Finite Group Databasebeta

← #37510042

Subgroups — #37510042

Order 22912.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#19153138C212yesmaximal, complemented
2C2×#19153040C212yesmaximal, complemented
3#26038340C211yesmaximal, characteristic, complemented
4#26038208C211yesmaximal, characteristic, complemented
5#537×C179C211yesmaximal, characteristic, complemented
6C22×D716C2211yescharacteristic
7#19153138—24no—
8#19153138—24no—
9C2×#15668246—22no—
10C2×#15668244—22no—
11C2×#15668243C2212yes—
12C2×(C2×D179)⋊C4—22no—
13C2×C4×Dic179C2211yescharacteristic
14#19153040—24no—
15C2×#41×C179C2212yes—
16C22⋊Q8×C179—22no—
17#121×C179—22no—
18#121×C179—22no—
19C2×C4⋊C4×C179C2211yescharacteristic
20C23×D179—21no—
21C2×D716—22no—
22C2×D716—22no—
23C2×D716—21no—
24C2×D716D412yescomplemented
25C22×Dic179—21no—
26#15668246—28no—
27#15668244—28no—
28#15668243—22no—
29#15668243D414yescomplemented
30C2×C179⋊C8—22no—
31(C2×D179)⋊C4—44no—
32C4×Dic179—22no—
33#41×C179—22no—
34#41×C179—22no—
35#41×C179—22no—
36#41×C179—22no—
37C2×Q8×C179D412yes—
38C2×D4×C179—22no—
39C2×D4×C179—22no—
40C2×D4×C179D412yes—
41C22×C4×C179—22no—
42C22×C4×C179D412yes—
43C22×C4×C179C2311yescharacteristic
44C4⋊C4×C179—22no—
45C4⋊C4×C179—22no—
46C22⋊C4×C179—24no—
47C22×D179—22no—
48C22×D179—42no—
49C22×D179—42no—
50C22×D179—41no—
51C22×D179—41no—
52D716—24no—
53D716—42no—
54C2×Dic179—22no—
55C2×Dic179—41no—
56C2×Dic179—41no—
57C179⋊C8—24no—
58C23×C179—22no—
59C23×C179C2×D411yescharacteristic
60Q8×C179—22no—
61Q8×C179—22no—
62D4×C179—22no—
63D4×C179—22no—
64D4×C179—42no—
65D4×C179—42no—
66D4×C179—82no—
67C2×C4×C179—22no—
68C2×C4×C179—22no—
69C2×C4×C179—42no—
70C2×C4×C179—42no—
71C2×C4×C179—42no—
72C2×C4×C179C2×D411yescharacteristic
73C2×C4×C179C2×D412yes—
74C2×C4×C179C2×D412yes—
75C2×C4×C179#4111yescharacteristic
76C22×C179—21no—
77C22×C179—21no—
78C22×C179—42no—
79C22×C179—42no—
80C22×C179—42no—
81C22×C179#12011yescharacteristic
82C22×C179#12111yescharacteristic
83C22×C179#12711yescharacteristic
84C2×D179—44no—
85C2×D179—42no—
86C2×D179—82no—
87C2×D179—81no—
88C2×D179—81no—
89C4×C179—42no—
90C4×C179—42no—
91C4×C179#12012yes—
92C4×C179#12112yes—
93Dic179—42no—
94C2×C179—21no—
95C2×C179—21no—
96C2×C179—82no—
97C2×C179#39211yescharacteristic
98C2×C179#45311yescharacteristic
99C2×C179#45411yescharacteristic
100D179—82no—
101C179#200611yescharacteristic, complemented, Sylow
102#2006—1791nomaximal, Sylow
103C2×AGL(1,ℤ/8)—1792no—
104#537D17911yescharacteristic, complemented
105C2×#104—1792no—
106#390—1791no—
107#343—1791no—
108C2×#41C2×D17912yes—
109C22×D4—1791no—
110AGL(1,ℤ/8)—3584no—
111AGL(1,ℤ/8)—3584no—
112C2×SD16—3582no—
113C2×D8—3582no—
114C2×M16—1792no—
115C22⋊Q8—22no—
116#121—22no—
117#121—22no—
118C2×C4⋊C4C2×D17911yescharacteristic
119C2×C22⋊C4—3582no—
120C2×C42—1791no—
121#104—3584no—
122C24—3581no—
123#41—22no—
124#41—22no—
125#41—22no—
126#41—22no—
127C2×Q8C22⋊D17912yes—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—3582no—
131C2×D4—3582no—
132C2×D4—3581no—
133C2×D4—1792no—
134C2×D4C22⋊D17912yes—
135C22×C4—22no—
136C22×C4—3581no—
137C22×C4C22⋊D17912yes—
138C22×C4C22×D17911yescharacteristic
139SD16—3588no—
140D8—3588no—
141M16—3582no—
142M16—1794no—
143C2×C8—3582no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C22⋊C4—24no—
147C22⋊C4—7164no—
148C42—3582no—
149C23—22no—
150C23—3582no—
151C23—7162no—
152C23—7162no—
153C23—7161no—
154C23—7161no—
155C23C2×C22⋊D17911yescharacteristic
156Q8—22no—
157Q8—22no—
158D4—22no—
159D4—22no—
160D4—42no—
161D4—42no—
162D4—82no—
163D4—3584no—
164D4—7162no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—3582no—
171C2×C4—7161no—
172C2×C4—7161no—
173C2×C4D4×D17911yescharacteristic
174C2×C4D4×D17912yes—
175C2×C4C2×C22⋊D17912yes—
176C2×C4#1566825811yescharacteristic
177C8—3584no—
178C22—21no—
179C22—21no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—14322no—
184C22—14321no—
185C22—14321no—
186C22—7164no—
187C22—7162no—
188C22#1915308211yescharacteristic
189C22#1915308411yescharacteristic
190C22#1915310911yescharacteristic
191C4—42no—
192C4—42no—
193C4—7162no—
194C4#1915308112yes—
195C4#1915308512yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—14322no—
200C2#2603834711yescharacteristic
201C2#2603860711yescharacteristic
202C2#2603861211yescharacteristic
subgroupG/H
centerC22#19153082
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#537×C179C2
socleC22×C179#120
hypercenter#537D179
hypocenterC179#2006
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#19153138C2semidirect
2. C2×#19153040C2semidirect
3. #26038340C2semidirect
4. #26038208C2semidirect
5. #537×C179C2semidirect
24. C2×D716D4semidirect
29. #15668243D4semidirect
101. C179#2006semidirect
104. #537D179semidirect

Maximal subgroups

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

subgroupindex
C2×#191531382
C2×#191530402
#260383402
#260382082
#537×C1792
#2006179