Finite Group Databasebeta

← #31901621

Subgroups — #31901621

Order 18176.

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

normal subgroups only

This group has 402 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, complemented
2#22587075C212yesmaximal, complemented
3#22586791C212yesmaximal, complemented
4C71⋊#1309C211yesmaximal, characteristic, complemented
5C71×#3253C211yesmaximal, characteristic
6C2×#14556063—22no—
7#17986023C2211yescharacteristic
8#17986023C2212yes—
9C2×C4×C71⋊C8C2211yescharacteristic
10#17985748—22no—
11#17985743—22no—
12D4⋊(C71⋊C8)—24no—
13#532×C71C2212yes—
14#476×C71—22no—
15#469×C71—22no—
16#459×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×#41×C71—22no—
28C2×D8×C71—22no—
29C2×M16×C71D412yes—
30C4⋊Q8×C71D412yes—
31#127×C71D412yes—
32#124×C71—22no—
33#121×C71—22no—
34#121×C71—22no—
35C4×D4×C71—22no—
36C4×D4×C71—22no—
37C2×C42×C71C2311yescharacteristic
38#106×C71—22no—
39C4⋊C8×C71—22no—
40#102×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—
48#41×C71—82no—
49#41×C71—82no—
50C2×Q8×C71—22no—
51C2×D4×C71—22no—
52C2×D4×C71—22no—
53C2×D4×C71—22no—
54C2×D4×C71—42no—
55C22×C4×C71—42no—
56C22×C4×C71C2×D411yescharacteristic
57C22×C4×C71#4112yes—
58D8×C71—82no—
59M16×C71—42no—
60C2×C8×C71—22no—
61C4⋊C4×C71—22no—
62C4⋊C4×C71—22no—
63C22⋊C4×C71—42no—
64C22⋊C4×C71—42no—
65C42×C71C2×D411yescharacteristic
66C42×C71C2×D412yes—
67C42×C71#4111yescharacteristic
68C23×C71—42no—
69C23×C71#12411yescharacteristic
70Q8×C71—82no—
71D4×C71—42no—
72D4×C71—42no—
73D4×C71—42no—
74D4×C71—42no—
75D4×C71—82no—
76D4×C71—84no—
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—42no—
82C2×C4×C71—82no—
83C2×C4×C71—82no—
84C2×C4×C71#12011yescharacteristic
85C2×C4×C71#12111yescharacteristic
86C2×C4×C71#12112yes—
87C2×C4×C71#12312yes—
88C8×C71—42no—
89C71⋊C8—24no—
90C71⋊C8—81no—
91C71⋊C8—82no—
92C22×C71—22no—
93C22×C71—21no—
94C22×C71—82no—
95C22×C71—82no—
96C22×C71—82no—
97C22×C71#39411yescharacteristic
98C4×C71—22no—
99C4×C71—21no—
100C4×C71—82no—
101C4×C71#45311yescharacteristic
102C4×C71#45611yescharacteristic
103C4×C71#47112yes—
104C4×C71#47212yes—
105#10530—711nomaximal, Sylow
106C2×C71—41no—
107C2×C71—162no—
108C2×C71#203911yescharacteristic
109C2×C71#205112yes—
110#3253D7111yescharacteristic
111#1848—711no—
112#1560—712no—
113#1492—712no—
114#1309—711no—
115C71#1053011yescharacteristic, complemented, Sylow
116#532C2×D7112yes—
117#476—22no—
118#469—22no—
119#459—22no—
120C2×#108—1422no—
121#423—711no—
122#423—712no—
123#423C2×D7111yescharacteristic
124C2×C4×C8—711no—
125#333—1422no—
126#331—1422no—
127D4⋊C8—1424no—
128C2×#41—22no—
129C2×D8—22no—
130C2×M16—1421no—
131C2×M16—1422no—
132C2×M16C22⋊D7112yes—
133C22×C8—1421no—
134C4⋊Q8C22⋊D7112yes—
135#127C22⋊D7112yes—
136#124—22no—
137#121—22no—
138#121—22no—
139C4×D4—22no—
140C4×D4—22no—
141C2×C42C22×D7111yescharacteristic
142#108—2844no—
143#106—22no—
144C4⋊C8—22no—
145C4⋊C8—1422no—
146C4⋊C8—1422no—
147C4⋊C8—1422no—
148#102—24no—
149C4×C8—712no—
150C4×C8—1421no—
151#41—82no—
152#41—82no—
153C2×Q8—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—42no—
158C22×C4—42no—
159C22×C4C2×C22⋊D7111yescharacteristic
160C22×C4#1176586412yes—
161D8—82no—
162M16—42no—
163M16—2842no—
164M16—5682no—
165C2×C8—22no—
166C2×C8—2841no—
167C2×C8—2842no—
168C2×C8—1422no—
169C2×C8—1422no—
170C2×C8—1422no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C42D4×D7111yescharacteristic
176C42C2×C22⋊D7112yes—
177C42#1176586411yescharacteristic
178C23—42no—
179C23#1455611511yescharacteristic
180Q8—82no—
181D4—42no—
182D4—42no—
183D4—42no—
184D4—42no—
185D4—82no—
186D4—84no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4#1455609711yescharacteristic
195C2×C4#1455609911yescharacteristic
196C2×C4#1455610012yes—
197C2×C4#1455610912yes—
198C8—42no—
199C8—5681no—
200C8—5682no—
201C8—1424no—
202C22—21no—
203C22—22no—
204C22—82no—
205C22—82no—
206C22—82no—
207C22#1798591311yescharacteristic
208C4—21no—
209C4—22no—
210C4—82no—
211C4#1798616711yescharacteristic
212C4#1798618211yescharacteristic
213C4#1798626012yes—
214C4#1798626712yes—
215C2—41no—
216C2—162no—
217C2#2258956111yescharacteristic
218C2#2258962912yes—
subgroupG/H
centerC22#17985913
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
Fitting, prime radicalC71×#3253C2
socleC22×C71#394
hypercenter#3253D71
hypocenterC71#10530
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3253

Complements

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

NH ≅ G/Ntype
1. #22588511C2semidirect
2. #22587075C2semidirect
3. #22586791C2semidirect
4. C71⋊#1309C2semidirect
115. C71#10530semidirect

Maximal subgroups

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

subgroupindex
#225885112
#225870752
#225867912
C71⋊#13092
C71×#32532
#1053071