Finite Group Databasebeta

← #31901952

Subgroups — #31901952

Order 18176.

374 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 95 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#22596612C211yesmaximal, characteristic
2#22588475C212yesmaximal
3#22586222C212yesmaximal
4#22586221C211yesmaximal, characteristic
5C71×#1847C211yesmaximal, characteristic
6C2×#14556039C2212yes—
7#17985941—22no—
8#17985938—21no—
9#17985920—22no—
10#17985919—21no—
11#17985865C2212yes—
12#17985761—22no—
13#17985761—22no—
14C2×#107×C71—21no—
15C2×#106×C71—21no—
16C2×#97×C71C2211yescharacteristic
17#384×C71C2211yescharacteristic
18#382×C71C2211yescharacteristic
19C22×Dic142—21no—
20C22×C71⋊C8—22no—
21C2×C4⋊Dic71—21no—
22C2×C4⋊Dic71D412yes—
23C2×C4⋊Dic71D412yes—
24C2×Dic71⋊C4—22no—
25#14556039—22no—
26#14556039Q814yescomplemented
27C71⋊#95—21no—
28C71⋊#95—22no—
29C71⋊#95—21no—
30C71⋊#95—22no—
31C71⋊#95—22no—
32C22×C8×C71—21no—
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71—22no—
35C2×C4⋊C4×C71D411yescharacteristic
36C2×C4⋊C4×C71D411yescharacteristic
37C2×C42×C71C2311yescharacteristic
38#107×C71—42no—
39#106×C71—42no—
40#97×C71—21no—
41#97×C71Q812yescomplemented
42#95×C71—22no—
43C2×Dic142—81no—
44C2×Dic142—82no—
45C22×Dic71—22no—
46C22×Dic71—22no—
47C2×C71⋊C8—24no—
48C2×C71⋊C8—42no—
49C2×C71⋊C8—42no—
50C4⋊Dic71—42no—
51C4⋊Dic71—42no—
52C4⋊Dic71—81no—
53Dic71⋊C4—82no—
54C22×C4×C71—21no—
55C22×C4×C71—21no—
56C22×C4×C71—41no—
57C22×C4×C71C2×D412yes—
58C22×C4×C71C2×D411yescharacteristic
59C2×C8×C71—22no—
60C2×C8×C71—22no—
61C2×C8×C71—22no—
62C4⋊C4×C71—41no—
63C4⋊C4×C71—41no—
64C4⋊C4×C71—41no—
65C4⋊C4×C71—41no—
66C4⋊C4×C71—44no—
67C42×C71C2×Q812yes—
68C42×C71C2×Q811yescharacteristic
69C42×C71#4111yescharacteristic
70C23×C71#12011yescharacteristic
71C2×C4×C71—21no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71—21no—
75C2×C4×C71—41no—
76C2×C4×C71—41no—
77C2×C4×C71—42no—
78C2×C4×C71—42no—
79C2×C4×C71—81no—
80C2×C4×C71—82no—
81C2×C4×C71C22⋊Q812yes—
82C2×C4×C71C22⋊Q811yescharacteristic
83C2×C4×C71C4⋊Q811yescharacteristic
84C2×C4×C71C4⋊Q812yes—
85C2×Dic71—42no—
86C2×Dic71—42no—
87C2×Dic71—42no—
88C2×Dic71—42no—
89C2×Dic71—42no—
90C2×Dic71—42no—
91Dic142—84no—
92C8×C71—24no—
93C71⋊C8—44no—
94C22×C71#39511yescharacteristic
95C22×C71#45311yescharacteristic
96C22×C71#45412yes—
97C22×C71#45512yes—
98C22×C71#45611yescharacteristic
99C4×C71—21no—
100C4×C71—21no—
101C4×C71—22no—
102C4×C71—22no—
103C4×C71—81no—
104C4×C71—81no—
105C4×C71—161no—
106Dic71—82no—
107Dic71—82no—
108#10584—711nomaximal, Sylow
109C2×C71#165711yescharacteristic
110C2×C71#165911yescharacteristic
111C2×C71#166412yes—
112C2×C71#206011yescharacteristic
113C2×C71#206112yes—
114#2844—711no—
115#1847D7111yescharacteristic
116#1836—712no—
117#1295—712no—
118#1295—711no—
119C71#1058411yescharacteristic, complemented, Sylow
120C2×#107—21no—
121C2×#106—21no—
122C2×#97—712no—
123C2×#97C2×D7111yescharacteristic
124#400—1422no—
125#400—1421no—
126#395—1422no—
127#395—1421no—
128#384C2×D7111yescharacteristic
129#382—712no—
130#382C2×D7111yescharacteristic
131#340—1422no—
132#340—1422no—
133C22×Q8—1421no—
134C22×C8—21no—
135C22×C8—1422no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—712no—
139C2×C4⋊C4—712no—
140C2×C4⋊C4—1422no—
141C2×C4⋊C4—1421no—
142C2×C4⋊C4C22⋊D7111yescharacteristic
143C2×C4⋊C4C22⋊D7111yescharacteristic
144C2×C42C22×D7111yescharacteristic
145#107—42no—
146#106—42no—
147#97—21no—
148#97—714no—
149#97—1422no—
150#97Dic14212yescomplemented
151#95—22no—
152#95—1422no—
153#95—1421no—
154#95—1421no—
155#95—1422no—
156#95—1422no—
157C2×Q8—5681no—
158C2×Q8—5682no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—41no—
162C22×C4—1422no—
163C22×C4—1422no—
164C22×C4D4×D7112yes—
165C22×C4C2×C22⋊D7111yescharacteristic
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—2842no—
170C2×C8—2842no—
171C2×C8—1424no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—44no—
177C4⋊C4—2842no—
178C4⋊C4—2842no—
179C4⋊C4—5682no—
180C4⋊C4—5681no—
181C42Q8×D7112yes—
182C42C2×Dic14211yescharacteristic
183C42#1176586511yescharacteristic
184C23#1455609611yescharacteristic
185Q8—5684no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—81no—
195C2×C4—82no—
196C2×C4—2842no—
197C2×C4—2842no—
198C2×C4—2842no—
199C2×C4—2842no—
200C2×C4—2842no—
201C2×C4—2842no—
202C2×C4#1455610612yes—
203C2×C4#1455610711yescharacteristic
204C2×C4#1455612611yescharacteristic
205C2×C4#1455612712yes—
206C8—24no—
207C8—2844no—
208C22#1798591911yescharacteristic
209C22#1798616611yescharacteristic
210C22#1798617012yes—
211C22#1798617512yes—
212C22#1798618111yescharacteristic
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—81no—
218C4—81no—
219C4—161no—
220C4—5682no—
221C4—5682no—
222C2#2258763611yescharacteristic
223C2#2258764411yescharacteristic
224C2#2258767212yes—
225C2#2258968511yescharacteristic
226C2#2258968812yes—
subgroupG/H
centerC23#14556096
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1847C2
socle, prime radicalC23×C71#120
hypercenter#1847D71
hypocenterC71#10584
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
26. #14556039Q8semidirect
41. #97×C71Q8semidirect
119. C71#10584semidirect
150. #97Dic142semidirect

Maximal subgroups

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

subgroupindex
#225966122
#225884752
#225862222
#225862212
C71×#18472
#1058471