Finite Group Databasebeta

← #31909475

Subgroups — #31909475

Order 18176.

226 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 69 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#22587255C211yesmaximal, characteristic, complemented
3#22586405C211yesmaximal, characteristic, complemented
4#22586307C211yesmaximal, characteristic, complemented
5#22586306C211yesmaximal, characteristic, complemented
6#22586261C211yesmaximal, characteristic, complemented
7C71×#1594C211yesmaximal, characteristic, complemented
8C2×#14556069C411yescharacteristic
9C4×D568C2211yescharacteristic
10#17986078—21no—
11#17985826C411yescharacteristic
12#17985800C411yescharacteristic
13#17985799C411yescharacteristic
14#17985775C2211yescharacteristic
15#17985775C2211yescharacteristic
16#17985750C2211yescharacteristic
17#17985728—21no—
18C4×D8×C71C2211yescharacteristic
19#436×C71—21no—
20#334×C71C2211yescharacteristic
21D4⋊C8×C71—21no—
22#322×C71C2211yescharacteristic
23C2×D568C2×C411yescharacteristic
24C2×#11765848—41no—
25C4×D284—21no—
26#14556069—22no—
27C2×C71⋊C16—21no—
28C2×C71⋊C16C2×C411yescharacteristic
29C2×C71⋊C16C2×C411yescharacteristic
30#14556062C2×C411yescharacteristic
31C4⋊(C71⋊C8)—21no—
32#14556050—21no—
33(C2×D71)⋊C8—41no—
34C2×D8×C71C2×C411yescharacteristic
35C2×M16×C71—41no—
36C4×D4×C71—21no—
37#107×C71C2×C411yescharacteristic
38C4⋊C8×C71—21no—
39#102×C71—21no—
40C22⋊C8×C71—41no—
41C4×C8×C71D411yescharacteristic
42C4×C8×C71D411yescharacteristic
43C4×C8×C71C2311yescharacteristic
44C2×D284—21no—
45C2×C4×D71—41no—
46D568—21no—
47D568—21no—
48#11765848—82no—
49C2×C71⋊C8—41no—
50C4⋊Dic71—21no—
51(C2×D71)⋊C4—41no—
52C71⋊C16—21no—
53C71⋊C16—41no—
54C71⋊C16M1612yescomplemented
55C2×D4×C71—21no—
56C22×C4×C71—41no—
57D8×C71—21no—
58D8×C71—21no—
59M16×C71—82no—
60C2×C8×C71—21no—
61C2×C8×C71—21no—
62C2×C8×C71—41no—
63C2×C8×C71C22×C411yescharacteristic
64C2×C8×C71#4111yescharacteristic
65C4⋊C4×C71—21no—
66C22⋊C4×C71—41no—
67C42×C71C2×D411yescharacteristic
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—41no—
85C8×C71—41no—
86C8×C71—81no—
87C8×C71C2×M1611yescharacteristic
88C8×C71#13111yescharacteristic
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#1594D7111yescharacteristic, complemented
112#1368—711no—
113#1333—711no—
114#1333—711no—
115#1313—711no—
116C71#1208911yescharacteristic, complemented, Sylow
117C2×D16—711no—
118C4×D8—711no—
119C4×D8C2×D7111yescharacteristic
120#436—21no—
121#436—1421no—
122#367—711no—
123#357—711no—
124#357—711no—
125#346—711no—
126#346—711no—
127#334—711no—
128#334C2×D7111yescharacteristic
129D4⋊C8—21no—
130D4⋊C8—1421no—
131#322C2×D7111yescharacteristic
132C2×D8—711no—
133C2×D8C4×D7111yescharacteristic
134C2×M16—41no—
135C2×M16—2841no—
136C4×D4—21no—
137C4×D4—1421no—
138D16—1422no—
139C2×C16—711no—
140C2×C16—711no—
141C2×C16—1421no—
142#107—711no—
143#107C4×D7111yescharacteristic
144C4⋊C8—21no—
145C4⋊C8—1421no—
146#102—21no—
147#102—1421no—
148C22⋊C8—41no—
149C22⋊C8—2841no—
150C4×C8C22⋊D7111yescharacteristic
151C4×C8C22⋊D7111yescharacteristic
152C4×C8C22×D7111yescharacteristic
153C2×D4—21no—
154C2×D4—1421no—
155C22×C4—41no—
156C22×C4—2841no—
157D8—21no—
158D8—21no—
159D8—1421no—
160D8—1421no—
161M16—82no—
162M16—5682no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—41no—
166C2×C8—2841no—
167C2×C8C2×C4×D7111yescharacteristic
168C2×C8#1176586511yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—1421no—
171C22⋊C4—41no—
172C22⋊C4—2841no—
173C42C2×C22⋊D7111yescharacteristic
174C16—2841no—
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×C4C4×C22⋊D7111yescharacteristic
190C2×C4#1455615311yescharacteristic
191C2×C4#1455615411yescharacteristic
192C8—21no—
193C8—41no—
194C8—41no—
195C8—81no—
196C8—5681no—
197C8M16×D7111yescharacteristic
198C8#1455613611yescharacteristic
199C22—81no—
200C22—81no—
201C22—81no—
202C22—5681no—
203C22—5681no—
204C22—5681no—
205C22#1798610911yescharacteristic
206C4—21no—
207C4—21no—
208C4—81no—
209C4—5681no—
210C4#1798607511yescharacteristic
211C4#1798611911yescharacteristic
212C2—161no—
213C2—11361no—
214C2#2258725211yescharacteristic
215C2#2259023611yescharacteristic
216C2#2259023811yescharacteristic
subgroupG/H
centerC22#17986109
commutatorC2×C8×C71C22×C4
FrattiniC4×C8C22×D71
FittingC71×#1594C2
socleC22×C71#442
prime radicalC2×#14556069C4
cosocleC4×C8×C71C23
hypercenter#1594D71
hypocenterC71#12089
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #322 ◁ #1594

Complements

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

NH ≅ G/Ntype
2. #22587255C2semidirect
3. #22586405C2semidirect
4. #22586307C2semidirect
5. #22586306C2semidirect
6. #22586261C2semidirect
7. C71×#1594C2semidirect
54. C71⋊C16M16semidirect
111. #1594D71semidirect
116. C71#12089semidirect

Maximal subgroups

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

subgroupindex
#225902142
#225872552
#225864052
#225863072
#225863062
#225862612
C71×#15942
#1208971