Finite Group Databasebeta

← #31911875

Subgroups — #31911875

Order 18176.

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

normal subgroups only

This group has 234 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#22590563C211yesmaximal, characteristic, complemented
2#22590159C211yesmaximal, characteristic, complemented
3#22587740C211yesmaximal, characteristic, complemented
4#22587692C211yesmaximal, characteristic
5#22586413C211yesmaximal, characteristic, complemented
6#22586397C211yesmaximal, characteristic, complemented
7C71×#1361C211yesmaximal, characteristic, complemented
8C2×C16×D71—21no—
9#17986404C2211yescharacteristic
10#17986290—21no—
11#17986191—21no—
12C8×D284C2211yescharacteristic
13#17985800—21no—
14#17985800—21no—
15(C2×D71)⋊C16—21no—
16C4×C71⋊C16C2211yescharacteristic
17#17985752C2211yescharacteristic
18#17985747—21no—
19#17985743—21no—
20#493×C71C2211yescharacteristic
21C4⋊C16×C71C2211yescharacteristic
22#335×C71C2211yescharacteristic
23C2×#11765850—41no—
24C2×C8×D71—21no—
25#14556128—21no—
26C4×D284D411yescharacteristic
27C2×C71⋊C16D411yescharacteristic
28C2×C71⋊C16D411yescharacteristic
29C16×D71—42no—
30#14556062—21no—
31#14556062—21no—
32#14556060—41no—
33C4⋊(C71⋊C8)—21no—
34C4⋊(C71⋊C8)D411yescharacteristic
35#14556050—41no—
36#14556049—41no—
37(C2×D71)⋊C8—21no—
38C2×D8×C71—21no—
39C2×D8×C71—21no—
40#127×C71—21no—
41C2×C16×C71—21no—
42C4⋊C8×C71—21no—
43C4×C8×C71C2311yescharacteristic
44C2×Dic142—41no—
45C2×D284D811yescharacteristic
46C2×C4×D71—21no—
47#11765850—82no—
48C2×C71⋊C8—21no—
49C2×C71⋊C8—41no—
50C8×D71—41no—
51C8×D71—41no—
52C4⋊Dic71—41no—
53C4⋊Dic71—41no—
54C4⋊Dic71D811yescharacteristic
55(C2×D71)⋊C4—21no—
56C71⋊C16—21no—
57C71⋊C16—21no—
58C2×D4×C71—41no—
59C2×D4×C71—41no—
60C2×D4×C71—41no—
61D8×C71—41no—
62D8×C71—41no—
63D8×C71—81no—
64C2×C8×C71—41no—
65C2×C8×C71C2×D411yescharacteristic
66C2×C8×C71#4111yescharacteristic
67C42×C71C2×D411yescharacteristic
68C16×C71—41no—
69C23×C71—81no—
70C22×D71—21no—
71D4×C71—81no—
72D4×C71—81no—
73D4×C71—81no—
74D4×C71—81no—
75D4×C71—161no—
76C2×C4×C71#12111yescharacteristic
77C2×C4×C71C2×D811yescharacteristic
78C2×C4×C71#13511yescharacteristic
79C2×Dic71—21no—
80C2×Dic71—81no—
81D284—21no—
82D284SD3212yes—
83C4×D71—41no—
84C4×D71—41no—
85Dic142—161no—
86C8×C71—21no—
87C8×C71—81no—
88C8×C71#13511yescharacteristic
89C8×C71AGL(1,ℤ/8)11yescharacteristic
90C71⋊C8—41no—
91C71⋊C8—81no—
92C22×C71—161no—
93C22×C71—161no—
94C22×C71—161no—
95C22×C71#46611yescharacteristic
96C2×D71—22no—
97C2×D71—41no—
98C2×D71—41no—
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—321no—
108C2×C71#168111yescharacteristic
109C2×C71#221711yescharacteristic
110C2×C71#222111yescharacteristic
111D71—42no—
112#2240—711no—
113D4×C16—711no—
114#1675—711no—
115#1667—711no—
116#1372—711no—
117#1365—711no—
118#1361D7111yescharacteristic, complemented
119C71#1246811yescharacteristic, complemented, Sylow
120C22×C16—1421no—
121#500—711no—
122#493C2×D7111yescharacteristic
123#476—1421no—
124#459—1421no—
125C8×D4—711no—
126C4⋊C16C2×D7111yescharacteristic
127#357—1421no—
128#357—1421no—
129C22⋊C16—1421no—
130C4×C16—711no—
131#335—711no—
132#335C2×D7111yescharacteristic
133#333—1421no—
134#331—1421no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—2841no—
138C22×C8—1421no—
139C4⋊Q8—1421no—
140#127—21no—
141C4×D4—711no—
142C2×C16—21no—
143C2×C16—2842no—
144C2×C16—711no—
145C2×C16—711no—
146#107—1421no—
147#107—1421no—
148#106—2841no—
149C4⋊C8—21no—
150C4⋊C8—711no—
151C4⋊C8—1421no—
152#102—2841no—
153#102—2841no—
154C22⋊C8—1421no—
155C4×C8C22×D7111yescharacteristic
156C2×Q8—2841no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—711no—
161C22×C4—1421no—
162D8—41no—
163D8—41no—
164D8—81no—
165D8—5682no—
166C2×C8—41no—
167C2×C8—2841no—
168C2×C8—2841no—
169C2×C8—2841no—
170C2×C8—1421no—
171C2×C8D4×D7111yescharacteristic
172C2×C8#1176586511yescharacteristic
173C4⋊C4—2841no—
174C4⋊C4—2841no—
175C4⋊C4—711no—
176C22⋊C4—1421no—
177C42D4×D7111yescharacteristic
178C16—41no—
179C16—1421no—
180C16—1421no—
181C23—81no—
182C23—1421no—
183Q8—11361no—
184D4—81no—
185D4—81no—
186D4—81no—
187D4—81no—
188D4—161no—
189D4—712no—
190D4—1421no—
191C2×C4—2841no—
192C2×C4—2841no—
193C2×C4—5681no—
194C2×C4—1421no—
195C2×C4#1455610111yescharacteristic
196C2×C4D8×D7111yescharacteristic
197C2×C4#1455614911yescharacteristic
198C8—21no—
199C8—81no—
200C8—2841no—
201C8—5681no—
202C8#1455614911yescharacteristic
203C8#1455615211yescharacteristic
204C22—161no—
205C22—161no—
206C22—161no—
207C22—2841no—
208C22—2841no—
209C22—1422no—
210C22#1798623411yescharacteristic
211C4—21no—
212C4—2841no—
213C4—11361no—
214C4#1798621311yescharacteristic
215C4#1798626111yescharacteristic
216C4SD32×D7111yescharacteristic
217C4#1798642811yescharacteristic
218C2—2842no—
219C2—321no—
220C2#2258777611yescharacteristic
221C2#2259043511yescharacteristic
222C2#2259045911yescharacteristic
subgroupG/H
centerC22#17986234
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1361C2
socleC22×C71#466
prime radical#22587692C2
hypercenter#1361D71
hypocenterC71#12468
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
1. #22590563C2semidirect
2. #22590159C2semidirect
3. #22587740C2semidirect
5. #22586413C2semidirect
6. #22586397C2semidirect
7. C71×#1361C2semidirect
118. #1361D71semidirect
119. C71#12468semidirect

Maximal subgroups

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

subgroupindex
#225905632
#225901592
#225877402
#225876922
#225864132
#225863972
C71×#13612
#1246871