Finite Group Databasebeta

← #31891838

Subgroups — #31891838

Order 18176.

396 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 119 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#22591526C211yesmaximal, characteristic
2#22589310C211yesmaximal, characteristic
3#22588474C212yesmaximal
4#22586222C212yesmaximal
5C2×#330×C71C211yesmaximal, characteristic
6C2×#14556039C2211yescharacteristic
7#17985898—22no—
8#17985885—22no—
9#17985885—22no—
10#17985869C2211yescharacteristic
11#17985865C2212yes—
12#17985762—22no—
13#17985762—22no—
14#17985761—22no—
15C2×#125×C71C2211yescharacteristic
16C2×#97×C71C2212yes—
17#330×C71C414yescomplemented
18C22×C71⋊C8—21no—
19C2×C4⋊Dic71—22no—
20C2×C4⋊Dic71—21no—
21C2×C4⋊Dic71—22no—
22C2×C4⋊Dic71D412yes—
23C2×C4⋊Dic71D412yes—
24C2×C4×Dic71—22no—
25#14556039—22no—
26C71⋊#95—22no—
27C71⋊#95—22no—
28C71⋊#95—22no—
29C71⋊#95—22no—
30C22×C8×C71—22no—
31#125×C71—22no—
32#125×C71C2×C412yes—
33#125×C71Q812yes—
34C2×C4⋊C4×C71—22no—
35C2×C4⋊C4×C71D412yes—
36C2×C42×C71C2311yescharacteristic
37#97×C71—22no—
38#97×C71C2×C414yes—
39C22×Dic71—22no—
40C22×Dic71—22no—
41C22×Dic71—41no—
42C2×C71⋊C8—22no—
43C2×C71⋊C8—41no—
44C2×C71⋊C8—41no—
45C4⋊Dic71—42no—
46C4⋊Dic71—42no—
47C4⋊Dic71—44no—
48C4⋊Dic71—42no—
49C4⋊Dic71—82no—
50C4×Dic71—82no—
51C22×C4×C71—22no—
52C22×C4×C71C2×D412yes—
53C22×C4×C71C2×D411yescharacteristic
54C2×C8×C71—24no—
55C2×C8×C71—42no—
56C2×C8×C71—42no—
57C4⋊C4×C71—24no—
58C4⋊C4×C71—24no—
59C4⋊C4×C71—22no—
60C4⋊C4×C71C4⋊C414yes—
61C42×C71C22×C411yescharacteristic
62C42×C71C2×Q811yescharacteristic
63C42×C71#4112yes—
64C23×C71#12011yescharacteristic
65C2×C4×C71—22no—
66C2×C4×C71—21no—
67C2×C4×C71—22no—
68C2×C4×C71—21no—
69C2×C4×C71—24no—
70C2×C4×C71—42no—
71C2×C4×C71—42no—
72C2×C4×C71C2×C4⋊C411yescharacteristic
73C2×C4×C71C4×D412yes—
74C2×C4×C71C22⋊Q812yes—
75C2×C4×C71#12311yescharacteristic
76C2×Dic71—42no—
77C2×Dic71—42no—
78C2×Dic71—42no—
79C2×Dic71—42no—
80C2×Dic71—42no—
81C2×Dic71—42no—
82C2×Dic71—82no—
83C2×Dic71—81no—
84C8×C71—44no—
85C71⋊C8—42no—
86C22×C71#38511yescharacteristic
87C22×C71#45412yes—
88C22×C71#45411yescharacteristic
89C22×C71#45512yes—
90C22×C71#45611yescharacteristic
91C4×C71—21no—
92C4×C71—21no—
93C4×C71—22no—
94C4×C71—22no—
95C4×C71—44no—
96Dic71—82no—
97Dic71—82no—
98Dic71—161no—
99#8915—711nomaximal, Sylow
100C2×C71#165811yescharacteristic
101C2×C71#166211yescharacteristic
102C2×C71#166412yes—
103C2×C71#185711yescharacteristic
104C2×C71#186012yes—
105#2382—711no—
106#1992—711no—
107#1836—712no—
108C2×#330D7111yescharacteristic
109#1295—712no—
110C71#891511yescharacteristic, complemented, Sylow
111C2×#125C2×D7111yescharacteristic
112C2×#97—711no—
113C2×#97C2×D7112yes—
114#389—1422no—
115#387—1422no—
116#387—1422no—
117#384—711no—
118#382—712no—
119#340—1422no—
120#340—1422no—
121#340—1422no—
122#330Dic7114yescomplemented
123C22×C8—22no—
124C22×C8—1421no—
125#125—22no—
126#125Dic14212yes—
127#125C2×Dic7112yes—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—712no—
130C2×C4⋊C4—712no—
131C2×C4⋊C4—1422no—
132C2×C4⋊C4—1421no—
133C2×C4⋊C4—1422no—
134C2×C4⋊C4D28412yes—
135C2×C42—1422no—
136C2×C42C22×D7111yescharacteristic
137#97—22no—
138#97—1422no—
139#97C2×Dic7114yes—
140#95—1422no—
141#95—1422no—
142#95—1422no—
143#95—1422no—
144C22×C4—22no—
145C22×C4—2841no—
146C22×C4—1422no—
147C22×C4—1422no—
148C22×C4D4×D7112yes—
149C22×C4C2×D28411yescharacteristic
150C2×C8—24no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—2841no—
154C2×C8—2841no—
155C2×C8—1422no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—2842no—
160C4⋊C4—2842no—
161C4⋊C4—2844no—
162C4⋊C4—2842no—
163C4⋊C4—5682no—
164C4⋊C4C4⋊Dic7114yes—
165C42—5682no—
166C42C22×Dic7111yescharacteristic
167C42C2×Dic14211yescharacteristic
168C42#1176586412yes—
169C23#1455609811yescharacteristic
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—2842no—
178C2×C4—2842no—
179C2×C4—2842no—
180C2×C4—2842no—
181C2×C4—2842no—
182C2×C4—2842no—
183C2×C4—5682no—
184C2×C4—5681no—
185C2×C4C2×C4⋊Dic7111yescharacteristic
186C2×C4D4×Dic7112yes—
187C2×C4#1455610512yes—
188C2×C4#1455611211yescharacteristic
189C8—44no—
190C8—2842no—
191C22#1798587411yescharacteristic
192C22#1798617312yes—
193C22#1798617411yescharacteristic
194C22#1798617812yes—
195C22#1798618411yescharacteristic
196C4—21no—
197C4—21no—
198C4—22no—
199C4—22no—
200C4—44no—
201C4—5682no—
202C4—5682no—
203C4—11361no—
204C2#2258764311yescharacteristic
205C2#2258766411yescharacteristic
206C2#2258767612yes—
207C2#2258855711yescharacteristic
208C2#2258857512yes—
subgroupG/H
centerC23#14556098
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC2×#330×C71C2
socle, prime radicalC23×C71#120
cosocleC2×C42×C71C23
hypercenterC2×#330D71
hypocenterC71#8915
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
17. #330×C71C4semidirect
110. C71#8915semidirect
122. #330Dic71semidirect

Maximal subgroups

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

subgroupindex
#225915262
#225893102
#225884742
#225862222
C2×#330×C712
#891571