Finite Group Databasebeta

← #31909474

Subgroups — #31909474

Order 18176.

226 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 67 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#22590214C211yesmaximal, characteristic
2#22587254C211yesmaximal, characteristic, complemented
3#22587253C211yesmaximal, characteristic, complemented
4#22586405C211yesmaximal, characteristic, complemented
5#22586308C211yesmaximal, characteristic, complemented
6#22586262C211yesmaximal, characteristic, complemented
7C71×#1333C211yesmaximal, characteristic, complemented
8C2×#14556069C411yescharacteristic
9C4×D568C2211yescharacteristic
10#17986077—21no—
11#17986076—21no—
12#17985826C411yescharacteristic
13#17985800C411yescharacteristic
14#17985799C411yescharacteristic
15#17985775C2211yescharacteristic
16#17985750C2211yescharacteristic
17D4⋊(C71⋊C8)—21no—
18#17985726—21no—
19#17985720C2211yescharacteristic
20C4×D8×C71C2211yescharacteristic
21#346×C71C2211yescharacteristic
22#334×C71C2211yescharacteristic
23C2×D568C2×C411yescharacteristic
24C2×#11765849—41no—
25C2×#11765848—41no—
26C4×D284—21no—
27#14556069—22no—
28C2×C71⋊C16C2×C411yescharacteristic
29C2×C71⋊C16C2×C411yescharacteristic
30#14556062C2×C411yescharacteristic
31C4⋊(C71⋊C8)—21no—
32#14556050—21no—
33C22⋊(C71⋊C8)—41no—
34(C2×D71)⋊C8—41no—
35C4×C71⋊C8D411yescharacteristic
36C4×C71⋊C8D411yescharacteristic
37C2×D8×C71C2×C411yescharacteristic
38C4×D4×C71—21no—
39C2×C16×C71—21no—
40#107×C71C2×C411yescharacteristic
41C4⋊C8×C71—21no—
42#102×C71—21no—
43C4×C8×C71C2311yescharacteristic
44C2×D284—21no—
45C2×C4×D71—41no—
46D568—21no—
47D568—21no—
48#11765849—82no—
49#11765848—82no—
50C2×C71⋊C8—21no—
51C2×C71⋊C8—21no—
52C2×C71⋊C8—41no—
53C4⋊Dic71—21no—
54(C2×D71)⋊C4—41no—
55C71⋊C16—21no—
56C71⋊C16M1612yescomplemented
57C2×D4×C71—21no—
58C22×C4×C71—41no—
59D8×C71—21no—
60D8×C71—21no—
61C2×C8×C71—41no—
62C2×C8×C71C22×C411yescharacteristic
63C2×C8×C71#4111yescharacteristic
64C4⋊C4×C71—21no—
65C22⋊C4×C71—41no—
66C42×C71C2×D411yescharacteristic
67C16×C71—41no—
68C23×C71—41no—
69C22×D71—41no—
70D4×C71—41no—
71D4×C71—41no—
72C2×C4×C71—41no—
73C2×C4×C71—81no—
74C2×C4×C71—81no—
75C2×C4×C71C4×D411yescharacteristic
76C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
77C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
78C2×Dic71—41no—
79D284—41no—
80D284—41no—
81C4×D71—81no—
82C4×D71—81no—
83C8×C71—21no—
84C8×C71—81no—
85C8×C71C2×M1611yescharacteristic
86C8×C71#13111yescharacteristic
87C71⋊C8—41no—
88C71⋊C8—41no—
89C71⋊C8—81no—
90C22×C71—81no—
91C22×C71—81no—
92C22×C71—81no—
93C22×C71#44211yescharacteristic
94C2×D71—81no—
95C2×D71—81no—
96C2×D71—81no—
97C4×C71—21no—
98C4×C71—21no—
99C4×C71—81no—
100C4×C71#43611yescharacteristic
101C4×C71#44411yescharacteristic
102Dic71—81no—
103#12089—711nomaximal, Sylow
104C2×C71—161no—
105C2×C71#159411yescharacteristic
106C2×C71AGL(1,ℤ/16)11yescharacteristic
107C2×C71AGL(1,ℤ/16)11yescharacteristic
108D71—161no—
109#2177—711no—
110#1594—711no—
111#1594—711no—
112#1368—711no—
113#1333—711no—
114#1333D7111yescharacteristic, complemented
115#1313—711no—
116C71#1208911yescharacteristic, complemented, Sylow
117C2×D16—711no—
118C4×D8—711no—
119C4×D8C2×D7111yescharacteristic
120#436—1421no—
121#436—1421no—
122#367—711no—
123#357—711no—
124#357—711no—
125#346—711no—
126#346C2×D7111yescharacteristic
127#334—711no—
128#334C2×D7111yescharacteristic
129D4⋊C8—1421no—
130D4⋊C8—1421no—
131#322—711no—
132C2×D8—711no—
133C2×D8C4×D7111yescharacteristic
134C2×M16—2841no—
135C2×M16—2841no—
136C4×D4—21no—
137C4×D4—1421no—
138D16—1422no—
139C2×C16—21no—
140C2×C16—711no—
141C2×C16—711no—
142#107—711no—
143#107C4×D7111yescharacteristic
144C4⋊C8—21no—
145C4⋊C8—1421no—
146#102—21no—
147#102—1421no—
148C22⋊C8—2841no—
149C22⋊C8—2841no—
150C4×C8—711no—
151C4×C8—711no—
152C4×C8C22×D7111yescharacteristic
153C2×D4—21no—
154C2×D4—1421no—
155C22×C4—41no—
156C22×C4—2841no—
157D8—21no—
158D8—21no—
159D8—1421no—
160D8—1421no—
161M16—5682no—
162M16—5682no—
163C2×C8—41no—
164C2×C8—2841no—
165C2×C8—1421no—
166C2×C8—1421no—
167C2×C8C2×C4×D7111yescharacteristic
168C2×C8#1176586411yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—1421no—
171C22⋊C4—41no—
172C22⋊C4—2841no—
173C42D4×D7111yescharacteristic
174C16—41no—
175C16—712no—
176C16—1421no—
177C23—41no—
178C23—2841no—
179D4—41no—
180D4—41no—
181D4—2841no—
182D4—2841no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—2841no—
187C2×C4—5681no—
188C2×C4—5681no—
189C2×C4(C22⋊D71)⋊C411yescharacteristic
190C2×C4#1455615211yescharacteristic
191C2×C4#1455615511yescharacteristic
192C8—21no—
193C8—81no—
194C8—2841no—
195C8—2841no—
196C8—5681no—
197C8M16×D7111yescharacteristic
198C8#1455613411yescharacteristic
199C22—81no—
200C22—81no—
201C22—81no—
202C22—5681no—
203C22—5681no—
204C22—5681no—
205C22#1798610811yescharacteristic
206C4—21no—
207C4—21no—
208C4—81no—
209C4—5681no—
210C4#1798607411yescharacteristic
211C4#1798611811yescharacteristic
212C2—161no—
213C2—11361no—
214C2#2258725111yescharacteristic
215C2#2259023511yescharacteristic
216C2#2259024111yescharacteristic
subgroupG/H
centerC22#17986108
commutatorC2×C8×C71C22×C4
FrattiniC4×C8C22×D71
FittingC71×#1333C2
socleC22×C71#442
prime radicalC2×#14556069C4
cosocleC4×C8×C71C23
hypercenter#1333D71
hypocenterC71#12089
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1333

Complements

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

NH ≅ G/Ntype
2. #22587254C2semidirect
3. #22587253C2semidirect
4. #22586405C2semidirect
5. #22586308C2semidirect
6. #22586262C2semidirect
7. C71×#1333C2semidirect
56. C71⋊C16M16semidirect
114. #1333D71semidirect
116. C71#12089semidirect

Maximal subgroups

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

subgroupindex
#225902142
#225872542
#225872532
#225864052
#225863082
#225862622
C71×#13332
#1208971