Finite Group Databasebeta

← #36355215

Subgroups — #36355215

Order 22144.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#26012063C211yesmaximal, characteristic
2#26011472C211yesmaximal, characteristic
3#26011463C211yesmaximal, characteristic
4#26011282C211yesmaximal, characteristic, complemented
5#26011281C211yesmaximal, characteristic, complemented
6#26011280C211yesmaximal, characteristic, complemented
7#352×C173C211yesmaximal, characteristic, complemented
8C2×#14583825C2211yescharacteristic
9C22×Dic346C411yescharacteristic, complemented
10M16×D173—21no—
11#19149678—21no—
12#19149665—21no—
13#19149663C411yescharacteristic, complemented
14#19149662C411yescharacteristic, complemented
15#19149660C2211yescharacteristic
16#19149657—21no—
17#19149656—21no—
18#19149635—21no—
19#19149635C411yescharacteristic, complemented
20#19149597—21no—
21#19149596C2211yescharacteristic
22#19149594C2211yescharacteristic
23#19149593—21no—
24#123×C173C2211yescharacteristic
25#100×C173C2211yescharacteristic
26#99×C173C2211yescharacteristic
27#14583825—41no—
28#14583825—41no—
29C2×Dic346—21no—
30C2×Dic346—21no—
31C2×Dic346—41no—
32C2×Dic346—41no—
33C2×Dic346D411yescharacteristic
34C2×C22⋊D173C2×C411yescharacteristic
35C2×C22⋊D173D411yescharacteristic
36C22×Dic173C2×C411yescharacteristic
37C22×Dic173C2×C411yescharacteristic
38C22×Dic173D411yescharacteristic
39C2×C4×D173D411yescharacteristic
40#14583810—21no—
41#14583809—41no—
42C8×D173—41no—
43C4⋊Dic173—21no—
44#14583805—21no—
45#14583805—21no—
46#14583805—21no—
47#14583805—41no—
48C22⋊Dic173—21no—
49C22⋊Dic173—21no—
50C22⋊Dic173C2×C411yescharacteristic
51#14583803—21no—
52#14583803—21no—
53C4×Dic173—21no—
54C4×Dic173—41no—
55C2×D4×C173C2311yescharacteristic
56C22×C4×C173C2×C411yescharacteristic
57M16×C173—21no—
58C4⋊C4×C173—21no—
59C22⋊C4×C173—21no—
60C22⋊C4×C173—21no—
61C22⋊C4×C173C2×C411yescharacteristic
62C22×D173C22⋊C411yescharacteristic
63Dic346—41no—
64Dic346—42no—
65Dic346—41no—
66Dic346—41no—
67C22⋊D173—41no—
68C22⋊D173—41no—
69C2×Dic173—21no—
70C2×Dic173—21no—
71C2×Dic173—21no—
72C2×Dic173—21no—
73C2×Dic173—21no—
74C2×Dic173—21no—
75C2×Dic173—41no—
76C2×Dic173—41no—
77C2×Dic173C22⋊C411yescharacteristic
78C2×Dic173C22⋊C411yescharacteristic
79C2×Dic173C22⋊C411yescharacteristic
80C4×D173—21no—
81C4×D173—21no—
82#13063436—41no—
83C23×C173C22×C411yescharacteristic
84C23×C173C2×D411yescharacteristic
85D4×C173—41no—
86C2×C4×C173—21no—
87C2×C4×C173—21no—
88C2×C4×C173—41no—
89C2×C4×C173—41no—
90C2×C4×C173C2×D411yescharacteristic
91C8×C173—41no—
92C22×C173—21no—
93C22×C173—21no—
94C22×C173—41no—
95C22×C173—41no—
96C22×C173C2×C22⋊C411yescharacteristic
97C2×D173—21no—
98C2×D173—41no—
99C4×C173—21no—
100C4×C173—41no—
101C4×C173—41no—
102C4×C173—81no—
103Dic173—21no—
104Dic173—21no—
105Dic173—41no—
106Dic173—41no—
107Dic173—81no—
108Dic173#9911yescharacteristic
109Dic173#9911yescharacteristic
110C2×C173—21no—
111C2×C173—41no—
112C2×C173—41no—
113C2×C173C2×#9911yescharacteristic
114D173—41no—
115C173#212311yescharacteristic, complemented, Sylow
116#2123—1731nomaximal, Sylow
117#536—1731no—
118#413—1731no—
119#410—1731no—
120#352—1731no—
121#352—1731no—
122#352—1731no—
123#352D17311yescharacteristic, complemented
124C2×#41—1731no—
125C22×Q8—1731no—
126C2×M16—3461no—
127C4⋊Q8—3461no—
128#124—3461no—
129#123—1731no—
130#123—1731no—
131#123—1731no—
132#123C2×D17311yescharacteristic
133C22⋊Q8—3461no—
134C22⋊Q8—3461no—
135#117—3461no—
136#117—1731no—
137#101—3461no—
138#100—1731no—
139#100C2×D17311yescharacteristic
140#99—3461no—
141#99—1731no—
142#99C2×D17311yescharacteristic
143#41—6921no—
144#41—6921no—
145C2×Q8—3461no—
146C2×Q8—3461no—
147C2×Q8—1731no—
148C2×Q8—6921no—
149C2×Q8—6921no—
150C2×D4—1731no—
151C2×D4—1731no—
152C2×D4C22×D17311yescharacteristic
153C22×C4—1731no—
154C22×C4—1731no—
155C22×C4—1731no—
156C22×C4—1731no—
157C22×C4C4×D17311yescharacteristic
158M16—21no—
159M16—3461no—
160M16—6921no—
161C2×C8—6921no—
162C4⋊C4—21no—
163C4⋊C4—3461no—
164C4⋊C4—3461no—
165C4⋊C4—3461no—
166C4⋊C4—3461no—
167C4⋊C4—6921no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—3461no—
171C22⋊C4—3461no—
172C22⋊C4—3461no—
173C22⋊C4—3461no—
174C22⋊C4—1731no—
175C22⋊C4C4×D17311yescharacteristic
176C42—3461no—
177C42—6921no—
178C23—1731no—
179C23C2×C4×D17311yescharacteristic
180C23D4×D17311yescharacteristic
181Q8—6921no—
182Q8—6922no—
183Q8—6921no—
184Q8—6921no—
185D4—41no—
186D4—6921no—
187D4—6921no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—3461no—
193C2×C4—3461no—
194C2×C4—3461no—
195C2×C4—3461no—
196C2×C4—3461no—
197C2×C4—3461no—
198C2×C4—3461no—
199C2×C4—3461no—
200C2×C4—1731no—
201C2×C4—1731no—
202C2×C4—1731no—
203C2×C4—6921no—
204C2×C4—6921no—
205C2×C4D4×D17311yescharacteristic
206C8—41no—
207C8—6921no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—3461no—
213C22—6921no—
214C22C22⋊C4×D17311yescharacteristic
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—3461no—
220C4—3461no—
221C4—13841no—
222C4—1731no—
223C4—1731no—
224C4—6921no—
225C4—6921no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2—6921no—
230C2#99×D17311yescharacteristic
subgroupG/H
centerC2#99×D173
commutatorC23×C173C22×C4
FrattiniC2×D4C22×D173
Fitting#352×C173C2
socleC2×C173C2×#99
prime radicalC2×#14583825C22
cosocleC2×D4×C173C23
hypercenter#352D173
hypocenterC173#2123
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #26011282C2semidirect
5. #26011281C2semidirect
6. #26011280C2semidirect
7. #352×C173C2semidirect
9. C22×Dic346C4semidirect
13. #19149663C4semidirect
14. #19149662C4semidirect
19. #19149635C4semidirect
115. C173#2123semidirect
123. #352D173semidirect

Maximal subgroups

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

subgroupindex
#260120632
#260114722
#260114632
#260112822
#260112812
#260112802
#352×C1732
#2123173