Finite Group Databasebeta

← #22589697

Subgroups — #22589697

Order 9088.

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

normal subgroups only

This group has 228 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
1C2×#14556063C211yesmaximal, characteristic, complemented
2#17986026C211yesmaximal, characteristic, complemented
3#17986003C211yesmaximal, characteristic
4C2×#14556048C211yesmaximal, characteristic
5C2×#14556044C211yesmaximal, characteristic
6#17985768C211yesmaximal, characteristic, complemented
7#428×C71C211yesmaximal, characteristic, complemented
8C22×D284C2211yescharacteristic
9C2×#11765849C2211yescharacteristic
10C2×#11765849C2211yescharacteristic
11C22×C71⋊C8C2211yescharacteristic
12#14556063—22no—
13C4⋊(C71⋊C8)—22no—
14#14556048—22no—
15#14556048—21no—
16#14556048—21no—
17#14556044—22no—
18C2×M16×C71C2211yescharacteristic
19#117×C71C2211yescharacteristic
20C2×C4⋊C4×C71C2211yescharacteristic
21#107×C71—21no—
22#106×C71—21no—
23C23×D71—21no—
24C2×D284—21no—
25C2×D284—21no—
26C2×D284—21no—
27C2×D284—21no—
28C2×D284—21no—
29C2×D284Q812yes—
30#11765849—21no—
31#11765849—21no—
32#11765849—21no—
33#11765849—21no—
34C2×C71⋊C8—21no—
35C2×C71⋊C8—21no—
36C2×C71⋊C8—21no—
37C2×C71⋊C8—21no—
38C2×C71⋊C8D412yes—
39C22×C4×C71—21no—
40C22×C4×C71C2311yescharacteristic
41M16×C71—21no—
42M16×C71D412yes—
43C2×C8×C71—21no—
44C4⋊C4×C71—21no—
45C4⋊C4×C71—21no—
46C4⋊C4×C71D411yescharacteristic
47C4⋊C4×C71D411yescharacteristic
48C22⋊C4×C71—21no—
49C42×C71—21no—
50C23×C71#4111yescharacteristic
51C22×D71—22no—
52C22×D71—41no—
53C22×D71—41no—
54C22×D71—81no—
55C22×D71—81no—
56C2×C4×C71—21no—
57C2×C4×C71—21no—
58C2×C4×C71—41no—
59C2×C4×C71—81no—
60C2×C4×C71C2×D411yescharacteristic
61C2×C4×C71C2×D411yescharacteristic
62C2×C4×C71C2×D411yescharacteristic
63C2×C4×C71C2×Q811yescharacteristic
64C2×C4×C71#4111yescharacteristic
65C2×C4×C71#4111yescharacteristic
66D284—22no—
67D284—22no—
68D284—41no—
69D284—41no—
70C8×C71—22no—
71C71⋊C8—22no—
72C71⋊C8—41no—
73C71⋊C8—41no—
74C22×C71—21no—
75C22×C71—21no—
76C22×C71#12111yescharacteristic
77C22×C71C22⋊Q811yescharacteristic
78C22×C71#12611yescharacteristic
79C2×D71—42no—
80C2×D71—82no—
81C2×D71—81no—
82C2×D71—81no—
83C2×D71—82no—
84C4×C71—41no—
85C4×C71—41no—
86C4×C71—81no—
87C4×C71#12111yescharacteristic
88C4×C71#12111yescharacteristic
89C4×C71C22⋊Q811yescharacteristic
90C4×C71C22⋊Q811yescharacteristic
91C2×C71—21no—
92C2×C71—21no—
93C2×C71#39611yescharacteristic
94C2×C71#47111yescharacteristic
95C2×C71#47211yescharacteristic
96D71—82no—
97#2063—711nomaximal, Sylow
98C71#206311yescharacteristic, complemented, Sylow
99C2×#108—711no—
100#428D7111yescharacteristic, complemented
101#424—711no—
102#418—711no—
103C2×#102—711no—
104C2×#100—711no—
105#343—711no—
106C22×D4—711no—
107C2×M16—711no—
108C2×M16—711no—
109C2×M16C2×D7111yescharacteristic
110C22×C8—711no—
111#117C2×D7111yescharacteristic
112C2×C4⋊C4C2×D7111yescharacteristic
113#108—1422no—
114#107—21no—
115#106—21no—
116C4⋊C8—1422no—
117#102—1422no—
118#102—1421no—
119#102—1421no—
120#100—1422no—
121C24—1421no—
122C2×D4—712no—
123C2×D4—1421no—
124C2×D4—1421no—
125C2×D4—1421no—
126C2×D4—1421no—
127C2×D4—1421no—
128C22×C4—21no—
129C22×C4C22×D7111yescharacteristic
130M16—21no—
131M16—1421no—
132M16—1421no—
133M16—1421no—
134M16—1421no—
135M16D28412yes—
136C2×C8—21no—
137C2×C8—712no—
138C2×C8—1421no—
139C2×C8—1421no—
140C2×C8—1421no—
141C2×C8—1421no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4C22⋊D7111yescharacteristic
145C4⋊C4C22⋊D7111yescharacteristic
146C22⋊C4—21no—
147C42—21no—
148C23—2841no—
149C23—2841no—
150C23—5681no—
151C23—5681no—
152C23—1422no—
153C23#1176586411yescharacteristic
154D4—2841no—
155D4—2841no—
156D4—1422no—
157D4—1422no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—81no—
162C2×C4D4×D7111yescharacteristic
163C2×C4C2×C22⋊D7111yescharacteristic
164C2×C4C2×D28411yescharacteristic
165C2×C4Q8×D7111yescharacteristic
166C2×C4#1176586511yescharacteristic
167C2×C4#1176586611yescharacteristic
168C8—22no—
169C8—2841no—
170C8—2841no—
171C8—1422no—
172C22—21no—
173C22—21no—
174C22—2842no—
175C22—5682no—
176C22—5682no—
177C22—5681no—
178C22—5681no—
179C22#1455610011yescharacteristic
180C22#1455610811yescharacteristic
181C22#1455612311yescharacteristic
182C4—41no—
183C4—41no—
184C4—81no—
185C4#1455610211yescharacteristic
186C4#1455610311yescharacteristic
187C4#1455610411yescharacteristic
188C4#1455610611yescharacteristic
189C2—21no—
190C2—21no—
191C2—5682no—
192C2#1798592111yescharacteristic
193C2#1798626411yescharacteristic
194C2#1798626711yescharacteristic
subgroupG/H
centerC22#14556100
commutatorC22×C4×C71C23
FrattiniC22×C4C22×D71
Fitting#428×C71C2
socleC22×C71#121
hypercenter#428D71
hypocenterC71#2063
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#14556063C2semidirect
2. #17986026C2semidirect
6. #17985768C2semidirect
7. #428×C71C2semidirect
98. C71#2063semidirect
100. #428D71semidirect