Finite Group Databasebeta

← #22587484

Subgroups — #22587484

Order 9088.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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#17986714C211yesmaximal, characteristic
2#17986378C211yesmaximal, characteristic, complemented
3#17986286C211yesmaximal, characteristic, complemented
4#17985747C211yesmaximal, characteristic, complemented
5D4⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
6#17985726C211yesmaximal, characteristic, complemented
7#476×C71C211yesmaximal, characteristic, complemented
8C2×#11765855—22no—
9#14556128C2211yescharacteristic
10#14556114—22no—
11#14556102—21no—
12#14556099—21no—
13C4×D284C2211yescharacteristic
14#14556087—21no—
15#14556086—21no—
16#14556059—21no—
17C4⋊(C71⋊C8)C2211yescharacteristic
18#14556048—21no—
19C4×C71⋊C8C2211yescharacteristic
20C4⋊Q8×C71C2211yescharacteristic
21C4×D4×C71C2211yescharacteristic
22#106×C71—21no—
23C4⋊C8×C71C2211yescharacteristic
24#102×C71—21no—
25C2×Dic142D412yes—
26C2×D284D411yescharacteristic
27C2×C22⋊D71—22no—
28C2×C4×D71—21no—
29#11765855—44no—
30C2×C71⋊C8—21no—
31C2×C71⋊C8—21no—
32C4⋊Dic71—21no—
33C4⋊Dic71—21no—
34C4⋊Dic71D411yescharacteristic
35C22⋊Dic71—22no—
36(C2×D71)⋊C4—22no—
37(C2×D71)⋊C4—21no—
38C4×Dic71—22no—
39C2×Q8×C71—21no—
40C2×D4×C71D411yescharacteristic
41C22×C4×C71—21no—
42C2×C8×C71—21no—
43C4⋊C4×C71—21no—
44C4⋊C4×C71—21no—
45C4⋊C4×C71D411yescharacteristic
46C22⋊C4×C71—21no—
47C42×C71C2311yescharacteristic
48C23×C71—21no—
49C22×D71—21no—
50Q8×C71—42no—
51D4×C71—21no—
52D4×C71—21no—
53C2×C4×C71—21no—
54C2×C4×C71—41no—
55C2×C4×C71—41no—
56C2×C4×C71—41no—
57C2×C4×C71C2×D411yescharacteristic
58C2×C4×C71C2×D411yescharacteristic
59C2×C4×C71C2×D411yescharacteristic
60C22⋊D71—82no—
61C2×Dic71—21no—
62C2×Dic71—22no—
63D284—21no—
64D284—21no—
65C4×D71—41no—
66C4×D71—41no—
67Dic142—42no—
68C8×C71—41no—
69C71⋊C8—22no—
70C71⋊C8—41no—
71C22×C71—41no—
72C22×C71—41no—
73C22×C71—41no—
74C22×C71#12011yescharacteristic
75C2×D71—41no—
76C2×D71—41no—
77C2×D71—41no—
78C4×C71—21no—
79C4×C71—41no—
80C4×C71—81no—
81C4×C71#13511yescharacteristic
82C4×C71#13511yescharacteristic
83C4×C71#13711yescharacteristic
84C4×C71#13711yescharacteristic
85Dic71—41no—
86Dic71—42no—
87C2×C71—81no—
88C2×C71#45211yescharacteristic
89C2×C71#45211yescharacteristic
90C2×C71#45311yescharacteristic
91D71—81no—
92#1634—711nomaximal, Sylow
93C71#163411yescharacteristic, complemented, Sylow
94#555—711no—
95#494—711no—
96#476—711no—
97#476D7111yescharacteristic, complemented
98#333—711no—
99D4⋊C8—711no—
100D4⋊C8—711no—
101C2×Q16—1422no—
102C4⋊Q8—711no—
103C4⋊Q8C2×D7111yescharacteristic
104#124—1422no—
105#121—1421no—
106#121—1421no—
107C4×D4—711no—
108C4×D4C2×D7111yescharacteristic
109#117—1421no—
110#117—1421no—
111#106—21no—
112#106—1421no—
113C4⋊C8—711no—
114C4⋊C8C2×D7111yescharacteristic
115#102—21no—
116#102—1421no—
117C4×C8—711no—
118C2×Q8—21no—
119C2×Q8—712no—
120C2×D4—711no—
121C2×D4—1422no—
122C2×D4C22⋊D7111yescharacteristic
123C22×C4—21no—
124C22×C4—1421no—
125Q16—2844no—
126C2×C8—21no—
127C2×C8—1421no—
128C2×C8—1421no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—711no—
132C4⋊C4—1421no—
133C4⋊C4—1421no—
134C4⋊C4C22⋊D7111yescharacteristic
135C22⋊C4—21no—
136C22⋊C4—1422no—
137C22⋊C4—1422no—
138C22⋊C4—1421no—
139C42—1422no—
140C42C22×D7111yescharacteristic
141C23—21no—
142C23—1421no—
143Q8—42no—
144Q8—2842no—
145D4—21no—
146D4—21no—
147D4—5682no—
148D4—1421no—
149D4—1421no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—2841no—
155C2×C4—2841no—
156C2×C4—1421no—
157C2×C4—1422no—
158C2×C4D4×D7111yescharacteristic
159C2×C4D4×D7111yescharacteristic
160C2×C4C2×C22⋊D7111yescharacteristic
161C8—41no—
162C8—2841no—
163C8—1422no—
164C22—41no—
165C22—41no—
166C22—41no—
167C22—2841no—
168C22—2841no—
169C22—2841no—
170C22#1455609611yescharacteristic
171C4—21no—
172C4—41no—
173C4—81no—
174C4—2841no—
175C4—2842no—
176C4#1455614811yescharacteristic
177C4#1455614911yescharacteristic
178C4#1455615711yescharacteristic
179C4#1455615911yescharacteristic
180C2—81no—
181C2—5681no—
182C2#1798615811yescharacteristic
183C2#1798615911yescharacteristic
184C2#1798616511yescharacteristic
subgroupG/H
centerC22#14556096
commutatorC42×C71C23
FrattiniC42C22×D71
Fitting#476×C71C2
socleC22×C71#120
hypercenter#476D71
hypocenterC71#1634
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #17986378C2semidirect
3. #17986286C2semidirect
4. #17985747C2semidirect
5. D4⋊(C71⋊C8)C2semidirect
6. #17985726C2semidirect
7. #476×C71C2semidirect
93. C71#1634semidirect
97. #476D71semidirect