Finite Group Databasebeta

← #31895077

Subgroups — #31895077

Order 18176.

434 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 103 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#22591446C211yesmaximal, characteristic
2#22586967C211yesmaximal, characteristic
3#22586852C212yesmaximal
4#22586234C212yesmaximal
5C71×#3166C211yesmaximal, characteristic
6#17986056C2212yes—
7#17985885—22no—
8#17985885—22no—
9C4⋊C4×Dic71—22no—
10#17985845C2212yes—
11#17985741—22no—
12#17985736—22no—
13#17985736—22no—
14#533×C71C2211yescharacteristic
15C2×#125×C71C2211yescharacteristic
16#501×C71C414yescomplemented
17#499×C71—22no—
18#428×C71—22no—
19C4×M16×C71C2211yescharacteristic
20C2×C4⋊Dic71—22no—
21C2×C4×Dic71—22no—
22C2×C4×Dic71—22no—
23C4⋊(C71⋊C8)—22no—
24C22⋊(C71⋊C8)—22no—
25#14556039—22no—
26C71⋊#95—22no—
27C71⋊#95—22no—
28C71⋊#95—22no—
29C71⋊#95D412yes—
30C71⋊#95D412yes—
31C2×M16×C71—21no—
32C4⋊Q8×C71C2×C412yes—
33#125×C71—21no—
34#125×C71—22no—
35#125×C71C2×C412yes—
36#125×C71D412yes—
37C22⋊Q8×C71—22no—
38C4×Q8×C71—22no—
39#117×C71—21no—
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71D412yes—
42C2×C42×C71C2311yescharacteristic
43#107×C71—24no—
44#106×C71—24no—
45#97×C71C2×C412yes—
46C4×C8×C71—21no—
47C22×Dic71—22no—
48C22×Dic71—22no—
49C2×C71⋊C8—42no—
50C4⋊Dic71—44no—
51C4×Dic71—44no—
52C4×Dic71—44no—
53C2×Q8×C71—22no—
54C22×C4×C71—22no—
55C22×C4×C71C2×D412yes—
56C22×C4×C71C2×D411yescharacteristic
57M16×C71—24no—
58C2×C8×C71—22no—
59C4⋊C4×C71—24no—
60C4⋊C4×C71—24no—
61C4⋊C4×C71—22no—
62C4⋊C4×C71—22no—
63C4⋊C4×C71—22no—
64C4⋊C4×C71—42no—
65C4⋊C4×C71C22⋊C414yes—
66C22⋊C4×C71—41no—
67C42×C71—41no—
68C42×C71C22×C411yescharacteristic
69C42×C71C2×D411yescharacteristic
70C42×C71#4112yes—
71C23×C71#12011yescharacteristic
72Q8×C71—82no—
73C2×C4×C71—22no—
74C2×C4×C71—24no—
75C2×C4×C71—42no—
76C2×C4×C71—42no—
77C2×C4×C71—42no—
78C2×C4×C71—42no—
79C2×C4×C71C2×C22⋊C411yescharacteristic
80C2×C4×C71C4×D412yes—
81C2×C4×C71#12112yes—
82C2×C4×C71#12311yescharacteristic
83C2×Dic71—24no—
84C2×Dic71—42no—
85C2×Dic71—42no—
86C2×Dic71—42no—
87C2×Dic71—42no—
88C2×Dic71—42no—
89C8×C71—24no—
90C71⋊C8—82no—
91C22×C71—22no—
92C22×C71#38611yescharacteristic
93C22×C71#45612yes—
94C4×C71—21no—
95C4×C71—21no—
96C4×C71—22no—
97C4×C71—22no—
98C4×C71—44no—
99C4×C71—82no—
100Dic71—44no—
101Dic71—82no—
102#9440—711nomaximal, Sylow
103C2×C71—22no—
104C2×C71#188611yescharacteristic
105C2×C71#190612yes—
106#3166D7111yescharacteristic
107#2371—711no—
108#1534—711no—
109#1506—712no—
110#1300—712no—
111C71#944011yescharacteristic, complemented, Sylow
112#533C2×D7111yescharacteristic
113C2×#125C2×D7111yescharacteristic
114#501Dic7114yescomplemented
115#499—22no—
116#432—712no—
117#428—22no—
118C4×M16C2×D7111yescharacteristic
119#387—1422no—
120#387—1422no—
121C4×C4⋊C4—1422no—
122#376—712no—
123#330—1422no—
124#328—1422no—
125#328—1422no—
126C2×M16—21no—
127C4⋊Q8C2×Dic7112yes—
128#125—21no—
129#125—22no—
130#125C2×Dic7112yes—
131#125C22⋊D7112yes—
132C22⋊Q8—22no—
133C4×Q8—22no—
134#117—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—1422no—
137C2×C4⋊C4C22⋊D7112yes—
138C2×C42—1422no—
139C2×C42—1422no—
140C2×C42C22×D7111yescharacteristic
141#107—24no—
142#106—24no—
143C4⋊C8—1422no—
144C22⋊C8—1422no—
145#97—1422no—
146#97C2×Dic7112yes—
147C4×C8—21no—
148#95—712no—
149#95—712no—
150#95—1422no—
151#95—1422no—
152#95—1422no—
153C2×Q8—22no—
154C22×C4—22no—
155C22×C4—1422no—
156C22×C4—1422no—
157C22×C4D4×D7112yes—
158C22×C4C2×C22⋊D7111yescharacteristic
159M16—24no—
160C2×C8—22no—
161C2×C8—2842no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—42no—
168C4⋊C4—2844no—
169C4⋊C4C22⋊Dic7114yes—
170C22⋊C4—41no—
171C42—41no—
172C42—2844no—
173C42—2844no—
174C42C22×Dic7111yescharacteristic
175C42C2×C22⋊D7111yescharacteristic
176C42#1176586412yes—
177C23#1455609611yescharacteristic
178Q8—82no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—2842no—
186C2×C4—2842no—
187C2×C4—2842no—
188C2×C4—2842no—
189C2×C4—2842no—
190C2×C4—1424no—
191C2×C4C2×C22⋊Dic7111yescharacteristic
192C2×C4D4×Dic7112yes—
193C2×C4#1455610012yes—
194C2×C4#1455610911yescharacteristic
195C8—24no—
196C8—5682no—
197C22—22no—
198C22#1798587911yescharacteristic
199C22#1798618112yes—
200C4—21no—
201C4—21no—
202C4—22no—
203C4—22no—
204C4—44no—
205C4—82no—
206C4—2844no—
207C4—5682no—
208C2—22no—
209C2#2258873411yescharacteristic
210C2#2258884312yes—
subgroupG/H
centerC22#17985879
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#3166C2
socleC22×C71#386
prime radicalC23×C71#120
cosocleC2×C42×C71C23
hypercenter#3166D71
hypocenterC71#9440
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3166

Complements

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

NH ≅ G/Ntype
16. #501×C71C4semidirect
111. C71#9440semidirect
114. #501Dic71semidirect

Maximal subgroups

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

subgroupindex
#225914462
#225869672
#225868522
#225862342
C71×#31662
#944071