Finite Group Databasebeta

← #37508004

Subgroups — #37508004

Order 22912.

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

normal subgroups only

This group has 254 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#26038753C211yesmaximal, characteristic, complemented
3#26038750C211yesmaximal, characteristic, complemented
4#26038371C211yesmaximal, 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⋊Dic179—21no—
16C2×C4⋊Dic179C2211yescharacteristic
17C2×Dic179⋊C4C2211yescharacteristic
18#19153047—21no—
19#19153046—21no—
20#19153035—21no—
21#19153034—21no—
22C22⋊(C179⋊C8)C2211yescharacteristic
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⋊Dic179—42no—
43C4⋊Dic179D411yescharacteristic
44Dic179⋊C4—21no—
45Dic179⋊C4—21no—
46Dic179⋊C4—21no—
47C22⋊Dic179—21no—
48(C2×D179)⋊C4—21no—
49(C2×D179)⋊C4—21no—
50C2×D4×C179—21no—
51C2×D4×C179D411yescharacteristic
52C22×C4×C179—21no—
53C22×C4×C179C2311yescharacteristic
54C2×C8×C179—21no—
55C4⋊C4×C179D411yescharacteristic
56C22⋊C4×C179—21no—
57C22×D179—21no—
58Dic358—42no—
59D716—41no—
60C22⋊D179—42no—
61C22⋊D179—41no—
62C22⋊D179—41no—
63C22⋊D179—81no—
64C2×Dic179—21no—
65C2×Dic179—21no—
66C2×Dic179—21no—
67C2×Dic179—21no—
68C2×Dic179—42no—
69C2×Dic179—41no—
70C2×Dic179—41no—
71C2×Dic179—41no—
72C2×Dic179—41no—
73C2×Dic179D812yes—
74C4×D179—42no—
75C179⋊C8—41no—
76C23×C179—21no—
77C23×C179C2×D411yescharacteristic
78D4×C179—41no—
79D4×C179—41no—
80D4×C179—41no—
81C2×C4×C179—22no—
82C2×C4×C179—21no—
83C2×C4×C179—41no—
84C2×C4×C179—41no—
85C2×C4×C179—41no—
86C2×C4×C179C2×D411yescharacteristic
87C2×C4×C179C2×D411yescharacteristic
88C8×C179—41no—
89C22×C179—21no—
90C22×C179—21no—
91C22×C179—41no—
92C22×C179—41no—
93C22×C179—41no—
94C22×C179#12011yescharacteristic
95C22×C179C2×D811yescharacteristic
96C22×C179AGL(1,ℤ/8)11yescharacteristic
97C2×D179—41no—
98C2×D179—41no—
99C2×D179—41no—
100C4×C179—21no—
101C4×C179—21no—
102C4×C179—42no—
103C4×C179—41no—
104Dic179—22no—
105Dic179—41no—
106Dic179—41no—
107Dic179—81no—
108C2×C179—21no—
109C2×C179—21no—
110C2×C179—81no—
111C2×C179#44711yescharacteristic
112C2×C179#45611yescharacteristic
113C2×C179#45811yescharacteristic
114D179—81no—
115C179#161011yescharacteristic, complemented, Sylow
116#1610—1791nomaximal, Sylow
117#537—1791no—
118#480—1791no—
119#480—1791no—
120#398—1791no—
121#327—1791no—
122#327D17911yescharacteristic, complemented
123#324—1791no—
124C2×#41—3581no—
125C22⋊Q8—3581no—
126C22⋊Q8—3581no—
127#121—3581no—
128#121—3581no—
129#121—3581no—
130#121—1791no—
131#121C2×D17911yescharacteristic
132C2×C4⋊C4—3581no—
133C2×C4⋊C4—1791no—
134C2×C4⋊C4—1791no—
135#107—3581no—
136#107—3581no—
137#102—3581no—
138#102—3581no—
139C22⋊C8—1791no—
140C22⋊C8C2×D17911yescharacteristic
141#95—3581no—
142#95C2×D17911yescharacteristic
143#41—7162no—
144#41—7162no—
145C2×Q8—3581no—
146C2×D4—21no—
147C2×D4—3581no—
148C2×D4—3581no—
149C2×D4—3581no—
150C2×D4—1791no—
151C2×D4D71611yescharacteristic
152C22×C4—21no—
153C22×C4—3581no—
154C22×C4—3581no—
155C22×C4—3581no—
156C22×C4—1791no—
157C22×C4—1791no—
158C22×C4C22×D17911yescharacteristic
159C2×C8—21no—
160C2×C8—3581no—
161C4⋊C4—3581no—
162C4⋊C4—3581no—
163C4⋊C4—3581no—
164C4⋊C4—3581no—
165C4⋊C4—3581no—
166C4⋊C4—1791no—
167C4⋊C4—7162no—
168C4⋊C4D71611yescharacteristic
169C22⋊C4—21no—
170C22⋊C4—3581no—
171C22⋊C4—3581no—
172C22⋊C4—3581no—
173C23—21no—
174C23—3581no—
175C23D4×D17911yescharacteristic
176Q8—7162no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—14321no—
181D4—7161no—
182D4—7162no—
183D4—7161no—
184D4—7161no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—3581no—
191C2×C4—3581no—
192C2×C4—3581no—
193C2×C4—3581no—
194C2×C4—1792no—
195C2×C4—7162no—
196C2×C4—7162no—
197C2×C4—7161no—
198C2×C4—7161no—
199C2×C4—7161no—
200C2×C4—7161no—
201C2×C4D4×D17911yescharacteristic
202C2×C4C2×D71611yescharacteristic
203C8—41no—
204C8—7161no—
205C22—21no—
206C22—21no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—7161no—
211C22—7161no—
212C22—7161no—
213C22#1915308311yescharacteristic
214C22D8×D17911yescharacteristic
215C22#1915314011yescharacteristic
216C4—21no—
217C4—21no—
218C4—42no—
219C4—41no—
220C4—3582no—
221C4—14321no—
222C4—7161no—
223C4—7161no—
224C2—21no—
225C2—21no—
226C2—81no—
227C2—14321no—
228C2#2603857411yescharacteristic
229C2#2603862311yescharacteristic
230C2#2603863011yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#327×C179C2
socleC22×C179#120
prime radical#26039016C2
hypercenter#327D179
hypocenterC179#1610
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. #26038753C2semidirect
3. #26038750C2semidirect
4. #26038371C2semidirect
5. #26038172C2semidirect
6. #26038164C2semidirect
7. #327×C179C2semidirect
115. C179#1610semidirect
122. #327D179semidirect

Maximal subgroups

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

subgroupindex
#260390162
#260387532
#260387502
#260383712
#260381722
#260381642
#327×C1792
#1610179