Finite Group Databasebeta

← #22599561

Subgroups — #22599561

Order 9088.

362 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 125 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#17986533C211yesmaximal, characteristic
2#17986405C212yesmaximal, complemented
3#17986361C212yesmaximal, complemented
4#17986329C212yesmaximal, complemented
5#17986312C212yesmaximal
6#17986291C212yesmaximal, complemented
7#17986286C212yesmaximal, complemented
8#17986056C211yesmaximal, characteristic, complemented
9C2×C4⋊Q8×C71C211yesmaximal, characteristic, complemented
10#14556122—21no—
11#14556120C2211yescharacteristic
12#14556116C2211yescharacteristic
13#14556111—21no—
14#14556107C2211yescharacteristic
15#14556102C2211yescharacteristic
16C4×Dic142C2211yescharacteristic
17C4×D284C2211yescharacteristic
18C4×C22⋊D71—21no—
19#14556085—21no—
20#14556061C2214yes—
21#14556059C2214yes—
22C4⋊(C71⋊C8)C2212yes—
23#14556051C2214yes—
24#14556048C2214yes—
25C22⋊(C71⋊C8)C2212yes—
26#14556039C2212yes—
27C22×Q8×C71—21no—
28C4⋊Q8×C71—21no—
29C4⋊Q8×C71—21no—
30C4⋊Q8×C71C2212yes—
31C4⋊Q8×C71C2212yes—
32C2×C4⋊C4×C71—21no—
33C2×C4⋊C4×C71C2212yes—
34C2×C42×C71C2211yescharacteristic
35C2×Dic142C2311yescharacteristic
36C2×D284C2311yescharacteristic
37C2×C22⋊D71—21no—
38C2×C4×D71—21no—
39C2×C71⋊C8C2314yes—
40C4⋊Dic71C2311yescharacteristic
41C4⋊Dic71C2311yescharacteristic
42Dic71⋊C4—21no—
43Dic71⋊C4—21no—
44Dic71⋊C4—21no—
45Dic71⋊C4—21no—
46C22⋊Dic71—21no—
47(C2×D71)⋊C4—21no—
48(C2×D71)⋊C4—21no—
49(C2×D71)⋊C4—21no—
50(C2×D71)⋊C4—21no—
51C4×Dic71—21no—
52C2×Q8×C71—22no—
53C2×Q8×C71—22no—
54C2×Q8×C71—44no—
55C22×C4×C71—22no—
56C22×C4×C71D411yescharacteristic
57C22×C4×C71D411yescharacteristic
58C22×C4×C71C2311yescharacteristic
59C4⋊C4×C71—22no—
60C4⋊C4×C71—22no—
61C4⋊C4×C71—22no—
62C4⋊C4×C71C2314yes—
63C42×C71D411yescharacteristic
64C42×C71D411yescharacteristic
65C42×C71C2311yescharacteristic
66C42×C71C2311yescharacteristic
67C23×C71C2×D411yescharacteristic
68C22×D71—21no—
69Q8×C71—44no—
70Q8×C71—44no—
71C2×C4×C71—22no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71—24no—
75C2×C4×C71—44no—
76C2×C4×C71C2×D411yescharacteristic
77C2×C4×C71C2×D411yescharacteristic
78C2×C4×C71C2×D411yescharacteristic
79C2×C4×C71C2×D411yescharacteristic
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2411yescharacteristic
82C22⋊D71—42no—
83C2×Dic71—21no—
84C2×Dic71—21no—
85C2×Dic71—21no—
86D284—22no—
87C4×D71—42no—
88Dic142—22no—
89C71⋊C8—24no—
90C22×C71—21no—
91C22×C71—21no—
92C22×C71AGL(1,ℤ/8)12yes—
93C22×C71C22×D411yescharacteristic
94C2×D71—41no—
95C2×D71—41no—
96C2×D71—41no—
97C4×C71—21no—
98C4×C71—21no—
99C4×C71—21no—
100C4×C71—21no—
101C4×C71—44no—
102C4×C71#13712yes—
103C4×C712−1+412yes—
104Dic71—41no—
105Dic71—41no—
106Dic71—41no—
107C2×C71—22no—
108C2×C71#53611yescharacteristic
109C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C71C2×#13711yescharacteristic
111D71—81no—
112#3238—711nomaximal, Sylow
113C71#323811yescharacteristic, complemented, Sylow
114C2×C4⋊Q8D7111yescharacteristic, complemented
115#529—711no—
116#501—712no—
117#489—712no—
118#483—712no—
119#480—712no—
120#477—712no—
121#476—712no—
122#432—711no—
123C22×Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8C2×D7112yes—
127C4⋊Q8C2×D7112yes—
128#126—1421no—
129#125—711no—
130#124—711no—
131#123—1421no—
132C22⋊Q8—711no—
133#121—711no—
134C4×Q8—711no—
135C4×D4—711no—
136C4×D4—1421no—
137#117—1421no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4C2×D7112yes—
140C2×C42C2×D7111yescharacteristic
141#107—714no—
142#106—714no—
143C4⋊C8—712no—
144#103—714no—
145#102—714no—
146C22⋊C8—712no—
147#97—712no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—44no—
151C2×Q8—711no—
152C2×D4—711no—
153C2×D4—1421no—
154C22×C4—22no—
155C22×C4—1421no—
156C22×C4C22⋊D7111yescharacteristic
157C22×C4C22⋊D7111yescharacteristic
158C22×C4C22×D7111yescharacteristic
159C2×C8—714no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—711no—
164C4⋊C4—711no—
165C4⋊C4—1421no—
166C4⋊C4—1421no—
167C4⋊C4—1421no—
168C4⋊C4—1421no—
169C4⋊C4C22×D7114yes—
170C22⋊C4—1421no—
171C22⋊C4—1421no—
172C22⋊C4—1421no—
173C22⋊C4—1421no—
174C22⋊C4—1421no—
175C42—1421no—
176C42C22⋊D7111yescharacteristic
177C42C22⋊D7111yescharacteristic
178C42C22×D7111yescharacteristic
179C42C22×D7111yescharacteristic
180C23—1421no—
181C23C2×C22⋊D7111yescharacteristic
182Q8—44no—
183Q8—44no—
184Q8—1422no—
185D4—2842no—
186D4—1422no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—44no—
192C2×C4—2842no—
193C2×C4—1421no—
194C2×C4—1421no—
195C2×C4—1421no—
196C2×C4C2×C22⋊D7111yescharacteristic
197C2×C4C2×C22⋊D7111yescharacteristic
198C2×C4C2×C22⋊D7111yescharacteristic
199C2×C4C2×C22⋊D7111yescharacteristic
200C2×C4C2×C22⋊D7111yescharacteristic
201C2×C4C23×D7111yescharacteristic
202C8—1424no—
203C22—21no—
204C22—21no—
205C22—2841no—
206C22—2841no—
207C22—2841no—
208C22#1455615412yes—
209C22C22×C22⋊D7111yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—44no—
215C4—2841no—
216C4—2841no—
217C4—2841no—
218C4#1455615912yes—
219C4#1455617612yes—
220C2—22no—
221C2—5681no—
222C2#1798657311yescharacteristic
223C2C2×#1455615411yescharacteristic
224C2C2×#1455615911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×C4⋊Q8×C71C2
socleC22×C71C22×D4
hypercenterC2×C4⋊Q8D71
hypocenterC71#3238
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #17986405C2semidirect
3. #17986361C2semidirect
4. #17986329C2semidirect
6. #17986291C2semidirect
7. #17986286C2semidirect
8. #17986056C2semidirect
9. C2×C4⋊Q8×C71C2semidirect
113. C71#3238semidirect
114. C2×C4⋊Q8D71semidirect