Finite Group Databasebeta

← #36353842

Subgroups — #36353842

Order 22144.

374 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 83 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#26011964C211yesmaximal, characteristic
2C2×#19149593C211yesmaximal, characteristic
3#26011370C211yesmaximal, characteristic
4#26011364C211yesmaximal, characteristic, complemented
5#26011258C211yesmaximal, characteristic, complemented
6#26011257C211yesmaximal, characteristic, complemented
7#385×C173C211yesmaximal, characteristic, complemented
8C22×C22⋊D173C2211yescharacteristic
9C4×C22⋊D173—21no—
10C4×C22⋊D173—21no—
11#19149639—21no—
12#19149639—21no—
13#19149635—21no—
14C2×#14583805—21no—
15C2×C22⋊Dic173C2211yescharacteristic
16C2×C22⋊Dic173C2211yescharacteristic
17C2×C22⋊Dic173C2211yescharacteristic
18C2×#14583803—21no—
19C22⋊C4×D173—21no—
20#19149593C414yescomplemented
21#19149586—21no—
22#19149586—21no—
23C23×C4×C173C2211yescharacteristic
24C2×C4⋊C4×C173—21no—
25C2×C22⋊C4×C173C2211yescharacteristic
26C2×C22⋊C4×C173C2211yescharacteristic
27#95×C173—21no—
28C23×D173D411yescharacteristic
29C2×C22⋊D173—21no—
30C2×C22⋊D173—21no—
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173C2×C412yes—
34C2×C22⋊D173Q812yes—
35C22×Dic173—21no—
36C22×Dic173—21no—
37C22×Dic173D411yescharacteristic
38C22×Dic173D411yescharacteristic
39C22×Dic173D411yescharacteristic
40C2×C4×D173—21no—
41C2×C4×D173—21no—
42#14583805—21no—
43#14583805—21no—
44#14583805—42no—
45C22⋊Dic173—21no—
46C22⋊Dic173—21no—
47C22⋊Dic173—21no—
48C22⋊Dic173—21no—
49C22⋊Dic173—21no—
50C22⋊Dic173C2×C412yescomplemented
51#14583803—21no—
52#14583803—21no—
53#14583803—42no—
54C4×Dic173—21no—
55C4×Dic173—21no—
56C24×C173C2311yescharacteristic
57C22×C4×C173—21no—
58C22×C4×C173—21no—
59C22×C4×C173—21no—
60C22×C4×C173—21no—
61C22×C4×C173—42no—
62C22×C4×C173D411yescharacteristic
63C22×C4×C173D411yescharacteristic
64C4⋊C4×C173—42no—
65C22⋊C4×C173—21no—
66C22⋊C4×C173—21no—
67C22⋊C4×C173—21no—
68C22⋊C4×C173C2×C412yescomplemented
69C22×D173—21no—
70C22×D173—21no—
71C22×D173—41no—
72C22×D173—41no—
73C22×D173C4⋊C412yescomplemented
74C22⋊D173—22no—
75C22⋊D173—22no—
76C22⋊D173—22no—
77C22⋊D173—22no—
78C2×Dic173—21no—
79C2×Dic173—21no—
80C2×Dic173—22no—
81C2×Dic173—21no—
82C2×Dic173—21no—
83C2×Dic173—41no—
84C2×Dic173—41no—
85C2×Dic173—41no—
86C2×Dic173—41no—
87C2×Dic173—41no—
88C2×Dic173—41no—
89C2×Dic173—41no—
90C2×Dic173C4⋊C412yescomplemented
91C4×D173—81no—
92C4×D173—81no—
93C23×C173—41no—
94C23×C173—41no—
95C23×C173C22×C411yescharacteristic
96C23×C173C2×D411yescharacteristic
97C23×C173C2×D411yescharacteristic
98C23×C173C2×D411yescharacteristic
99C23×C173C2×Q811yescharacteristic
100C23×C173#4111yescharacteristic
101C23×C173#4111yescharacteristic
102C2×C4×C173—22no—
103C2×C4×C173—22no—
104C2×C4×C173—22no—
105C2×C4×C173—22no—
106C2×C4×C173—21no—
107C2×C4×C173—21no—
108C2×C4×C173—41no—
109C2×C4×C173—41no—
110C2×C4×C173—41no—
111C2×C4×C173—41no—
112C2×C4×C173—41no—
113C2×C4×C173—41no—
114C2×C4×C173—41no—
115C2×C4×C173—41no—
116C22×C173—21no—
117C22×C173—21no—
118C22×C173—21no—
119C22×C173—21no—
120C22×C173—21no—
121C22×C173—21no—
122C22×C173—41no—
123C22×C173—41no—
124C22×C173—41no—
125C22×C173—41no—
126C22×C173C2×C4⋊C411yescharacteristic
127C22×C173C4×D411yescharacteristic
128C22×C173C4×D411yescharacteristic
129C22×C173#12011yescharacteristic
130C22×C173C22⋊Q811yescharacteristic
131C22×C173C22⋊Q811yescharacteristic
132C22×C173#12311yescharacteristic
133C2×D173—22no—
134C2×D173—41no—
135C2×D173—41no—
136C2×D173—42no—
137C2×D173—81no—
138C4×C173—42no—
139C4×C173—42no—
140C4×C173—41no—
141C4×C173—41no—
142Dic173—22no—
143Dic173—42no—
144Dic173—41no—
145Dic173—41no—
146Dic173—81no—
147C2×C173—21no—
148C2×C173—21no—
149C2×C173—21no—
150C2×C173—21no—
151C2×C173—21no—
152C2×C173—21no—
153C2×C173#38511yescharacteristic
154C2×C173B(4,𝔽2)11yescharacteristic
155C2×C173#45811yescharacteristic
156D173—42no—
157C173#186711yescharacteristic, complemented, Sylow
158#1867—1731nomaximal, Sylow
159#518—1731no—
160C2×#99—1731no—
161#386—1731no—
162#385—1731no—
163#385D17311yescharacteristic, complemented
164#342—1731no—
165#342—1731no—
166C22×D4—1731no—
167C23×C4C2×D17311yescharacteristic
168C4×D4—3461no—
169C4×D4—3461no—
170C4×D4—3461no—
171C4×D4—3461no—
172#117—3461no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—3461no—
175C2×C22⋊C4—3461no—
176C2×C22⋊C4—3461no—
177C2×C22⋊C4—1731no—
178C2×C22⋊C4—1731no—
179C2×C22⋊C4—1731no—
180C2×C22⋊C4C2×D17311yescharacteristic
181C2×C22⋊C4C2×D17311yescharacteristic
182#99—1734no—
183#95—21no—
184#95—3461no—
185#95—3461no—
186C24—1731no—
187C24C22×D17311yescharacteristic
188C2×D4—3461no—
189C2×D4—3461no—
190C2×D4—3461no—
191C2×D4—3461no—
192C2×D4—1732no—
193C2×D4—1732no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—42no—
199C22×C4—3461no—
200C22×C4—3461no—
201C22×C4—3461no—
202C22×C4—3461no—
203C22×C4—1731no—
204C22×C4—1731no—
205C22×C4—1731no—
206C22×C4C22⋊D17311yescharacteristic
207C22×C4C22⋊D17311yescharacteristic
208C4⋊C4—42no—
209C4⋊C4—3461no—
210C4⋊C4—3461no—
211C4⋊C4—6922no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—3461no—
216C22⋊C4—3461no—
217C22⋊C4—3461no—
218C22⋊C4—3461no—
219C22⋊C4—3461no—
220C22⋊C4—3461no—
221C22⋊C4—3461no—
222C22⋊C4—1732no—
223C22⋊C4—6922no—
224C22⋊C4C4×D17312yescomplemented
225C42—3461no—
226C42—3461no—
227C23—41no—
228C23—41no—
229C23—3461no—
230C23—3461no—
231C23—1732no—
232C23—6921no—
233C23—6921no—
234C23C2×C4×D17311yescharacteristic
235C23D4×D17311yescharacteristic
236C23D4×D17311yescharacteristic
237C23C2×C22⋊D17311yescharacteristic
238C23Q8×D17311yescharacteristic
239C23#1458382611yescharacteristic
240C23#1458382711yescharacteristic
241D4—3462no—
242D4—3462no—
243D4—3462no—
244D4—3462no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—3461no—
260C2×C4—3461no—
261C2×C4—3462no—
262C2×C4—3461no—
263C2×C4—3461no—
264C2×C4—13841no—
265C2×C4—13841no—
266C2×C4—1732no—
267C2×C4—6921no—
268C2×C4—6921no—
269C2×C4—6921no—
270C2×C4—6921no—
271C2×C4—6921no—
272C2×C4—6921no—
273C2×C4—6921no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—3462no—
285C22—13841no—
286C22—6922no—
287C22—6921no—
288C22—6921no—
289C22C4⋊C4×D17311yescharacteristic
290C22C4×C22⋊D17311yescharacteristic
291C22#1914964111yescharacteristic
292C22#1914964711yescharacteristic
293C22#1914965511yescharacteristic
294C22#1914965711yescharacteristic
295C22#1914966411yescharacteristic
296C4—42no—
297C4—42no—
298C4—41no—
299C4—41no—
300C4—3462no—
301C4—13841no—
302C4—6922no—
303C4—6921no—
304C4—6921no—
305C2—21no—
306C2—21no—
307C2—21no—
308C2—21no—
309C2—21no—
310C2—21no—
311C2—6922no—
312C2#2601136211yescharacteristic
313C2#2601167711yescharacteristic
314C2#2601168011yescharacteristic
subgroupG/H
centerC22#19149647
commutatorC23×C173C22×C4
FrattiniC24C22×D173
Fitting#385×C173C2
socleC22×C173#120
prime radicalC22×C22⋊D173C22
cosocleC24×C173C23
hypercenter#385D173
hypocenterC173#1867
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C24 ◁ #385

Complements

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

NH ≅ G/Ntype
4. #26011364C2semidirect
5. #26011258C2semidirect
6. #26011257C2semidirect
7. #385×C173C2semidirect
20. #19149593C4semidirect
50. C22⋊Dic173C2×C4semidirect
68. C22⋊C4×C173C2×C4semidirect
73. C22×D173C4⋊C4semidirect
90. C2×Dic173C4⋊C4semidirect
157. C173#1867semidirect
163. #385D173semidirect
224. C22⋊C4C4×D173semidirect

Maximal subgroups

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

subgroupindex
#260119642
C2×#191495932
#260113702
#260113642
#260112582
#260112572
#385×C1732
#1867173