Finite Group Databasebeta

← #22599081

Subgroups — #22599081

Order 9088.

358 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 129 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#17986556C211yesmaximal, characteristic
2#17986395C211yesmaximal, characteristic, complemented
3#17986378C211yesmaximal, characteristic, complemented
4#17986351C211yesmaximal, characteristic, complemented
5#17986349C211yesmaximal, characteristic, complemented
6#17986255C214yesmaximal
7#17986104C214yesmaximal, complemented
8C4×#11765849C211yesmaximal, characteristic
9#533×C71C211yesmaximal, characteristic
10C2×#11765855C2214yes—
11C2×#11765849C2212yes—
12#14556128C2211yescharacteristic
13#14556128C2211yescharacteristic
14#14556120—21no—
15#14556107—21no—
16#14556107C2212yes—
17C4×Dic142—21no—
18C4×Dic142C2212yes—
19#14556087—21no—
20#14556059C2214yes—
21#14556052C2214yes—
22#14556051C2214yes—
23#14556039C2212yes—
24C4×C71⋊C8C2211yescharacteristic
25C4×C71⋊C8C2211yescharacteristic
26C4⋊Q8×C71C2211yescharacteristic
27C4⋊Q8×C71C2211yescharacteristic
28#125×C71—21no—
29C22⋊Q8×C71—21no—
30C22⋊Q8×C71C2212yes—
31C4×Q8×C71—21no—
32C4×Q8×C71C2212yes—
33#117×C71—21no—
34C2×C42×C71C2211yescharacteristic
35C2×Dic142—21no—
36C2×Dic142C2312yes—
37#11765855—28no—
38#11765849—24no—
39C2×C71⋊C8C2312yes—
40C2×C71⋊C8C2312yes—
41C4⋊Dic71—21no—
42C4⋊Dic71—21no—
43C4⋊Dic71—21no—
44C4⋊Dic71C2312yes—
45Dic71⋊C4—22no—
46Dic71⋊C4—22no—
47C22⋊Dic71—22no—
48C4×Dic71—22no—
49C2×Q8×C71—21no—
50C2×Q8×C71C2312yes—
51C22×C4×C71D411yescharacteristic
52C22×C4×C71D411yescharacteristic
53C22×C4×C71C2311yescharacteristic
54C4⋊C4×C71—21no—
55C4⋊C4×C71—21no—
56C4⋊C4×C71—21no—
57C4⋊C4×C71—22no—
58C4⋊C4×C71—22no—
59C4⋊C4×C71C2312yes—
60C22⋊C4×C71—22no—
61C42×C71—22no—
62C42×C71D411yescharacteristic
63C42×C71D411yescharacteristic
64C42×C71C2311yescharacteristic
65C42×C71C2311yescharacteristic
66C23×C71C2×D411yescharacteristic
67Q8×C71—24no—
68Q8×C71—42no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71C2×D411yescharacteristic
75C2×C4×C71C2×D411yescharacteristic
76C2×C4×C71C2×D411yescharacteristic
77C2×C4×C71C2×D411yescharacteristic
78C2×C4×C71C2×D411yescharacteristic
79C2×C4×C71C2411yescharacteristic
80C2×Dic71—22no—
81C2×Dic71—22no—
82Dic142—24no—
83Dic142—42no—
84C71⋊C8—22no—
85C71⋊C8#4114yes—
86C22×C71—21no—
87C22×C71—21no—
88C22×C71—21no—
89C22×C71C22×D411yescharacteristic
90C4×C71—21no—
91C4×C71—21no—
92C4×C71—21no—
93C4×C71—42no—
94C4×C71—42no—
95C4×C71#13712yes—
96C4×C71#13712yes—
97C4×C71C2×#4112yes—
98Dic71—42no—
99Dic71—42no—
100C2×C71—41no—
101C2×C71#53211yescharacteristic
102C2×C71C2×#13711yescharacteristic
103C2×C71C2×#13711yescharacteristic
104#3185—711nomaximal, Sylow
105C71#318511yescharacteristic, complemented, Sylow
106#533—711no—
107#533D7111yescharacteristic
108#498—711no—
109#494—711no—
110#487—711no—
111#487—711no—
112#470—714no—
113#441—714no—
114C4×M16—711no—
115C2×Q16—714no—
116C2×M16—712no—
117C4⋊Q8—711no—
118C4⋊Q8—711no—
119C4⋊Q8C2×D7111yescharacteristic
120C4⋊Q8C2×D7111yescharacteristic
121#125—21no—
122#125—1421no—
123C22⋊Q8—21no—
124C22⋊Q8—712no—
125C22⋊Q8—1421no—
126C22⋊Q8C2×D7112yes—
127C4×Q8—21no—
128C4×Q8—712no—
129C4×Q8—1421no—
130C4×Q8C2×D7112yes—
131#117—21no—
132#117—1421no—
133C2×C42C2×D7111yescharacteristic
134#106—714no—
135#103—714no—
136#103—714no—
137#97—712no—
138C4×C8—711no—
139C4×C8—711no—
140C2×Q8—21no—
141C2×Q8—712no—
142C2×Q8—1421no—
143C2×Q8C22×D7112yes—
144C22×C4C22⋊D7111yescharacteristic
145C22×C4C22⋊D7111yescharacteristic
146C22×C4C22×D7111yescharacteristic
147Q16—1428no—
148M16—1424no—
149C2×C8—712no—
150C2×C8—712no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—712no—
157C4⋊C4—1422no—
158C4⋊C4—1422no—
159C4⋊C4—1421no—
160C4⋊C4—1421no—
161C4⋊C4—1421no—
162C4⋊C4C22×D7112yes—
163C22⋊C4—22no—
164C22⋊C4—1422no—
165C42—22no—
166C42—1422no—
167C42C22⋊D7111yescharacteristic
168C42C22⋊D7111yescharacteristic
169C42C22×D7111yescharacteristic
170C42C22×D7111yescharacteristic
171C23C2×C22⋊D7111yescharacteristic
172Q8—24no—
173Q8—42no—
174Q8—2842no—
175Q8—1424no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—1422no—
182C2×C4—1422no—
183C2×C4C2×C22⋊D7111yescharacteristic
184C2×C4C2×C22⋊D7111yescharacteristic
185C2×C4C2×C22⋊D7111yescharacteristic
186C2×C4C2×C22⋊D7111yescharacteristic
187C2×C4C2×C22⋊D7111yescharacteristic
188C2×C4C23×D7111yescharacteristic
189C8—714no—
190C8—1422no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22C22×C22⋊D7111yescharacteristic
195C4—21no—
196C4—21no—
197C4—21no—
198C4—42no—
199C4—42no—
200C4—2842no—
201C4—2842no—
202C4#1455615812yes—
203C4#1455615912yes—
204C4#41×D7112yes—
205C2—41no—
206C2#1798654511yescharacteristic
207C2C2×#1455615811yescharacteristic
208C2C2×#1455615911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#533×C71C2
socleC22×C71C22×D4
hypercenter#533D71
hypocenterC71#3185
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #17986395C2semidirect
3. #17986378C2semidirect
4. #17986351C2semidirect
5. #17986349C2semidirect
7. #17986104C2semidirect
105. C71#3185semidirect