Finite Group Databasebeta

← #22589610

Subgroups — #22589610

Order 9088.

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

normal subgroups only

This group has 274 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#17986047C211yesmaximal, characteristic, complemented
2#17986016C211yesmaximal, characteristic, complemented
3#17986004C211yesmaximal, characteristic, complemented
4#17985983C211yesmaximal, characteristic, complemented
5C2×#14556045C211yesmaximal, characteristic, complemented
6#17985771C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C71C211yesmaximal, characteristic
8C2×#11765849C2211yescharacteristic
9C2×#11765849C2211yescharacteristic
10C2×#11765849C2211yescharacteristic
11#14556087C2211yescharacteristic
12#14556063—22no—
13#14556055—22no—
14#14556049—22no—
15#14556047—21no—
16#14556045—22no—
17#14556045—21no—
18C2×#41×C71C2211yescharacteristic
19C22×D4×C71C2211yescharacteristic
20AGL(1,ℤ/8)×C71—22no—
21AGL(1,ℤ/8)×C71—22no—
22C2×SD16×C71—21no—
23C2×D8×C71—21no—
24C2×M16×C71C2211yescharacteristic
25#11765849—21no—
26#11765849—21no—
27#11765849—21no—
28#11765849—21no—
29#11765849—21no—
30#11765849D412yes—
31C2×C71⋊C8—21no—
32C2×C71⋊C8—21no—
33C2×C71⋊C8—21no—
34C4⋊Dic71—21no—
35C22⋊Dic71—21no—
36C4×Dic71—21no—
37C24×C71—21no—
38#41×C71—21no—
39#41×C71—21no—
40#41×C71—21no—
41#41×C71—21no—
42C2×Q8×C71D411yescharacteristic
43C2×D4×C71—21no—
44C2×D4×C71—21no—
45C2×D4×C71—21no—
46C2×D4×C71—21no—
47C2×D4×C71—21no—
48C2×D4×C71—21no—
49C2×D4×C71D411yescharacteristic
50C2×D4×C71D412yes—
51C22×C4×C71—21no—
52C22×C4×C71C2311yescharacteristic
53SD16×C71—24no—
54D8×C71—24no—
55M16×C71—21no—
56M16×C71D412yes—
57C2×C8×C71—21no—
58C23×C71—22no—
59C23×C71—21no—
60C23×C71—41no—
61C23×C71—41no—
62C23×C71—82no—
63C23×C71#4111yescharacteristic
64Q8×C71—21no—
65Q8×C71—21no—
66D4×C71—21no—
67D4×C71—22no—
68D4×C71—21no—
69D4×C71—22no—
70D4×C71—41no—
71D4×C71—41no—
72D4×C71—41no—
73D4×C71—41no—
74C2×C4×C71—21no—
75C2×C4×C71—41no—
76C2×C4×C71—41no—
77C2×C4×C71C2×D411yescharacteristic
78C2×C4×C71C2×D411yescharacteristic
79C2×C4×C71C2×D411yescharacteristic
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71#4111yescharacteristic
82C2×C4×C71#4111yescharacteristic
83C2×Dic71—41no—
84C8×C71—22no—
85C71⋊C8—22no—
86C71⋊C8—41no—
87C71⋊C8—41no—
88C22×C71—41no—
89C22×C71—41no—
90C22×C71—42no—
91C22×C71—81no—
92C22×C71—82no—
93C22×C71—82no—
94C22×C71—82no—
95C22×C71#12111yescharacteristic
96C22×C71#12311yescharacteristic
97C22×C71#12411yescharacteristic
98C4×C71—41no—
99C4×C71#12011yescharacteristic
100C4×C71#12111yescharacteristic
101C4×C71#12111yescharacteristic
102C4×C71#12311yescharacteristic
103Dic71—81no—
104C2×C71—21no—
105C2×C71—21no—
106C2×C71—21no—
107C2×C71—81no—
108C2×C71—82no—
109C2×C71#39411yescharacteristic
110#2048—711nomaximal, Sylow
111C71#204811yescharacteristic, complemented, Sylow
112C2×AGL(1,ℤ/8)D7111yescharacteristic
113#430—711no—
114#421—711no—
115#418—711no—
116#413—711no—
117C2×#100—711no—
118#344—711no—
119C2×#41C2×D7111yescharacteristic
120C22×D4C2×D7111yescharacteristic
121AGL(1,ℤ/8)—22no—
122AGL(1,ℤ/8)—22no—
123C2×SD16—21no—
124C2×D8—21no—
125C2×M16—711no—
126C2×M16—711no—
127C2×M16—711no—
128C2×M16C2×D7111yescharacteristic
129#117—711no—
130#108—1422no—
131#104—1422no—
132#102—1422no—
133#101—1421no—
134#100—1422no—
135#100—1421no—
136C24—21no—
137#41—21no—
138#41—21no—
139#41—21no—
140#41—21no—
141C2×Q8C22⋊D7111yescharacteristic
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4C22⋊D7111yescharacteristic
149C2×D4C22⋊D7112yes—
150C22×C4—21no—
151C22×C4C22×D7111yescharacteristic
152SD16—24no—
153D8—24no—
154M16—21no—
155M16—712no—
156M16—1421no—
157M16—1421no—
158M16—1421no—
159M16—1421no—
160M16—1421no—
161M16C22⋊D7112yes—
162C2×C8—21no—
163C2×C8—1421no—
164C2×C8—1421no—
165C2×C8—1421no—
166C4⋊C4—1421no—
167C22⋊C4—1421no—
168C42—1421no—
169C23—22no—
170C23—21no—
171C23—41no—
172C23—41no—
173C23—82no—
174C23#1176586411yescharacteristic
175Q8—21no—
176Q8—21no—
177D4—21no—
178D4—22no—
179D4—21no—
180D4—22no—
181D4—41no—
182D4—41no—
183D4—41no—
184D4—41no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—2841no—
189C2×C4D4×D7111yescharacteristic
190C2×C4C2×C22⋊D7111yescharacteristic
191C2×C4C2×C22⋊D7111yescharacteristic
192C2×C4C2×C22⋊D7111yescharacteristic
193C2×C4#1176586411yescharacteristic
194C2×C4#1176586411yescharacteristic
195C8—22no—
196C8—2841no—
197C8—2841no—
198C8—1422no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—82no—
203C22—81no—
204C22—82no—
205C22—82no—
206C22#1455610011yescharacteristic
207C22#1455610911yescharacteristic
208C22#1455611511yescharacteristic
209C4—41no—
210C4—5681no—
211C4#1455609711yescharacteristic
212C4#1455609911yescharacteristic
213C4#1455610011yescharacteristic
214C4#1455610911yescharacteristic
215C2—21no—
216C2—21no—
217C2—21no—
218C2—82no—
219C2—81no—
220C2#1798591311yescharacteristic
subgroupG/H
centerC2#17985913
commutatorC22×C4×C71C23
FrattiniC22×C4C22×D71
FittingC2×AGL(1,ℤ/8)×C71C2
socleC2×C71#394
hypercenterC2×AGL(1,ℤ/8)D71
hypocenterC71#2048
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #17986047C2semidirect
2. #17986016C2semidirect
3. #17986004C2semidirect
4. #17985983C2semidirect
5. C2×#14556045C2semidirect
6. #17985771C2semidirect
111. C71#2048semidirect