Finite Group Databasebeta

← #37510041

Subgroups — #37510041

Order 22912.

406 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 73 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×#19153139C212yesmaximal, complemented
2#26039020C211yesmaximal, characteristic, complemented
3C2×#19153041C212yesmaximal, complemented
4#26038208C211yesmaximal, characteristic, complemented
5#390×C179C211yesmaximal, characteristic, complemented
6C2×#15668259C2212yes—
7#19153139—24no—
8#19153139—24no—
9C2×#15668247—22no—
10C2×#15668244—22no—
11C2×#15668243C2212yes—
12#19153093—22no—
13#19153088—22no—
14#19153087—22no—
15#19153041—24no—
16C22×D4×C179C2211yescharacteristic
17C2×C4⋊C4×C179C2211yescharacteristic
18C2×C22⋊C4×C179—22no—
19C2×C42×C179C2211yescharacteristic
20#15668259—22no—
21#15668259—22no—
22#15668259—22no—
23#15668259—22no—
24C2×Dic358D412yes—
25C2×D716—22no—
26C2×D716D412yes—
27C2×C22⋊D179—22no—
28C2×C4×D179—22no—
29#15668247—28no—
30#15668244—28no—
31#15668243—22no—
32#15668243D414yescomplemented
33C2×C179⋊C8—22no—
34C4⋊Dic179—22no—
35(C2×D179)⋊C4—24no—
36C24×C179—21no—
37C2×D4×C179—22no—
38C2×D4×C179—22no—
39C2×D4×C179—21no—
40C2×D4×C179D412yescomplemented
41C22×C4×C179—21no—
42C22×C4×C179D412yes—
43C22×C4×C179C2311yescharacteristic
44C4⋊C4×C179—22no—
45C22⋊C4×C179—44no—
46C42×C179—22no—
47C22×D179—22no—
48Dic358—22no—
49Dic358—22no—
50D716—22no—
51D716—22no—
52D716—82no—
53C22⋊D179—42no—
54C22⋊D179—42no—
55C2×Dic179—22no—
56C4×D179—42no—
57C4×D179—42no—
58C179⋊C8—24no—
59C23×C179—22no—
60C23×C179—42no—
61C23×C179—42no—
62C23×C179—41no—
63C23×C179—41no—
64C23×C179C2×D411yescharacteristic
65D4×C179—24no—
66D4×C179—42no—
67C2×C4×C179—22no—
68C2×C4×C179—22no—
69C2×C4×C179—41no—
70C2×C4×C179—41no—
71C2×C4×C179—42no—
72C2×C4×C179C2×D412yes—
73C2×C4×C179C2×D412yes—
74C2×C4×C179C2×D411yescharacteristic
75C2×C4×C179#4111yescharacteristic
76C22×C179—21no—
77C22×C179—21no—
78C22×C179—44no—
79C22×C179—42no—
80C22×C179—82no—
81C22×C179—81no—
82C22×C179—81no—
83C22×C179#12011yescharacteristic
84C22×C179#12111yescharacteristic
85C22×C179#12711yescharacteristic
86C2×D179—42no—
87C2×D179—42no—
88C2×D179—42no—
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#537—1791no—
105C2×#104—1792no—
106#390D17911yescharacteristic, complemented
107#343—1791no—
108C2×#41—1792no—
109C22×D4C2×D17911yescharacteristic
110AGL(1,ℤ/8)—3584no—
111AGL(1,ℤ/8)—3584no—
112C2×SD16—3582no—
113C2×D8—3582no—
114C2×M16—1792no—
115C22⋊Q8—3582no—
116#121—3582no—
117#121—3582no—
118C2×C4⋊C4C2×D17911yescharacteristic
119C2×C22⋊C4—22no—
120C2×C42C2×D17911yescharacteristic
121#104—3584no—
122C24—21no—
123#41—3582no—
124#41—3582no—
125#41—3582no—
126#41—3582no—
127C2×Q8—1792no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—3582no—
132C2×D4—3582no—
133C2×D4—1792no—
134C2×D4D71612yescomplemented
135C22×C4—21no—
136C22×C4—3582no—
137C22×C4C22⋊D17912yes—
138C22×C4C22×D17911yescharacteristic
139SD16—3588no—
140D8—3588no—
141M16—3582no—
142M16—1794no—
143C2×C8—3582no—
144C4⋊C4—22no—
145C4⋊C4—3582no—
146C22⋊C4—44no—
147C22⋊C4—3584no—
148C42—22no—
149C23—22no—
150C23—42no—
151C23—42no—
152C23—41no—
153C23—41no—
154C23—3582no—
155C23C2×C22⋊D17911yescharacteristic
156Q8—3582no—
157Q8—3582no—
158D4—24no—
159D4—42no—
160D4—3582no—
161D4—3582no—
162D4—14322no—
163D4—7162no—
164D4—7162no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—3582no—
171C2×C4—7162no—
172C2×C4—7162no—
173C2×C4D4×D17912yes—
174C2×C4D4×D17912yes—
175C2×C4C2×D71611yescharacteristic
176C2×C4#1566825911yescharacteristic
177C8—3584no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—42no—
182C22—82no—
183C22—81no—
184C22—81no—
185C22—7162no—
186C22—7162no—
187C22—7162no—
188C22#1915308111yescharacteristic
189C22#1915308711yescharacteristic
190C22#1915310911yescharacteristic
191C4—42no—
192C4—42no—
193C4—7162no—
194C4#1915308312yes—
195C4#1915308612yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—14322no—
200C2#2603834811yescharacteristic
201C2#2603860611yescharacteristic
202C2#2603861111yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#390×C179C2
socleC22×C179#120
hypercenter#390D179
hypocenterC179#2006
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#19153139C2semidirect
2. #26039020C2semidirect
3. C2×#19153041C2semidirect
4. #26038208C2semidirect
5. #390×C179C2semidirect
32. #15668243D4semidirect
40. C2×D4×C179D4semidirect
101. C179#2006semidirect
106. #390D179semidirect
134. C2×D4D716semidirect

Maximal subgroups

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

subgroupindex
C2×#191531392
#260390202
C2×#191530412
#260382082
#390×C1792
#2006179