Finite Group Databasebeta

← #36369415

Subgroups — #36369415

Order 22144.

536 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 82 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#26012490C212yesmaximal, complemented
2#26012484C212yesmaximal, complemented
3#26012477C211yesmaximal, characteristic, complemented
4#26012470C211yesmaximal, characteristic, complemented
5C2×#19149727C211yesmaximal, characteristic
6D173⋊M16C2212yes—
7#19149758C2212yes—
8#19149744—24no—
9#19149742—24no—
10#19149740—21no—
11#19149739—21no—
12#19149738—21no—
13#19149737—21no—
14#19149727—24no—
15#19149727—24no—
16C2×#14583827—21no—
17C2×#14583827C411yescharacteristic, complemented
18C2×#14583827C2211yescharacteristic
19C2×#14583826—21no—
20C2×#14583826—21no—
21C2×#14583826C411yescharacteristic, complemented
22C2×#14583826C2211yescharacteristic
23C2×Q8×D173—21no—
24C2×Q8×D173C411yescharacteristic, complemented
25C2×Q8×D173C2211yescharacteristic
26C22×Q8×C173C411yescharacteristic, complemented
27#14583836—22no—
28#14583835—22no—
29D173⋊C8—22no—
30C4⋊#10621924—22no—
31C22⋊#10621924—22no—
32C4×#10621924—22no—
33#14583827—28no—
34#14583827—22no—
35#14583827—22no—
36#14583827—22no—
37#14583827D414yes—
38#14583826—28no—
39#14583826—28no—
40#14583826—22no—
41#14583826—22no—
42#14583826—22no—
43#14583826—22no—
44C2×Dic346—21no—
45C2×Dic346—21no—
46C2×Dic346C2×C411yescharacteristic
47C2×Dic346D411yescharacteristic
48Q8×D173—28no—
49Q8×D173—22no—
50Q8×D173—22no—
51Q8×D173—22no—
52Q8×D173D414yes—
53C2×D692—21no—
54C2×D692—21no—
55C2×D692C2×C411yescharacteristic
56C2×D692D411yescharacteristic
57C2×C22⋊D173—21no—
58C2×C22⋊D173D411yescharacteristic
59C2×C22⋊D173D411yescharacteristic
60C2×C4×D173—21no—
61C2×C4×D173—21no—
62C2×C4×D173—21no—
63C2×C4×D173—21no—
64C2×C4×D173—21no—
65C2×C4×D173C2×C411yescharacteristic
66C2×C4×D173C2311yescharacteristic
67C2×Q8×C173—22no—
68C2×Q8×C173—22no—
69C2×Q8×C173—21no—
70C2×Q8×C173C2×C411yescharacteristic
71C2×Q8×C173C2×C411yescharacteristic
72C22×C4×C173—21no—
73C22×C4×C173C2×C411yescharacteristic
74C2×#10621924—42no—
75#13063443—42no—
76C22×D173—21no—
77C22×D173C22⋊C411yescharacteristic
78C22×D173C2×D411yescharacteristic
79Dic346—24no—
80Dic346—21no—
81Dic346—22no—
82Dic346—24no—
83Dic346C22⋊C412yes—
84D692—24no—
85D692—24no—
86D692—22no—
87D692—21no—
88D692C22⋊C412yes—
89C22⋊D173—24no—
90C22⋊D173—22no—
91C22⋊D173—22no—
92C2×Dic173—21no—
93C2×Dic173C22⋊C411yescharacteristic
94C2×Dic173C2×D411yescharacteristic
95C4×D173—24no—
96C4×D173—24no—
97C4×D173—24no—
98C4×D173—24no—
99C4×D173—21no—
100C4×D173—21no—
101C4×D173—24no—
102C4×D173C2×D412yes—
103C4×D173C2×D412yes—
104C23×C173C22⋊C411yescharacteristic
105Q8×C173—24no—
106Q8×C173—21no—
107Q8×C173—21no—
108Q8×C173C22⋊C412yes—
109Q8×C173C22⋊C412yes—
110C2×C4×C173—24no—
111C2×C4×C173—21no—
112C2×C4×C173—21no—
113C2×C4×C173—21no—
114C2×C4×C173—21no—
115C2×C4×C173C22⋊C411yescharacteristic
116C2×C4×C173C22×C411yescharacteristic
117C22×C173—21no—
118C22×C173—41no—
119C22×C173C2×C22⋊C411yescharacteristic
120C2×D173—21no—
121C2×D173—22no—
122C2×D173—41no—
123C2×D173—42no—
124C2×D173—41no—
125C2×D173#12012yes—
126#10621924—82no—
127C4×C173—22no—
128C4×C173—22no—
129C4×C173—21no—
130C4×C173C2×C22⋊C411yescharacteristic
131C4×C173C2×C22⋊C411yescharacteristic
132Dic173—22no—
133Dic173—21no—
134Dic173#12012yes—
135C2×C173—21no—
136C2×C173—41no—
137C2×C173#37911yescharacteristic
138D173—22no—
139D173—41no—
140D173—42no—
141C173#179611yescharacteristic, complemented, Sylow
142#1796—1731nomaximal, Sylow
143C2×2−1+4—1731no—
144#421—1732no—
145#419—1732no—
146#413—1731no—
147#410—1731no—
1482−1+4—3464no—
1492−1+4—3464no—
150C2×#41—3461no—
151C2×#41—3461no—
152C2×#41—3461no—
153C2×#41—1731no—
154C2×#41—1731no—
155C2×#41—1731no—
156C2×#41—1731no—
157C22×Q8—3461no—
158C22×Q8—1731no—
159C22×Q8—1731no—
160C22×Q8#1062192411yescharacteristic, complemented
161C2×M16—1732no—
162#117—1732no—
163#104—3464no—
164#103—3464no—
165#101—3461no—
166#100—3461no—
167#99—3461no—
168#99—3461no—
169#41—3468no—
170#41—3468no—
171#41—3462no—
172#41—3462no—
173#41—3462no—
174#41—3462no—
175#41—3468no—
176#41—3462no—
177#41—3462no—
178#41—3462no—
179#41—1734no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—21no—
183C2×Q8—3461no—
184C2×Q8—3461no—
185C2×Q8—3468no—
186C2×Q8—3462no—
187C2×Q8—3462no—
188C2×Q8—3462no—
189C2×Q8—1734no—
190C2×Q8—1731no—
191C2×Q8—1731no—
192C2×Q8C2×#1062192411yescharacteristic
193C2×Q8C2×#1062192411yescharacteristic
194C2×D4—3461no—
195C2×D4—3461no—
196C2×D4—3461no—
197C2×D4—1731no—
198C2×D4—1731no—
199C2×D4—1731no—
200C2×D4—1731no—
201C22×C4—21no—
202C22×C4—3461no—
203C22×C4—3461no—
204C22×C4—3461no—
205C22×C4—3461no—
206C22×C4—3461no—
207C22×C4—1731no—
208C22×C4—1731no—
209C22×C4C2×#1062192411yescharacteristic
210M16—3462no—
211M16—3462no—
212C2×C8—3462no—
213C4⋊C4—3462no—
214C22⋊C4—3462no—
215C42—3462no—
216C23—3461no—
217C23—1731no—
218C23—1731no—
219C23C22⋊#1062192411yescharacteristic
220Q8—24no—
221Q8—21no—
222Q8—21no—
223Q8—3464no—
224Q8—3464no—
225Q8—3462no—
226Q8—3461no—
227Q8—1732no—
228Q8C22⋊#1062192412yes—
229Q8C22⋊#1062192412yes—
230D4—3464no—
231D4—3464no—
232D4—3462no—
233D4—3464no—
234D4—3462no—
235D4—3462no—
236D4—3461no—
237D4—1732no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—3464no—
244C2×C4—3464no—
245C2×C4—3464no—
246C2×C4—3464no—
247C2×C4—3464no—
248C2×C4—3461no—
249C2×C4—3461no—
250C2×C4—3461no—
251C2×C4—1731no—
252C2×C4—1731no—
253C2×C4—1732no—
254C2×C4—1732no—
255C2×C4—6922no—
256C2×C4C22⋊#1062192411yescharacteristic
257C2×C4C22×#1062192411yescharacteristic
258C8—6922no—
259C22—21no—
260C22—41no—
261C22—3461no—
262C22—3462no—
263C22—1732no—
264C22—6921no—
265C22—6921no—
266C22—6922no—
267C22C2×C22⋊#1062192411yescharacteristic
268C4—22no—
269C4—22no—
270C4—21no—
271C4—3462no—
272C4—3461no—
273C4—13842no—
274C4—1732no—
275C4C2×C22⋊#1062192411yescharacteristic
276C4C2×C22⋊#1062192411yescharacteristic
277C2—21no—
278C2—41no—
279C2—3462no—
280C2—6921no—
281C2—6922no—
282C2#2601243211yescharacteristic
subgroupG/H
centerC2#26012432
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC22×Q8×C173C4
socleC2×C173#379
prime radicalC2×#19149727C2
cosocleC2×C4×D173C23
hypercenterC22×Q8#10621924
hypocenterC173#1796
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. #26012490C2semidirect
2. #26012484C2semidirect
3. #26012477C2semidirect
4. #26012470C2semidirect
17. C2×#14583827C4semidirect
21. C2×#14583826C4semidirect
24. C2×Q8×D173C4semidirect
26. C22×Q8×C173C4semidirect
141. C173#1796semidirect
160. C22×Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124902
#260124842
#260124772
#260124702
C2×#191497272
#1796173