Finite Group Databasebeta

← #31911877

Subgroups — #31911877

Order 18176.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#22590564C211yesmaximal, characteristic, complemented
2C16×D284C211yesmaximal, characteristic, complemented
3#22587740C211yesmaximal, characteristic, complemented
4#22587696C211yesmaximal, characteristic
5#22586399C211yesmaximal, characteristic, complemented
6#22586385C211yesmaximal, characteristic, complemented
7C71×#1372C211yesmaximal, characteristic, complemented
8C2×C16×D71—21no—
9#17986404C2211yescharacteristic
10#17986376C2211yescharacteristic
11#17986290—21no—
12#17986195—21no—
13C8×D284C2211yescharacteristic
14C4⋊(C71⋊C16)C2211yescharacteristic
15#17985801—21no—
16#17985801—21no—
17(C2×D71)⋊C16—21no—
18#17985747—21no—
19#17985742—21no—
20C4×C16×C71C2211yescharacteristic
21#335×C71C2211yescharacteristic
22#335×C71C2211yescharacteristic
23C2×D568—21no—
24C2×D568—21no—
25C2×D568—41no—
26C2×C8×D71—21no—
27#14556128—21no—
28#14556125—21no—
29C4×D284D411yescharacteristic
30C2×C71⋊C16—21no—
31C16×D71—42no—
32#14556062—21no—
33#14556062—21no—
34#14556060—41no—
35C4⋊(C71⋊C8)D411yescharacteristic
36#14556050—41no—
37#14556050—41no—
38(C2×D71)⋊C8—21no—
39C2×C16×C71D411yescharacteristic
40C2×C16×C71D411yescharacteristic
41C4⋊C8×C71—21no—
42C4⋊C8×C71—21no—
43C4×C8×C71C2311yescharacteristic
44C2×Dic142—41no—
45C2×D284—41no—
46C2×D284—41no—
47C2×D284—41no—
48C2×D284D811yescharacteristic
49C2×C4×D71—21no—
50D568—41no—
51D568—41no—
52D568—81no—
53D568—82no—
54C2×C71⋊C8—21no—
55C8×D71—41no—
56C8×D71—41no—
57C4⋊Dic71—41no—
58C4⋊Dic71—41no—
59C4⋊Dic71D811yescharacteristic
60(C2×D71)⋊C4—21no—
61C71⋊C16—41no—
62C2×C8×C71—41no—
63C2×C8×C71—41no—
64C2×C8×C71C2×D411yescharacteristic
65C2×C8×C71#4111yescharacteristic
66C42×C71C2×D411yescharacteristic
67C16×C71—21no—
68C16×C71—21no—
69C22×D71—21no—
70C22×D71—81no—
71C2×C4×C71#12111yescharacteristic
72C2×C4×C71C2×D811yescharacteristic
73C2×C4×C71#13511yescharacteristic
74C2×Dic71—21no—
75C2×Dic71—81no—
76D284—21no—
77D284—81no—
78D284—81no—
79D284—81no—
80D284—81no—
81D284—161no—
82D284SD3212yes—
83C4×D71—41no—
84C4×D71—41no—
85Dic142—161no—
86C8×C71—21no—
87C8×C71—81no—
88C8×C71—81no—
89C8×C71#13511yescharacteristic
90C8×C71AGL(1,ℤ/8)11yescharacteristic
91C71⋊C8—41no—
92C22×C71#46611yescharacteristic
93C2×D71—22no—
94C2×D71—41no—
95C2×D71—41no—
96C2×D71—161no—
97C2×D71—161no—
98C2×D71—161no—
99C4×C71—21no—
100C4×C71#46311yescharacteristic
101C4×C71#47111yescharacteristic
102C4×C71C2×SD3211yescharacteristic
103C4×C71#50811yescharacteristic
104Dic71—41no—
105Dic71—161no—
106#12468—711nomaximal, Sylow
107C2×C71#168111yescharacteristic
108C2×C71#221711yescharacteristic
109C2×C71#222111yescharacteristic
110D71—42no—
111D71—321no—
112#2240—711no—
113D4×C16—711no—
114#1675—711no—
115#1667—711no—
116#1372D7111yescharacteristic, complemented
117#1365—711no—
118#1361—711no—
119C71#1246811yescharacteristic, complemented, Sylow
120C22×C16—1421no—
121#500—711no—
122#493—711no—
123#476—1421no—
124#459—1421no—
125C8×D4—711no—
126C4⋊C16—711no—
127#357—1421no—
128#357—1421no—
129C22⋊C16—1421no—
130C4×C16C2×D7111yescharacteristic
131#335C2×D7111yescharacteristic
132#335C2×D7111yescharacteristic
133#333—1421no—
134#331—1421no—
135C2×D8—2841no—
136C2×D8—1421no—
137C2×D8—1421no—
138C22×C8—1421no—
139C4⋊Q8—1421no—
140#127—1421no—
141C4×D4—711no—
142C2×C16—2842no—
143C2×C16—1421no—
144C2×C16D28411yescharacteristic
145C2×C16D28411yescharacteristic
146#107—1421no—
147#107—1421no—
148#106—2841no—
149C4⋊C8—21no—
150C4⋊C8—21no—
151C4⋊C8—711no—
152#102—2841no—
153#102—2841no—
154C22⋊C8—1421no—
155C4×C8C22×D7111yescharacteristic
156C2×Q8—2841no—
157C2×D4—2841no—
158C2×D4—2841no—
159C2×D4—2841no—
160C2×D4—711no—
161C22×C4—1421no—
162D8—2841no—
163D8—2841no—
164D8—5681no—
165D8—5682no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—2841no—
169C2×C8—2841no—
170C2×C8—1421no—
171C2×C8C2×D28411yescharacteristic
172C2×C8#1176586611yescharacteristic
173C4⋊C4—2841no—
174C4⋊C4—2841no—
175C4⋊C4—711no—
176C22⋊C4—1421no—
177C42D4×D7111yescharacteristic
178C16—21no—
179C16—21no—
180C16—2841no—
181C23—5681no—
182C23—1421no—
183Q8—11361no—
184D4—5681no—
185D4—5681no—
186D4—5681no—
187D4—5681no—
188D4—712no—
189D4—11361no—
190D4—1421no—
191C2×C4—2841no—
192C2×C4—2841no—
193C2×C4—5681no—
194C2×C4—1421no—
195C2×C4#1455610311yescharacteristic
196C2×C4D8×D7111yescharacteristic
197C2×C4#1455615111yescharacteristic
198C8—21no—
199C8—81no—
200C8—81no—
201C8—2841no—
202C8#1455615011yescharacteristic
203C8#1455615611yescharacteristic
204C22—2841no—
205C22—2841no—
206C22—11361no—
207C22—11361no—
208C22—11361no—
209C22—1422no—
210C22#1798623611yescharacteristic
211C4—21no—
212C4—2841no—
213C4—11361no—
214C4#1798621511yescharacteristic
215C4#1798626311yescharacteristic
216C4SD32×D7111yescharacteristic
217C4#1798643011yescharacteristic
218C2—2842no—
219C2—22721no—
220C2#2258777811yescharacteristic
221C2#2259043711yescharacteristic
222C2#2259046111yescharacteristic
subgroupG/H
centerC22#17986236
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1372C2
socleC22×C71#466
prime radical#22587696C2
hypercenter#1372D71
hypocenterC71#12468
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1372

Complements

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

NH ≅ G/Ntype
1. #22590564C2semidirect
2. C16×D284C2semidirect
3. #22587740C2semidirect
5. #22586399C2semidirect
6. #22586385C2semidirect
7. C71×#1372C2semidirect
116. #1372D71semidirect
119. C71#12468semidirect

Maximal subgroups

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

subgroupindex
#225905642
C16×D2842
#225877402
#225876962
#225863992
#225863852
C71×#13722
#1246871