Finite Group Databasebeta

← #36355548

Subgroups — #36355548

Order 22144.

354 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 41 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#26012123C211yesmaximal, characteristic, complemented
2#26011679C211yesmaximal, characteristic, complemented
3#26011677C211yesmaximal, characteristic, complemented
4#26011661C211yesmaximal, characteristic, complemented
5#26011283C211yesmaximal, characteristic, complemented
6#26011279C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C173C211yesmaximal, characteristic, complemented
8#19149724C2211yescharacteristic, complemented
9C22×C22⋊D173—21no—
10C2×D4×D173—21no—
11#19149707—21no—
12#19149675C2211yescharacteristic
13#19149652—21no—
14#19149651—21no—
15#19149649—21no—
16#19149649—21no—
17#19149648—21no—
18#19149647C2211yescharacteristic
19#19149639—21no—
20#19149605—21no—
21#19149594C2211yescharacteristic, complemented
22#19149593—21no—
23#19149593—21no—
24#120×C173C2211yescharacteristic
25#100×C173C2211yescharacteristic, complemented
26#99×C173C2211yescharacteristic, complemented
27C23×D173—21no—
28C23×D173D411yescharacteristic, complemented
29#14583826—21no—
30#14583825—41no—
31C2×D692—21no—
32C2×D692—41no—
33C2×D692D411yescharacteristic, complemented
34C2×C22⋊D173—21no—
35C2×C22⋊D173—21no—
36C2×C22⋊D173—21no—
37C2×C22⋊D173—21no—
38C2×C22⋊D173—21no—
39C2×C22⋊D173—21no—
40C2×C22⋊D173—21no—
41C2×C22⋊D173—44no—
42C2×C22⋊D173—41no—
43D4×D173—44no—
44D4×D173—41no—
45C22×Dic173—21no—
46C2×C4×D173—21no—
47#14583815—41no—
48D1384—41no—
49#14583805—21no—
50C22⋊Dic173—21no—
51C22⋊Dic173—41no—
52C22⋊Dic173D411yescharacteristic, complemented
53#14583803—21no—
54#14583803—21no—
55#14583803—41no—
56C4×Dic173D411yescharacteristic, complemented
57C24×C173D411yescharacteristic, complemented
58C2×D4×C173—21no—
59C2×D4×C173C2311yescharacteristic
60M16×C173—21no—
61C22⋊C4×C173—21no—
62C22⋊C4×C173—21no—
63C22⋊C4×C173D411yescharacteristic, complemented
64C22×D173—21no—
65C22×D173—21no—
66C22×D173—21no—
67C22×D173—21no—
68C22×D173—42no—
69C22×D173—42no—
70C22×D173—41no—
71C22×D173—42no—
72C22×D173—81no—
73Dic346—21no—
74D692—21no—
75D692—41no—
76D692—42no—
77D692—82no—
78C22⋊D173—24no—
79C22⋊D173—42no—
80C22⋊D173—42no—
81C22⋊D173—42no—
82C22⋊D173—41no—
83C22⋊D173—41no—
84C22⋊D173—42no—
85C22⋊D173—41no—
86C22⋊D173—82no—
87C2×Dic173—21no—
88C2×Dic173—21no—
89C2×Dic173—21no—
90C2×Dic173—21no—
91C2×Dic173—42no—
92C2×Dic173—41no—
93C4×D173—42no—
94C4×D173—41no—
95C23×C173—21no—
96C23×C173—42no—
97C23×C173—41no—
98C23×C173C2×D411yescharacteristic
99C23×C173C2×D411yescharacteristic
100D4×C173—41no—
101D4×C173—42no—
102C2×C4×C173—21no—
103C2×C4×C173—41no—
104C2×C4×C173C2×D411yescharacteristic
105C8×C173—41no—
106C22×C173—22no—
107C22×C173—21no—
108C22×C173—21no—
109C22×C173—41no—
110C22×C173—41no—
111C22×C173—41no—
112C22×C173—41no—
113C22×C173—42no—
114C22×C173—81no—
115C22×C173#12011yescharacteristic
116C2×D173—22no—
117C2×D173—41no—
118C2×D173—41no—
119C2×D173—42no—
120C2×D173—41no—
121C2×D173—41no—
122C2×D173—41no—
123C2×D173—41no—
124C2×D173—81no—
125C2×D173—84no—
126C2×D173—81no—
127C2×D173—81no—
128C4×C173—21no—
129C4×C173—41no—
130C4×C173—81no—
131Dic173—22no—
132Dic173—41no—
133Dic173—41no—
134Dic173—81no—
135C2×C173—21no—
136C2×C173—41no—
137C2×C173—41no—
138C2×C173—42no—
139C2×C173B(4,𝔽2)11yescharacteristic
140D173—42no—
141D173—81no—
142D173—81no—
143C173#219611yescharacteristic, complemented, Sylow
144#2196—1731nomaximal, Sylow
145D42—1731no—
146B(4,𝔽2)—1731no—
147B(4,𝔽2)—1731no—
148#453—1731no—
149#353—1731no—
150AΣL(1,𝔽16)—1731no—
151AΣL(1,𝔽16)D17311yescharacteristic, complemented
152H(5,𝔽2)—1731no—
153C22×D4—3461no—
154C22×D4—3461no—
155AGL(1,ℤ/8)—3461no—
156#127—1731no—
157#121—3461no—
158#121—3461no—
159#120—3461no—
160#120—3461no—
161#120—3461no—
162#120—1731no—
163#120C2×D17311yescharacteristic
164C4×D4—3461no—
165#104—3461no—
166#100C2×D17311yescharacteristic, complemented
167#99—3461no—
168#99—3461no—
169#99—1731no—
170#99C2×D17311yescharacteristic, complemented
171C24—3461no—
172C24—1731no—
173C24D69211yescharacteristic, complemented
174#41—3461no—
175#41—6921no—
176C2×D4—21no—
177C2×D4—3461no—
178C2×D4—3461no—
179C2×D4—3461no—
180C2×D4—3461no—
181C2×D4—3461no—
182C2×D4—3461no—
183C2×D4—3461no—
184C2×D4—3461no—
185C2×D4—1731no—
186C2×D4—6921no—
187C2×D4—6921no—
188C2×D4—6924no—
189C2×D4—6924no—
190C2×D4—6921no—
191C2×D4C22×D17311yescharacteristic
192C22×C4—3461no—
193C22×C4—3461no—
194SD16—6921no—
195D8—6921no—
196M16—21no—
197C4⋊C4—3461no—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—3461no—
201C22⋊C4—3461no—
202C22⋊C4—3461no—
203C22⋊C4—1731no—
204C22⋊C4—6921no—
205C22⋊C4—6921no—
206C22⋊C4D69211yescharacteristic, complemented
207C42—1731no—
208C23—21no—
209C23—41no—
210C23—42no—
211C23—3461no—
212C23—3461no—
213C23—3461no—
214C23—3461no—
215C23—13841no—
216C23—6922no—
217C23—6921no—
218C23—6922no—
219C23—6922no—
220C23D4×D17311yescharacteristic
221C23C2×D69211yescharacteristic
222Q8—3461no—
223D4—41no—
224D4—42no—
225D4—3461no—
226D4—3464no—
227D4—13842no—
228D4—13842no—
229D4—6921no—
230D4—6922no—
231D4—6922no—
232D4—6922no—
233D4—6922no—
234D4—6921no—
235D4—6921no—
236D4—6922no—
237D4—6921no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—3461no—
241C2×C4—3461no—
242C2×C4—3461no—
243C2×C4—3461no—
244C2×C4—6921no—
245C2×C4—6922no—
246C2×C4—6922no—
247C2×C4—6921no—
248C2×C4D4×D17311yescharacteristic
249C8—41no—
250C22—22no—
251C22—21no—
252C22—21no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—42no—
257C22—41no—
258C22—81no—
259C22—3462no—
260C22—13841no—
261C22—13841no—
262C22—13844no—
263C22—13841no—
264C22—6921no—
265C22—6921no—
266C22—6921no—
267C22—6922no—
268C22—6921no—
269C22—6921no—
270C22—6921no—
271C22#1914964911yescharacteristic
272C4—21no—
273C4—41no—
274C4—81no—
275C4—3462no—
276C4—13841no—
277C4—6921no—
278C4—6921no—
279C2—21no—
280C2—41no—
281C2—41no—
282C2—42no—
283C2—13841no—
284C2—13841no—
285C2—6922no—
286C2#2601167911yescharacteristic
subgroupG/H
centerC2#26011679
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
FittingAΣL(1,𝔽16)×C173C2
socleC2×C173B(4,𝔽2)
hypercenterAΣL(1,𝔽16)D173
hypocenterC173#2196
LCSG ▷ C2×D4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
1. #26012123C2semidirect
2. #26011679C2semidirect
3. #26011677C2semidirect
4. #26011661C2semidirect
5. #26011283C2semidirect
6. #26011279C2semidirect
7. AΣL(1,𝔽16)×C173C2semidirect
8. #19149724C22semidirect
21. #19149594C22semidirect
25. #100×C173C22semidirect
26. #99×C173C22semidirect
28. C23×D173D4semidirect
33. C2×D692D4semidirect
52. C22⋊Dic173D4semidirect
56. C4×Dic173D4semidirect
57. C24×C173D4semidirect
63. C22⋊C4×C173D4semidirect
143. C173#2196semidirect
151. AΣL(1,𝔽16)D173semidirect
166. #100C2×D173semidirect
170. #99C2×D173semidirect
173. C24D692semidirect
206. C22⋊C4D692semidirect

Maximal subgroups

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

subgroupindex
#260121232
#260116792
#260116772
#260116612
#260112832
#260112792
AΣL(1,𝔽16)×C1732
#2196173