Finite Group Databasebeta

← #37508015

Subgroups — #37508015

Order 22912.

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

normal subgroups only

This group has 232 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
2#26038770C211yesmaximal, characteristic, complemented
3#26038750C211yesmaximal, characteristic, complemented
4#26038366C211yesmaximal, characteristic, complemented
5#26038175C211yesmaximal, characteristic, complemented
6#26038162C211yesmaximal, characteristic, complemented
7#327×C179C211yesmaximal, characteristic, complemented
8#19153108—21no—
9#19153099—21no—
10#19153097—21no—
11#19153092C2211yescharacteristic
12#19153085—21no—
13#19153081—21no—
14D4×Dic179—21no—
15C2×C4⋊Dic179C2211yescharacteristic
16C2×(C2×D179)⋊C4—21no—
17C2×(C2×D179)⋊C4C2211yescharacteristic
18#19153046—21no—
19#19153045—21no—
20#19153038—21no—
21#19153034—21no—
22C22⋊(C179⋊C8)C2211yescharacteristic
23C179⋊#95—21no—
24#121×C179C2211yescharacteristic
25C22⋊C8×C179C2211yescharacteristic
26#95×C179C2211yescharacteristic
27C23×D179D411yescharacteristic
28C2×Dic358D411yescharacteristic
29C2×C22⋊D179—21no—
30C2×C22⋊D179—21no—
31C22×Dic179—21no—
32C22×Dic179—21no—
33C22×Dic179D411yescharacteristic
34C2×C179⋊C8—21no—
35C4⋊Dic179—21no—
36C4⋊Dic179—21no—
37C4⋊Dic179—21no—
38C4⋊Dic179D411yescharacteristic
39Dic179⋊C4—21no—
40C22⋊Dic179—21no—
41C22⋊Dic179—21no—
42(C2×D179)⋊C4—21no—
43(C2×D179)⋊C4—21no—
44(C2×D179)⋊C4—21no—
45(C2×D179)⋊C4—21no—
46(C2×D179)⋊C4—42no—
47C4×Dic179—21no—
48C2×D4×C179—21no—
49C2×D4×C179D411yescharacteristic
50C22×C4×C179—21no—
51C22×C4×C179C2311yescharacteristic
52C2×C8×C179—21no—
53C4⋊C4×C179D411yescharacteristic
54C22⋊C4×C179—21no—
55C22×D179—21no—
56C22×D179—21no—
57C22×D179—21no—
58C22×D179—41no—
59C22×D179—41no—
60Dic358—41no—
61C22⋊D179—81no—
62C22⋊D179—81no—
63C2×Dic179—21no—
64C2×Dic179—21no—
65C2×Dic179—21no—
66C2×Dic179—41no—
67C2×Dic179—41no—
68C2×Dic179—41no—
69C2×Dic179—41no—
70C2×Dic179—81no—
71C179⋊C8—41no—
72C23×C179—21no—
73C23×C179C2×D411yescharacteristic
74D4×C179—41no—
75D4×C179—41no—
76D4×C179—41no—
77C2×C4×C179—21no—
78C2×C4×C179—41no—
79C2×C4×C179—41no—
80C2×C4×C179—41no—
81C2×C4×C179—41no—
82C2×C4×C179C2×D411yescharacteristic
83C2×C4×C179C2×D411yescharacteristic
84C8×C179—41no—
85C22×C179—21no—
86C22×C179—21no—
87C22×C179—41no—
88C22×C179—41no—
89C22×C179—41no—
90C22×C179#12011yescharacteristic
91C22×C179#13511yescharacteristic
92C22×C179AGL(1,ℤ/8)11yescharacteristic
93C2×D179—42no—
94C2×D179—41no—
95C2×D179—41no—
96C2×D179—41no—
97C2×D179—81no—
98C4×C179—21no—
99C4×C179—21no—
100C4×C179—41no—
101C4×C179—81no—
102Dic179—41no—
103Dic179—41no—
104Dic179—41no—
105Dic179—81no—
106C2×C179—21no—
107C2×C179—21no—
108C2×C179—81no—
109C2×C179#44911yescharacteristic
110C2×C179#45511yescharacteristic
111C2×C179#45811yescharacteristic
112D179—81no—
113C179#161211yescharacteristic, complemented, Sylow
114#1612—1791nomaximal, Sylow
115#538—1791no—
116#483—1791no—
117#480—1791no—
118#397—1791no—
119#328—1791no—
120#327D17911yescharacteristic, complemented
121#323—1791no—
122#126—3581no—
123#124—3581no—
124#123—3581no—
125C22⋊Q8—1791no—
126#121—3581no—
127#121C2×D17911yescharacteristic
128#120—3581no—
129C4×D4—3581no—
130C2×C4⋊C4—1791no—
131C2×C22⋊C4—3581no—
132C2×C22⋊C4—1791no—
133#107—3581no—
134#106—3581no—
135#103—3581no—
136#102—3581no—
137C22⋊C8—1791no—
138C22⋊C8C2×D17911yescharacteristic
139#95—3581no—
140#95C2×D17911yescharacteristic
141C24—1791no—
142C2×Q8—1791no—
143C2×D4—21no—
144C2×D4—3581no—
145C2×D4—3581no—
146C2×D4D71611yescharacteristic
147C22×C4—21no—
148C22×C4—3581no—
149C22×C4—3581no—
150C22×C4—1791no—
151C22×C4C22×D17911yescharacteristic
152C2×C8—21no—
153C2×C8—3581no—
154C4⋊C4—3581no—
155C4⋊C4—3581no—
156C4⋊C4—3581no—
157C4⋊C4—3581no—
158C4⋊C4—1791no—
159C4⋊C4D71611yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—3581no—
162C22⋊C4—3581no—
163C22⋊C4—3581no—
164C22⋊C4—3581no—
165C22⋊C4—3581no—
166C22⋊C4—3581no—
167C22⋊C4—7162no—
168C42—3581no—
169C23—21no—
170C23—3581no—
171C23—3581no—
172C23—3581no—
173C23—7161no—
174C23—7161no—
175C23D4×D17911yescharacteristic
176Q8—7161no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—14321no—
181D4—14321no—
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—14321no—
191C2×C4—7161no—
192C2×C4—7161no—
193C2×C4—7161no—
194C2×C4—7161no—
195C2×C4D4×D17911yescharacteristic
196C2×C4C2×D71611yescharacteristic
197C8—41no—
198C8—7161no—
199C22—21no—
200C22—21no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—14321no—
205C22—7162no—
206C22—7161no—
207C22—7161no—
208C22—7161no—
209C22#1915308311yescharacteristic
210C22#1915313411yescharacteristic
211C22#1915313711yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—14321no—
217C4—7161no—
218C4—7161no—
219C4—7161no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—14321no—
224C2#2603858511yescharacteristic
225C2#2603861811yescharacteristic
226C2#2603863011yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#327×C179C2
socleC22×C179#120
prime radical#26039026C2
hypercenter#327D179
hypocenterC179#1612
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. #26038770C2semidirect
3. #26038750C2semidirect
4. #26038366C2semidirect
5. #26038175C2semidirect
6. #26038162C2semidirect
7. #327×C179C2semidirect
113. C179#1612semidirect
120. #327D179semidirect

Maximal subgroups

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

subgroupindex
#260390262
#260387702
#260387502
#260383662
#260381752
#260381622
#327×C1792
#1612179