Finite Group Databasebeta

← #31901626

Subgroups — #31901626

Order 18176.

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

normal subgroups only

This group has 386 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#22588477C211yesmaximal, characteristic, complemented
2#22587075C212yesmaximal, complemented
3#22586791C212yesmaximal, complemented
4C71⋊#1309C211yesmaximal, characteristic, complemented
5C71×#3253C211yesmaximal, characteristic
6#17986023C2211yescharacteristic
7#17986023C2212yes—
8C2×C4×C71⋊C8C2211yescharacteristic
9#17985763—22no—
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—
21C4⋊(C71⋊C8)—22no—
22C4⋊(C71⋊C8)—22no—
23C4⋊(C71⋊C8)—22no—
24C4×C71⋊C8—21no—
25C4×C71⋊C8D412yes—
26C2×#41×C71—22no—
27C2×D8×C71—22no—
28C2×M16×C71D412yes—
29C4⋊Q8×C71D412yes—
30#127×C71D412yes—
31#124×C71—22no—
32#121×C71—22no—
33#121×C71—22no—
34C4×D4×C71—22no—
35C4×D4×C71—22no—
36C2×C42×C71C2311yescharacteristic
37#106×C71—22no—
38C4⋊C8×C71—22no—
39#102×C71—24no—
40#11765849—42no—
41#11765849—82no—
42C2×C71⋊C8—22no—
43C2×C71⋊C8—41no—
44C2×C71⋊C8—41no—
45C2×C71⋊C8—41no—
46C2×C71⋊C8—42no—
47#41×C71—82no—
48#41×C71—82no—
49C2×Q8×C71—22no—
50C2×D4×C71—22no—
51C2×D4×C71—22no—
52C2×D4×C71—22no—
53C2×D4×C71—42no—
54C22×C4×C71—42no—
55C22×C4×C71C2×D411yescharacteristic
56C22×C4×C71#4112yes—
57D8×C71—82no—
58M16×C71—42no—
59C2×C8×C71—22no—
60C4⋊C4×C71—22no—
61C4⋊C4×C71—22no—
62C22⋊C4×C71—42no—
63C22⋊C4×C71—42no—
64C42×C71C2×D411yescharacteristic
65C42×C71C2×D412yes—
66C42×C71#4111yescharacteristic
67C23×C71—42no—
68C23×C71#12411yescharacteristic
69Q8×C71—82no—
70D4×C71—42no—
71D4×C71—42no—
72D4×C71—42no—
73D4×C71—42no—
74D4×C71—82no—
75D4×C71—84no—
76C2×C4×C71—21no—
77C2×C4×C71—21no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—42no—
81C2×C4×C71—82no—
82C2×C4×C71—82no—
83C2×C4×C71#12011yescharacteristic
84C2×C4×C71#12111yescharacteristic
85C2×C4×C71#12112yes—
86C2×C4×C71#12312yes—
87C8×C71—42no—
88C71⋊C8—42no—
89C71⋊C8—81no—
90C71⋊C8—82no—
91C22×C71—22no—
92C22×C71—21no—
93C22×C71—82no—
94C22×C71—82no—
95C22×C71—82no—
96C22×C71#39411yescharacteristic
97C4×C71—22no—
98C4×C71—21no—
99C4×C71—82no—
100C4×C71#45411yescharacteristic
101C4×C71#45511yescharacteristic
102C4×C71#47112yes—
103C4×C71#47212yes—
104#10531—711nomaximal, Sylow
105C2×C71—41no—
106C2×C71—162no—
107C2×C71#204011yescharacteristic
108C2×C71#205112yes—
109#3253D7111yescharacteristic
110#1837—711no—
111#1560—712no—
112#1492—712no—
113#1309—711no—
114C71#1053111yescharacteristic, complemented, Sylow
115#532C2×D7112yes—
116#476—22no—
117#469—22no—
118#459—22no—
119#423—711no—
120#423—712no—
121#423C2×D7111yescharacteristic
122C2×C4×C8—711no—
123#341—1422no—
124#333—1422no—
125#331—1422no—
126D4⋊C8—1424no—
127C2×#41—22no—
128C2×D8—22no—
129C2×M16—1421no—
130C2×M16—1422no—
131C2×M16C22⋊D7112yes—
132C22×C8—1421no—
133C4⋊Q8C22⋊D7112yes—
134#127C22⋊D7112yes—
135#124—22no—
136#121—22no—
137#121—22no—
138C4×D4—22no—
139C4×D4—22no—
140C2×C42C22×D7111yescharacteristic
141#106—22no—
142C4⋊C8—22no—
143C4⋊C8—1422no—
144C4⋊C8—1422no—
145C4⋊C8—1422no—
146#102—24no—
147C4×C8—712no—
148C4×C8—1421no—
149#41—82no—
150#41—82no—
151C2×Q8—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—42no—
156C22×C4—42no—
157C22×C4C2×C22⋊D7111yescharacteristic
158C22×C4#1176586412yes—
159D8—82no—
160M16—42no—
161M16—2842no—
162M16—5682no—
163C2×C8—22no—
164C2×C8—2841no—
165C2×C8—2841no—
166C2×C8—2841no—
167C2×C8—2842no—
168C2×C8—1422no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C22⋊C4—42no—
172C22⋊C4—42no—
173C42D4×D7111yescharacteristic
174C42C2×C22⋊D7112yes—
175C42#1176586411yescharacteristic
176C23—42no—
177C23#1455611511yescharacteristic
178Q8—82no—
179D4—42no—
180D4—42no—
181D4—42no—
182D4—42no—
183D4—82no—
184D4—84no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4#1455609711yescharacteristic
193C2×C4#1455609911yescharacteristic
194C2×C4#1455610012yes—
195C2×C4#1455610912yes—
196C8—42no—
197C8—2842no—
198C8—5681no—
199C8—5682no—
200C22—21no—
201C22—22no—
202C22—82no—
203C22—82no—
204C22—82no—
205C22#1798591311yescharacteristic
206C4—21no—
207C4—22no—
208C4—82no—
209C4#1798617211yescharacteristic
210C4#1798617711yescharacteristic
211C4#1798626012yes—
212C4#1798626712yes—
213C2—41no—
214C2—162no—
215C2#2258956611yescharacteristic
216C2#2258962912yes—
subgroupG/H
centerC22#17985913
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
Fitting, prime radicalC71×#3253C2
socleC22×C71#394
hypercenter#3253D71
hypocenterC71#10531
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. #22588477C2semidirect
2. #22587075C2semidirect
3. #22586791C2semidirect
4. C71⋊#1309C2semidirect
114. C71#10531semidirect

Maximal subgroups

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

subgroupindex
#225884772
#225870752
#225867912
C71⋊#13092
C71×#32532
#1053171