Finite Group Databasebeta

← #22601139

Subgroups — #22601139

Order 9088.

326 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 123 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#17986727C211yesmaximal, characteristic
2#17986367C212yesmaximal
3#17986349C211yesmaximal, characteristic
4#17986306C212yesmaximal
5#17986284C211yesmaximal, characteristic
6#17986206C211yesmaximal, characteristic
7#17986206C212yesmaximal
8#17986130C211yesmaximal, characteristic
9#17986104C212yesmaximal
10C4×#11765855C211yesmaximal, characteristic
11#557×C71C211yesmaximal, characteristic
12C2×#11765855C2211yescharacteristic
13C2×#11765855C2212yes—
14#14556128C2211yescharacteristic
15#14556126—21no—
16#14556120C2212yes—
17#14556118—22no—
18C4×Dic142C2211yescharacteristic
19C4×Dic142C2212yes—
20#14556093—21no—
21#14556061C2211yescharacteristic
22#14556059C2212yes—
23C4⋊(C71⋊C8)C2212yes—
24C4⋊(C71⋊C8)C2211yescharacteristic
25#14556052C2212yes—
26#14556052C2212yes—
27#14556052C2211yescharacteristic
28#14556052C2212yes—
29#14556052C2212yes—
30#14556051C2211yescharacteristic
31#14556039C2212yes—
32C4×C71⋊C8C2211yescharacteristic
33C4⋊Q8×C71C2211yescharacteristic
34C4⋊Q8×C71C2212yes—
35#125×C71—21no—
36#125×C71—22no—
37C4×Q8×C71—21no—
38C4×Q8×C71C2212yes—
39C4×Q8×C71C2211yescharacteristic
40C4×Q8×C71C2211yescharacteristic
41C2×Dic142—21no—
42C2×Dic142C2311yescharacteristic
43#11765855—22no—
44#11765855—24no—
45C2×C71⋊C8C2311yescharacteristic
46C2×C71⋊C8C2311yescharacteristic
47C2×C71⋊C8C2312yes—
48C4⋊Dic71—21no—
49C4⋊Dic71C2312yes—
50C4⋊Dic71C2311yescharacteristic
51Dic71⋊C4—22no—
52Dic71⋊C4—21no—
53Dic71⋊C4—22no—
54Dic71⋊C4—22no—
55C4×Dic71—22no—
56C4×Dic71—21no—
57C2×Q8×C71D411yescharacteristic
58C2×Q8×C71C2312yes—
59C2×Q8×C71C2311yescharacteristic
60C4⋊C4×C71—22no—
61C4⋊C4×C71—21no—
62C4⋊C4×C71—22no—
63C4⋊C4×C71—21no—
64C4⋊C4×C71—22no—
65C4⋊C4×C71—21no—
66C4⋊C4×C71D411yescharacteristic
67C4⋊C4×C71D412yes—
68C4⋊C4×C71C2311yescharacteristic
69C42×C71—22no—
70C42×C71—21no—
71C42×C71C2311yescharacteristic
72C42×C71C2312yes—
73Q8×C71—22no—
74Q8×C71—22no—
75Q8×C71—21no—
76Q8×C71—22no—
77Q8×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—22no—
80C2×C4×C71—21no—
81C2×C4×C71C2×D412yes—
82C2×C4×C71C2×D412yes—
83C2×C4×C71C2×D411yescharacteristic
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71C2411yescharacteristic
86C2×Dic71—21no—
87C2×Dic71—21no—
88C2×Dic71—22no—
89Dic142—21no—
90Dic142—42no—
91Dic142#4112yes—
92C71⋊C8—21no—
93C71⋊C8—22no—
94C71⋊C8—21no—
95C22×C71C22×D411yescharacteristic
96C4×C71—22no—
97C4×C71—22no—
98C4×C71—21no—
99C4×C71—42no—
100C4×C71—41no—
101C4×C71—41no—
102C4×C71#13712yes—
103C4×C71C2×#4111yescharacteristic
104C4×C712−1+411yescharacteristic
105Dic71—22no—
106Dic71—42no—
107Dic71—41no—
108C2×C71#54811yescharacteristic
109C2×C71C2×#13711yescharacteristic
110C2×C71#57811yescharacteristic
111#3374—711nomaximal, Sylow
112C71#337411yescharacteristic, complemented, Sylow
113#557—711no—
114#557D7111yescharacteristic
115#491—712no—
116#487—711no—
117#479—712no—
118#475—711no—
119#462—712no—
120#462—711no—
121#446—711no—
122#441—712no—
123C4×Q16—711no—
124C2×Q16—711no—
125C2×Q16—712no—
126C4⋊Q8—711no—
127C4⋊Q8—1421no—
128C4⋊Q8C2×D7111yescharacteristic
129C4⋊Q8C2×D7112yes—
130#125—21no—
131#125—22no—
132#125—712no—
133#125—1422no—
134C4×Q8—21no—
135C4×Q8—711no—
136C4×Q8—712no—
137C4×Q8—1421no—
138C4×Q8C2×D7111yescharacteristic
139C4×Q8C2×D7112yes—
140C4×Q8C2×D7111yescharacteristic
141#107—711no—
142#106—712no—
143C4⋊C8—711no—
144C4⋊C8—712no—
145#103—711no—
146#103—712no—
147#103—712no—
148#103—712no—
149#103—712no—
150#103—711no—
151#97—712no—
152C4×C8—711no—
153C2×Q8—711no—
154C2×Q8—1421no—
155C2×Q8C22⋊D7111yescharacteristic
156C2×Q8C22×D7111yescharacteristic
157C2×Q8C22×D7112yes—
158Q16—1422no—
159Q16—1424no—
160C2×C8—711no—
161C2×C8—711no—
162C2×C8—712no—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—22no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—712no—
170C4⋊C4—711no—
171C4⋊C4—1421no—
172C4⋊C4—1422no—
173C4⋊C4—1422no—
174C4⋊C4—1422no—
175C4⋊C4—1421no—
176C4⋊C4C22⋊D7111yescharacteristic
177C4⋊C4C22⋊D7112yes—
178C4⋊C4C22×D7111yescharacteristic
179C42—22no—
180C42—21no—
181C42—1422no—
182C42—1421no—
183C42C22×D7111yescharacteristic
184C42C22×D7112yes—
185Q8—22no—
186Q8—21no—
187Q8—22no—
188Q8—21no—
189Q8—22no—
190Q8—2842no—
191Q8—712no—
192Q8—1421no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—1421no—
197C2×C4—1422no—
198C2×C4—1421no—
199C2×C4C2×C22⋊D7112yes—
200C2×C4C2×C22⋊D7112yes—
201C2×C4C2×C22⋊D7111yescharacteristic
202C2×C4C2×C22⋊D7111yescharacteristic
203C2×C4C23×D7111yescharacteristic
204C8—1421no—
205C8—1422no—
206C8—1421no—
207C22C22×C22⋊D7111yescharacteristic
208C4—22no—
209C4—22no—
210C4—21no—
211C4—42no—
212C4—41no—
213C4—41no—
214C4—2842no—
215C4—2841no—
216C4—1422no—
217C4#1455615912yes—
218C4#41×D7111yescharacteristic
219C4#1455617511yescharacteristic
220C2#1798665611yescharacteristic
221C2C2×#1455615911yescharacteristic
222C2#1798681411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#557×C71C2
socleC22×C71C22×D4
hypercenter#557D71
hypocenterC71#3374
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
112. C71#3374semidirect