Finite Group Databasebeta

← #353×D173 #36355219

Subgroups — #353×D173 #36355219

Order 22144.

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

normal subgroups only

This group has 368 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#127×D173C211yesmaximal, characteristic
2#99×D173C212yesmaximal, complemented
3#26011284C211yesmaximal, characteristic, complemented
4#26011283C212yesmaximal, complemented
5#353×C173C211yesmaximal, characteristic, complemented
6C2×D4×D173—22no—
7C2×D4×D173C411yescharacteristic, complemented
8C2×D4×D173C2211yescharacteristic
9#19149676C411yescharacteristic, complemented
10#19149675—22no—
11#19149675C411yescharacteristic, complemented
12#19149675C2211yescharacteristic
13C22⋊C4×D173—22no—
14C42×D173C411yescharacteristic, complemented
15#19149594C2212yescomplemented
16#19149593—22no—
17#127×C173C2211yescharacteristic
18#99×C173C2212yescomplemented
19C23×D173—21no—
20C23×D173D412yes—
21C2×D692—22no—
22C2×D692C2×C411yescharacteristic
23C2×D692C2×C411yescharacteristic, complemented
24C2×C22⋊D173—22no—
25C2×C22⋊D173—21no—
26C2×C22⋊D173—22no—
27C2×C22⋊D173D412yes—
28D4×D173—41no—
29D4×D173—41no—
30D4×D173—48no—
31D4×D173—41no—
32D4×D173—41no—
33C2×C4×D173—21no—
34C2×C4×D173—42no—
35C2×C4×D173C2×C411yescharacteristic
36C22⋊Dic173—22no—
37#14583803—42no—
38C4×Dic173—21no—
39C4×Dic173C2×C411yescharacteristic, complemented
40C2×D4×C173—22no—
41C2×D4×C173C2×C411yescharacteristic, complemented
42C2×D4×C173C2311yescharacteristic
43C22⋊C4×C173—22no—
44C42×C173C2×C411yescharacteristic, complemented
45C22×D173—22no—
46C22×D173—22no—
47C22×D173—21no—
48C22×D173—44no—
49C22×D173—44no—
50C22×D173—41no—
51C22×D173—41no—
52C22×D173C22⋊C411yescharacteristic
53C22×D173C22⋊C412yes—
54D692—41no—
55D692—41no—
56D692—44no—
57C22⋊D173—42no—
58C22⋊D173—44no—
59C22⋊D173—44no—
60C22⋊D173—42no—
61C2×Dic173—21no—
62C2×Dic173—42no—
63C2×Dic173C22⋊C411yescharacteristic
64C4×D173—24no—
65C4×D173—21no—
66C4×D173—21no—
67C4×D173—84no—
68C23×C173—21no—
69C23×C173C2×D412yes—
70D4×C173—41no—
71D4×C173—41no—
72D4×C173—44no—
73C2×C4×C173—21no—
74C2×C4×C173—42no—
75C2×C4×C173C22×C411yescharacteristic
76C22×C173—22no—
77C22×C173—42no—
78C22×C173—42no—
79C22×C173—41no—
80C22×C173C2×C22⋊C411yescharacteristic
81C2×D173—22no—
82C2×D173—22no—
83C2×D173—44no—
84C2×D173—42no—
85C2×D173—42no—
86C2×D173—41no—
87C2×D173—82no—
88C2×D173—82no—
89C2×D173—82no—
90C2×D173#9911yescharacteristic
91C2×D173#9911yescharacteristic
92C4×C173—22no—
93C4×C173—21no—
94C4×C173—82no—
95Dic173—22no—
96Dic173—21no—
97Dic173—82no—
98C2×C173—21no—
99C2×C173—42no—
100C2×C173—81no—
101C2×C173C2×#9911yescharacteristic
102D173—21no—
103D173—42no—
104D173—81no—
105D173#35312yesdirect product factor
106C173C2×#35311yescharacteristic, complemented, Sylow
107C2×#353—1731nomaximal, Sylow
108C2×#127—1731no—
109C2×#99—1732no—
110#353—1731no—
111#353—1732no—
112#353D17311yescharacteristic, direct product factor
113C22×D4—3462no—
114C22×D4—1731no—
115C22×D4—1731no—
116#127—3462no—
117#127—1731no—
118#127—1731no—
119#127—1731no—
120#127C2×D17311yescharacteristic
121C2×C22⋊C4—3462no—
122C2×C42—1731no—
123#99—3462no—
124#99—1732no—
125#99C2×D17312yescomplemented
126C24—3461no—
127C24—1732no—
128C2×D4—22no—
129C2×D4—3462no—
130C2×D4—3462no—
131C2×D4—3462no—
132C2×D4—3461no—
133C2×D4—1732no—
134C2×D4—1731no—
135C2×D4—1731no—
136C2×D4—6921no—
137C2×D4—6921no—
138C2×D4—6921no—
139C2×D4—6921no—
140C2×D4—6928no—
141C2×D4C4×D17311yescharacteristic, complemented
142C2×D4C22×D17311yescharacteristic
143C22×C4—3461no—
144C22×C4—1731no—
145C22×C4—6922no—
146C22⋊C4—22no—
147C22⋊C4—3462no—
148C22⋊C4—6922no—
149C42—3461no—
150C42—1731no—
151C42C4×D17311yescharacteristic, complemented
152C23—21no—
153C23—3462no—
154C23—3462no—
155C23—3461no—
156C23—1731no—
157C23—1732no—
158C23—6924no—
159C23—6924no—
160C23—6921no—
161C23—6921no—
162C23D4×D17312yes—
163D4—41no—
164D4—41no—
165D4—44no—
166D4—6921no—
167D4—6921no—
168D4—6924no—
169D4—6922no—
170D4—6924no—
171D4—6924no—
172D4—6922no—
173C2×C4—21no—
174C2×C4—42no—
175C2×C4—3464no—
176C2×C4—3461no—
177C2×C4—3461no—
178C2×C4—3461no—
179C2×C4—13844no—
180C2×C4—1731no—
181C2×C4—6922no—
182C2×C4C2×C4×D17311yescharacteristic
183C22—22no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—3462no—
188C22—3462no—
189C22—13842no—
190C22—13842no—
191C22—13842no—
192C22—1731no—
193C22—1731no—
194C22—6922no—
195C22—6922no—
196C22—6924no—
197C22—6921no—
198C22C22⋊C4×D17311yescharacteristic
199C4—22no—
200C4—21no—
201C4—82no—
202C4—3462no—
203C4—3461no—
204C4—13842no—
205C2—21no—
206C2—42no—
207C2—81no—
208C2—3461no—
209C2—13841no—
210C2—1732no—
211C2—6922no—
212C2#99×D17311yescharacteristic
subgroupG/H
centerC2#99×D173
commutatorC2×C4×C173C22×C4
FrattiniC2×D4C22×D173
Fitting#353×C173C2
socleC2×C173C2×#99
prime radical#127×D173C2
cosocleC2×D4×C173C23
hypercenter#353D173
hypocenterC173C2×#353
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #99×D173C2semidirect
3. #26011284C2semidirect
4. #26011283C2semidirect
5. #353×C173C2semidirect
7. C2×D4×D173C4semidirect
9. #19149676C4semidirect
11. #19149675C4semidirect
14. C42×D173C4semidirect
15. #19149594C22semidirect
18. #99×C173C22semidirect
23. C2×D692C2×C4semidirect
39. C4×Dic173C2×C4semidirect
41. C2×D4×C173C2×C4semidirect
44. C42×C173C2×C4semidirect
105. D173#353direct (N × H)
106. C173C2×#353semidirect
112. #353D173direct (N × H)
125. #99C2×D173semidirect
141. C2×D4C4×D173semidirect
151. C42C4×D173semidirect

Maximal subgroups

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

subgroupindex
#127×D1732
#99×D1732
#260112842
#260112832
#353×C1732
C2×#353173