Finite Group Databasebeta

← #36352725

Subgroups — #36352725

Order 22144.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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#26012156C211yesmaximal, characteristic
2#26011783C211yesmaximal, characteristic, complemented
3#26011778C211yesmaximal, characteristic
4#26011237C211yesmaximal, characteristic, complemented
5#26011223C211yesmaximal, characteristic, complemented
6#26011220C211yesmaximal, characteristic
7#487×C173C211yesmaximal, characteristic, complemented
8C2×Q8×D173—21no—
9#19149678—21no—
10#19149657—22no—
11#19149655—21no—
12C4×Dic346C2211yescharacteristic
13C4×D692C2211yescharacteristic
14#19149641—21no—
15#19149612—21no—
16#19149610—21no—
17C4⋊#13063436C2211yescharacteristic
18C4⋊#13063436C2211yescharacteristic
19#19149602—21no—
20#19149599—21no—
21C4⋊Q8×C173C2211yescharacteristic
22C4⋊Q8×C173C2211yescharacteristic
23#103×C173—22no—
24C4×C8×C173C2211yescharacteristic
25C2×Dic346—22no—
26C2×Dic346D411yescharacteristic
27Q8×D173—42no—
28Q8×D173—42no—
29C2×D692D411yescharacteristic
30C2×C4×D173—22no—
31C2×C4×D173—21no—
32C2×#13063436—21no—
33C2×#13063436—21no—
34C4⋊Dic173D411yescharacteristic
35C4⋊Dic173D411yescharacteristic
36#14583805—22no—
37#14583805—21no—
38#14583803—22no—
39#14583803—21no—
40C4×Dic173—21no—
41C2×Q8×C173—21no—
42C2×Q8×C173—21no—
43C2×C8×C173—21no—
44C4⋊C4×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173—21no—
47C4⋊C4×C173D412yes—
48C42×C173C2311yescharacteristic
49C22×D173—21no—
50Dic346—21no—
51Dic346—44no—
52Dic346SD1612yes—
53D692—21no—
54D692Q1612yes—
55C2×Dic173—21no—
56C2×Dic173—21no—
57C2×Dic173—21no—
58C4×D173—44no—
59C4×D173—41no—
60C4×D173—41no—
61#13063436—41no—
62#13063436—41no—
63Q8×C173—42no—
64Q8×C173—41no—
65Q8×C173—41no—
66C2×C4×C173—22no—
67C2×C4×C173—41no—
68C2×C4×C173C2×D411yescharacteristic
69C2×C4×C173C2×D411yescharacteristic
70C2×C4×C173C2×D411yescharacteristic
71C8×C173—41no—
72C22×C173#12011yescharacteristic
73C2×D173—22no—
74C2×D173—41no—
75C2×D173—41no—
76C4×C173—21no—
77C4×C173—42no—
78C4×C173—81no—
79C4×C173C2×SD1611yescharacteristic
80C4×C173C2×Q1611yescharacteristic
81C4×C173AGL(1,ℤ/8)11yescharacteristic
82C4×C173#13711yescharacteristic
83Dic173—22no—
84Dic173—41no—
85Dic173—41no—
86C2×C173#45011yescharacteristic
87C2×C173#45111yescharacteristic
88C2×C173#45411yescharacteristic
89D173—42no—
90C173#163711yescharacteristic, complemented, Sylow
91#1637—1731nomaximal, Sylow
92D4×Q8—1731no—
93#487D17311yescharacteristic, complemented
94#477—1731no—
95#476—1731no—
96#332—1731no—
97Q8⋊C8—1731no—
98D4⋊C8—1731no—
99C22×Q8—3461no—
100C4⋊Q8—3461no—
101C4⋊Q8C2×D17311yescharacteristic
102C4⋊Q8C2×D17311yescharacteristic
103C22⋊Q8—3461no—
104C22⋊Q8—3462no—
105C4×Q8—1731no—
106C4×D4—3461no—
107C4×D4—1731no—
108#107—3461no—
109#106—3461no—
110C4⋊C8—1731no—
111C4⋊C8—1731no—
112#103—22no—
113#103—3461no—
114#102—3461no—
115C4×C8C2×D17311yescharacteristic
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—3462no—
119C2×Q8—1731no—
120C2×Q8—6922no—
121C2×Q8—6922no—
122C2×D4—1731no—
123C22×C4—3462no—
124C22×C4—3461no—
125C2×C8—21no—
126C2×C8—3461no—
127C2×C8—3461no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—3462no—
132C4⋊C4—3461no—
133C4⋊C4—1731no—
134C4⋊C4—1731no—
135C4⋊C4C22⋊D17312yes—
136C22⋊C4—3462no—
137C22⋊C4—3461no—
138C42—3461no—
139C42C22×D17311yescharacteristic
140C23—3461no—
141Q8—42no—
142Q8—41no—
143Q8—41no—
144Q8—3461no—
145Q8—1732no—
146Q8—6924no—
147D4—3461no—
148D4—1732no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—3461no—
152C2×C4—3461no—
153C2×C4—3461no—
154C2×C4—6924no—
155C2×C4—6921no—
156C2×C4—6921no—
157C2×C4D4×D17311yescharacteristic
158C2×C4D4×D17311yescharacteristic
159C2×C4C2×C22⋊D17311yescharacteristic
160C8—41no—
161C8—6921no—
162C8—6921no—
163C22—3462no—
164C22—6921no—
165C22—6921no—
166C22#1914964711yescharacteristic
167C4—21no—
168C4—42no—
169C4—81no—
170C4—3462no—
171C4—6921no—
172C4—6921no—
173C4SD16×D17311yescharacteristic
174C4Q16×D17311yescharacteristic
175C4#1914970611yescharacteristic
176C4#1914971111yescharacteristic
177C2—6922no—
178C2#2601164011yescharacteristic
179C2#2601164511yescharacteristic
180C2#2601166311yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#487×C173C2
socleC22×C173#120
prime radicalC2×D692D4
hypercenter#487D173
hypocenterC173#1637
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #26011783C2semidirect
4. #26011237C2semidirect
5. #26011223C2semidirect
7. #487×C173C2semidirect
90. C173#1637semidirect
93. #487D173semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#260121562
#260117832
#260117782
#260112372
#260112232
#260112202
#487×C1732
#1637173