Finite Group Databasebeta

← #31911735

Subgroups — #31911735

Order 18176.

228 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 63 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#22590595C211yesmaximal, characteristic
2#22590245C211yesmaximal, characteristic, complemented
3#22587711C211yesmaximal, characteristic, complemented
4#22587697C211yesmaximal, characteristic
5#22586411C211yesmaximal, characteristic, complemented
6#22586390C211yesmaximal, characteristic, complemented
7C71×#1361C211yesmaximal, characteristic
8C2×#14556072—21no—
9C2×#14556072—21no—
10#17986379C2211yescharacteristic
11#17986285—21no—
12#17986196—21no—
13C8×Dic142C2211yescharacteristic
14Dic71⋊C16—21no—
15#17985774—21no—
16C4×C71⋊C16C2211yescharacteristic
17#17985752C2211yescharacteristic
18#17985744—21no—
19#17985743—21no—
20#493×C71C2211yescharacteristic
21C4⋊C16×C71C2211yescharacteristic
22#335×C71C2211yescharacteristic
23C2×Dic284—21no—
24C2×Dic284—21no—
25C2×#11765853—41no—
26#14556128—21no—
27C4×Dic142D411yescharacteristic
28#14556072—42no—
29#14556072—42no—
30C2×C71⋊C16D411yescharacteristic
31C2×C71⋊C16D411yescharacteristic
32#14556060—41no—
33C4⋊(C71⋊C8)—21no—
34C4⋊(C71⋊C8)D411yescharacteristic
35Dic71⋊C8—21no—
36#14556053—41no—
37#14556049—41no—
38C8×Dic71—21no—
39C2×D8×C71—21no—
40C2×D8×C71—21no—
41#127×C71—21no—
42C2×C16×C71—21no—
43C4⋊C8×C71—21no—
44C4×C8×C71C2311yescharacteristic
45C2×Dic142—41no—
46C2×Dic142D811yescharacteristic
47Dic284—41no—
48Dic284—41no—
49Dic284—81no—
50#11765853—82no—
51C2×C71⋊C8—21no—
52C2×C71⋊C8—41no—
53C4⋊Dic71—41no—
54C4⋊Dic71—41no—
55C4⋊Dic71D811yescharacteristic
56Dic71⋊C4—21no—
57C4×Dic71—21no—
58C71⋊C16—21no—
59C71⋊C16SD1612yescomplemented
60C2×D4×C71—41no—
61C2×D4×C71—41no—
62C2×D4×C71—41no—
63D8×C71—41no—
64D8×C71—41no—
65D8×C71—81no—
66C2×C8×C71—41no—
67C2×C8×C71C2×D411yescharacteristic
68C2×C8×C71#4111yescharacteristic
69C42×C71C2×D411yescharacteristic
70C16×C71—41no—
71C23×C71—81no—
72D4×C71—81no—
73D4×C71—81no—
74D4×C71—81no—
75D4×C71—81no—
76D4×C71—161no—
77C2×C4×C71#12111yescharacteristic
78C2×C4×C71C2×D811yescharacteristic
79C2×C4×C71#13511yescharacteristic
80C2×Dic71—21no—
81C2×Dic71—21no—
82C2×Dic71—81no—
83Dic142—21no—
84Dic142—21no—
85Dic142—81no—
86Dic142—81no—
87C8×C71—21no—
88C8×C71—81no—
89C8×C71C2×SD1611yescharacteristic
90C8×C71AGL(1,ℤ/8)11yescharacteristic
91C71⋊C8—41no—
92C71⋊C8—81no—
93C22×C71—161no—
94C22×C71—161no—
95C22×C71—161no—
96C22×C71#46611yescharacteristic
97C4×C71—21no—
98C4×C71#46011yescharacteristic
99C4×C71#47211yescharacteristic
100C4×C71#50911yescharacteristic
101C4×C71#51011yescharacteristic
102Dic71—41no—
103Dic71—41no—
104Dic71—161no—
105#12444—711nomaximal, Sylow
106C2×C71—321no—
107C2×C71#166811yescharacteristic
108C2×C71#222011yescharacteristic
109C2×C71#222311yescharacteristic
110#2248—711no—
111#2183—711no—
112#1670—711no—
113#1668—711no—
114#1371—711no—
115#1362—711no—
116#1361D7111yescharacteristic
117C71#1244411yescharacteristic, complemented, Sylow
118C2×SD32—1421no—
119C2×SD32—1421no—
120#494—711no—
121#493C2×D7111yescharacteristic
122#475—1421no—
123#460—1421no—
124C8×Q8—711no—
125C4⋊C16—1421no—
126C4⋊C16C2×D7111yescharacteristic
127#346—1421no—
128C4×C16—711no—
129#335—711no—
130#335C2×D7111yescharacteristic
131#332—1421no—
132#331—1421no—
133C2×Q16—1421no—
134C2×Q16—1421no—
135C2×SD16—2841no—
136C2×D8—21no—
137C2×D8—21no—
138C4⋊Q8—1421no—
139#127—21no—
140C4×Q8—711no—
141SD32—2842no—
142SD32—2842no—
143C2×C16—21no—
144C2×C16—711no—
145C2×C16—711no—
146#106—2841no—
147C4⋊C8—21no—
148C4⋊C8—711no—
149C4⋊C8—1421no—
150C4⋊C8—1421no—
151#103—2841no—
152#102—2841no—
153C4×C8—1421no—
154C4×C8C22×D7111yescharacteristic
155C2×Q8—2841no—
156C2×Q8—711no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160Q16—2841no—
161Q16—2841no—
162Q16—5681no—
163SD16—5682no—
164D8—41no—
165D8—41no—
166D8—81no—
167C2×C8—41no—
168C2×C8—2841no—
169C2×C8—1421no—
170C2×C8D4×D7111yescharacteristic
171C2×C8#1176586511yescharacteristic
172C4⋊C4—2841no—
173C4⋊C4—2841no—
174C4⋊C4—711no—
175C4⋊C4—1421no—
176C42—1421no—
177C42D4×D7111yescharacteristic
178C16—41no—
179C16—712no—
180C16—1421no—
181C23—81no—
182Q8—5681no—
183Q8—5681no—
184Q8—1421no—
185Q8—1421no—
186D4—81no—
187D4—81no—
188D4—81no—
189D4—81no—
190D4—161no—
191C2×C4—5681no—
192C2×C4—1421no—
193C2×C4—1421no—
194C2×C4#1455610111yescharacteristic
195C2×C4D8×D7111yescharacteristic
196C2×C4#1455614911yescharacteristic
197C8—21no—
198C8—81no—
199C8—2841no—
200C8—5681no—
201C8SD16×D7111yescharacteristic
202C8#1455615211yescharacteristic
203C22—161no—
204C22—161no—
205C22—161no—
206C22#1798623411yescharacteristic
207C4—21no—
208C4—2841no—
209C4—2841no—
210C4—11361no—
211C4#1798619811yescharacteristic
212C4#1798626811yescharacteristic
213C4#1798643311yescharacteristic
214C4#1798643811yescharacteristic
215C2—321no—
216C2#2258769911yescharacteristic
217C2#2259045211yescharacteristic
218C2#2259046911yescharacteristic
subgroupG/H
centerC22#17986234
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1361C2
socleC22×C71#466
prime radical#493×C71C22
hypercenter#1361D71
hypocenterC71#12444
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1361

Complements

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

NH ≅ G/Ntype
2. #22590245C2semidirect
3. #22587711C2semidirect
5. #22586411C2semidirect
6. #22586390C2semidirect
59. C71⋊C16SD16semidirect
117. C71#12444semidirect

Maximal subgroups

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

subgroupindex
#225905952
#225902452
#225877112
#225876972
#225864112
#225863902
C71×#13612
#1244471