Finite Group Databasebeta

← #37508038

Subgroups — #37508038

Order 22912.

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

normal subgroups only

This group has 246 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#26039016C211yesmaximal, characteristic
2#26038758C211yesmaximal, characteristic, complemented
3#26038755C211yesmaximal, characteristic, complemented
4#26038379C211yesmaximal, characteristic, complemented
5#26038172C211yesmaximal, characteristic, complemented
6#26038164C211yesmaximal, characteristic, complemented
7#327×C179C211yesmaximal, characteristic, complemented
8C2×#15668258—21no—
9#19153091—21no—
10#19153090—21no—
11#19153087C2211yescharacteristic
12#19153086—21no—
13#19153085—21no—
14#19153085—21no—
15C2×C4⋊Dic179C2211yescharacteristic
16C2×Dic179⋊C4C2211yescharacteristic
17#19153045—21no—
18#19153044—21no—
19#19153035—21no—
20#19153034—21no—
21C22⋊(C179⋊C8)C2211yescharacteristic
22C179⋊#95—21no—
23C179⋊#95—21no—
24#121×C179C2211yescharacteristic
25C22⋊C8×C179C2211yescharacteristic
26#95×C179C2211yescharacteristic
27#15668258—42no—
28#15668258—42no—
29C2×Dic358—21no—
30C2×D716D411yescharacteristic
31C2×C22⋊D179—21no—
32C2×C22⋊D179—21no—
33C2×C22⋊D179—21no—
34C22×Dic179—21no—
35C22×Dic179—21no—
36C22×Dic179D411yescharacteristic
37C22×Dic179D411yescharacteristic
38C2×C4×D179—21no—
39C2×C179⋊C8—21no—
40C4⋊Dic179—21no—
41C4⋊Dic179—21no—
42C4⋊Dic179D411yescharacteristic
43Dic179⋊C4—21no—
44Dic179⋊C4—21no—
45Dic179⋊C4—21no—
46C22⋊Dic179—21no—
47(C2×D179)⋊C4—21no—
48(C2×D179)⋊C4—21no—
49C2×D4×C179—21no—
50C2×D4×C179D411yescharacteristic
51C22×C4×C179—21no—
52C22×C4×C179C2311yescharacteristic
53C2×C8×C179—21no—
54C4⋊C4×C179D411yescharacteristic
55C22⋊C4×C179—21no—
56C22×D179—21no—
57Dic358—42no—
58D716—41no—
59C22⋊D179—42no—
60C22⋊D179—41no—
61C22⋊D179—41no—
62C22⋊D179—81no—
63C2×Dic179—21no—
64C2×Dic179—21no—
65C2×Dic179—21no—
66C2×Dic179—21no—
67C2×Dic179—42no—
68C2×Dic179—41no—
69C2×Dic179—42no—
70C2×Dic179—41no—
71C2×Dic179SD1612yes—
72C4×D179—42no—
73C179⋊C8—41no—
74C23×C179—21no—
75C23×C179C2×D411yescharacteristic
76D4×C179—41no—
77D4×C179—41no—
78D4×C179—41no—
79C2×C4×C179—21no—
80C2×C4×C179—41no—
81C2×C4×C179—41no—
82C2×C4×C179—41no—
83C2×C4×C179—41no—
84C2×C4×C179C2×D411yescharacteristic
85C2×C4×C179C2×D411yescharacteristic
86C8×C179—41no—
87C22×C179—21no—
88C22×C179—21no—
89C22×C179—41no—
90C22×C179—41no—
91C22×C179—41no—
92C22×C179#12011yescharacteristic
93C22×C179C2×SD1611yescharacteristic
94C22×C179AGL(1,ℤ/8)11yescharacteristic
95C2×D179—41no—
96C2×D179—41no—
97C2×D179—41no—
98C4×C179—21no—
99C4×C179—21no—
100C4×C179—41no—
101C4×C179—81no—
102Dic179—22no—
103Dic179—41no—
104Dic179—41no—
105Dic179—81no—
106C2×C179—21no—
107C2×C179—21no—
108C2×C179—81no—
109C2×C179#45011yescharacteristic
110C2×C179#45411yescharacteristic
111C2×C179#45811yescharacteristic
112D179—81no—
113C179#161611yescharacteristic, complemented, Sylow
114#1616—1791nomaximal, Sylow
115#537—1791no—
116#481—1791no—
117#481—1791no—
118#400—1791no—
119#327—1791no—
120#327D17911yescharacteristic, complemented
121#324—1791no—
122C2×#41—3581no—
123C22⋊Q8—3581no—
124C22⋊Q8—3581no—
125#121—3581no—
126#121—3581no—
127#121—3581no—
128#121—1791no—
129#121C2×D17911yescharacteristic
130C2×C4⋊C4—1791no—
131C2×C4⋊C4—1791no—
132#106—3581no—
133#106—3581no—
134#102—3581no—
135#102—3581no—
136C22⋊C8—1791no—
137C22⋊C8C2×D17911yescharacteristic
138#95—3581no—
139#95—3581no—
140#95C2×D17911yescharacteristic
141#41—7162no—
142#41—7162no—
143C2×Q8—3581no—
144C2×D4—21no—
145C2×D4—3581no—
146C2×D4—3581no—
147C2×D4—3581no—
148C2×D4—1791no—
149C2×D4D71611yescharacteristic
150C22×C4—21no—
151C22×C4—3581no—
152C22×C4—3581no—
153C22×C4—3581no—
154C22×C4—1791no—
155C22×C4—1791no—
156C22×C4C22×D17911yescharacteristic
157C2×C8—21no—
158C2×C8—3581no—
159C4⋊C4—3581no—
160C4⋊C4—3581no—
161C4⋊C4—3581no—
162C4⋊C4—3581no—
163C4⋊C4—3581no—
164C4⋊C4—1791no—
165C4⋊C4D71611yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—3581no—
168C22⋊C4—3581no—
169C22⋊C4—3581no—
170C23—21no—
171C23—3581no—
172C23D4×D17911yescharacteristic
173Q8—7162no—
174D4—41no—
175D4—41no—
176D4—41no—
177D4—14321no—
178D4—7161no—
179D4—7162no—
180D4—7161no—
181D4—7161no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—3581no—
188C2×C4—3581no—
189C2×C4—3581no—
190C2×C4—3581no—
191C2×C4—1792no—
192C2×C4—7162no—
193C2×C4—7162no—
194C2×C4—7162no—
195C2×C4—7161no—
196C2×C4—7161no—
197C2×C4D4×D17911yescharacteristic
198C2×C4C2×D71611yescharacteristic
199C8—41no—
200C8—7161no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—7161no—
207C22—7161no—
208C22—7161no—
209C22#1915308311yescharacteristic
210C22SD16×D17911yescharacteristic
211C22#1915313711yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—3582no—
217C4—14321no—
218C4—7161no—
219C4—7161no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—14321no—
224C2#2603859111yescharacteristic
225C2#2603861311yescharacteristic
226C2#2603863011yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#327×C179C2
socleC22×C179#120
prime radical#26039016C2
hypercenter#327D179
hypocenterC179#1616
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
2. #26038758C2semidirect
3. #26038755C2semidirect
4. #26038379C2semidirect
5. #26038172C2semidirect
6. #26038164C2semidirect
7. #327×C179C2semidirect
113. C179#1616semidirect
120. #327D179semidirect

Maximal subgroups

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

subgroupindex
#260390162
#260387582
#260387552
#260383792
#260381722
#260381642
#327×C1792
#1616179