Finite Group Databasebeta

← #36369569

Subgroups — #36369569

Order 22144.

324 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 68 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#26012559C211yesmaximal, characteristic
2#26012484C212yesmaximal
3D173⋊#101C211yesmaximal, characteristic
4#26012362C212yesmaximal
5C4⋊Q8×D173C211yesmaximal, characteristic
6D173⋊M16C2212yes—
7Q8×#10621924—24no—
8#19149758—22no—
9#19149758C2212yes—
10#19149742—24no—
11#19149740—22no—
12C2×Q8×D173C2211yescharacteristic
13C2×Q8×D173C2211yescharacteristic
14#19149679C411yescharacteristic, complemented
15#19149678—22no—
16#19149677C411yescharacteristic, complemented
17#19149677C411yescharacteristic, complemented
18C4⋊C4×D173—22no—
19C42×D173C2211yescharacteristic
20C4⋊Q8×C173C411yescharacteristic, complemented
21#14583836—22no—
22#14583835—22no—
23D173⋊C8—22no—
24C4⋊#10621924—24no—
25C4⋊#10621924D414yes—
26C22⋊#10621924—22no—
27C4×#10621924—24no—
28C4×#10621924—22no—
29C2×Dic346—21no—
30C2×Dic346C2×C411yescharacteristic
31C2×Dic346C2×C411yescharacteristic
32C2×Dic346D412yes—
33Q8×D173—22no—
34Q8×D173—22no—
35Q8×D173—22no—
36Q8×D173—22no—
37C2×C4×D173—21no—
38C2×C4×D173—21no—
39C2×C4×D173D412yes—
40C2×C4×D173C2311yescharacteristic
41C4⋊Dic173—22no—
42#14583805—22no—
43#14583805—22no—
44C4×Dic173—21no—
45C4×Dic173C2×C411yescharacteristic
46C2×Q8×C173C2×C411yescharacteristic
47C2×Q8×C173C2×C411yescharacteristic
48C4⋊C4×C173—22no—
49C42×C173C2×C411yescharacteristic
50C2×#10621924—24no—
51#13063443—42no—
52C22×D173C2×D411yescharacteristic
53Dic346—22no—
54Dic346—24no—
55Dic346—22no—
56Dic346—42no—
57C2×Dic173—21no—
58C2×Dic173—21no—
59C2×Dic173C22⋊C412yes—
60C2×Dic173C2×D411yescharacteristic
61C4×D173—24no—
62C4×D173—22no—
63C4×D173—22no—
64C4×D173—41no—
65C4×D173—41no—
66C4×D173C2×D412yes—
67C4×D173#4112yes—
68Q8×C173—22no—
69Q8×C173—22no—
70C2×C4×C173—21no—
71C2×C4×C173—21no—
72C2×C4×C173C22⋊C412yes—
73C2×C4×C173C22×C411yescharacteristic
74C22×C173C2×C22⋊C411yescharacteristic
75C2×D173—22no—
76C2×D173#12011yescharacteristic
77C2×D173#12311yescharacteristic
78#10621924—44no—
79C4×C173—22no—
80C4×C173—22no—
81C4×C173—41no—
82C4×C173C4×D412yes—
83Dic173—22no—
84Dic173—22no—
85Dic173—41no—
86Dic173#12112yes—
87C2×C173—21no—
88C2×C173#38611yescharacteristic
89D173—21no—
90D173#45612yes—
91C173#188611yescharacteristic, complemented, Sylow
92#1886—1731nomaximal, Sylow
93C2×C4⋊Q8—1731no—
94#519—1731no—
95#419—1732no—
96C2×#101—1731no—
97#339—1732no—
98C22×Q8—1731no—
99C22×Q8—1731no—
100C2×M16—1732no—
101C4⋊Q8—3462no—
102C4⋊Q8—1731no—
103C4⋊Q8—1731no—
104C4⋊Q8—1731no—
105C4⋊Q8#1062192411yescharacteristic, complemented
106C4×Q8—3464no—
107#117—3462no—
108#117—1732no—
109C2×C4⋊C4—3462no—
110C2×C42—1731no—
111#103—3464no—
112#101—3462no—
113C2×Q8—3461no—
114C2×Q8—3462no—
115C2×Q8—3462no—
116C2×Q8—3462no—
117C2×Q8—3462no—
118C2×Q8—1731no—
119C2×Q8—1732no—
120C2×Q8—1731no—
121C2×Q8C2×#1062192411yescharacteristic
122C2×Q8C2×#1062192411yescharacteristic
123C22×C4—3461no—
124C22×C4—3461no—
125C22×C4—1732no—
126C22×C4—1731no—
127M16—3462no—
128M16—3462no—
129C2×C8—3462no—
130C4⋊C4—22no—
131C4⋊C4—3462no—
132C4⋊C4—3462no—
133C4⋊C4—3464no—
134C4⋊C4—3462no—
135C4⋊C4—1734no—
136C22⋊C4—3462no—
137C42—3461no—
138C42—3464no—
139C42—3462no—
140C42—1731no—
141C42C2×#1062192411yescharacteristic
142C23—1731no—
143Q8—22no—
144Q8—22no—
145Q8—3464no—
146Q8—3462no—
147Q8—3462no—
148Q8—6922no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—3464no—
152C2×C4—3462no—
153C2×C4—3462no—
154C2×C4—3461no—
155C2×C4—3464no—
156C2×C4—3461no—
157C2×C4—1732no—
158C2×C4—1731no—
159C2×C4—1732no—
160C2×C4—1732no—
161C2×C4—6921no—
162C2×C4—6921no—
163C2×C4C22⋊#1062192412yes—
164C2×C4C22×#1062192411yescharacteristic
165C8—6922no—
166C22—3462no—
167C22—1731no—
168C22—1731no—
169C22C2×C22⋊#1062192411yescharacteristic
170C4—22no—
171C4—22no—
172C4—41no—
173C4—3462no—
174C4—3462no—
175C4—1732no—
176C4—6924no—
177C4—6921no—
178C4D4×#1062192412yes—
179C2—21no—
180C2—3461no—
181C2—1732no—
182C2#2601243911yescharacteristic
subgroupG/H
centerC2#26012439
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC4⋊Q8×C173C4
socleC2×C173#386
prime radicalC22×D173C2×D4
cosocleC2×C4×D173C23
hypercenterC4⋊Q8#10621924
hypocenterC173#1886
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
14. #19149679C4semidirect
16. #19149677C4semidirect
17. #19149677C4semidirect
20. C4⋊Q8×C173C4semidirect
91. C173#1886semidirect
105. C4⋊Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125592
#260124842
D173⋊#1012
#260123622
C4⋊Q8×D1732
#1886173