Finite Group Databasebeta

← #37507947

Subgroups — #37507947

Order 22912.

320 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 49 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#26039026C211yesmaximal, characteristic, complemented
2#26038592C211yesmaximal, characteristic, complemented
3#26038572C211yesmaximal, characteristic, complemented
4#26038349C211yesmaximal, characteristic, complemented
5#26038175C211yesmaximal, characteristic, complemented
6#26038162C211yesmaximal, characteristic, complemented
7#327×C179C211yesmaximal, characteristic, complemented
8C22×D716—21no—
9C22×D716C2211yescharacteristic
10C2×#15668248—21no—
11C2×#15668244—21no—
12#19153108—21no—
13#19153099—21no—
14#19153097—21no—
15#19153092C2211yescharacteristic
16#19153085—21no—
17#19153081—21no—
18D4×Dic179—21no—
19C2×(C2×D179)⋊C4—21no—
20C2×(C2×D179)⋊C4C2211yescharacteristic
21#19153035—21no—
22#19153033—21no—
23C22⋊(C179⋊C8)C2211yescharacteristic
24#121×C179C2211yescharacteristic
25C22⋊C8×C179C2211yescharacteristic
26#95×C179C2211yescharacteristic
27C23×D179—21no—
28C23×D179D411yescharacteristic
29C2×Dic358D411yescharacteristic, complemented
30C2×D716—21no—
31C2×D716—21no—
32C2×D716—42no—
33C2×D716—42no—
34C2×D716—42no—
35C2×D716—42no—
36C2×C22⋊D179—21no—
37C2×C22⋊D179—21no—
38C22×Dic179—21no—
39C22×Dic179D411yescharacteristic
40#15668248—42no—
41#15668244—42no—
42C2×C179⋊C8—21no—
43C4⋊Dic179—21no—
44C4⋊Dic179D411yescharacteristic, complemented
45Dic179⋊C4—21no—
46C22⋊Dic179—21no—
47C22⋊Dic179—21no—
48(C2×D179)⋊C4—21no—
49(C2×D179)⋊C4—21no—
50(C2×D179)⋊C4—21no—
51(C2×D179)⋊C4—21no—
52(C2×D179)⋊C4—42no—
53C4×Dic179—21no—
54C2×D4×C179—21no—
55C2×D4×C179D411yescharacteristic, complemented
56C22×C4×C179—21no—
57C22×C4×C179C2311yescharacteristic
58C2×C8×C179—21no—
59C4⋊C4×C179D411yescharacteristic, complemented
60C22⋊C4×C179—21no—
61C22×D179—22no—
62C22×D179—21no—
63C22×D179—21no—
64C22×D179—21no—
65C22×D179—41no—
66C22×D179—41no—
67C22×D179—41no—
68C22×D179—41no—
69C22×D179—41no—
70C22×D179—41no—
71C22×D179—41no—
72C22×D179—41no—
73Dic358—41no—
74D716—22no—
75D716—22no—
76D716—41no—
77D716—41no—
78D716—48no—
79C22⋊D179—81no—
80C22⋊D179—81no—
81C2×Dic179—21no—
82C2×Dic179—21no—
83C2×Dic179—21no—
84C2×Dic179—41no—
85C2×Dic179—81no—
86C179⋊C8—41no—
87C23×C179—21no—
88C23×C179C2×D411yescharacteristic
89D4×C179—41no—
90D4×C179—41no—
91D4×C179—41no—
92C2×C4×C179—22no—
93C2×C4×C179—21no—
94C2×C4×C179—41no—
95C2×C4×C179—41no—
96C2×C4×C179—41no—
97C2×C4×C179C2×D411yescharacteristic
98C2×C4×C179C2×D411yescharacteristic
99C8×C179—41no—
100C22×C179—21no—
101C22×C179—21no—
102C22×C179—41no—
103C22×C179—41no—
104C22×C179—41no—
105C22×C179#12011yescharacteristic
106C22×C179#13511yescharacteristic
107C22×C179AGL(1,ℤ/8)11yescharacteristic
108C2×D179—42no—
109C2×D179—42no—
110C2×D179—42no—
111C2×D179—41no—
112C2×D179—41no—
113C2×D179—41no—
114C2×D179—42no—
115C2×D179—81no—
116C2×D179—81no—
117C2×D179—81no—
118C2×D179—81no—
119C2×D179—81no—
120C4×C179—21no—
121C4×C179—21no—
122C4×C179—42no—
123C4×C179—41no—
124Dic179—41no—
125Dic179—41no—
126Dic179—41no—
127C2×C179—21no—
128C2×C179—21no—
129C2×C179—81no—
130C2×C179#44911yescharacteristic
131C2×C179#45311yescharacteristic
132C2×C179B(4,𝔽2)11yescharacteristic
133D179—81no—
134D179—82no—
135C179#160011yescharacteristic, complemented, Sylow
136#1600—1791nomaximal, Sylow
137#538—1791no—
138#450—1791no—
139#447—1791no—
140#392—1791no—
141#328—1791no—
142#327D17911yescharacteristic, complemented
143#323—1791no—
144C22×D4—3581no—
145C22×D4—1791no—
146C2×SD16—3581no—
147C2×D8—3581no—
148#126—3581no—
149#124—3581no—
150#123—3581no—
151C22⋊Q8—1791no—
152#121—3581no—
153#121C2×D17911yescharacteristic
154#120—3581no—
155C4×D4—3581no—
156C2×C22⋊C4—3581no—
157C2×C22⋊C4—1791no—
158#102—3581no—
159#102—3581no—
160C22⋊C8—1791no—
161C22⋊C8C2×D17911yescharacteristic
162#95C2×D17911yescharacteristic
163C24—3581no—
164C24—1791no—
165C2×Q8—1791no—
166C2×D4—21no—
167C2×D4—3581no—
168C2×D4—3581no—
169C2×D4—3581no—
170C2×D4—3581no—
171C2×D4—7162no—
172C2×D4—7162no—
173C2×D4—7162no—
174C2×D4—7162no—
175C2×D4D71611yescharacteristic, complemented
176C22×C4—21no—
177C22×C4—3581no—
178C22×C4—1791no—
179C22×C4C22×D17911yescharacteristic
180SD16—7162no—
181D8—7162no—
182C2×C8—21no—
183C2×C8—3581no—
184C4⋊C4—3581no—
185C4⋊C4—3581no—
186C4⋊C4—1791no—
187C4⋊C4D71611yescharacteristic, complemented
188C22⋊C4—21no—
189C22⋊C4—3581no—
190C22⋊C4—3581no—
191C22⋊C4—3581no—
192C22⋊C4—3581no—
193C22⋊C4—3581no—
194C22⋊C4—3581no—
195C22⋊C4—7162no—
196C42—3581no—
197C23—21no—
198C23—3582no—
199C23—3581no—
200C23—3581no—
201C23—3581no—
202C23—7161no—
203C23—7161no—
204C23—7161no—
205C23—7161no—
206C23—7161no—
207C23—7161no—
208C23—7161no—
209C23—7161no—
210C23D4×D17911yescharacteristic
211Q8—7161no—
212D4—41no—
213D4—41no—
214D4—41no—
215D4—3582no—
216D4—3582no—
217D4—14321no—
218D4—14321no—
219D4—7161no—
220D4—7161no—
221D4—7168no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—3581no—
228C2×C4—3581no—
229C2×C4—3581no—
230C2×C4—14321no—
231C2×C4—7161no—
232C2×C4D4×D17911yescharacteristic
233C2×C4C2×D71611yescharacteristic
234C8—41no—
235C8—7161no—
236C22—21no—
237C22—21no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—14321no—
242C22—14321no—
243C22—14321no—
244C22—14321no—
245C22—14321no—
246C22—7162no—
247C22—7162no—
248C22—7162no—
249C22—7161no—
250C22—7161no—
251C22—7162no—
252C22—7161no—
253C22#1915308311yescharacteristic
254C22#1915313411yescharacteristic
255C22#1915313711yescharacteristic
256C4—21no—
257C4—21no—
258C4—42no—
259C4—41no—
260C4—7161no—
261C4—7161no—
262C4—7161no—
263C2—21no—
264C2—21no—
265C2—81no—
266C2—14322no—
267C2—14321no—
268C2#2603858511yescharacteristic
269C2#2603860811yescharacteristic
270C2#2603862711yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#327×C179C2
socleC22×C179#120
hypercenter#327D179
hypocenterC179#1600
LCSG ▷ C22×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
1. #26039026C2semidirect
2. #26038592C2semidirect
3. #26038572C2semidirect
4. #26038349C2semidirect
5. #26038175C2semidirect
6. #26038162C2semidirect
7. #327×C179C2semidirect
29. C2×Dic358D4semidirect
44. C4⋊Dic179D4semidirect
55. C2×D4×C179D4semidirect
59. C4⋊C4×C179D4semidirect
135. C179#1600semidirect
142. #327D179semidirect
175. C2×D4D716semidirect
187. C4⋊C4D716semidirect

Maximal subgroups

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

subgroupindex
#260390262
#260385922
#260385722
#260383492
#260381752
#260381622
#327×C1792
#1600179