Finite Group Databasebeta

← #36369608

Subgroups — #36369608

Order 22144.

362 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 66 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#26012558C211yesmaximal, characteristic, complemented
2#26012490C212yesmaximal, complemented
3D173⋊#100C211yesmaximal, characteristic, complemented
4#26012358C212yesmaximal, complemented
5#26012059C211yesmaximal, characteristic
6D173⋊M16C2212yes—
7D4×#10621924—24no—
8#19149758C2212yes—
9C2×C22⋊#10621924—22no—
10#19149744—24no—
11#19149739—22no—
12C2×#14583827C2211yescharacteristic
13C2×D4×D173C2211yescharacteristic
14#19149676C411yescharacteristic, complemented
15#19149675C411yescharacteristic, complemented
16#19149668C411yescharacteristic, complemented
17#19149652—22no—
18#19149649—22no—
19#19149636C2211yescharacteristic
20#124×C173C411yescharacteristic, complemented
21C22×#10621924—24no—
22#14583836—22no—
23#14583835—22no—
24D173⋊C8—22no—
25C4⋊#10621924—22no—
26C22⋊#10621924—24no—
27C22⋊#10621924—22no—
28C4×#10621924D414yescomplemented
29C23×D173D412yes—
30#14583827—22no—
31#14583827—22no—
32C2×D692—21no—
33C2×D692—22no—
34C2×D692C2×C411yescharacteristic
35C2×D692C2×C411yescharacteristic, complemented
36C2×C22⋊D173—22no—
37C2×C22⋊D173D412yes—
38D4×D173—22no—
39D4×D173—22no—
40C2×C4×D173—21no—
41C2×C4×D173C2311yescharacteristic
42#14583805—21no—
43#14583803—22no—
44#14583803—21no—
45C4×Dic173C2×C411yescharacteristic, complemented
46C2×Q8×C173C2×C411yescharacteristic, complemented
47C2×D4×C173C2×C411yescharacteristic
48C22⋊C4×C173—22no—
49C42×C173C2×C411yescharacteristic, complemented
50C2×#10621924—24no—
51C2×#10621924—84no—
52#13063443—42no—
53C22×D173—22no—
54C22×D173—22no—
55C22×D173—21no—
56C22×D173—82no—
57C22×D173C22⋊C412yes—
58C22×D173C2×D411yescharacteristic
59D692—22no—
60D692—22no—
61D692—41no—
62D692—41no—
63D692—82no—
64C22⋊D173—24no—
65C22⋊D173—82no—
66C2×Dic173—21no—
67C2×Dic173C2×D411yescharacteristic
68C4×D173—42no—
69C4×D173C2×D412yes—
70C4×D173#4112yes—
71C23×C173C22⋊C412yes—
72Q8×C173—22no—
73D4×C173—22no—
74C2×C4×C173—21no—
75C2×C4×C173—21no—
76C2×C4×C173C22×C411yescharacteristic
77C22×C173—22no—
78C22×C173—42no—
79C22×C173C2×C22⋊C411yescharacteristic
80C2×D173—22no—
81C2×D173—41no—
82C2×D173—44no—
83C2×D173—42no—
84C2×D173—42no—
85C2×D173—41no—
86C2×D173—82no—
87C2×D173#12011yescharacteristic
88C2×D173#12311yescharacteristic
89#10621924—44no—
90C4×C173—41no—
91C4×C173—41no—
92C4×C173C4×D412yes—
93Dic173—41no—
94Dic173#12112yes—
95C2×C173—21no—
96C2×C173—42no—
97C2×C173#38611yescharacteristic
98D173—21no—
99D173—21no—
100D173—42no—
101D173—81no—
102C173#191111yescharacteristic, complemented, Sylow
103#1911—1731nomaximal, Sylow
104#535—1731no—
105#518—1731no—
106#421—1732no—
107C2×#100—1731no—
108H(3,ℤ/4)—1732no—
109C2×#41—1731no—
110C22×D4—1731no—
111C2×M16—1732no—
112#127—1731no—
113#127—1731no—
114#124—1731no—
115#124#1062192411yescharacteristic, complemented
116#121—3462no—
117#120—3462no—
118C4×D4—3464no—
119#117—1732no—
120#117—1731no—
121C2×C22⋊C4—3462no—
122#104—3464no—
123#100—3462no—
124C24—1732no—
125#41—3462no—
126#41—3462no—
127C2×Q8C2×#1062192411yescharacteristic, complemented
128C2×D4—3462no—
129C2×D4—3461no—
130C2×D4—3462no—
131C2×D4—3462no—
132C2×D4—3462no—
133C2×D4—1732no—
134C2×D4—1731no—
135C2×D4—1731no—
136C2×D4C2×#1062192411yescharacteristic
137C22×C4—3461no—
138C22×C4—3464no—
139C22×C4—1731no—
140M16—3462no—
141M16—3462no—
142C2×C8—3462no—
143C4⋊C4—3461no—
144C4⋊C4—3462no—
145C22⋊C4—22no—
146C22⋊C4—3461no—
147C22⋊C4—3462no—
148C22⋊C4—3464no—
149C22⋊C4—3462no—
150C42—1734no—
151C42—1731no—
152C42C2×#1062192411yescharacteristic, complemented
153C23—3462no—
154C23—3462no—
155C23—3461no—
156C23—13842no—
157C23—1732no—
158C23—1731no—
159C23C22⋊#1062192412yes—
160Q8—22no—
161D4—22no—
162D4—3464no—
163D4—3462no—
164D4—3462no—
165D4—13842no—
166D4—13842no—
167D4—6921no—
168D4—6921no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—3464no—
172C2×C4—3461no—
173C2×C4—13844no—
174C2×C4—1731no—
175C2×C4—1732no—
176C2×C4—1732no—
177C2×C4—6922no—
178C2×C4C22×#1062192411yescharacteristic
179C8—6922no—
180C22—22no—
181C22—42no—
182C22—3462no—
183C22—13842no—
184C22—1731no—
185C22—1731no—
186C22—6921no—
187C22—6922no—
188C22—6922no—
189C22—6924no—
190C22—6921no—
191C22C2×C22⋊#1062192411yescharacteristic
192C4—41no—
193C4—41no—
194C4—1732no—
195C4—6921no—
196C4—6924no—
197C4D4×#1062192412yes—
198C2—21no—
199C2—42no—
200C2—3461no—
201C2—3461no—
202C2—13841no—
203C2—6922no—
204C2#2601243911yescharacteristic
subgroupG/H
centerC2#26012439
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#124×C173C4
socleC2×C173#386
prime radical#26012059C2
cosocleC2×C4×D173C23
hypercenter#124#10621924
hypocenterC173#1911
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #26012558C2semidirect
2. #26012490C2semidirect
3. D173⋊#100C2semidirect
4. #26012358C2semidirect
14. #19149676C4semidirect
15. #19149675C4semidirect
16. #19149668C4semidirect
20. #124×C173C4semidirect
28. C4×#10621924D4semidirect
35. C2×D692C2×C4semidirect
45. C4×Dic173C2×C4semidirect
46. C2×Q8×C173C2×C4semidirect
49. C42×C173C2×C4semidirect
102. C173#1911semidirect
115. #124#10621924semidirect
127. C2×Q8C2×#10621924semidirect
152. C42C2×#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125582
#260124902
D173⋊#1002
#260123582
#260120592
#1911173