Finite Group Databasebeta

← #22589661

Subgroups — #22589661

Order 9088.

288 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C22×#11765855C211yesmaximal, characteristic
2C2×Dic71⋊C8C211yesmaximal, characteristic
3C2×#14556053C211yesmaximal, characteristic
4#17985906C211yesmaximal, characteristic
5#17985904C211yesmaximal, characteristic
6#17985761C211yesmaximal, characteristic
7C2×#103×C71C211yesmaximal, characteristic
8C22×Dic142C2211yescharacteristic
9C2×#11765855—22no—
10C2×#11765855—22no—
11C2×#11765855—22no—
12C22×C71⋊C8C2211yescharacteristic
13C2×C4⋊Dic71C2211yescharacteristic
14C2×C4×Dic71C2211yescharacteristic
15Dic71⋊C8—22no—
16#14556053—22no—
17C71⋊#95—21no—
18C71⋊#95—21no—
19C71⋊#95—21no—
20C71⋊#95—21no—
21C22×Q8×C71C2211yescharacteristic
22C22×C8×C71C2211yescharacteristic
23C2×C4⋊C4×C71C2211yescharacteristic
24#103×C71—22no—
25C2×Dic142—21no—
26C2×Dic142—21no—
27C2×Dic142—21no—
28C2×Dic142—21no—
29C2×Dic142—21no—
30C2×Dic142D412yes—
31C22×Dic71—21no—
32C22×Dic71—21no—
33C22×Dic71D411yescharacteristic
34C22×Dic71D411yescharacteristic
35#11765855—28no—
36C2×C71⋊C8—21no—
37C2×C71⋊C8—21no—
38C2×C71⋊C8D412yes—
39C4⋊Dic71—21no—
40C4⋊Dic71—21no—
41C4×Dic71—21no—
42C4×Dic71—21no—
43C2×Q8×C71—21no—
44C2×Q8×C71—21no—
45C2×Q8×C71—21no—
46C2×Q8×C71—21no—
47C2×Q8×C71—21no—
48C2×Q8×C71D412yes—
49C22×C4×C71—21no—
50C22×C4×C71—21no—
51C22×C4×C71C2311yescharacteristic
52C2×C8×C71—21no—
53C2×C8×C71—21no—
54C2×C8×C71—21no—
55C4⋊C4×C71—21no—
56C4⋊C4×C71—21no—
57C23×C71C2×D411yescharacteristic
58Q8×C71—22no—
59Q8×C71—22no—
60Q8×C71—41no—
61Q8×C71—41no—
62C2×C4×C71—22no—
63C2×C4×C71—41no—
64C2×C4×C71—41no—
65C2×C4×C71—41no—
66C2×C4×C71—41no—
67C2×C4×C71—41no—
68C2×C4×C71C2×D411yescharacteristic
69C2×C4×C71C2×D411yescharacteristic
70C2×C4×C71C2×D411yescharacteristic
71C2×C4×C71#4111yescharacteristic
72C2×C4×C71#4111yescharacteristic
73C2×C4×C71#4111yescharacteristic
74C2×Dic71—21no—
75C2×Dic71—21no—
76C2×Dic71—21no—
77C2×Dic71—22no—
78C2×Dic71—21no—
79C2×Dic71—21no—
80C2×Dic71—41no—
81C2×Dic71—41no—
82C2×Dic71—41no—
83C2×Dic71—41no—
84C2×Dic71—41no—
85C2×Dic71Q1612yescomplemented
86Dic142—22no—
87Dic142—22no—
88Dic142—41no—
89Dic142—41no—
90C8×C71—41no—
91C71⋊C8—22no—
92C22×C71#12111yescharacteristic
93C22×C71#12111yescharacteristic
94C22×C71#12111yescharacteristic
95C22×C71C2×Q1611yescharacteristic
96C22×C71#13511yescharacteristic
97C22×C71#13711yescharacteristic
98C22×C71#13711yescharacteristic
99C4×C71—42no—
100C4×C71—81no—
101C4×C71#12011yescharacteristic
102C4×C71#12311yescharacteristic
103C4×C71#12311yescharacteristic
104C4×C71#12411yescharacteristic
105Dic71—22no—
106Dic71—42no—
107Dic71—41no—
108Dic71—81no—
109C2×C71#39411yescharacteristic
110C2×C71#46211yescharacteristic
111C2×C71#46211yescharacteristic
112C2×C71#46411yescharacteristic
113C2×C71#46411yescharacteristic
114C2×C71#46711yescharacteristic
115C2×C71#47011yescharacteristic
116#2056—711nomaximal, Sylow
117C71#205611yescharacteristic, complemented, Sylow
118C22×Q16—711no—
119C2×C4⋊C8—711no—
120C2×#103—711no—
121C2×#103D7111yescharacteristic
122#391—711no—
123#391—711no—
124#340—711no—
125C22×Q8—711no—
126C22×Q8C2×D7111yescharacteristic
127C2×Q16—1422no—
128C2×Q16—1422no—
129C2×Q16—1422no—
130C22×C8—711no—
131C22×C8C2×D7111yescharacteristic
132C2×C4⋊C4—711no—
133C2×C4⋊C4C2×D7111yescharacteristic
134C2×C42—711no—
135C4⋊C8—1422no—
136#103—22no—
137#103—1422no—
138#95—1421no—
139#95—1421no—
140#95—1421no—
141#95—1421no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—712no—
148C2×Q8—1421no—
149C2×Q8—1421no—
150C2×Q8—1421no—
151C2×Q8—1421no—
152C2×Q8—1421no—
153C2×Q8C22⋊D7112yes—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—711no—
157C22×C4—711no—
158C22×C4—1421no—
159C22×C4—1421no—
160C22×C4C22×D7111yescharacteristic
161Q16—1428no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—712no—
166C2×C8—1421no—
167C2×C8—1421no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—1421no—
171C4⋊C4—1421no—
172C42—1421no—
173C42—1421no—
174C23D4×D7111yescharacteristic
175Q8—22no—
176Q8—22no—
177Q8—41no—
178Q8—41no—
179Q8—2841no—
180Q8—2841no—
181Q8—1422no—
182Q8—1422no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—2841no—
190C2×C4—2841no—
191C2×C4—2841no—
192C2×C4—2841no—
193C2×C4—2841no—
194C2×C4—712no—
195C2×C4—1421no—
196C2×C4—1421no—
197C2×C4—1422no—
198C2×C4—1421no—
199C2×C4—1421no—
200C2×C4—1421no—
201C2×C4D4×D7111yescharacteristic
202C2×C4D4×D7111yescharacteristic
203C2×C4C2×C22⋊D7111yescharacteristic
204C2×C4#1176586411yescharacteristic
205C2×C4#1176586511yescharacteristic
206C2×C4#1176586511yescharacteristic
207C8—41no—
208C8—1422no—
209C22#1455610011yescharacteristic
210C22#1455610111yescharacteristic
211C22#1455610111yescharacteristic
212C22Q16×D7111yescharacteristic
213C22#1455614911yescharacteristic
214C22#1455615711yescharacteristic
215C22#1455616011yescharacteristic
216C4—42no—
217C4—81no—
218C4—2842no—
219C4—2841no—
220C4—5681no—
221C4—1422no—
222C4#1455609611yescharacteristic
223C4#1455611011yescharacteristic
224C4#1455611111yescharacteristic
225C4#1455611411yescharacteristic
226C2#1798591411yescharacteristic
227C2#1798620711yescharacteristic
228C2#1798620811yescharacteristic
229C2#1798621911yescharacteristic
230C2#1798622411yescharacteristic
231C2#1798623911yescharacteristic
232C2#1798625611yescharacteristic
subgroupG/H
centerC23D4×D71
commutatorC22×C4×C71C23
FrattiniC22×C4C22×D71
FittingC2×#103×C71C2
socleC23×C71C2×D4
hypercenterC2×#103D71
hypocenterC71#2056
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
85. C2×Dic71Q16semidirect
117. C71#2056semidirect