic
187C2×C4#1455611412yes—
188C2×C4#1455611611yescharacteristic
189C2×C4C2×#1176585411yescharacteristic
190C2×C4C2×Dic28411yescharacteristic
191C2×C4#1455615611yescharacteristic
192C2×C4#1455616111yescharacteristic
193C8—82no—
194C8—5681no—
195C22—22no—
196C22#1798591211yescharacteristic
197C22#1798615211yescharacteristic
198C22#1798615711yescharacteristic
199C4—24no—
200C4—22no—
201C4—22no—
202C4—42no—
203C4—5684no—
204C4—5682no—
205C4#1798628511yescharacteristic
206C4#1798629511yescharacteristic
207C4#1798635011yescharacteristic
208C4#1798635911yescharacteristic
209C2—21no—
210C2—21no—
211C2#2258948211yescharacteristic
212C2#2258951312yes—
subgroupG/H
centerC22#17985912
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1276C2
socleC22×C71#393
prime radicalC23×C71#120
hypercenter#1276D71
hypocenterC71#10363
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
21. #14556128Q8semidirect
112. C71#10363semidirect

Maximal subgroups

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

subgroupindex
#225967182
#225893172
#225869052
#225861992
C71×#12762
#1036371