Finite Group Databasebeta

← #31901645

Subgroups — #31901645

Order 18176.

358 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 85 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#22588511C211yesmaximal, characteristic
2#22587078C212yesmaximal
3#22586795C212yesmaximal
4C71⋊#1309C211yesmaximal, characteristic
5C71×#3255C211yesmaximal, characteristic
6C2×#14556063—22no—
7#17986023C2211yescharacteristic
8#17986023C2212yes—
9C2×C4×C71⋊C8C2211yescharacteristic
10#17985745—22no—
11#17985745—22no—
12Q8⋊(C71⋊C8)—24no—
13#533×C71C2212yes—
14#475×C71—22no—
15#470×C71—22no—
16#462×C71—22no—
17#423×C71C2211yescharacteristic
18C2×#11765849—21no—
19C2×#11765849—22no—
20C22×C71⋊C8—21no—
21#14556063—44no—
22C4⋊(C71⋊C8)—22no—
23C4⋊(C71⋊C8)—22no—
24C4⋊(C71⋊C8)—22no—
25C4×C71⋊C8—21no—
26C4×C71⋊C8D412yes—
27C2×Q16×C71—22no—
28C2×M16×C71D412yes—
29C4⋊Q8×C71D412yes—
30C4⋊Q8×C71D412yes—
31#125×C71—22no—
32C22⋊Q8×C71—22no—
33C22⋊Q8×C71—22no—
34C4×Q8×C71—22no—
35C4×Q8×C71—22no—
36#117×C71—22no—
37C2×C42×C71C2311yescharacteristic
38#106×C71—22no—
39C4⋊C8×C71—22no—
40#103×C71—24no—
41#11765849—42no—
42#11765849—82no—
43C2×C71⋊C8—22no—
44C2×C71⋊C8—22no—
45C2×C71⋊C8—22no—
46C2×C71⋊C8—41no—
47C2×C71⋊C8—42no—
48C2×Q8×C71—22no—
49C2×Q8×C71—22no—
50C22×C4×C71C2×D411yescharacteristic
51C22×C4×C71#4112yes—
52Q16×C71—82no—
53M16×C71—42no—
54C2×C8×C71—22no—
55C4⋊C4×C71—22no—
56C4⋊C4×C71—22no—
57C4⋊C4×C71—22no—
58C4⋊C4×C71—22no—
59C4⋊C4×C71—42no—
60C4⋊C4×C71—42no—
61C22⋊C4×C71—42no—
62C42×C71—42no—
63C42×C71C2×D411yescharacteristic
64C42×C71C2×D412yes—
65C42×C71#4111yescharacteristic
66C23×C71#12411yescharacteristic
67Q8×C71—42no—
68Q8×C71—42no—
69Q8×C71—42no—
70Q8×C71—42no—
71C2×C4×C71—21no—
72C2×C4×C71—21no—
73C2×C4×C71—22no—
74C2×C4×C71—22no—
75C2×C4×C71—42no—
76C2×C4×C71—42no—
77C2×C4×C71#12011yescharacteristic
78C2×C4×C71#12111yescharacteristic
79C2×C4×C71#12112yes—
80C2×C4×C71#12312yes—
81C8×C71—42no—
82C71⋊C8—24no—
83C71⋊C8—81no—
84C71⋊C8—82no—
85C22×C71—22no—
86C22×C71—21no—
87C22×C71#39411yescharacteristic
88C4×C71—22no—
89C4×C71—21no—
90C4×C71—82no—
91C4×C71—82no—
92C4×C71#45311yescharacteristic
93C4×C71#45611yescharacteristic
94C4×C71#47112yes—
95C4×C71#47312yes—
96#10534—711nomaximal, Sylow
97C2×C71—41no—
98C2×C71#203911yescharacteristic
99C2×C71#205212yes—
100#3255D7111yescharacteristic
101#1848—711no—
102#1561—712no—
103#1493—712no—
104#1309—711no—
105C71#1053411yescharacteristic, complemented, Sylow
106#533C2×D7112yes—
107#475—22no—
108#470—22no—
109#462—22no—
110C2×#108—1422no—
111#423—711no—
112#423—712no—
113#423C2×D7111yescharacteristic
114C2×C4×C8—711no—
115#332—1422no—
116#332—1422no—
117Q8⋊C8—1424no—
118C2×Q16—22no—
119C2×M16—1421no—
120C2×M16—1422no—
121C2×M16C22⋊D7112yes—
122C22×C8—1421no—
123C4⋊Q8C22⋊D7112yes—
124C4⋊Q8C22⋊D7112yes—
125#125—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130#117—22no—
131C2×C42C22×D7111yescharacteristic
132#108—2844no—
133#106—22no—
134C4⋊C8—22no—
135C4⋊C8—1422no—
136C4⋊C8—1422no—
137C4⋊C8—1422no—
138#103—24no—
139C4×C8—712no—
140C4×C8—1421no—
141C2×Q8—22no—
142C2×Q8—22no—
143C22×C4C2×C22⋊D7111yescharacteristic
144C22×C4#1176586412yes—
145Q16—82no—
146M16—42no—
147M16—2842no—
148M16—5682no—
149C2×C8—22no—
150C2×C8—2841no—
151C2×C8—2842no—
152C2×C8—1422no—
153C2×C8—1422no—
154C2×C8—1422no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C22⋊C4—42no—
162C42—42no—
163C42D4×D7111yescharacteristic
164C42C2×C22⋊D7112yes—
165C42#1176586411yescharacteristic
166C23#1455611511yescharacteristic
167Q8—42no—
168Q8—42no—
169Q8—42no—
170Q8—42no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4#1455609711yescharacteristic
178C2×C4#1455609911yescharacteristic
179C2×C4#1455610012yes—
180C2×C4#1455610912yes—
181C8—42no—
182C8—5681no—
183C8—5682no—
184C8—1424no—
185C22—21no—
186C22—22no—
187C22#1798591311yescharacteristic
188C4—21no—
189C4—22no—
190C4—82no—
191C4—82no—
192C4#1798616711yescharacteristic
193C4#1798618211yescharacteristic
194C4#1798626412yes—
195C4#1798627212yes—
196C2—41no—
197C2#2258956111yescharacteristic
198C2#2258963612yes—
subgroupG/H
centerC22#17985913
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#3255C2
socleC22×C71#394
prime radicalC23×C71#124
hypercenter#3255D71
hypocenterC71#10534
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3255

Complements

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

NH ≅ G/Ntype
105. C71#10534semidirect

Maximal subgroups

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

subgroupindex
#225885112
#225870782
#225867952
C71⋊#13092
C71×#32552
#1053471