Finite Group Databasebeta

← #36352482

Subgroups — #36352482

Order 22144.

368 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 51 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#26012049C211yesmaximal, characteristic, complemented
2#26011627C211yesmaximal, characteristic, complemented
3#26011624C211yesmaximal, characteristic, complemented
4#26011401C211yesmaximal, characteristic, complemented
5#26011224C211yesmaximal, characteristic, complemented
6#26011214C211yesmaximal, characteristic, complemented
7#327×C173C211yesmaximal, characteristic, complemented
8C22×D692—21no—
9C22×D692C2211yescharacteristic, complemented
10C2×D4×D173—21no—
11C2×D1384—21no—
12C2×#14583811—21no—
13#19149664—21no—
14#19149661—21no—
15#19149653C2211yescharacteristic
16#19149651—21no—
17#19149649—21no—
18#19149647—21no—
19C2×#14583803—21no—
20C2×#14583803C2211yescharacteristic
21#19149601—21no—
22#19149599—21no—
23C22⋊#13063436C2211yescharacteristic, complemented
24#121×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic, complemented
26#95×C173C2211yescharacteristic, complemented
27C23×D173—21no—
28C23×D173—21no—
29C23×D173D411yescharacteristic
30C2×D692—21no—
31C2×D692—21no—
32C2×D692—21no—
33C2×D692—42no—
34C2×D692—42no—
35C2×D692—42no—
36C2×D692—42no—
37C2×D692D411yescharacteristic, complemented
38C2×C22⋊D173—21no—
39C2×C22⋊D173—21no—
40C2×C22⋊D173—21no—
41D4×D173—42no—
42D4×D173—42no—
43C22×Dic173D411yescharacteristic
44C2×C4×D173—21no—
45D1384—42no—
46#14583811—42no—
47C2×#13063436—21no—
48C4⋊Dic173D411yescharacteristic, complemented
49#14583805—21no—
50C22⋊Dic173—21no—
51#14583803—21no—
52#14583803—21no—
53#14583803—21no—
54#14583803—21no—
55#14583803—42no—
56C2×D4×C173—21no—
57C2×D4×C173D411yescharacteristic, complemented
58C22×C4×C173—21no—
59C22×C4×C173C2311yescharacteristic
60C2×C8×C173—21no—
61C4⋊C4×C173D411yescharacteristic, complemented
62C22⋊C4×C173—21no—
63C22×D173—21no—
64C22×D173—21no—
65C22×D173—22no—
66C22×D173—21no—
67C22×D173—42no—
68C22×D173—41no—
69C22×D173—41no—
70C22×D173—41no—
71C22×D173—41no—
72C22×D173—41no—
73C22×D173—41no—
74C22×D173—41no—
75C22×D173—41no—
76C22×D173—81no—
77C22×D173—81no—
78C22×D173D812yescomplemented
79D692—22no—
80D692—22no—
81D692—41no—
82D692—41no—
83D692—41no—
84D692—42no—
85D692—48no—
86C22⋊D173—42no—
87C22⋊D173—41no—
88C22⋊D173—41no—
89C22⋊D173—81no—
90C2×Dic173—21no—
91C2×Dic173—21no—
92C2×Dic173—41no—
93C4×D173—42no—
94#13063436—41no—
95C23×C173—21no—
96C23×C173C2×D411yescharacteristic
97D4×C173—41no—
98D4×C173—41no—
99D4×C173—41no—
100C2×C4×C173—22no—
101C2×C4×C173—21no—
102C2×C4×C173—41no—
103C2×C4×C173—41no—
104C2×C4×C173—41no—
105C2×C4×C173C2×D411yescharacteristic
106C2×C4×C173C2×D411yescharacteristic
107C8×C173—41no—
108C22×C173—21no—
109C22×C173—21no—
110C22×C173—41no—
111C22×C173—41no—
112C22×C173—41no—
113C22×C173#12011yescharacteristic
114C22×C173C2×D811yescharacteristic
115C22×C173AGL(1,ℤ/8)11yescharacteristic
116C2×D173—22no—
117C2×D173—41no—
118C2×D173—41no—
119C2×D173—42no—
120C2×D173—42no—
121C2×D173—42no—
122C2×D173—41no—
123C2×D173—41no—
124C2×D173—42no—
125C2×D173—41no—
126C2×D173—81no—
127C2×D173—84no—
128C2×D173—81no—
129C2×D173—81no—
130C2×D173—81no—
131C2×D173—81no—
132C4×C173—21no—
133C4×C173—21no—
134C4×C173—42no—
135C4×C173—41no—
136Dic173—41no—
137Dic173—41no—
138C2×C173—21no—
139C2×C173—21no—
140C2×C173—81no—
141C2×C173#44711yescharacteristic
142C2×C173#45311yescharacteristic
143C2×C173B(4,𝔽2)11yescharacteristic
144D173—42no—
145D173—82no—
146D173—81no—
147C173#159511yescharacteristic, complemented, Sylow
148#1595—1731nomaximal, Sylow
149#534—1731no—
150#447—1731no—
151#447—1731no—
152#392—1731no—
153#327—1731no—
154#327D17311yescharacteristic, complemented
155#323—1731no—
156C22×D4—3461no—
157C22×D4—3461no—
158C22×D4—1731no—
159C2×D8—3461no—
160C2×D8—3461no—
161#123—3461no—
162#123—3461no—
163#121—3461no—
164#121—1731no—
165#121C2×D17311yescharacteristic
166#120—3461no—
167#120—3461no—
168C2×C22⋊C4—3461no—
169C2×C22⋊C4—1731no—
170#102—3461no—
171#102—3461no—
172C22⋊C8—1731no—
173C22⋊C8C2×D17311yescharacteristic, complemented
174#95C2×D17311yescharacteristic, complemented
175C24—3461no—
176C24—3461no—
177C24—1731no—
178C2×D4—21no—
179C2×D4—3461no—
180C2×D4—3461no—
181C2×D4—3461no—
182C2×D4—3461no—
183C2×D4—3461no—
184C2×D4—3461no—
185C2×D4—1731no—
186C2×D4—6922no—
187C2×D4—6922no—
188C2×D4—6922no—
189C2×D4—6922no—
190C2×D4—6922no—
191C2×D4—6922no—
192C2×D4D69211yescharacteristic, complemented
193C22×C4—21no—
194C22×C4—3461no—
195C22×C4—1731no—
196C22×C4C22×D17311yescharacteristic
197D8—6922no—
198D8—6922no—
199C2×C8—21no—
200C2×C8—3461no—
201C4⋊C4—3461no—
202C4⋊C4—1731no—
203C4⋊C4D69211yescharacteristic, complemented
204C22⋊C4—21no—
205C22⋊C4—3461no—
206C22⋊C4—3461no—
207C22⋊C4—3461no—
208C22⋊C4—3461no—
209C22⋊C4—3461no—
210C22⋊C4—6922no—
211C23—21no—
212C23—3462no—
213C23—3461no—
214C23—3461no—
215C23—3461no—
216C23—13841no—
217C23—13841no—
218C23—1732no—
219C23—6921no—
220C23—6921no—
221C23—6921no—
222C23—6921no—
223C23—6921no—
224C23—6922no—
225C23—6921no—
226C23—6921no—
227C23—6921no—
228C23D4×D17311yescharacteristic
229D4—41no—
230D4—41no—
231D4—41no—
232D4—3462no—
233D4—3462no—
234D4—13841no—
235D4—6921no—
236D4—6921no—
237D4—6921no—
238D4—6922no—
239D4—6928no—
240D4—6922no—
241D4—6921no—
242D4—6921no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—3461no—
249C2×C4—3461no—
250C2×C4—6922no—
251C2×C4—6921no—
252C2×C4D4×D17311yescharacteristic
253C2×C4C2×D69211yescharacteristic
254C8—41no—
255C8—6921no—
256C22—21no—
257C22—21no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—3462no—
262C22—13841no—
263C22—13841no—
264C22—13841no—
265C22—13844no—
266C22—13841no—
267C22—13841no—
268C22—6922no—
269C22—6922no—
270C22—6922no—
271C22—6921no—
272C22—6922no—
273C22—6921no—
274C22—6921no—
275C22—6921no—
276C22—6921no—
277C22#1914964911yescharacteristic
278C22D8×D17311yescharacteristic
279C22#1914970611yescharacteristic
280C4—21no—
281C4—21no—
282C4—42no—
283C4—41no—
284C4—6921no—
285C4—6921no—
286C2—21no—
287C2—21no—
288C2—81no—
289C2—13842no—
290C2—13841no—
291C2—6922no—
292C2#2601162611yescharacteristic
293C2#2601166011yescharacteristic
294C2#2601167911yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#327×C173C2
socleC22×C173#120
hypercenter#327D173
hypocenterC173#1595
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
1. #26012049C2semidirect
2. #26011627C2semidirect
3. #26011624C2semidirect
4. #26011401C2semidirect
5. #26011224C2semidirect
6. #26011214C2semidirect
7. #327×C173C2semidirect
9. C22×D692C22semidirect
23. C22⋊#13063436C22semidirect
25. C22⋊C8×C173C22semidirect
26. #95×C173C22semidirect
37. C2×D692D4semidirect
48. C4⋊Dic173D4semidirect
57. C2×D4×C173D4semidirect
61. C4⋊C4×C173D4semidirect
78. C22×D173D8semidirect
147. C173#1595semidirect
154. #327D173semidirect
173. C22⋊C8C2×D173semidirect
174. #95C2×D173semidirect
192. C2×D4D692semidirect
203. C4⋊C4D692semidirect

Maximal subgroups

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

subgroupindex
#260120492
#260116272
#260116242
#260114012
#260112242
#260112142
#327×C1732
#1595173