Finite Group Databasebeta

← #22589499

Subgroups — #22589499

Order 9088.

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

normal subgroups only

This group has 246 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#17986627C211yesmaximal, characteristic
2#17986008C211yesmaximal, characteristic
3#17985906C211yesmaximal, characteristic
4#17985767C211yesmaximal, characteristic
5#17985760C211yesmaximal, characteristic
6#17985758C211yesmaximal, characteristic
7#419×C71C211yesmaximal, characteristic
8C2×#11765849C2211yescharacteristic
9#14556128—21no—
10#14556127—21no—
11#14556120—21no—
12#14556119—21no—
13#14556105—22no—
14#14556087C2211yescharacteristic
15C2×C4⋊Dic71C2211yescharacteristic
16C2×C4×Dic71C2211yescharacteristic
17#14556052—21no—
18#14556052—21no—
19C71⋊#95—22no—
20C22×Q8×C71C2211yescharacteristic
21C2×M16×C71C2211yescharacteristic
22#117×C71C2211yescharacteristic
23#103×C71—21no—
24#103×C71—21no—
25C2×Dic142—22no—
26C22×Dic71—21no—
27C22×Dic71D412yes—
28#11765849—21no—
29#11765849—21no—
30C2×C71⋊C8—21no—
31C4⋊Dic71—22no—
32C4⋊Dic71—21no—
33C4⋊Dic71—21no—
34C4⋊Dic71D411yescharacteristic
35C4⋊Dic71D411yescharacteristic
36Dic71⋊C4—22no—
37Dic71⋊C4—22no—
38C22⋊Dic71—21no—
39C4×Dic71—21no—
40C4×Dic71—21no—
41C4×Dic71—21no—
42C2×Q8×C71—21no—
43C2×Q8×C71—21no—
44C2×Q8×C71—21no—
45C2×Q8×C71—21no—
46C2×Q8×C71—21no—
47C2×Q8×C71Q812yescomplemented
48C22×C4×C71—21no—
49C22×C4×C71C2311yescharacteristic
50M16×C71—21no—
51M16×C71—21no—
52C2×C8×C71—21no—
53C4⋊C4×C71D411yescharacteristic
54C4⋊C4×C71D411yescharacteristic
55C22⋊C4×C71—21no—
56C42×C71—21no—
57C23×C71C2×D411yescharacteristic
58Q8×C71—22no—
59Q8×C71—22no—
60Q8×C71—41no—
61Q8×C71—41no—
62C2×C4×C71—21no—
63C2×C4×C71—21no—
64C2×C4×C71—22no—
65C2×C4×C71—41no—
66C2×C4×C71—41no—
67C2×C4×C71C2×D411yescharacteristic
68C2×C4×C71C2×D411yescharacteristic
69C2×C4×C71C2×Q811yescharacteristic
70C2×C4×C71#4111yescharacteristic
71C2×C4×C71#4111yescharacteristic
72C2×C4×C71#4111yescharacteristic
73C2×Dic71—22no—
74C2×Dic71—21no—
75C2×Dic71—21no—
76C2×Dic71—22no—
77C2×Dic71—41no—
78C2×Dic71—41no—
79C2×Dic71—42no—
80Dic142—82no—
81C8×C71—41no—
82C71⋊C8—41no—
83C22×C71—21no—
84C22×C71—21no—
85C22×C71#12011yescharacteristic
86C22×C71C22⋊Q811yescharacteristic
87C22×C71#12411yescharacteristic
88C4×C71—42no—
89C4×C71—41no—
90C4×C71—41no—
91C4×C71C22⋊Q811yescharacteristic
92C4×C71C22⋊Q811yescharacteristic
93C4×C71#12411yescharacteristic
94C4×C71#12411yescharacteristic
95Dic71—42no—
96Dic71—41no—
97Dic71—42no—
98Dic71—41no—
99C2×C71—21no—
100C2×C71—21no—
101C2×C71#39311yescharacteristic
102C2×C71#45511yescharacteristic
103C2×C71#45611yescharacteristic
104#2028—711nomaximal, Sylow
105C71#202811yescharacteristic, complemented, Sylow
106#544—711no—
107#419—711no—
108#419D7111yescharacteristic
109#391—711no—
110#343—711no—
111#339—711no—
112#339—711no—
113C22×Q8C2×D7111yescharacteristic
114C2×M16—711no—
115C2×M16C2×D7111yescharacteristic
116C4⋊Q8—1421no—
117C4⋊Q8—1421no—
118#125—1421no—
119#125—1421no—
120C22⋊Q8—1422no—
121#117—711no—
122#117C2×D7111yescharacteristic
123C2×C4⋊C4—711no—
124C2×C42—711no—
125#103—21no—
126#103—21no—
127#103—1421no—
128#103—1421no—
129#95—1422no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—1422no—
136C2×Q8Dic14212yescomplemented
137C22×C4—21no—
138C22×C4—712no—
139C22×C4—1421no—
140C22×C4C22×D7111yescharacteristic
141M16—21no—
142M16—21no—
143M16—1421no—
144M16—1421no—
145C2×C8—21no—
146C2×C8—1421no—
147C4⋊C4—711no—
148C4⋊C4—711no—
149C4⋊C4—1422no—
150C4⋊C4—1422no—
151C4⋊C4—1422no—
152C4⋊C4—1421no—
153C4⋊C4—1421no—
154C4⋊C4C22⋊D7111yescharacteristic
155C4⋊C4C22⋊D7111yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—1421no—
158C42—21no—
159C42—1421no—
160C42—1421no—
161C42—1421no—
162C23D4×D7111yescharacteristic
163Q8—22no—
164Q8—22no—
165Q8—41no—
166Q8—41no—
167Q8—5682no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—2841no—
174C2×C4—2841no—
175C2×C4—2842no—
176C2×C4—1422no—
177C2×C4—1421no—
178C2×C4—1421no—
179C2×C4—1422no—
180C2×C4D4×D7111yescharacteristic
181C2×C4C2×C22⋊D7111yescharacteristic
182C2×C4C2×Dic14211yescharacteristic
183C2×C4#1176586411yescharacteristic
184C2×C4#1176586411yescharacteristic
185C2×C4#1176586511yescharacteristic
186C8—41no—
187C8—2841no—
188C22—21no—
189C22—21no—
190C22#1455609611yescharacteristic
191C22#1455610511yescharacteristic
192C22#1455611411yescharacteristic
193C4—42no—
194C4—41no—
195C4—41no—
196C4—2842no—
197C4—2842no—
198C4—2841no—
199C4—2841no—
200C4#1455610511yescharacteristic
201C4#1455610711yescharacteristic
202C4#1455611411yescharacteristic
203C4#1455611511yescharacteristic
204C2—21no—
205C2—21no—
206C2#1798591111yescharacteristic
207C2#1798617511yescharacteristic
208C2#1798618011yescharacteristic
subgroupG/H
centerC22#14556096
commutatorC22×C4×C71C23
FrattiniC22×C4C22×D71
Fitting#419×C71C2
socleC22×C71#120
hypercenter#419D71
hypocenterC71#2028
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
47. C2×Q8×C71Q8semidirect
105. C71#2028semidirect
136. C2×Q8Dic142semidirect