Finite Group Databasebeta

← #37507962

Subgroups — #37507962

Order 22912.

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

normal subgroups only

This group has 244 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#26039038C211yesmaximal, characteristic
2#26038578C211yesmaximal, characteristic
3#26038358C211yesmaximal, characteristic, complemented
4#26038176C211yesmaximal, characteristic, complemented
5#26038174C211yesmaximal, characteristic
6#26038164C211yesmaximal, characteristic, complemented
7#451×C179C211yesmaximal, characteristic, complemented
8C2×#15668246—21no—
9#19153108—21no—
10#19153104—21no—
11#19153098—21no—
12#19153097—21no—
13#19153089—21no—
14#19153087C2211yescharacteristic
15(C22⋊D179)⋊C4—21no—
16C2×Dic179⋊C4—21no—
17C2×Dic179⋊C4C2211yescharacteristic
18#19153037—21no—
19C22⋊(C179⋊C8)C2211yescharacteristic
20C179⋊#95—21no—
21C179⋊#95C2211yescharacteristic
22C22×Q8×C179C2211yescharacteristic
23C2×Q16×C179—21no—
24C22⋊Q8×C179C2211yescharacteristic
25#103×C179—21no—
26C22⋊C8×C179C2211yescharacteristic
27C2×D716D411yescharacteristic
28C2×C22⋊D179—21no—
29C22×Dic179—21no—
30C22×Dic179D411yescharacteristic
31C22×Dic179D411yescharacteristic
32C2×C4×D179—21no—
33#15668246—42no—
34C2×C179⋊C8—21no—
35C4⋊Dic179—21no—
36C4⋊Dic179—21no—
37C4⋊Dic179D411yescharacteristic
38Dic179⋊C4—21no—
39Dic179⋊C4—21no—
40Dic179⋊C4—21no—
41Dic179⋊C4—42no—
42(C2×D179)⋊C4—21no—
43(C2×D179)⋊C4—21no—
44(C2×D179)⋊C4—21no—
45(C2×D179)⋊C4—21no—
46C4×Dic179—21no—
47C2×Q8×C179—21no—
48C2×Q8×C179—21no—
49C2×Q8×C179—42no—
50C2×Q8×C179D411yescharacteristic
51C22×C4×C179—21no—
52C22×C4×C179C2311yescharacteristic
53Q16×C179—42no—
54C2×C8×C179—21no—
55C4⋊C4×C179—21no—
56C4⋊C4×C179D411yescharacteristic
57C22⋊C4×C179—21no—
58C22×D179—21no—
59D716—41no—
60C22⋊D179—41no—
61C22⋊D179—41no—
62C2×Dic179—22no—
63C2×Dic179—21no—
64C2×Dic179—21no—
65C2×Dic179—21no—
66C2×Dic179—21no—
67C2×Dic179—21no—
68C2×Dic179—41no—
69C2×Dic179—41no—
70C2×Dic179—41no—
71C4×D179—81no—
72C179⋊C8—41no—
73C23×C179C2×D411yescharacteristic
74Q8×C179—22no—
75Q8×C179—22no—
76Q8×C179—41no—
77Q8×C179—41no—
78Q8×C179—41no—
79C2×C4×C179—22no—
80C2×C4×C179—21no—
81C2×C4×C179—21no—
82C2×C4×C179—41no—
83C2×C4×C179—41no—
84C2×C4×C179—41no—
85C2×C4×C179C2×D411yescharacteristic
86C2×C4×C179C2×D411yescharacteristic
87C8×C179—41no—
88C22×C179—21no—
89C22×C179—21no—
90C22×C179#12011yescharacteristic
91C22×C179#13511yescharacteristic
92C22×C179#13711yescharacteristic
93C2×D179—41no—
94C2×D179—41no—
95C2×D179—41no—
96C4×C179—21no—
97C4×C179—21no—
98C4×C179—41no—
99C4×C179—42no—
100C4×C179—41no—
101Dic179—42no—
102Dic179—41no—
103Dic179—41no—
104Dic179—41no—
105C2×C179—21no—
106C2×C179—21no—
107C2×C179#45211yescharacteristic
108C2×C179#45611yescharacteristic
109C2×C179B(4,𝔽2)11yescharacteristic
110D179—81no—
111C179#160311yescharacteristic, complemented, Sylow
112#1603—1791nomaximal, Sylow
113#539—1791no—
114#451D17911yescharacteristic, complemented
115#448—1791no—
116#395—1791no—
117#328—1791no—
118#327—1791no—
119#324—1791no—
120C22×Q8C2×D17911yescharacteristic
121C2×Q16—21no—
122C2×SD16—3581no—
123#126—3581no—
124#125—3581no—
125#123—3581no—
126#123—3581no—
127C22⋊Q8—3581no—
128C22⋊Q8C2×D17911yescharacteristic
129#121—1791no—
130C4×D4—3581no—
131C2×C4⋊C4—3581no—
132C2×C4⋊C4—1791no—
133#103—21no—
134#103—3581no—
135C22⋊C8—1791no—
136C22⋊C8C2×D17911yescharacteristic
137#95—3581no—
138#95—1791no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—42no—
142C2×Q8C22⋊D17911yescharacteristic
143C2×D4—3581no—
144C2×D4—1791no—
145C22×C4—21no—
146C22×C4—3581no—
147C22×C4—3581no—
148C22×C4—1791no—
149C22×C4—1791no—
150C22×C4C22×D17911yescharacteristic
151Q16—42no—
152SD16—7162no—
153C2×C8—21no—
154C2×C8—3581no—
155C4⋊C4—21no—
156C4⋊C4—3581no—
157C4⋊C4—3581no—
158C4⋊C4—3581no—
159C4⋊C4—3581no—
160C4⋊C4—3581no—
161C4⋊C4—1791no—
162C4⋊C4—7162no—
163C4⋊C4C22⋊D17911yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—3581no—
166C22⋊C4—3581no—
167C22⋊C4—3581no—
168C22⋊C4—3581no—
169C42—3581no—
170C23—3581no—
171C23D4×D17911yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—41no—
177D4—7161no—
178D4—7161no—
179D4—7161no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—3582no—
187C2×C4—3581no—
188C2×C4—3581no—
189C2×C4—3581no—
190C2×C4—3581no—
191C2×C4—3581no—
192C2×C4—14321no—
193C2×C4—7161no—
194C2×C4—7161no—
195C2×C4—7161no—
196C2×C4D4×D17911yescharacteristic
197C2×C4C2×C22⋊D17911yescharacteristic
198C8—41no—
199C8—7161no—
200C22—21no—
201C22—21no—
202C22—7161no—
203C22—7161no—
204C22—7161no—
205C22#1915308111yescharacteristic
206C22#1915313611yescharacteristic
207C22#1915314511yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—42no—
212C4—41no—
213C4—7162no—
214C4—7161no—
215C4—7161no—
216C4—7161no—
217C2—21no—
218C2—21no—
219C2—14321no—
220C2#2603860011yescharacteristic
221C2#2603862011yescharacteristic
222C2#2603862511yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#451×C179C2
socleC22×C179#120
prime radical#19153087C22
hypercenter#451D179
hypocenterC179#1603
LCSG ▷ C22×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #26038358C2semidirect
4. #26038176C2semidirect
6. #26038164C2semidirect
7. #451×C179C2semidirect
111. C179#1603semidirect
114. #451D179semidirect

Maximal subgroups

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

subgroupindex
#260390382
#260385782
#260383582
#260381762
#260381742
#260381642
#451×C1792
#1603179